# bibtab

bibtype: article
filename: fparith.bib
label: Seo:2015:MMS
author: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
title: Montgomery multiplication and squaring for Optimal Prime Fields
journal: j-COMPUT-SECUR
volume: 52
pages: 276--291
month: jul
monthnumber: 7
year: 2015
CODEN: CPSEDU
ISSN: 0167-4048 (print), 1872-6208 (electronic) OR 01674048 (print), 18726208 (electronic)
ISSNL: 0167-4048 OR 01674048
bibdate: Mon Sep 23 09:46:42 MDT 2019
bibtimestamp: 2019.09.23 09:46:42 MDT
URL: https://www.sciencedirect.com/science/article/pii/S0167404815000371
fjournal: Computers & Security
acknowledgement: ack-nhfb
journalURL: https://www.sciencedirect.com/science/journal/01674048
rawauthor: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
rawtitle: {Montgomery} multiplication and squaring for Optimal Prime Fields
bibsource:
https://www.math.utah.edu/pub/tex/bib/computsecur2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Seo:2015:MMS,
  author =       "Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and
                 Jongseok Choi and Howon Kim",
  title =        "{Montgomery} multiplication and squaring for Optimal
                 Prime Fields",
  journal =      j-COMPUT-SECUR,
  volume =       "52",
  number =       "??",
  pages =        "276--291",
  month =        jul,
  year =         "2015",
  CODEN =        "CPSEDU",
  ISSN =         "0167-4048 (print), 1872-6208 (electronic)",
  ISSN-L =       "0167-4048",
  bibdate =      "Mon Sep 23 09:46:42 MDT 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computsecur2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S0167404815000371",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers \& Security",
  journal-URL =  "https://www.sciencedirect.com/science/journal/01674048",
}


bibtype: inproceedings
filename: fparith.bib
label: Singh:2015:DIQ
author: Arun Kumar Singh and Madhav Kumar Singh and Kailash Chandra Ray
booktitle: 2015 IEEE International Symposium on Nanoelectronic and Information Systems
title: Design and Implementation of Quadruple Floating-Point CORDIC
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 286--290
year: 2015
DOI: https://doi.org/10.1109/iNIS.2015.23
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
acknowledgement: ack-nhfb
rawauthor: Arun Kumar Singh and Madhav Kumar Singh and Kailash Chandra Ray
rawbooktitle: {2015 IEEE International Symposium on Nanoelectronic and Information Systems}
rawtitle: Design and Implementation of Quadruple Floating-Point {CORDIC}
keywords:
Computer architecture; CORDIC; Dynamic range; FPGA Implementation; Hardware design languages; Heuristic algorithms; Mathematical model; Quadruple Floating-Point; Signal processing algorithms; Standards; VLSI Architecture

entry:
@InProceedings{Singh:2015:DIQ,
  author =       "Arun Kumar Singh and Madhav Kumar Singh and Kailash
                 Chandra Ray",
  booktitle =    "{2015 IEEE International Symposium on Nanoelectronic
                 and Information Systems}",
  title =        "Design and Implementation of Quadruple Floating-Point
                 {CORDIC}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "286--290",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/iNIS.2015.23",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Computer architecture; CORDIC; Dynamic range; FPGA
                 Implementation; Hardware design languages; Heuristic
                 algorithms; Mathematical model; Quadruple
                 Floating-Point; Signal processing algorithms;
                 Standards; VLSI Architecture",
}


bibtype: techreport
filename: fparith.bib
label: Solovyev:2015:REFa
author: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamaric and Ganesh Gopalakrishnan
title: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions
type: Technical Report
number: UUCS-15-001
institution: School of Computing, University of Utah
address: Salt Lake City, UT 84112 USA
pages: 31
day: 6
month: apr
monthnumber: 4
year: 2015
bibdate: Fri Apr 17 18:02:17 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:02:17 ???
URL: http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf
acknowledgement: ack-nhfb
rawauthor: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamari{\'c} and Ganesh Gopalakrishnan
rawtitle: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic {Taylor} Expansions
entry:
@TechReport{Solovyev:2015:REFa,
  author =       "Alexey Solovyev and Charles Jacobsen and Zvonimir
                 Rakamari{\'c} and Ganesh Gopalakrishnan",
  title =        "Rigorous Estimation of Floating-Point Round-off Errors
                 with Symbolic {Taylor} Expansions",
  type =         "Technical Report",
  number =       "UUCS-15-001",
  institution =  "School of Computing, University of Utah",
  address =      "Salt Lake City, UT 84112 USA",
  pages =        "31",
  day =          "6",
  month =        apr,
  year =         "2015",
  bibdate =      "Fri Apr 17 18:02:17 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Solovyev:2015:REFb
author: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamaric and Ganesh Gopalakrishnan
booktitle: 20th International Symposium on Formal Methods (FM 2015), Oslo, Norway
title: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions
publisher: pub-SV
address: pub-SV:adr
pages: 532--550
day: 6
month: jun
monthnumber: 6
year: 2015
bibdate: Fri Apr 17 18:02:17 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.04.17 18:02:17 ???
note: Extended version available in cite Solovyev:2015:REFa.
URL: http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf
acknowledgement: ack-nhfb
rawauthor: Alexey Solovyev and Charles Jacobsen and Zvonimir Rakamari{\'c} and Ganesh Gopalakrishnan
rawbooktitle: {20th International Symposium on Formal Methods (FM 2015), Oslo, Norway}
rawnote: Extended version available in \cite{Solovyev:2015:REFa}.
rawtitle: Rigorous Estimation of Floating-Point Round-off Errors with Symbolic {Taylor} Expansions
entry:
@InProceedings{Solovyev:2015:REFb,
  author =       "Alexey Solovyev and Charles Jacobsen and Zvonimir
                 Rakamari{\'c} and Ganesh Gopalakrishnan",
  booktitle =    "{20th International Symposium on Formal Methods (FM
                 2015), Oslo, Norway}",
  title =        "Rigorous Estimation of Floating-Point Round-off Errors
                 with Symbolic {Taylor} Expansions",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "532--550",
  day =          "6",
  month =        jun,
  year =         "2015",
  bibdate =      "Fri Apr 17 18:02:17 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  note =         "Extended version available in
                 \cite{Solovyev:2015:REFa}.",
  URL =          "http://formalverification.cs.utah.edu/papers/fptaylor_report.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Sullivan:2015:LCD
author: Michael B. Sullivan and Earl E. Swartzlander
title: Low-Cost Duplicate Multiplication
crossref: Muller:2015:ISC
pages: 2--9
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.29
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH 2015; ARITH-22
acknowledgement: ack-nhfb
rawauthor: Michael B. Sullivan and Earl E. Swartzlander
rawtitle: Low-Cost Duplicate Multiplication
entry:
@InProceedings{Sullivan:2015:LCD,
  author =       "Michael B. Sullivan and Earl E. Swartzlander",
  title =        "Low-Cost Duplicate Multiplication",
  crossref =     "Muller:2015:ISC",
  pages =        "2--9",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.29",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2015; ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Thomas:2015:GPM
author: David B. Thomas
title: A General-Purpose Method for Faithfully Rounded Floating-Point Function Approximation in FPGAs
crossref: Muller:2015:ISC
pages: 42--49
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.27
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH 2015; ARITH-22
acknowledgement: ack-nhfb
rawauthor: David B. Thomas
rawtitle: A General-Purpose Method for Faithfully Rounded Floating-Point Function Approximation in {FPGAs}
entry:
@InProceedings{Thomas:2015:GPM,
  author =       "David B. Thomas",
  title =        "A General-Purpose Method for Faithfully Rounded
                 Floating-Point Function Approximation in {FPGAs}",
  crossref =     "Muller:2015:ISC",
  pages =        "42--49",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.27",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2015; ARITH-22",
}


bibtype: article
filename: fparith.bib
label: Trefethen:2015:CNF
author: Lloyd N. Trefethen
title: Computing numerically with functions instead of numbers
journal: j-CACM
volume: 58
number: 10
pages: 91--97
month: oct
monthnumber: 10
year: 2015
CODEN: CACMA2
DOI: https://doi.org/10.1145/2814847
ISSN: 0001-0782 (print), 1557-7317 (electronic) OR 00010782 (print), 15577317 (electronic)
ISSNL: 0001-0782 OR 00010782
bibdate: Wed Sep 30 07:29:42 MDT 2015
bibtimestamp: 2015.09.30 07:29:42 MDT
URL: http://cacm.acm.org/magazines/2015/10/192390/fulltext
fjournal: Communications of the ACM
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/cacm
rawauthor: Lloyd N. Trefethen
rawtitle: Computing numerically with functions instead of numbers
bibsource:
http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib; https://www.math.utah.edu/pub/tex/bib/cacm2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib

abstract:
Science and engineering depend upon computation of functions such as flow fields, charge distributions, and quantum states. Ultimately, such computations require some kind of discretization, but in recent years, it has become possible in many cases to hide the discretizations from the user. We present the Chebfun system for numerical computation with functions, which is based on a key idea: an analogy of floating-point arithmetic for functions rather than numbers.

keywords:
Chebfun; Chebyshev approximation; floating-point arithmetic; Matlab; rational arithmetic; symbolic algebra

rawabstract:
Science and engineering depend upon computation of functions such as flow fields, charge distributions, and quantum states. Ultimately, such computations require some kind of discretization, but in recent years, it has become possible in many cases to hide the discretizations from the user. We present the Chebfun system for numerical computation with functions, which is based on a key idea: an analogy of floating-point arithmetic for functions rather than numbers.

entry:
@Article{Trefethen:2015:CNF,
  author =       "Lloyd N. Trefethen",
  title =        "Computing numerically with functions instead of
                 numbers",
  journal =      j-CACM,
  volume =       "58",
  number =       "10",
  pages =        "91--97",
  month =        oct,
  year =         "2015",
  CODEN =        "CACMA2",
  DOI =          "https://doi.org/10.1145/2814847",
  ISSN =         "0001-0782 (print), 1557-7317 (electronic)",
  ISSN-L =       "0001-0782",
  bibdate =      "Wed Sep 30 07:29:42 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/cacm/;
                 https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib;
                 https://www.math.utah.edu/pub/tex/bib/cacm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib",
  URL =          "http://cacm.acm.org/magazines/2015/10/192390/fulltext",
  abstract =     "Science and engineering depend upon computation of
                 functions such as flow fields, charge distributions,
                 and quantum states. Ultimately, such computations
                 require some kind of discretization, but in recent
                 years, it has become possible in many cases to hide the
                 discretizations from the user. We present the Chebfun
                 system for numerical computation with functions, which
                 is based on a key idea: an analogy of floating-point
                 arithmetic for functions rather than numbers.",
  acknowledgement = ack-nhfb,
  fjournal =     "Communications of the ACM",
  journal-URL =  "https://dl.acm.org/loi/cacm",
  keywords =     "Chebfun; Chebyshev approximation; floating-point
                 arithmetic; Matlab; rational arithmetic; symbolic
                 algebra",
}


bibtype: inproceedings
filename: fparith.bib
label: vanderHoeven:2015:FFM
author: Joris van der Hoeven and Gregoire Lecerf
title: Faster FFTs in Medium Precision
crossref: Muller:2015:ISC
pages: 75--82
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.10
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH 2015; ARITH-22
acknowledgement: ack-nhfb
rawauthor: Joris van der Hoeven and Gr{\'e}goire Lecerf
rawtitle: Faster {FFTs} in Medium Precision
entry:
@InProceedings{vanderHoeven:2015:FFM,
  author =       "Joris van der Hoeven and Gr{\'e}goire Lecerf",
  title =        "Faster {FFTs} in Medium Precision",
  crossref =     "Muller:2015:ISC",
  pages =        "75--82",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.10",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2015; ARITH-22",
}


bibtype: inproceedings
filename: fparith.bib
label: Volkova:2015:REW
author: Anastasia Volkova and Thibault Hilaire and Christoph Lauter
title: Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision
crossref: Muller:2015:ISC
pages: 96--103
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.14
bibdate: Sat Aug 01 08:05:52 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH 2015; ARITH-22
acknowledgement: ack-nhfb
rawauthor: Anastasia Volkova and Thibault Hilaire and Christoph Lauter
rawtitle: Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision
entry:
@InProceedings{Volkova:2015:REW,
  author =       "Anastasia Volkova and Thibault Hilaire and Christoph
                 Lauter",
  title =        "Reliable Evaluation of the Worst-Case Peak Gain Matrix
                 in Multiple Precision",
  crossref =     "Muller:2015:ISC",
  pages =        "96--103",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.14",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2015; ARITH-22",
}


bibtype: misc
filename: fparith.bib
label: Wilczak:2015:CAP
author: Daniel Wilczak and others
title: Computer Assisted Proofs in Dynamics
pages: 96--103
month: jun
monthnumber: 6
year: 2015
bibdate: Wed Jun 17 11:30:51 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.06.17 11:30:51 ???
URL: http://sourceforge.net/projects/capd/
acknowledgement: ack-nhfb
howpublished: Web site and software.
rawauthor: Daniel Wilczak and others
rawtitle: Computer Assisted Proofs in Dynamics
remark:
From the Web site: ``The CAPD library is a collection of flexible C++ modules which are mainly designed to computation of homology of sets and maps and nonrigorous and validated numerics for dynamical systems.'' From the announcement on the Reliable Computing mailing list on 14-Jun-2015: ``In 2015, the library has over 100,000 lines of code. It provides rigorous solvers for ODEs, variational equations for ODEs, differential inclusions, automatic computation of Poincare maps and their derivatives, computation of homology of spaces, maps and many other features. Its efficiency is confirmed by several nontrivial results in dynamics, including proving of chaos, presence of local and global bifurcations for ODEs, existence of attractors for ODEs, periodic orbits for PDEs, and many other results.par The library comes with a comprehensive tutorial with many examples illustrating very basic usage of CAPD as well as quite advanced, like a complete computer assisted proof of the existence of attractor in the Rossler system with uniformly hyperbolic and chaotic invariant set.''

entry:
@Misc{Wilczak:2015:CAP,
  author =       "Daniel Wilczak and others",
  title =        "Computer Assisted Proofs in Dynamics",
  howpublished = "Web site and software.",
  month =        jun,
  year =         "2015",
  bibdate =      "Wed Jun 17 11:30:51 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://sourceforge.net/projects/capd/",
  acknowledgement = ack-nhfb,
  remark =       "From the Web site: ``The CAPD library is a collection
                 of flexible C++ modules which are mainly designed to
                 computation of homology of sets and maps and
                 nonrigorous and validated numerics for dynamical
                 systems.'' From the announcement on the Reliable
                 Computing mailing list on 14-Jun-2015: ``In 2015, the
                 library has over 100,000 lines of code. It provides
                 rigorous solvers for ODEs, variational equations for
                 ODEs, differential inclusions, automatic computation of
                 Poincare maps and their derivatives, computation of
                 homology of spaces, maps and many other features. Its
                 efficiency is confirmed by several nontrivial results
                 in dynamics, including proving of chaos, presence of
                 local and global bifurcations for ODEs, existence of
                 attractors for ODEs, periodic orbits for PDEs, and many
                 other results.\par

                 The library comes with a comprehensive tutorial with
                 many examples illustrating very basic usage of CAPD as
                 well as quite advanced, like a complete computer
                 assisted proof of the existence of attractor in the
                 Rossler system with uniformly hyperbolic and chaotic
                 invariant set.''",
}


bibtype: article
filename: fparith.bib
label: Wittmann:2015:SNC
author: Markus Wittmann and Thomas Zeiser and Georg Hager and Gerhard Wellein
journal: arXiv.org
pages: ??--??
day: 12
month: jun
monthnumber: 6
year: 2015
bibdate: Wed Sep 30 12:43:49 MDT 2015
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2015.09.30 12:43:49 MDT
URL: http://arxiv.org/abs/1506.03997
subject: Performance (cs.PF)
acknowledgement: ack-nhfb
rawauthor: Markus Wittmann and Thomas Zeiser and Georg Hager and Gerhard Wellein
title:
Short Note on Costs of Floating Point Operations on current x86-64 Architectures: Denormals, Overflow, Underflow, and Division by Zero

abstract:
Simple floating point operations like addition or multiplication on normalized floating point values can be computed by current AMD and Intel processors in three to five cycles. This is different for denormalized numbers, which appear when an underflow occurs and the value can no longer be represented as a normalized floating-point value. Here the costs are about two magnitudes higher.

rawabstract:
Simple floating point operations like addition or multiplication on normalized floating point values can be computed by current AMD and Intel processors in three to five cycles. This is different for denormalized numbers, which appear when an underflow occurs and the value can no longer be represented as a normalized floating-point value. Here the costs are about two magnitudes higher.

rawtitle:
Short Note on Costs of Floating Point Operations on current x86-64 Architectures: Denormals, Overflow, Underflow, and Division by Zero

entry:
@Article{Wittmann:2015:SNC,
  author =       "Markus Wittmann and Thomas Zeiser and Georg Hager and
                 Gerhard Wellein",
  title =        "Short Note on Costs of Floating Point Operations on
                 current x86-64 Architectures: Denormals, Overflow,
                 Underflow, and Division by Zero",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "12",
  month =        jun,
  year =         "2015",
  CODEN =        "????",
  ISSN =         "????",
  ISSN-L =       "????",
  bibdate =      "Wed Sep 30 12:43:49 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1506.03997",
  abstract =     "Simple floating point operations like addition or
                 multiplication on normalized floating point values can
                 be computed by current AMD and Intel processors in
                 three to five cycles. This is different for
                 denormalized numbers, which appear when an underflow
                 occurs and the value can no longer be represented as a
                 normalized floating-point value. Here the costs are
                 about two magnitudes higher.",
  acknowledgement = ack-nhfb,
  subject =      "Performance (cs.PF)",
}


bibtype: article
filename: fparith.bib
label: Yamazaki:2015:MPC
author: Ichitaro Yamazaki and Stanimire Tomov and Jack Dongarra
journal: j-SIAM-J-SCI-COMP
volume: 37
number: 3
pages: C307--C330
year: 2015
CODEN: SJOCE3
DOI: https://doi.org/10.1137/14M0973773
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Sat Aug 8 06:48:44 MDT 2015
bibtimestamp: 2015.08.08 06:48:44 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
onlinedate: January 2015
rawauthor: Ichitaro Yamazaki and Stanimire Tomov and Jack Dongarra
title:
Mixed-Precision Cholesky $ Q R $ Factorization and Its Case Studies on Multicore CPU with Multiple GPUs

bibsource:
http://epubs.siam.org/toc/sjoce3/37/3; https://www.math.utah.edu/pub/bibnet/authors/d/dongarra-jack-j.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib

rawtitle:
Mixed-Precision {Cholesky} {$ Q R $} Factorization and Its Case Studies on Multicore {CPU} with Multiple {GPUs}

entry:
@Article{Yamazaki:2015:MPC,
  author =       "Ichitaro Yamazaki and Stanimire Tomov and Jack
                 Dongarra",
  title =        "Mixed-Precision {Cholesky} {$ Q R $} Factorization and
                 Its Case Studies on Multicore {CPU} with Multiple
                 {GPUs}",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "37",
  number =       "3",
  pages =        "C307--C330",
  month =        "????",
  year =         "2015",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/14M0973773",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Sat Aug 8 06:48:44 MDT 2015",
  bibsource =    "http://epubs.siam.org/toc/sjoce3/37/3;
                 https://www.math.utah.edu/pub/bibnet/authors/d/dongarra-jack-j.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
  onlinedate =   "January 2015",
}


bibtype: inproceedings
filename: fparith.bib
label: Yin:2015:NRT
booktitle: 2015 IEEE International Symposium on Circuits and Systems (ISCAS)
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 954--957
year: 2015
DOI: https://doi.org/10.1109/ISCAS.2015.7168793
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
acknowledgement: ack-nhfb
rawbooktitle: {2015 IEEE International Symposium on Circuits and Systems (ISCAS)}
author:
Hongxu Yin and Bah Hwee Gwee and Zhiping Lin and Anil Kumar and Sirajudeen Gulam Razul and Chong Meng Samson See

title:
Novel real-time system design for floating-point sub-Nyquist multi-coset signal blind reconstruction

keywords:
CORDIC; Field programmable gate arrays; FPGA; Jacobi; Jacobian matrices; Mathematical model; multi-coset sampling; Multiband; Multiple signal classification; MUSIC; Real-time systems; Reconstruction; Signal to noise ratio; sub-Nyquist

rawauthor:
Hongxu Yin and Bah Hwee Gwee and Zhiping Lin and Anil Kumar and Sirajudeen Gulam Razul and Chong Meng Samson See

rawtitle:
Novel real-time system design for floating-point sub-Nyquist multi-coset signal blind reconstruction

entry:
@InProceedings{Yin:2015:NRT,
  author =       "Hongxu Yin and Bah Hwee Gwee and Zhiping Lin and Anil
                 Kumar and Sirajudeen Gulam Razul and Chong Meng Samson
                 See",
  booktitle =    "{2015 IEEE International Symposium on Circuits and
                 Systems (ISCAS)}",
  title =        "Novel real-time system design for floating-point
                 sub-Nyquist multi-coset signal blind reconstruction",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "954--957",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ISCAS.2015.7168793",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "CORDIC; Field programmable gate arrays; FPGA; Jacobi;
                 Jacobian matrices; Mathematical model; multi-coset
                 sampling; Multiband; Multiple signal classification;
                 MUSIC; Real-time systems; Reconstruction; Signal to
                 noise ratio; sub-Nyquist",
}


bibtype: inproceedings
filename: fparith.bib
label: Ahmed:2016:ILM
author: Syed Ershad Ahmed and Sanket Kadam and M. B. Srinivas
title: An Iterative Logarithmic Multiplier with Improved Precision
crossref: Montuschi:2016:ISC
pages: 104--111
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.25
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Syed Ershad Ahmed and Sanket Kadam and M. B. Srinivas
rawtitle: An Iterative Logarithmic Multiplier with Improved Precision
entry:
@InProceedings{Ahmed:2016:ILM,
  author =       "Syed Ershad Ahmed and Sanket Kadam and M. B.
                 Srinivas",
  title =        "An Iterative Logarithmic Multiplier with Improved
                 Precision",
  crossref =     "Montuschi:2016:ISC",
  pages =        "104--111",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.25",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: techreport
filename: fparith.bib
label: Ahrens:2016:ERF
author: Peter Ahrens and Hong Diep Nguyen and James Demmel
title: Efficient Reproducible Floating Point Summation and BLAS
type: Report
number: UCB/EECS-2016-121
institution: EECS Department, UC Berkeley
address: Berkeley, CA, USA
pages: 104--111
day: 18
month: jun
monthnumber: 6
year: 2016
bibdate: Wed Oct 12 18:31:23 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.10.12 18:31:23 ???
URL: http://www.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-121.html
keywords: accurate dot product; accurate summation
acknowledgement: ack-nhfb
rawauthor: Peter Ahrens and Hong Diep Nguyen and James Demmel
rawtitle: Efficient Reproducible Floating Point Summation and {BLAS}
entry:
@TechReport{Ahrens:2016:ERF,
  author =       "Peter Ahrens and Hong Diep Nguyen and James Demmel",
  title =        "Efficient Reproducible Floating Point Summation and
                 {BLAS}",
  type =         "Report",
  number =       "UCB/EECS-2016-121",
  institution =  "EECS Department, UC Berkeley",
  address =      "Berkeley, CA, USA",
  day =          "18",
  month =        jun,
  year =         "2016",
  bibdate =      "Wed Oct 12 18:31:23 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-121.html",
  acknowledgement = ack-nhfb,
  keywords =     "accurate dot product; accurate summation",
}


bibtype: inproceedings
filename: fparith.bib
label: Almeida:2016:VCT
author: Jose Bacelar Almeida and Manuel Barbosa and Gilles Barthe and Francois Dupressoir and Michael Emmi
editor: T. Holz and S. Savage
booktitle: Proceedings of the 25th USENIX Security 2016, August 10--12, 2016, Austin, TX
title: Verifying Constant-Time Implementations
publisher: pub-USENIX
address: pub-USENIX:adr
pages: 53--70
year: 2016
bibdate: Tue Aug 27 06:44:12 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2024.08.27 06:44:12 ???
keywords: elliptic curve arithmetic; fixed-point arithmetic; libfixedtimefixedpoint
acknowledgement: ack-nhfb
rawbooktitle: Proceedings of the {25th USENIX Security 2016, August 10--12, 2016, Austin, TX}
raweditor: T. Holz and S. Savage
rawtitle: Verifying Constant-Time Implementations
URL:
https://github.com/kmowery/libfixedtimefixedpoint; https://www.usenix.org/system/files/conference/usenixsecurity16/sec16_paper_almeida.pdf

abstract:
The constant-time programming discipline is an effective countermeasure against timing attacks, which can lead to complete breaks of otherwise secure systems. However, adhering to constant-time programming is hard on its own, and extremely hard under additional efficiency and legacy constraints. This makes automated verification of constant-time code an essential component for building secure software.par We propose a novel approach for verifying constant-time security of real-world code. Our approach is able to validate implementations that locally and intentionally violate the constant-time policy, when such violations are benign and leak no more information than the public outputs of the computation. Such implementations, which are used in cryptographic libraries to obtain important speedups or to comply with legacy APIs, would be declared insecure by all prior solutions.par We implement our approach in a publicly available, cross-platform, and fully automated prototype, ct-verif, that leverages the SMACK and Boogie tools and verifies optimized LLVM implementations. We present verification results obtained over a wide range of constant-time components from the NaCl, OpenSSL, FourQ and other off-the-shelf libraries. The diversity and scale of our examples, as well as the fact that we deal with top-level APIs rather than being limited to low-level leaf functions, distinguishes ct-verif from prior tools.par Our approach is based on a simple reduction of constant-time security of a program $P$ to safety of a product program $Q$ that simulates two executions of $P$. We formalize and verify the reduction for a core high-level language using the Coq proof assistant.

rawabstract:
The constant-time programming discipline is an effective countermeasure against timing attacks, which can lead to complete breaks of otherwise secure systems. However, adhering to constant-time programming is hard on its own, and extremely hard under additional efficiency and legacy constraints. This makes automated verification of constant-time code an essential component for building secure software.\par We propose a novel approach for verifying constant-time security of real-world code. Our approach is able to validate implementations that locally and intentionally violate the constant-time policy, when such violations are benign and leak no more information than the public outputs of the computation. Such implementations, which are used in cryptographic libraries to obtain important speedups or to comply with legacy APIs, would be declared insecure by all prior solutions.\par We implement our approach in a publicly available, cross-platform, and fully automated prototype, ct-verif, that leverages the SMACK and Boogie tools and verifies optimized LLVM implementations. We present verification results obtained over a wide range of constant-time components from the NaCl, OpenSSL, FourQ and other off-the-shelf libraries. The diversity and scale of our examples, as well as the fact that we deal with top-level APIs rather than being limited to low-level leaf functions, distinguishes ct-verif from prior tools.\par Our approach is based on a simple reduction of constant-time security of a program $P$ to safety of a product program $Q$ that simulates two executions of $P$. We formalize and verify the reduction for a core high-level language using the Coq proof assistant.

rawauthor:
Jos{\'e} Bacelar Almeida and Manuel Barbosa and Gilles Barthe and Fran{\c{c}}ois Dupressoir and Michael Emmi

entry:
@InProceedings{Almeida:2016:VCT,
  author =       "Jos{\'e} Bacelar Almeida and Manuel Barbosa and Gilles
                 Barthe and Fran{\c{c}}ois Dupressoir and Michael Emmi",
  editor =       "T. Holz and S. Savage",
  booktitle =    "Proceedings of the {25th USENIX Security 2016, August
                 10--12, 2016, Austin, TX}",
  title =        "Verifying Constant-Time Implementations",
  publisher =    pub-USENIX,
  address =      pub-USENIX:adr,
  pages =        "53--70",
  year =         "2016",
  bibdate =      "Tue Aug 27 06:44:12 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://github.com/kmowery/libfixedtimefixedpoint;
                 https://www.usenix.org/system/files/conference/usenixsecurity16/sec16_paper_almeida.pdf",
  abstract =     "The constant-time programming discipline is an
                 effective countermeasure against timing attacks, which
                 can lead to complete breaks of otherwise secure
                 systems. However, adhering to constant-time programming
                 is hard on its own, and extremely hard under additional
                 efficiency and legacy constraints. This makes automated
                 verification of constant-time code an essential
                 component for building secure software.\par

                 We propose a novel approach for verifying constant-time
                 security of real-world code. Our approach is able to
                 validate implementations that locally and intentionally
                 violate the constant-time policy, when such violations
                 are benign and leak no more information than the public
                 outputs of the computation. Such implementations, which
                 are used in cryptographic libraries to obtain important
                 speedups or to comply with legacy APIs, would be
                 declared insecure by all prior solutions.\par

                 We implement our approach in a publicly available,
                 cross-platform, and fully automated prototype,
                 ct-verif, that leverages the SMACK and Boogie tools and
                 verifies optimized LLVM implementations. We present
                 verification results obtained over a wide range of
                 constant-time components from the NaCl, OpenSSL, FourQ
                 and other off-the-shelf libraries. The diversity and
                 scale of our examples, as well as the fact that we deal
                 with top-level APIs rather than being limited to
                 low-level leaf functions, distinguishes ct-verif from
                 prior tools.\par

                 Our approach is based on a simple reduction of
                 constant-time security of a program $P$ to safety of a
                 product program $Q$ that simulates two executions of
                 $P$. We formalize and verify the reduction for a core
                 high-level language using the Coq proof assistant.",
  acknowledgement = ack-nhfb,
  keywords =     "elliptic curve arithmetic; fixed-point arithmetic;
                 libfixedtimefixedpoint",
}


bibtype: article
filename: fparith.bib
label: Andrysco:2016:PFP
author: Marc Andrysco and Ranjit Jhala and Sorin Lerner
title: Printing floating-point numbers: a faster, always correct method
journal: j-SIGPLAN
volume: 51
number: 1
pages: 555--567
month: jan
monthnumber: 1
year: 2016
CODEN: SINODQ
DOI: https://doi.org/10.1145/2914770.2837654
ISSN: 0362-1340 (print), 1523-2867 (print), 1558-1160 OR 03621340 (print), 15232867 (print), 15581160
ISSNL: 0362-1340 OR 03621340
bibdate: Thu Jun 9 17:13:57 MDT 2016
bibtimestamp: 2016.06.09 17:13:57 MDT
fjournal: ACM SIGPLAN Notices
remark: POPL '16 conference proceedings.
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J706
rawauthor: Marc Andrysco and Ranjit Jhala and Sorin Lerner
rawtitle: Printing floating-point numbers: a faster, always correct method
abstract:
Floating-point numbers are an essential part of modern software, recently gaining particular prominence on the web as the exclusive numeric format of Javascript. To use floating-point numbers, we require a way to convert binary machine representations into human readable decimal outputs. Existing conversion algorithms make trade-offs between completeness and performance. The classic Dragon4 algorithm by Steele and White and its later refinements achieve completeness --- i.e. produce correct and optimal outputs on all inputs --- by using arbitrary precision integer (bignum) arithmetic which leads to a high performance cost. On the other hand, the recent Grisu3 algorithm by Loitsch shows how to recover performance by using native integer arithmetic but sacrifices optimality for 0.5% of all inputs. We present Errol, a new complete algorithm that is guaranteed to produce correct and optimal results for all inputs while simultaneously being 2x faster than the incomplete Grisu3 and 4x faster than previous complete methods.

rawabstract:
Floating-point numbers are an essential part of modern software, recently gaining particular prominence on the web as the exclusive numeric format of Javascript. To use floating-point numbers, we require a way to convert binary machine representations into human readable decimal outputs. Existing conversion algorithms make trade-offs between completeness and performance. The classic Dragon4 algorithm by Steele and White and its later refinements achieve completeness --- i.e. produce correct and optimal outputs on all inputs --- by using arbitrary precision integer (bignum) arithmetic which leads to a high performance cost. On the other hand, the recent Grisu3 algorithm by Loitsch shows how to recover performance by using native integer arithmetic but sacrifices optimality for 0.5\% of all inputs. We present Errol, a new complete algorithm that is guaranteed to produce correct and optimal results for all inputs while simultaneously being 2x faster than the incomplete Grisu3 and 4x faster than previous complete methods.

entry:
@Article{Andrysco:2016:PFP,
  author =       "Marc Andrysco and Ranjit Jhala and Sorin Lerner",
  title =        "Printing floating-point numbers: a faster, always
                 correct method",
  journal =      j-SIGPLAN,
  volume =       "51",
  number =       "1",
  pages =        "555--567",
  month =        jan,
  year =         "2016",
  CODEN =        "SINODQ",
  DOI =          "https://doi.org/10.1145/2914770.2837654",
  ISSN =         "0362-1340 (print), 1523-2867 (print), 1558-1160
                 (electronic)",
  ISSN-L =       "0362-1340",
  bibdate =      "Thu Jun 9 17:13:57 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigplan2010.bib",
  abstract =     "Floating-point numbers are an essential part of modern
                 software, recently gaining particular prominence on the
                 web as the exclusive numeric format of Javascript. To
                 use floating-point numbers, we require a way to convert
                 binary machine representations into human readable
                 decimal outputs. Existing conversion algorithms make
                 trade-offs between completeness and performance. The
                 classic Dragon4 algorithm by Steele and White and its
                 later refinements achieve completeness --- i.e. produce
                 correct and optimal outputs on all inputs --- by using
                 arbitrary precision integer (bignum) arithmetic which
                 leads to a high performance cost. On the other hand,
                 the recent Grisu3 algorithm by Loitsch shows how to
                 recover performance by using native integer arithmetic
                 but sacrifices optimality for 0.5\% of all inputs. We
                 present Errol, a new complete algorithm that is
                 guaranteed to produce correct and optimal results for
                 all inputs while simultaneously being 2x faster than
                 the incomplete Grisu3 and 4x faster than previous
                 complete methods.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGPLAN Notices",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J706",
  remark =       "POPL '16 conference proceedings.",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2016:KTS
author: Anonymous
title: Keynote talks and special sessions
crossref: Montuschi:2016:ISC
pages: xv--xxi
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.33
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Anonymous
rawtitle: {Keynote} talks and special sessions
entry:
@InProceedings{Anonymous:2016:KTS,
  author =       "Anonymous",
  title =        "{Keynote} talks and special sessions",
  crossref =     "Montuschi:2016:ISC",
  pages =        "xv--xxi",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.33",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Area:2016:ACS
author: Ivan Area and Dimitar K. Dimitrov and Eduardo Godoy and Vanessa G. Paschoa
title: Approximate Calculation of Sums II: Gaussian Type Quadrature
journal: j-SIAM-J-NUMER-ANAL
volume: 54
number: 4
pages: 2210--2227
year: 2016
CODEN: SJNAAM
DOI: https://doi.org/10.1137/140993752
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Fri Sep 23 11:00:53 MDT 2016
bibtimestamp: 2016.09.23 11:00:53 MDT
fjournal: SIAM Journal on Numerical Analysis
keywords: Chebfun
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
onlinedate: January 2016
rawauthor: Iv{\'a}n Area and Dimitar K. Dimitrov and Eduardo Godoy and Vanessa G. Paschoa
rawtitle: Approximate Calculation of Sums {II}: {Gaussian} Type Quadrature
bibsource:
http://epubs.siam.org/http://epubs.siam.org/toc/sjnaam/54/4; https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjnumeranal2010.bib

entry:
@Article{Area:2016:ACS,
  author =       "Iv{\'a}n Area and Dimitar K. Dimitrov and Eduardo
                 Godoy and Vanessa G. Paschoa",
  title =        "Approximate Calculation of Sums {II}: {Gaussian} Type
                 Quadrature",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "54",
  number =       "4",
  pages =        "2210--2227",
  month =        "????",
  year =         "2016",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/140993752",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Fri Sep 23 11:00:53 MDT 2016",
  bibsource =    "http://epubs.siam.org/http://epubs.siam.org/toc/sjnaam/54/4;
                 https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjnumeranal2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  keywords =     "Chebfun",
  onlinedate =   "January 2016",
}


bibtype: article
filename: fparith.bib
label: Avenel:2016:STM
author: Christophe Avenel and Pierre Fortin and Mourad Gouicem and Samia Zaidi
title: Solving the Table Maker's Dilemma on Current SIMD Architectures
journal: j-SCPE
volume: 17
number: 3
pages: 237--250
year: 2016
ISSN: 1895-1767 OR 18951767
ISSNL: 1895-1767 OR 18951767
bibdate: Mon Jan 7 06:46:48 MST 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/scpe.bib
bibtimestamp: 2019.01.07 06:46:48 MST
URL: https://www.scpe.org/index.php/scpe/article/view/1183
fjournal: Scalable Computing: Practice and Experience
acknowledgement: ack-nhfb
journalURL: http://www.scpe.org/
rawauthor: Christophe Avenel and Pierre Fortin and Mourad Gouicem and Samia Zaidi
rawtitle: Solving the {Table Maker}'s Dilemma on Current {SIMD} Architectures
entry:
@Article{Avenel:2016:STM,
  author =       "Christophe Avenel and Pierre Fortin and Mourad Gouicem
                 and Samia Zaidi",
  title =        "Solving the {Table Maker}'s Dilemma on Current {SIMD}
                 Architectures",
  journal =      j-SCPE,
  volume =       "17",
  number =       "3",
  pages =        "237--250",
  month =        "????",
  year =         "2016",
  CODEN =        "????",
  ISSN =         "1895-1767",
  ISSN-L =       "1895-1767",
  bibdate =      "Mon Jan 7 06:46:48 MST 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/scpe.bib",
  URL =          "https://www.scpe.org/index.php/scpe/article/view/1183",
  acknowledgement = ack-nhfb,
  fjournal =     "Scalable Computing: Practice and Experience",
  journal-URL =  "http://www.scpe.org/",
}


bibtype: article
filename: fparith.bib
label: Bagnara:2016:EBF
author: Roberto Bagnara and Matthieu Carlier and Roberta Gori and Arnaud Gotlieb
title: Exploiting binary floating-point representations for constraint propagation
journal: j-INFORMS-J-COMPUT
volume: 28
number: 1
pages: 31--46
month: Winter
year: 2016
DOI: https://doi.org/10.1287/ijoc.2015.0663
ISSN: 1091-9856 (print), 1526-5528 (electronic) OR 10919856 (print), 15265528 (electronic)
ISSNL: 1091-9856 OR 10919856
MRclass: 68M07 (65Y04 68W40)
MRnumber: 3461544
bibdate: Mon Apr 9 08:22:11 MDT 2018
bibtimestamp: 2018.04.09 08:22:11 MDT
URL: https://pubsonline.informs.org/doi/abs/10.1287/ijoc.2015.0663
fjournal: INFORMS Journal on Computing
acknowledgement: ack-nhfb
ajournal: INFORMS J. Comput.
journalURL: https://pubsonline.informs.org/journal/ijoc
onlinedate: January 21, 2016
rawauthor: Roberto Bagnara and Matthieu Carlier and Roberta Gori and Arnaud Gotlieb
rawtitle: Exploiting binary floating-point representations for constraint propagation
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/informs-j-comput.bib

entry:
@Article{Bagnara:2016:EBF,
  author =       "Roberto Bagnara and Matthieu Carlier and Roberta Gori
                 and Arnaud Gotlieb",
  title =        "Exploiting binary floating-point representations for
                 constraint propagation",
  journal =      j-INFORMS-J-COMPUT,
  volume =       "28",
  number =       "1",
  pages =        "31--46",
  month =        "Winter",
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1287/ijoc.2015.0663",
  ISSN =         "1091-9856 (print), 1526-5528 (electronic)",
  ISSN-L =       "1091-9856",
  MRclass =      "68M07 (65Y04 68W40)",
  MRnumber =     "3461544",
  bibdate =      "Mon Apr 9 08:22:11 MDT 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/informs-j-comput.bib",
  URL =          "https://pubsonline.informs.org/doi/abs/10.1287/ijoc.2015.0663",
  acknowledgement = ack-nhfb,
  ajournal =     "INFORMS J. Comput.",
  fjournal =     "INFORMS Journal on Computing",
  journal-URL =  "https://pubsonline.informs.org/journal/ijoc",
  onlinedate =   "January 21, 2016",
}


bibtype: incollection
filename: fparith.bib
label: Bailey:2016:FRS
author: David H. Bailey and Jonathan M. Borwein and Victoria Stodden
editor: Harald Atmanspacher and Sabine Maasen
booktitle: Reproducibility: Principles, Problems, Practices, and Prospects
title: Facilitating Reproducibility in Scientific Computing: Principles and Practice
publisher: pub-WILEY
address: pub-WILEY:adr
pages: 205--231
year: 2016
DOI: https://doi.org/10.1002/9781118865064.ch9
ISBN: 1-118-86506-5 OR 1118865065
ISBN13: 978-1-118-86506-4 OR 9781118865064
bibdate: Thu Aug 11 11:01:49 2016
bibtimestamp: 2016.08.11 11:01:49 ???
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: David H. Bailey and Jonathan M. Borwein and Victoria Stodden
rawbooktitle: Reproducibility: Principles, Problems, Practices, and Prospects
raweditor: Harald Atmanspacher and Sabine Maasen
rawtitle: Facilitating Reproducibility in Scientific Computing: Principles and Practice
ORCID: Bailey, David H./0000-0002-7574-8342; Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
This chapter discusses the roots of the reproducibility problem in scientific computing, and summarizes some possible solutions that have been suggested in the community. Statistical overfitting can be thought of as an instance of ``selection bias'', wherein one presents the results of only those tests that support well one's hypothesis. The problem of statistical overfitting in computational science is perhaps best illustrated in the field of mathematical finance. Present-day researchers are not as fully aware of the potential pitfalls of performance reporting. In any event, various high-performance computing researchers have noted a resurrection of some of these questionable practices. Using high-precision arithmetic (higher than the standard IEEE 64-bit arithmetic) is often quite useful in ameliorating numerical difficulties and enhancing reproducibility. Very high-precision floating-point arithmetic is essential to obtain reproducible results in experimental mathematics and in related mathematical physics applications. The chapter also discusses reproducibility in symbolic computing.

keywords:
high-performance computing; high-precision arithmetic; mathematical physics; numerical reproducibility; performance reporting; scientific computing; statistical overfitting; symbolic computing

rawabstract:
This chapter discusses the roots of the reproducibility problem in scientific computing, and summarizes some possible solutions that have been suggested in the community. Statistical overfitting can be thought of as an instance of ``selection bias'', wherein one presents the results of only those tests that support well one's hypothesis. The problem of statistical overfitting in computational science is perhaps best illustrated in the field of mathematical finance. Present-day researchers are not as fully aware of the potential pitfalls of performance reporting. In any event, various high-performance computing researchers have noted a resurrection of some of these questionable practices. Using high-precision arithmetic (higher than the standard IEEE 64-bit arithmetic) is often quite useful in ameliorating numerical difficulties and enhancing reproducibility. Very high-precision floating-point arithmetic is essential to obtain reproducible results in experimental mathematics and in related mathematical physics applications. The chapter also discusses reproducibility in symbolic computing.

entry:
@InCollection{Bailey:2016:FRS,
  author =       "David H. Bailey and Jonathan M. Borwein and Victoria
                 Stodden",
  editor =       "Harald Atmanspacher and Sabine Maasen",
  booktitle =    "Reproducibility: Principles, Problems, Practices, and
                 Prospects",
  title =        "Facilitating Reproducibility in Scientific Computing:
                 Principles and Practice",
  publisher =    pub-WILEY,
  address =      pub-WILEY:adr,
  pages =        "205--231",
  year =         "2016",
  DOI =          "https://doi.org/10.1002/9781118865064.ch9",
  ISBN =         "1-118-86506-5",
  ISBN-13 =      "978-1-118-86506-4",
  bibdate =      "Thu Aug 11 11:01:49 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This chapter discusses the roots of the
                 reproducibility problem in scientific computing, and
                 summarizes some possible solutions that have been
                 suggested in the community. Statistical overfitting can
                 be thought of as an instance of ``selection bias'',
                 wherein one presents the results of only those tests
                 that support well one's hypothesis. The problem of
                 statistical overfitting in computational science is
                 perhaps best illustrated in the field of mathematical
                 finance. Present-day researchers are not as fully aware
                 of the potential pitfalls of performance reporting. In
                 any event, various high-performance computing
                 researchers have noted a resurrection of some of these
                 questionable practices. Using high-precision arithmetic
                 (higher than the standard IEEE 64-bit arithmetic) is
                 often quite useful in ameliorating numerical
                 difficulties and enhancing reproducibility. Very
                 high-precision floating-point arithmetic is essential
                 to obtain reproducible results in experimental
                 mathematics and in related mathematical physics
                 applications. The chapter also discusses
                 reproducibility in symbolic computing.",
  acknowledgement = ack-nhfb,
  author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
  keywords =     "high-performance computing; high-precision arithmetic;
                 mathematical physics; numerical reproducibility;
                 performance reporting; scientific computing;
                 statistical overfitting; symbolic computing",
  ORCID-numbers = "Bailey, David H./0000-0002-7574-8342; Borwein,
                 Jonathan/0000-0002-1263-0646",
  xxnote =       "This book was awarded the 2017 Prose Award
                 (``Honorable Mention'') in the category ``Textbook /
                 Best in Physical Sciences and Mathematics''; see
                 \url{https://proseawards.com/winners/2017-award-winners/}
                 and
                 \url{http://experimentalmath.org/2017/02/reproducibility-principles-problems-practices-and-prospects/}.",
}


bibtype: inproceedings
filename: fparith.bib
label: Bajard:2016:MFA
author: Jean-Claude Bajard and Julien Eynard and Nabil Merkiche
title: Multi-fault Attack Detection for RNS Cryptographic Architecture
crossref: Montuschi:2016:ISC
pages: 16--23
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.16
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Jean-Claude Bajard and Julien Eynard and Nabil Merkiche
rawtitle: Multi-fault Attack Detection for {RNS} Cryptographic Architecture
bibsource:
https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@InProceedings{Bajard:2016:MFA,
  author =       "Jean-Claude Bajard and Julien Eynard and Nabil
                 Merkiche",
  title =        "Multi-fault Attack Detection for {RNS} Cryptographic
                 Architecture",
  crossref =     "Montuschi:2016:ISC",
  pages =        "16--23",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.16",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Ballard:2016:INS
author: Grey Ballard and Austin R. Benson and Alex Druinsky and Benjamin Lipshitz and Oded Schwartz
title: Improving the Numerical Stability of Fast Matrix Multiplication
journal: j-SIAM-J-MAT-ANA-APPL
volume: 37
number: 4
pages: 1382--1418
year: 2016
CODEN: SJMAEL
DOI: https://doi.org/10.1137/15M1032168
ISSN: 0895-4798 (print), 1095-7162 (electronic) OR 08954798 (print), 10957162 (electronic)
ISSNL: 0895-4798 OR 08954798
bibdate: Fri Aug 25 09:01:43 MDT 2017
bibtimestamp: 2017.08.25 09:01:43 MDT
fjournal: SIAM Journal on Matrix Analysis and Applications
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/simax
onlinedate: January 2016
rawauthor: Grey Ballard and Austin R. Benson and Alex Druinsky and Benjamin Lipshitz and Oded Schwartz
rawtitle: Improving the Numerical Stability of Fast Matrix Multiplication
bibsource:
http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib

entry:
@Article{Ballard:2016:INS,
  author =       "Grey Ballard and Austin R. Benson and Alex Druinsky
                 and Benjamin Lipshitz and Oded Schwartz",
  title =        "Improving the Numerical Stability of Fast Matrix
                 Multiplication",
  journal =      j-SIAM-J-MAT-ANA-APPL,
  volume =       "37",
  number =       "4",
  pages =        "1382--1418",
  month =        "????",
  year =         "2016",
  CODEN =        "SJMAEL",
  DOI =          "https://doi.org/10.1137/15M1032168",
  ISSN =         "0895-4798 (print), 1095-7162 (electronic)",
  ISSN-L =       "0895-4798",
  bibdate =      "Fri Aug 25 09:01:43 MDT 2017",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Matrix Analysis and Applications",
  journal-URL =  "http://epubs.siam.org/simax",
  onlinedate =   "January 2016",
}


bibtype: misc
filename: fparith.bib
label: BEBOP:2016:RRB
author: Berkeley Benchmarking and OPtimization Group (BEBOP)
title: ReproBLAS: Reproducible BLAS
pages: 1382--1418
month: jan
monthnumber: 1
year: 2016
bibdate: Sat Jan 02 18:46:46 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.01.02 18:46:46 ???
URL: http://bebop.cs.berkeley.edu/reproblas/
acknowledgement: ack-nhfb
howpublished: Web site with software downloads.
rawauthor: {Berkeley Benchmarking and OPtimization Group (BEBOP)}
rawtitle: {ReproBLAS: Reproducible BLAS}
entry:
@Misc{BEBOP:2016:RRB,
  author =       "{Berkeley Benchmarking and OPtimization Group
                 (BEBOP)}",
  title =        "{ReproBLAS: Reproducible BLAS}",
  howpublished = "Web site with software downloads.",
  month =        jan,
  year =         "2016",
  bibdate =      "Sat Jan 02 18:46:46 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://bebop.cs.berkeley.edu/reproblas/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Bigou:2016:BTP
author: Karim Bigou and Arnaud Tisserand
title: Binary-Ternary Plus-Minus Modular Inversion in RNS
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 11
pages: 3495--3501
month: nov
monthnumber: 11
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2016.2529625
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Oct 11 05:14:24 MDT 2016
bibtimestamp: 2016.10.11 05:14:24 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Karim Bigou and Arnaud Tisserand
rawtitle: Binary-Ternary Plus-Minus Modular Inversion in {RNS}
entry:
@Article{Bigou:2016:BTP,
  author =       "Karim Bigou and Arnaud Tisserand",
  title =        "Binary-Ternary Plus-Minus Modular Inversion in {RNS}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "11",
  pages =        "3495--3501",
  month =        nov,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2016.2529625",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Oct 11 05:14:24 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: Bigou:2016:HPR
author: Karim Bigou and Arnaud Tisserand
title: Hybrid Position-Residues Number System
crossref: Montuschi:2016:ISC
pages: 126--133
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.15
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Karim Bigou and Arnaud Tisserand
rawtitle: Hybrid Position-Residues Number System
entry:
@InProceedings{Bigou:2016:HPR,
  author =       "Karim Bigou and Arnaud Tisserand",
  title =        "Hybrid Position-Residues Number System",
  crossref =     "Montuschi:2016:ISC",
  pages =        "126--133",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.15",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Biham:2016:BA
author: Eli Biham and Yaniv Carmeli and Adi Shamir
title: Bug Attacks
journal: j-J-CRYPTOLOGY
volume: 29
number: 4
pages: 775--805
month: oct
monthnumber: 10
year: 2016
CODEN: JOCREQ
DOI: https://doi.org/10.1007/s00145-015-9209-1
ISSN: 0933-2790 (print), 1432-1378 (electronic) OR 09332790 (print), 14321378 (electronic)
ISSNL: 0933-2790 OR 09332790
bibdate: Mon Sep 12 07:07:07 MDT 2016
bibtimestamp: 2016.09.12 07:07:07 MDT
fjournal: Journal of Cryptology
keywords: Bug attack; ElGamal encryption; Fault attack; Pohlig Hellman; RSA
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/145
rawauthor: Eli Biham and Yaniv Carmeli and Adi Shamir
rawtitle: Bug Attacks
bibsource:
http://link.springer.com/journal/145/29/4; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcryptology.bib

URL:
http://link.springer.com/accesspage/article/10.1007/s00145-015-9209-1; http://link.springer.com/article/10.1007/s00145-015-9209-1

remark:
From the abstract: ``The best-known example of such a bug is the Intel division bug, which resulted in slightly inaccurate results for extremely rare inputs. ldots such bugs can be a security disaster: decrypting ciphertexts on any computer which [sic] multiplies even one pair of numbers incorrectly can lead to full leakage of the secret key, sometimes with a single well-chosen ciphertext.

entry:
@Article{Biham:2016:BA,
  author =       "Eli Biham and Yaniv Carmeli and Adi Shamir",
  title =        "Bug Attacks",
  journal =      j-J-CRYPTOLOGY,
  volume =       "29",
  number =       "4",
  pages =        "775--805",
  month =        oct,
  year =         "2016",
  CODEN =        "JOCREQ",
  DOI =          "https://doi.org/10.1007/s00145-015-9209-1",
  ISSN =         "0933-2790 (print), 1432-1378 (electronic)",
  ISSN-L =       "0933-2790",
  bibdate =      "Mon Sep 12 07:07:07 MDT 2016",
  bibsource =    "http://link.springer.com/journal/145/29/4;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcryptology.bib",
  URL =          "http://link.springer.com/accesspage/article/10.1007/s00145-015-9209-1;
                 http://link.springer.com/article/10.1007/s00145-015-9209-1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Cryptology",
  journal-URL =  "http://link.springer.com/journal/145",
  keywords =     "Bug attack; ElGamal encryption; Fault attack; Pohlig
                 Hellman; RSA",
  remark =       "From the abstract: ``The best-known example of such a
                 bug is the Intel division bug, which resulted in
                 slightly inaccurate results for extremely rare inputs.
                 \ldots{} such bugs can be a security disaster:
                 decrypting ciphertexts on any computer which [sic]
                 multiplies even one pair of numbers incorrectly can
                 lead to full leakage of the secret key, sometimes with
                 a single well-chosen ciphertext.",
}


bibtype: techreport
filename: fparith.bib
label: Boldo:2016:RFA
author: Sylvie Boldo and Stef Graillat and Jean-Michel Muller
title: On the robustness of the 2Sum and Fast2Sum algorithms
type: Report
number: ensl-01310023
institution: Inria, LRI, CNRS & Universite Paris-Sud, Universite Paris-Saclay, France
pages: 1 + 17
day: 1
month: may
monthnumber: 5
year: 2016
bibdate: Thu May 19 10:09:03 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.05.19 10:09:03 ???
URL: https://hal-ens-lyon.archives-ouvertes.fr/ensl-01310023
keywords: accurate floating-point summation
acknowledgement: ack-nhfb
rawauthor: Sylvie Boldo and Stef Graillat and Jean-Michel Muller
rawtitle: On the robustness of the {2Sum} and {Fast2Sum} algorithms
abstract:
The 2Sum and Fast2Sum algorithms are important building blocks in numerical computing. They are used (implicitly or explicitly) in many compensated algorithms (such as compensated summation or compensated polynomial evaluation). They are also used for manipulating floating-point expansions. We show that these algorithms are much more robust than it is usually believed: the returned result makes sense even when the rounding function is not round-to-nearest, and they are almost immune to overflow.

rawabstract:
The 2Sum and Fast2Sum algorithms are important building blocks in numerical computing. They are used (implicitly or explicitly) in many compensated algorithms (such as compensated summation or compensated polynomial evaluation). They are also used for manipulating floating-point expansions. We show that these algorithms are much more robust than it is usually believed: the returned result makes sense even when the rounding function is not round-to-nearest, and they are almost immune to overflow.

entry:
@TechReport{Boldo:2016:RFA,
  author =       "Sylvie Boldo and Stef Graillat and Jean-Michel
                 Muller",
  title =        "On the robustness of the {2Sum} and {Fast2Sum}
                 algorithms",
  type =         "Report",
  number =       "ensl-01310023",
  institution =  "Inria, LRI, CNRS \& Universit{\'e} Paris-Sud,
                 Universit{\'e} Paris-Saclay, France",
  pages =        "1 + 17",
  day =          "1",
  month =        may,
  year =         "2016",
  bibdate =      "Thu May 19 10:09:03 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal-ens-lyon.archives-ouvertes.fr/ensl-01310023",
  abstract =     "The 2Sum and Fast2Sum algorithms are important
                 building blocks in numerical computing. They are used
                 (implicitly or explicitly) in many compensated
                 algorithms (such as compensated summation or
                 compensated polynomial evaluation). They are also used
                 for manipulating floating-point expansions. We show
                 that these algorithms are much more robust than it is
                 usually believed: the returned result makes sense even
                 when the rounding function is not round-to-nearest, and
                 they are almost immune to overflow.",
  acknowledgement = ack-nhfb,
  keywords =     "accurate floating-point summation",
}


bibtype: article
filename: fparith.bib
label: Brisebarre:2016:CBB
author: Nicolas Brisebarre and Christoph Lauter and Marc Mezzarobba and Jean-Michel Muller
title: Comparison between Binary and Decimal Floating-Point Numbers
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 7
pages: 2032--2044
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2479602
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Jun 15 06:49:34 MDT 2016
bibtimestamp: 2016.06.15 06:49:34 MDT
fjournal: IEEE Transactions on Computers -
remark: Longer report in cite Brisebarre:2013:CBB.
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Nicolas Brisebarre and Christoph Lauter and Marc Mezzarobba and Jean-Michel Muller
rawtitle: Comparison between Binary and Decimal Floating-Point Numbers
entry:
@Article{Brisebarre:2016:CBB,
  author =       "Nicolas Brisebarre and Christoph Lauter and Marc
                 Mezzarobba and Jean-Michel Muller",
  title =        "Comparison between Binary and Decimal Floating-Point
                 Numbers",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "7",
  pages =        "2032--2044",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2479602",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Jun 15 06:49:34 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers -",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  remark =       "Longer report in \cite{Brisebarre:2013:CBB}.",
}


bibtype: inproceedings
filename: fparith.bib
label: Brzicova:2016:LMD
author: Marta Brzicova and Christiane Frougny and Edita Pelantova and Milena Svobodova
title: On-line Multiplication and Division in Real and Complex Bases
crossref: Montuschi:2016:ISC
pages: 134--141
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.13
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Marta Brzicov{\'a} and Christiane Frougny and Edita Pelantov{\'a} and Milena Svobodov{\'a}
rawtitle: On-line Multiplication and Division in Real and Complex Bases
entry:
@InProceedings{Brzicova:2016:LMD,
  author =       "Marta Brzicov{\'a} and Christiane Frougny and Edita
                 Pelantov{\'a} and Milena Svobodov{\'a}",
  title =        "On-line Multiplication and Division in Real and
                 Complex Bases",
  crossref =     "Montuschi:2016:ISC",
  pages =        "134--141",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.13",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: inproceedings
filename: fparith.bib
label: Cervantes:2016:IOC
booktitle: 2016 IEEE 36th Central American and Panama Convention (CONCAPAN XXXVI)
title: Implementation of an open core IEEE 754-based FPU with non-linear arithmetic support
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--6
year: 2016
DOI: https://doi.org/10.1109/CONCAPAN.2016.7942354
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
acknowledgement: ack-nhfb
rawbooktitle: {2016 IEEE 36th Central American and Panama Convention (CONCAPAN XXXVI)}
rawtitle: Implementation of an open core {IEEE 754}-based {FPU} with non-linear arithmetic support
author:
Adrian Cervantes and Francis Lopez and Jeffry Quiros and Diego Rodriguez and Carlos Salazar-Garcia and Carlos Meza and Alfonso Chacon-Rodriguez

keywords:
Adders; CORDIC; Field programmable gate arrays; floating point arithmetic; FPGA; Hardware; IEEE 754 floating point representation; Mathematical model; Process control; Signal processing algorithms; Standards; Verilog

rawauthor:
Adrian Cervantes and Francis Lopez and Jeffry Quiros and Diego Rodriguez and Carlos Salazar-Garcia and Carlos Meza and Alfonso Chacon-Rodriguez

entry:
@InProceedings{Cervantes:2016:IOC,
  author =       "Adrian Cervantes and Francis Lopez and Jeffry Quiros
                 and Diego Rodriguez and Carlos Salazar-Garcia and
                 Carlos Meza and Alfonso Chacon-Rodriguez",
  booktitle =    "{2016 IEEE 36th Central American and Panama Convention
                 (CONCAPAN XXXVI)}",
  title =        "Implementation of an open core {IEEE 754}-based {FPU}
                 with non-linear arithmetic support",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--6",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/CONCAPAN.2016.7942354",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Adders; CORDIC; Field programmable gate arrays;
                 floating point arithmetic; FPGA; Hardware; IEEE 754
                 floating point representation; Mathematical model;
                 Process control; Signal processing algorithms;
                 Standards; Verilog",
}


bibtype: article
filename: fparith.bib
label: Chen:2016:CFP
author: Jiyang Chen and Yuanwu Lei and Yuanxi Peng and Tingting He and Ziye Deng
title: Configurable Floating-Point FFT Accelerator on FPGA Based Multiple-Rotation CORDIC
journal: Chinese Journal of Electronics
volume: 25
number: 6
pages: 1063--1070
year: 2016
DOI: https://doi.org/10.1049/cje.2016.08.002
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
URL: https://onlinelibrary.wiley.com/doi/abs/10.1049/cje.2016.08.002
acknowledgement: ack-nhfb
rawauthor: Jiyang Chen and Yuanwu Lei and Yuanxi Peng and Tingting He and Ziye Deng
rawtitle: Configurable Floating-Point {FFT} Accelerator on {FPGA} Based Multiple-Rotation {CORDIC}
keywords:
clock frequency; configurable floating-point FFT accelerator; coordinate rotation digital computer; Coordinate rotation digital computer (CORDIC); CORDIC algorithm; CORDIC rotation; Costs; Digital computers; Fast Fourier transform (FFT); fast Fourier transform accelerator; fast Fourier transforms; Fast Fourier transforms; Field programmable gate arrays; floating point arithmetic; Floating-point; FPGA; FPGA based multiple-rotation CORDIC; hardware cost; Prediction algorithms; Random access memory; Real-time systems; redundant CORDIC; redundant number systems; segmented-parallel iteration; signal processing; Signal processing; Signal processing algorithms; Table lookup; twiddle angles; twiddle direction prediction

entry:
@Article{Chen:2016:CFP,
  author =       "Jiyang Chen and Yuanwu Lei and Yuanxi Peng and
                 Tingting He and Ziye Deng",
  title =        "Configurable Floating-Point {FFT} Accelerator on
                 {FPGA} Based Multiple-Rotation {CORDIC}",
  journal =      "Chinese Journal of Electronics",
  volume =       "25",
  number =       "6",
  pages =        "1063--1070",
  year =         "2016",
  DOI =          "https://doi.org/10.1049/cje.2016.08.002",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://onlinelibrary.wiley.com/doi/abs/10.1049/cje.2016.08.002",
  acknowledgement = ack-nhfb,
  eprint =       "https://onlinelibrary.wiley.com/doi/pdf/10.1049/cje.2016.08.002",
  keywords =     "clock frequency; configurable floating-point FFT
                 accelerator; coordinate rotation digital computer;
                 Coordinate rotation digital computer (CORDIC); CORDIC
                 algorithm; CORDIC rotation; Costs; Digital computers;
                 Fast Fourier transform (FFT); fast Fourier transform
                 accelerator; fast Fourier transforms; Fast Fourier
                 transforms; Field programmable gate arrays; floating
                 point arithmetic; Floating-point; FPGA; FPGA based
                 multiple-rotation CORDIC; hardware cost; Prediction
                 algorithms; Random access memory; Real-time systems;
                 redundant CORDIC; redundant number systems;
                 segmented-parallel iteration; signal processing; Signal
                 processing; Signal processing algorithms; Table lookup;
                 twiddle angles; twiddle direction prediction",
}


bibtype: article
filename: fparith.bib
label: Chen:2016:DAR
author: Linbin Chen and Jie Han and Weiqiang Liu and Fabrizio Lombardi
title: On the Design of Approximate Restoring Dividers for Error-Tolerant Applications
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 8
pages: 2522--2533
month: aug
monthnumber: 8
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2494005
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 11 18:58:52 MDT 2016
bibtimestamp: 2016.07.11 18:58:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Linbin Chen and Jie Han and Weiqiang Liu and Fabrizio Lombardi
rawtitle: On the Design of Approximate Restoring Dividers for Error-Tolerant Applications
entry:
@Article{Chen:2016:DAR,
  author =       "Linbin Chen and Jie Han and Weiqiang Liu and Fabrizio
                 Lombardi",
  title =        "On the Design of Approximate Restoring Dividers for
                 Error-Tolerant Applications",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "8",
  pages =        "2522--2533",
  month =        aug,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2494005",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 11 18:58:52 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Chen:2016:PSA
author: D. D. Chen and G. X. Yao and R. C. Cheung and D. Pao and C. K. Koc
title: Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 1
pages: 147--160
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2417553
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Dec 15 09:36:24 MST 2015
bibtimestamp: 2015.12.15 09:36:24 MST
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: D. D. Chen and G. X. Yao and R. C. Cheung and D. Pao and C. K. Koc
rawtitle: Parameter Space for the Architecture of {FFT}-Based {Montgomery} Modular Multiplication
entry:
@Article{Chen:2016:PSA,
  author =       "D. D. Chen and G. X. Yao and R. C. Cheung and D. Pao
                 and C. K. Koc",
  title =        "Parameter Space for the Architecture of {FFT}-Based
                 {Montgomery} Modular Multiplication",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "1",
  pages =        "147--160",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2417553",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Dec 15 09:36:24 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Coleman:2016:LCT
author: J. N. Coleman and R. Che Ismail
title: LNS with Co-Transformation Competes with Floating-Point
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 1
pages: 136--146
year: 2016
CODEN: ITCOB4
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Dec 15 09:36:24 MST 2015
bibtimestamp: 2015.12.15 09:36:24 MST
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. N. Coleman and R. Che Ismail
rawtitle: {LNS} with Co-Transformation Competes with Floating-Point
entry:
@Article{Coleman:2016:LCT,
  author =       "J. N. Coleman and R. Che Ismail",
  title =        "{LNS} with Co-Transformation Competes with
                 Floating-Point",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "1",
  pages =        "136--146",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Dec 15 09:36:24 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: Collange:2016:PFP
author: Caroline Collange and Mioara Joldes and Jean-Michel Muller and Valentina Popescu
editor: IEEE
title: Parallel floating-point expansions for extended-precision GPU computations
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 139--146
year: 2016
DOI: https://doi.org/10.1109/ASAP.2016.7760783
ISBN: 1-5090-1503-5 OR 1509015035
ISBN13: 978-1-5090-1503-0 OR 9781509015030
ISSN: 2160-0511 (print), 2160-052X (electronic) OR 21600511 (print), 2160052X (electronic)
ISSNL: 2160-0511 OR 21600511
bibdate: Fri Sep 29 10:56:29 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.09.29 10:56:29 ???
acknowledgement: ack-nhfb
rawauthor: Caroline Collange and Mioara Joldes and Jean-Michel Muller and Valentina Popescu
raweditor: {IEEE}
rawtitle: Parallel floating-point expansions for extended-precision {GPU} computations
booktitle:
2016 IEEE 27th International Conference on Application-Specific Systems, Architectures and Processors (ASAP). July 6--8, 2016. Imperial College London

rawbooktitle:
{2016 IEEE 27th International Conference on Application-Specific Systems, Architectures and Processors (ASAP). July 6--8, 2016. Imperial College London}

entry:
@InProceedings{Collange:2016:PFP,
  author =       "Caroline Collange and Mioara Joldes and Jean-Michel
                 Muller and Valentina Popescu",
  editor =       "{IEEE}",
  booktitle =    "{2016 IEEE 27th International Conference on
                 Application-Specific Systems, Architectures and
                 Processors (ASAP). July 6--8, 2016. Imperial College
                 London}",
  title =        "Parallel floating-point expansions for
                 extended-precision {GPU} computations",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "139--146",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ASAP.2016.7760783",
  ISBN =         "1-5090-1503-5",
  ISBN-13 =      "978-1-5090-1503-0",
  ISSN =         "2160-0511 (print), 2160-052X (electronic)",
  ISSN-L =       "2160-0511",
  bibdate =      "Fri Sep 29 10:56:29 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Cui:2016:PDM
author: Xiaoping Cui and Weiqiang Liu and Dong Wenwen and Fabrizio Lombardi
title: A Parallel Decimal Multiplier Using Hybrid Binary Coded Decimal (BCD) Codes
crossref: Montuschi:2016:ISC
pages: 150--155
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.8
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Xiaoping Cui and Weiqiang Liu and Dong Wenwen and Fabrizio Lombardi
rawtitle: A Parallel Decimal Multiplier Using Hybrid Binary Coded Decimal ({BCD}) Codes
entry:
@InProceedings{Cui:2016:PDM,
  author =       "Xiaoping Cui and Weiqiang Liu and Dong Wenwen and
                 Fabrizio Lombardi",
  title =        "A Parallel Decimal Multiplier Using Hybrid Binary
                 Coded Decimal ({BCD}) Codes",
  crossref =     "Montuschi:2016:ISC",
  pages =        "150--155",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.8",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: techreport
filename: fparith.bib
label: Damouche:2016:TSB
title: Toward a Standard Benchmark Format and Suite for Floating-Point Analysis
type: Report
institution: Universite de Perpignan Via Domitia and University of Washington
address: Perpignan, France and Seattle, WA, USA
pages: 15
month: jul
monthnumber: 7
year: 2016
bibdate: Thu Oct 17 05:26:46 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.17 05:26:46 ???
URL: https://fpbench.org/; https://homes.cs.washington.edu/~ztatlock/pubs/fpbench-damouche-nsv16.pdf
acknowledgement: ack-nhfb
rawtitle: Toward a Standard Benchmark Format and Suite for Floating-Point Analysis
author:
Nasrine Damouche and Matthieu Martel and Pavel Panchekha and Jason Qiu and Alex Sanchez-Stern and Zachary Tatlock

abstract:
We introduce FPBench, a standard benchmark format for validation and optimization of numerical accuracy in floating-point computations. FPBench is a first step toward addressing an increasing need in our community for comparisons and combinations of tools from different application domains. To this end, FPBench provides a basic floating-point benchmark format and accuracy measures for comparing different tools. The FPBench format and measures allow comparing and composing different floating-point tools. We describe the FPBench format and measures and show that FPBench expresses benchmarks from recent papers in the literature, by building an initial benchmark suite drawn from these papers. We intend for FPBench to grow into a standard benchmark suite for the members of the floating-point tools research community

rawabstract:
We introduce FPBench, a standard benchmark format for validation and optimization of numerical accuracy in floating-point computations. FPBench is a first step toward addressing an increasing need in our community for comparisons and combinations of tools from different application domains. To this end, FPBench provides a basic floating-point benchmark format and accuracy measures for comparing different tools. The FPBench format and measures allow comparing and composing different floating-point tools. We describe the FPBench format and measures and show that FPBench expresses benchmarks from recent papers in the literature, by building an initial benchmark suite drawn from these papers. We intend for FPBench to grow into a standard benchmark suite for the members of the floating-point tools research community

rawauthor:
Nasrine Damouche and Matthieu Martel and Pavel Panchekha and Jason Qiu and Alex Sanchez-Stern and Zachary Tatlock

entry:
@TechReport{Damouche:2016:TSB,
  author =       "Nasrine Damouche and Matthieu Martel and Pavel
                 Panchekha and Jason Qiu and Alex Sanchez-Stern and
                 Zachary Tatlock",
  title =        "Toward a Standard Benchmark Format and Suite for
                 Floating-Point Analysis",
  type =         "Report",
  number =       "??",
  institution =  "Universit{\'e} de Perpignan Via Domitia and University
                 of Washington",
  address =      "Perpignan, France and Seattle, WA, USA",
  pages =        "15",
  month =        jul,
  year =         "2016",
  bibdate =      "Thu Oct 17 05:26:46 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://fpbench.org/;
                 https://homes.cs.washington.edu/~ztatlock/pubs/fpbench-damouche-nsv16.pdf",
  abstract =     "We introduce FPBench, a standard benchmark format for
                 validation and optimization of numerical accuracy in
                 floating-point computations. FPBench is a first step
                 toward addressing an increasing need in our community
                 for comparisons and combinations of tools from
                 different application domains. To this end, FPBench
                 provides a basic floating-point benchmark format and
                 accuracy measures for comparing different tools. The
                 FPBench format and measures allow comparing and
                 composing different floating-point tools. We describe
                 the FPBench format and measures and show that FPBench
                 expresses benchmarks from recent papers in the
                 literature, by building an initial benchmark suite
                 drawn from these papers. We intend for FPBench to grow
                 into a standard benchmark suite for the members of the
                 floating-point tools research community",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: DelBarrio:2016:PCS
author: Alberto A. Del Barrio and Roman Hermida and Seda Ogrenci Memik
title: A Partial Carry-Save On-the-Fly Correction Multispeculative Multiplier
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 11
pages: 3251--3264
month: nov
monthnumber: 11
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2016.2529626
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Oct 11 05:14:24 MDT 2016
bibtimestamp: 2016.10.11 05:14:24 MDT
fjournal: IEEE Transactions on Computers
keywords: Adders; Benchmark testing; Computer science; Delays; Encoding; Mathematical model; Proposals
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Alberto A. {Del Barrio} and Rom{\'a}n Hermida and Seda Ogrenci Memik
rawtitle: A Partial Carry-Save On-the-Fly Correction Multispeculative Multiplier
entry:
@Article{DelBarrio:2016:PCS,
  author =       "Alberto A. {Del Barrio} and Rom{\'a}n Hermida and Seda
                 Ogrenci Memik",
  title =        "A Partial Carry-Save On-the-Fly Correction
                 Multispeculative Multiplier",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "11",
  pages =        "3251--3264",
  month =        nov,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2016.2529626",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Oct 11 05:14:24 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "Adders; Benchmark testing; Computer science; Delays;
                 Encoding; Mathematical model; Proposals",
}


bibtype: techreport
filename: fparith.bib
label: Demmel:2016:ERF
author: James Demmel and Peter Ahrens and Hong Diep Nguyen
title: Efficient reproducible floating point summation and BLAS
type: Technical Report
number: UCB/EECS-2016-121
institution: EECS Department, University of California, Berkeley
address: Berkeley, CA, USA
pages: 3251--3264
year: 2016
bibdate: Mon Feb 10 17:49:15 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 17:49:15 ???
URL: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-121.html
keywords: accurate floating-point summation; reproducible floating point summation
acknowledgement: ack-nhfb
rawauthor: James Demmel and Peter Ahrens and Hong Diep Nguyen
rawtitle: Efficient reproducible floating point summation and {BLAS}
abstract:
We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on a subset of the IEEE Floating Point Standard 754-2008. It is communication-optimal, in the sense that it does just one pass over the data in the sequential case, or one reduction operation in the parallel case, requiring an ``accumulator'' represented by just 6 floating point words (more can be used if higher precision is desired). The arithmetic cost with a 6-word accumulator is $ 7 n $ floating point additions to sum $n$ words, and (in IEEE double precision) the final error bound can be up to $ 10^( - 8)$ times smaller than the error bound for conventional summation. We describe the basic summation algorithm, the software infrastructure used to build reproducible BLAS (ReproBLAS), and performance results. For example, when computing the dot product of 4096 double precision floating point numbers, we get a $ 4 times $ slowdown compared to Intel Math Kernel Library (MKL) running on an Intel Core i7-2600 CPU operating at 3.4 GHz and 256 KB L2 Cache.

rawabstract:
We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on a subset of the IEEE Floating Point Standard 754-2008. It is communication-optimal, in the sense that it does just one pass over the data in the sequential case, or one reduction operation in the parallel case, requiring an ``accumulator'' represented by just 6 floating point words (more can be used if higher precision is desired). The arithmetic cost with a 6-word accumulator is $ 7 n $ floating point additions to sum $n$ words, and (in IEEE double precision) the final error bound can be up to $ 10^( - 8)$ times smaller than the error bound for conventional summation. We describe the basic summation algorithm, the software infrastructure used to build reproducible BLAS (ReproBLAS), and performance results. For example, when computing the dot product of 4096 double precision floating point numbers, we get a $ 4 \times $ slowdown compared to Intel Math Kernel Library (MKL) running on an Intel Core i7-2600 CPU operating at 3.4 GHz and 256 KB L2 Cache.

entry:
@TechReport{Demmel:2016:ERF,
  author =       "James Demmel and Peter Ahrens and Hong Diep Nguyen",
  title =        "Efficient reproducible floating point summation and
                 {BLAS}",
  type =         "Technical Report",
  number =       "UCB/EECS-2016-121",
  institution =  "EECS Department, University of California, Berkeley",
  address =      "Berkeley, CA, USA",
  year =         "2016",
  bibdate =      "Mon Feb 10 17:49:15 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-121.html",
  abstract =     "We define reproducibility to mean getting bitwise
                 identical results from multiple runs of the same
                 program, perhaps with different hardware resources or
                 other changes that should ideally not change the
                 answer. Many users depend on reproducibility for
                 debugging or correctness. However, dynamic scheduling
                 of parallel computing resources, combined with
                 nonassociativity of floating point addition, makes
                 attaining reproducibility a challenge even for simple
                 operations like summing a vector of numbers, or more
                 complicated operations like the Basic Linear Algebra
                 Subprograms (BLAS). We describe an algorithm that
                 computes a reproducible sum of floating point numbers,
                 independent of the order of summation. The algorithm
                 depends only on a subset of the IEEE Floating Point
                 Standard 754-2008. It is communication-optimal, in the
                 sense that it does just one pass over the data in the
                 sequential case, or one reduction operation in the
                 parallel case, requiring an ``accumulator'' represented
                 by just 6 floating point words (more can be used if
                 higher precision is desired). The arithmetic cost with
                 a 6-word accumulator is $ 7 n $ floating point
                 additions to sum $n$ words, and (in IEEE double
                 precision) the final error bound can be up to $ 10^( -
                 8)$ times smaller than the error bound for conventional
                 summation. We describe the basic summation algorithm,
                 the software infrastructure used to build reproducible
                 BLAS (ReproBLAS), and performance results. For example,
                 when computing the dot product of 4096 double precision
                 floating point numbers, we get a $ 4 \times $ slowdown
                 compared to Intel Math Kernel Library (MKL) running on
                 an Intel Core i7-2600 CPU operating at 3.4 GHz and 256
                 KB L2 Cache.",
  acknowledgement = ack-nhfb,
  keywords =     "accurate floating-point summation; reproducible
                 floating point summation",
}


bibtype: inproceedings
filename: fparith.bib
label: Denis:2016:VCF
author: Christophe Denis and Pablo de Oliveira Castro and Eric Petit
title: Verificarlo: Checking Floating Point Accuracy through Monte Carlo Arithmetic
crossref: Montuschi:2016:ISC
pages: 55--62
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.31
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Christophe Denis and Pablo de Oliveira Castro and Eric Petit
rawtitle: {Verificarlo}: Checking Floating Point Accuracy through {Monte Carlo} Arithmetic
entry:
@InProceedings{Denis:2016:VCF,
  author =       "Christophe Denis and Pablo de Oliveira Castro and Eric
                 Petit",
  title =        "{Verificarlo}: Checking Floating Point Accuracy
                 through {Monte Carlo} Arithmetic",
  crossref =     "Montuschi:2016:ISC",
  pages =        "55--62",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.31",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: DHollander:2016:HLS
author: Erik H. D'Hollander
title: High-Level Synthesis Optimization for Blocked Floating-Point Matrix Multiplication
journal: j-COMP-ARCH-NEWS
volume: 44
number: 4
pages: 74--79
month: sep
monthnumber: 9
year: 2016
CODEN: CANED2
DOI: https://doi.org/10.1145/3039902.3039916
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Mon Jun 5 18:01:57 MDT 2017
bibtimestamp: 2017.06.05 18:01:57 MDT
fjournal: ACM SIGARCH Computer Architecture News
remark: HEART '16 conference proceedings.
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/sigarch
rawauthor: Erik H. D'Hollander
rawtitle: High-Level Synthesis Optimization for Blocked Floating-Point Matrix Multiplication
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigarch.bib

abstract:
In the last decade floating-point matrix multiplication on FPGAs has been studied extensively and efficient architectures as well as detailed performance models have been developed. By design these IP cores take a fixed footprint which not necessarily optimizes the use of all available resources. Moreover, the low-level architectures are not easily amenable to a parameterized synthesis. In this paper high-level synthesis is used to fine-tune the configuration parameters in order to achieve the highest performance with maximal resource utilization. Anexploration strategy is presented to optimize the use of critical resources (DSPs, memory) for any given FPGA. To account for the limited memory size on the FPGA, a block-oriented matrix multiplication is organized such that the block summation is done on the CPU while the block multiplication occurs on the logic fabric simultaneously. The communication overhead between the CPU and the FPGA is minimized by streaming the blocks in a Gray code ordering scheme which maximizes the data reuse for consecutive block matrix product calculations. Using highlevel synthesis optimization, the programmable logic operates at 93% of the theoretical peak performance and the combined CPU-FPGA design achieves 76% of the available hardware processing speed for the floating-point multiplication of 2K by 2K matrices.

rawabstract:
In the last decade floating-point matrix multiplication on FPGAs has been studied extensively and efficient architectures as well as detailed performance models have been developed. By design these IP cores take a fixed footprint which not necessarily optimizes the use of all available resources. Moreover, the low-level architectures are not easily amenable to a parameterized synthesis. In this paper high-level synthesis is used to fine-tune the configuration parameters in order to achieve the highest performance with maximal resource utilization. An\ exploration strategy is presented to optimize the use of critical resources (DSPs, memory) for any given FPGA. To account for the limited memory size on the FPGA, a block-oriented matrix multiplication is organized such that the block summation is done on the CPU while the block multiplication occurs on the logic fabric simultaneously. The communication overhead between the CPU and the FPGA is minimized by streaming the blocks in a Gray code ordering scheme which maximizes the data reuse for consecutive block matrix product calculations. Using highlevel synthesis optimization, the programmable logic operates at 93\% of the theoretical peak performance and the combined CPU-FPGA design achieves 76\% of the available hardware processing speed for the floating-point multiplication of 2K by 2K matrices.

entry:
@Article{DHollander:2016:HLS,
  author =       "Erik H. D'Hollander",
  title =        "High-Level Synthesis Optimization for Blocked
                 Floating-Point Matrix Multiplication",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "44",
  number =       "4",
  pages =        "74--79",
  month =        sep,
  year =         "2016",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/3039902.3039916",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Mon Jun 5 18:01:57 MDT 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigarch.bib",
  abstract =     "In the last decade floating-point matrix
                 multiplication on FPGAs has been studied extensively
                 and efficient architectures as well as detailed
                 performance models have been developed. By design these
                 IP cores take a fixed footprint which not necessarily
                 optimizes the use of all available resources. Moreover,
                 the low-level architectures are not easily amenable to
                 a parameterized synthesis. In this paper high-level
                 synthesis is used to fine-tune the configuration
                 parameters in order to achieve the highest performance
                 with maximal resource utilization. An\ exploration
                 strategy is presented to optimize the use of critical
                 resources (DSPs, memory) for any given FPGA. To account
                 for the limited memory size on the FPGA, a
                 block-oriented matrix multiplication is organized such
                 that the block summation is done on the CPU while the
                 block multiplication occurs on the logic fabric
                 simultaneously. The communication overhead between the
                 CPU and the FPGA is minimized by streaming the blocks
                 in a Gray code ordering scheme which maximizes the data
                 reuse for consecutive block matrix product
                 calculations. Using highlevel synthesis optimization,
                 the programmable logic operates at 93\% of the
                 theoretical peak performance and the combined CPU-FPGA
                 design achieves 76\% of the available hardware
                 processing speed for the floating-point multiplication
                 of 2K by 2K matrices.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "https://dl.acm.org/loi/sigarch",
  remark =       "HEART '16 conference proceedings.",
}


bibtype: inproceedings
filename: fparith.bib
label: Du:2016:AEB
author: Peibing Du and Hao Jiang and Housen Li and Lizhi Cheng and Canqun Yang
editor: Hong Shen and Yingpeng Sang and Hui Tian
title: Accurate evaluation of bivariate polynomials
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 51--56
year: 2016
DOI: https://doi.org/10.1109/PDCAT.2016.026
ISBN: 1-5090-5081-7, 1-5090-5082-5 (print) OR 1509050817, 1509050825 (print)
ISBN13: 978-1-5090-5081-9, 978-1-5090-5082-6 (print) OR 9781509050819, 9781509050826 (print)
bibdate: Mon Mar 19 14:28:19 2018
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2018.03.19 14:28:19 ???
URL: https://ieeexplore.ieee.org/document/7943331/
remark: IEEE Computer Society Order Number E5982.
acknowledgement: ack-nhfb
rawauthor: Peibing Du and Hao Jiang and Housen Li and Lizhi Cheng and Canqun Yang
raweditor: Hong Shen and Yingpeng Sang and Hui Tian
rawtitle: Accurate evaluation of bivariate polynomials
booktitle:
Proceedings of the Seventeenth International Conference on Parallel and Distributed Computing, Applications and Technologies, PDCAT 2016, Guangzhou, China, December 16--18, 2016

rawbooktitle:
Proceedings of the {Seventeenth International Conference on Parallel and Distributed Computing, Applications and Technologies, PDCAT 2016, Guangzhou, China, December 16--18, 2016}

entry:
@InProceedings{Du:2016:AEB,
  author =       "Peibing Du and Hao Jiang and Housen Li and Lizhi Cheng
                 and Canqun Yang",
  editor =       "Hong Shen and Yingpeng Sang and Hui Tian",
  booktitle =    "Proceedings of the {Seventeenth International
                 Conference on Parallel and Distributed Computing,
                 Applications and Technologies, PDCAT 2016, Guangzhou,
                 China, December 16--18, 2016}",
  title =        "Accurate evaluation of bivariate polynomials",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "51--56",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/PDCAT.2016.026",
  ISBN =         "1-5090-5081-7, 1-5090-5082-5 (print)",
  ISBN-13 =      "978-1-5090-5081-9, 978-1-5090-5082-6 (print)",
  bibdate =      "Mon Mar 19 14:28:19 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://ieeexplore.ieee.org/document/7943331/",
  acknowledgement = ack-nhfb,
  remark =       "IEEE Computer Society Order Number E5982.",
}


bibtype: inproceedings
filename: fparith.bib
label: Dukhan:2016:WFP
author: M. Dukhan and R. Vuduc and J. Riedy
title: Wanted: Floating-Point Add Round-off Error instruction
pages: ??--??
year: 2016
bibdate: Wed Oct 12 18:36:50 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.10.12 18:36:50 ???
URL: http://arxiv.org/abs/1603.00491
acknowledgement: ack-nhfb
rawauthor: M. Dukhan and R. Vuduc and J. Riedy
rawtitle: Wanted: Floating-Point Add Round-off Error instruction
booktitle:
Proceedings of the 2nd International Workshop on Performance Modeling: Methods and Applications (PMMA16) at ISC High Performance, Frankfurt, Germany, [June or] July 2016

rawbooktitle:
Proceedings of the {2nd International Workshop on Performance Modeling: Methods and Applications (PMMA16) at ISC High Performance, Frankfurt, Germany, [June or] July 2016}

entry:
@InProceedings{Dukhan:2016:WFP,
  author =       "M. Dukhan and R. Vuduc and J. Riedy",
  editor =       "????",
  booktitle =    "Proceedings of the {2nd International Workshop on
                 Performance Modeling: Methods and Applications (PMMA16)
                 at ISC High Performance, Frankfurt, Germany, [June or]
                 July 2016}",
  title =        "Wanted: Floating-Point Add Round-off Error
                 instruction",
  publisher =    "????",
  address =      "????",
  pages =        "??--??",
  year =         "2016",
  LCCN =         "",
  bibdate =      "Wed Oct 12 18:36:50 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1603.00491",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: El-Razouk:2016:NAD
author: Hayssam El-Razouk and Arash Reyhani-Masoleh
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 8
pages: 2495--2509
month: aug
monthnumber: 8
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2481408
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 11 18:58:52 MDT 2016
bibtimestamp: 2016.07.11 18:58:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Hayssam El-Razouk and Arash Reyhani-Masoleh
title:
New Architectures for Digit-Level Single, Hybrid-Double, Hybrid-Triple Field Multiplications and Exponentiation Using Gaussian Normal Bases

rawtitle:
New Architectures for Digit-Level Single, Hybrid-Double, Hybrid-Triple Field Multiplications and Exponentiation Using {Gaussian} Normal Bases

entry:
@Article{El-Razouk:2016:NAD,
  author =       "Hayssam El-Razouk and Arash Reyhani-Masoleh",
  title =        "New Architectures for Digit-Level Single,
                 Hybrid-Double, Hybrid-Triple Field Multiplications and
                 Exponentiation Using {Gaussian} Normal Bases",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "8",
  pages =        "2495--2509",
  month =        aug,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2481408",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 11 18:58:52 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: Emmart:2016:OMM
author: Niall Emmart and Justin Luitjens and Charles Weems and Cliff Woolley
title: Optimizing Modular Multiplication for NVIDIA's Maxwell GPUs
crossref: Montuschi:2016:ISC
pages: 47--54
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.21
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Niall Emmart and Justin Luitjens and Charles Weems and Cliff Woolley
rawtitle: Optimizing Modular Multiplication for {NVIDIA}'s {Maxwell} {GPUs}
entry:
@InProceedings{Emmart:2016:OMM,
  author =       "Niall Emmart and Justin Luitjens and Charles Weems and
                 Cliff Woolley",
  title =        "Optimizing Modular Multiplication for {NVIDIA}'s
                 {Maxwell} {GPUs}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "47--54",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.21",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Fang:2016:OSV
author: Xin Fang and Miriam Leeser
title: Open-Source Variable-Precision Floating-Point Library for Major Commercial FPGAs
journal: j-TRETS
volume: 9
number: 3
pages: 1--17
month: jul
monthnumber: 7
year: 2016
DOI: https://doi.org/10.1145/2851507
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Sat Feb 8 10:53:20 2020
bibtimestamp: 2020.02.08 10:53:20 ???
URL: https://dl.acm.org/doi/abs/10.1145/2851507
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Xin Fang and Miriam Leeser
rawtitle: Open-Source Variable-Precision Floating-Point Library for Major Commercial {FPGAs}
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib; https://www.math.utah.edu/pub/tex/bib/trets.bib

abstract:
There is increased interest in implementing floating-point designs for different precisions that take advantage of the flexibility offered by Field-Programmable Gate Arrays (FPGAs). In this article, we present updates to the Variable-precision FLOATing Point Library (VFLOAT) developed at Northeastern University and highlight recent improvements in implementations for implementing reciprocal, division, and square root components that scale to double precision for FPGAs from the two major vendors: Altera and Xilinx. Our library is open source and flexible and provides the user with many options. A designer has many tradeoffs to consider including clock frequency, total latency, and resource usage as well as target architecture. We compare the generated cores to those produced by each vendor and to another popular open-source tool: FloPoCo. VFLOAT has the advantage of not tying the user's design to a specific target architecture and of providing the maximum flexibility for all options including clock frequency and latency compared to other alternatives. Our results show that variable-precision as well as double-precision designs can easily be accommodated and the resulting components are competitive and in many cases superior to the alternatives.

rawabstract:
There is increased interest in implementing floating-point designs for different precisions that take advantage of the flexibility offered by Field-Programmable Gate Arrays (FPGAs). In this article, we present updates to the Variable-precision FLOATing Point Library (VFLOAT) developed at Northeastern University and highlight recent improvements in implementations for implementing reciprocal, division, and square root components that scale to double precision for FPGAs from the two major vendors: Altera and Xilinx. Our library is open source and flexible and provides the user with many options. A designer has many tradeoffs to consider including clock frequency, total latency, and resource usage as well as target architecture. We compare the generated cores to those produced by each vendor and to another popular open-source tool: FloPoCo. VFLOAT has the advantage of not tying the user's design to a specific target architecture and of providing the maximum flexibility for all options including clock frequency and latency compared to other alternatives. Our results show that variable-precision as well as double-precision designs can easily be accommodated and the resulting components are competitive and in many cases superior to the alternatives.

entry:
@Article{Fang:2016:OSV,
  author =       "Xin Fang and Miriam Leeser",
  title =        "Open-Source Variable-Precision Floating-Point Library
                 for Major Commercial {FPGAs}",
  journal =      j-TRETS,
  volume =       "9",
  number =       "3",
  pages =        "1--17",
  month =        jul,
  year =         "2016",
  DOI =          "https://doi.org/10.1145/2851507",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Sat Feb 8 10:53:20 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/gnu.bib;
                 https://www.math.utah.edu/pub/tex/bib/trets.bib",
  URL =          "https://dl.acm.org/doi/abs/10.1145/2851507",
  abstract =     "There is increased interest in implementing
                 floating-point designs for different precisions that
                 take advantage of the flexibility offered by
                 Field-Programmable Gate Arrays (FPGAs). In this
                 article, we present updates to the Variable-precision
                 FLOATing Point Library (VFLOAT) developed at
                 Northeastern University and highlight recent
                 improvements in implementations for implementing
                 reciprocal, division, and square root components that
                 scale to double precision for FPGAs from the two major
                 vendors: Altera and Xilinx. Our library is open source
                 and flexible and provides the user with many options. A
                 designer has many tradeoffs to consider including clock
                 frequency, total latency, and resource usage as well as
                 target architecture. We compare the generated cores to
                 those produced by each vendor and to another popular
                 open-source tool: FloPoCo. VFLOAT has the advantage of
                 not tying the user's design to a specific target
                 architecture and of providing the maximum flexibility
                 for all options including clock frequency and latency
                 compared to other alternatives. Our results show that
                 variable-precision as well as double-precision designs
                 can easily be accommodated and the resulting components
                 are competitive and in many cases superior to the
                 alternatives.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: techreport
filename: fparith.bib
label: Fevotte:2016:VAF
author: Francois Fevotte and Bruno Lathuiliere
title: VERROU: Assessing Floating-Point Accuracy Without Recompiling
type: Working paper
pages: 1--17
month: oct
monthnumber: 10
year: 2016
bibdate: Thu Oct 17 05:30:32 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.17 05:30:32 ???
URL: https://hal.archives-ouvertes.fr/hal-01383417
acknowledgement: ack-nhfb
rawauthor: Fran{\c{c}}ois F{\'e}votte and Bruno Lathuili{\`e}re
rawtitle: {VERROU}: Assessing Floating-Point Accuracy Without Recompiling
entry:
@TechReport{Fevotte:2016:VAF,
  author =       "Fran{\c{c}}ois F{\'e}votte and Bruno Lathuili{\`e}re",
  title =        "{VERROU}: Assessing Floating-Point Accuracy Without
                 Recompiling",
  type =         "Working paper",
  number =       "??",
  institution =  "????",
  address =      "????",
  month =        oct,
  year =         "2016",
  bibdate =      "Thu Oct 17 05:30:32 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://hal.archives-ouvertes.fr/hal-01383417",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Fritz:2016:IPM
author: Christopher Fritz and Adly T. Fam
title: Interlaced Partition Multiplier
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 8
pages: 2672--2677
month: aug
monthnumber: 8
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2481379
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 11 18:58:52 MDT 2016
bibtimestamp: 2016.07.11 18:58:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Christopher Fritz and Adly T. Fam
rawtitle: Interlaced Partition Multiplier
entry:
@Article{Fritz:2016:IPM,
  author =       "Christopher Fritz and Adly T. Fam",
  title =        "Interlaced Partition Multiplier",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "8",
  pages =        "2672--2677",
  month =        aug,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2481379",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 11 18:58:52 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Garcia-Vega:2016:DMO
author: Carlos Garcia-Vega and Sonia Gonzalez-Navarro and Pedro Balboa-La Chica and Julio Villalba-Moreno
title: Decimal Multiformat Online Addition
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 10
pages: 3203--3209
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2016.2516009
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Sep 13 06:19:58 MDT 2016
bibtimestamp: 2016.09.13 06:19:58 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Carlos Garcia-Vega and Sonia Gonzalez-Navarro and Pedro Balboa-La Chica and Julio Villalba-Moreno
rawtitle: Decimal Multiformat Online Addition
entry:
@Article{Garcia-Vega:2016:DMO,
  author =       "Carlos Garcia-Vega and Sonia Gonzalez-Navarro and
                 Pedro Balboa-La Chica and Julio Villalba-Moreno",
  title =        "Decimal Multiformat Online Addition",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "10",
  pages =        "3203--3209",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2016.2516009",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Sep 13 06:19:58 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Garrido:2016:CIN
author: Mario Garrido and Petter Kallstrom and Martin Kumm and Oscar Gustafsson
title: CORDIC II: A New Improved CORDIC Algorithm
journal: j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS
volume: 63
number: 2
pages: 186--190
month: feb
monthnumber: 2
year: 2016
DOI: https://doi.org/10.1109/TCSII.2015.2483422
ISSN: 1549-7747 (print), 1558-3791 (electronic) OR 15497747 (print), 15583791 (electronic)
ISSNL: 1549-7747 OR 15497747
bibdate: Sat Feb 8 10:11:25 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.08 10:11:25 ???
fjournal: IEEE Transactions on Circuits and Systems II: Express Briefs
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/issues?punumber=8920
rawauthor: Mario Garrido and Petter K{\"a}llstr{\"o}m and Martin Kumm and Oscar Gustafsson
rawtitle: {CORDIC II}: A New Improved {CORDIC} Algorithm
keywords:
Adders; Convergence; CORDIC; CORDIC II; digital arithmetic; friend angles; Hardware; Kernel; Multiplexing; nano-rotation; nanorotation; rotation; rotation angle; Signal processing algorithms; uniformly scaled redundant (USR) CORDIC; USR

entry:
@Article{Garrido:2016:CIN,
  author =       "Mario Garrido and Petter K{\"a}llstr{\"o}m and Martin
                 Kumm and Oscar Gustafsson",
  title =        "{CORDIC II}: A New Improved {CORDIC} Algorithm",
  journal =      j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS,
  volume =       "63",
  number =       "2",
  pages =        "186--190",
  month =        feb,
  year =         "2016",
  DOI =          "https://doi.org/10.1109/TCSII.2015.2483422",
  ISSN =         "1549-7747 (print), 1558-3791 (electronic)",
  ISSN-L =       "1549-7747",
  bibdate =      "Sat Feb 8 10:11:25 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Circuits and Systems II: Express
                 Briefs",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/issues?punumber=8920",
  keywords =     "Adders; Convergence; CORDIC; CORDIC II; digital
                 arithmetic; friend angles; Hardware; Kernel;
                 Multiplexing; nano-rotation; nanorotation; rotation;
                 rotation angle; Signal processing algorithms; uniformly
                 scaled redundant (USR) CORDIC; USR",
}


bibtype: article
filename: fparith.bib
label: Garrido:2016:MUG
author: Mario Garrido and Rikard Andersson and Fahad Qureshi and Oscar Gustafsson
title: Multiplierless Unity-Gain SDF FFTs
journal: j-IEEE-TRANS-VLSI-SYST
volume: 24
number: 9
pages: 3003--3007
year: 2016
CODEN: IEVSE9
DOI: https://doi.org/10.1109/TVLSI.2016.2542583
ISSN: 1063-8210 (print), 1557-9999 (electronic) OR 10638210 (print), 15579999 (electronic)
ISSNL: 1063-8210 OR 10638210
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
fjournal: IEEE Transactions on Very Large Scale Integration (VLSI) Systems
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/issues?punumber=92
rawauthor: Mario Garrido and Rikard Andersson and Fahad Qureshi and Oscar Gustafsson
rawtitle: Multiplierless Unity-Gain {SDF} {FFTs}
keywords:
Adders; Algorithm design and analysis; Combined coefficient selection and shift-and-add implementation (CCSSI); Computer architecture; coordinate rotation digital computer (CORDIC); fast Fourier transform (FFT); Fast Fourier transforms; Hardware; multiplierless; pipelined architecture; single-delay feedback (SDF); Throughput; unity gain; Very large scale integration

entry:
@Article{Garrido:2016:MUG,
  author =       "Mario Garrido and Rikard Andersson and Fahad Qureshi
                 and Oscar Gustafsson",
  title =        "Multiplierless Unity-Gain {SDF} {FFTs}",
  journal =      j-IEEE-TRANS-VLSI-SYST,
  volume =       "24",
  number =       "9",
  pages =        "3003--3007",
  year =         "2016",
  CODEN =        "IEVSE9",
  DOI =          "https://doi.org/10.1109/TVLSI.2016.2542583",
  ISSN =         "1063-8210 (print), 1557-9999 (electronic)",
  ISSN-L =       "1063-8210",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Very Large Scale Integration
                 (VLSI) Systems",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/issues?punumber=92",
  keywords =     "Adders; Algorithm design and analysis; Combined
                 coefficient selection and shift-and-add implementation
                 (CCSSI); Computer architecture; coordinate rotation
                 digital computer (CORDIC); fast Fourier transform
                 (FFT); Fast Fourier transforms; Hardware;
                 multiplierless; pipelined architecture; single-delay
                 feedback (SDF); Throughput; unity gain; Very large
                 scale integration",
}


bibtype: inproceedings
filename: fparith.bib
label: Geran:2016:CBC
author: Amir Ali Kouzeh Geran and Arash Reyhani-Masoleh
title: A CRC-Based Concurrent Fault Detection Architecture for Galois / Counter Mode (GCM)
crossref: Montuschi:2016:ISC
pages: 24--31
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.19
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Amir Ali Kouzeh Geran and Arash Reyhani-Masoleh
rawtitle: A {CRC}-Based Concurrent Fault Detection Architecture for {Galois\slash Counter Mode (GCM)}
entry:
@InProceedings{Geran:2016:CBC,
  author =       "Amir Ali Kouzeh Geran and Arash Reyhani-Masoleh",
  title =        "A {CRC}-Based Concurrent Fault Detection Architecture
                 for {Galois\slash Counter Mode (GCM)}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "24--31",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.19",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: inproceedings
filename: fparith.bib
label: Gueron:2016:ABI
author: Shay Gueron and Vlad Krasnov
title: Accelerating Big Integer Arithmetic Using Intel IFMA Extensions
crossref: Montuschi:2016:ISC
pages: 32--38
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.22
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23; integer fused multiply-add (IFMA)
acknowledgement: ack-nhfb
rawauthor: Shay Gueron and Vlad Krasnov
rawtitle: Accelerating Big Integer Arithmetic Using {Intel} {IFMA} Extensions
entry:
@InProceedings{Gueron:2016:ABI,
  author =       "Shay Gueron and Vlad Krasnov",
  title =        "Accelerating Big Integer Arithmetic Using {Intel}
                 {IFMA} Extensions",
  crossref =     "Montuschi:2016:ISC",
  pages =        "32--38",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.22",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23; integer fused multiply-add
                 (IFMA)",
}


bibtype: inproceedings
filename: fparith.bib
label: Gueron:2016:HIA
author: Shay Gueron and Sanu Mathew
crossref: Montuschi:2016:ISC
pages: 112--117
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.32
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Shay Gueron and Sanu Mathew
title:
Hardware Implementation of AES Using Area-Optimal Polynomials for Composite-Field Representation $ mathrm GF(2^4)^2 $ of $ mathrm GF(2^8) $

rawtitle:
Hardware Implementation of {AES} Using Area-Optimal Polynomials for Composite-Field Representation {$ \mathrm {GF}(2^4)^2 $} of {$ \mathrm {GF}(2^8) $}

entry:
@InProceedings{Gueron:2016:HIA,
  author =       "Shay Gueron and Sanu Mathew",
  title =        "Hardware Implementation of {AES} Using Area-Optimal
                 Polynomials for Composite-Field Representation {$
                 \mathrm {GF}(2^4)^2 $} of {$ \mathrm {GF}(2^8) $}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "112--117",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.32",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: misc
filename: fparith.bib
label: Gustafson:2016:BFP
author: John L. Gustafson
title: Beyond Floating Point: Next Generation Computer Arithmetic
pages: 112--117
year: 2016
bibdate: Thu Dec 14 16:05:07 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.14 16:05:07 ???
acknowledgement: ack-nhfb
howpublished: Stanford seminar video.
rawauthor: John L. Gustafson
rawtitle: Beyond Floating Point: Next Generation Computer Arithmetic
URL:
http://www.johngustafson.net/pdfs/DebateTranscription.pdf; https://www.youtube.com/watch?v=aP0Y1uAA-2Y

entry:
@Misc{Gustafson:2016:BFP,
  author =       "John L. Gustafson",
  title =        "Beyond Floating Point: Next Generation Computer
                 Arithmetic",
  howpublished = "Stanford seminar video.",
  year =         "2016",
  bibdate =      "Thu Dec 14 16:05:07 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.johngustafson.net/pdfs/DebateTranscription.pdf;
                 https://www.youtube.com/watch?v=aP0Y1uAA-2Y",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Gustafson:2016:RAC
author: John L. Gustafson
title: A Radical Approach to Computation with Real Numbers
journal: j-SUPERFRI
volume: 3
number: 2
pages: 38--53
year: 2016
DOI: https://doi.org/10.14529/jsfi160203
ISSN: 2409-6008 (print), 2313-8734 (electronic) OR 24096008 (print), 23138734 (electronic)
ISSNL: 2313-8734 OR 23138734
bibdate: Sat Nov 11 07:15:27 MST 2017
bibtimestamp: 2017.11.11 07:15:27 MST
URL: http://superfri.org/superfri/article/view/94
fjournal: Supercomputing Frontiers and Innovations
acknowledgement: ack-nhfb
journalURL: http://superfri.org/superfri/issue/archive
rawauthor: John L. Gustafson
rawtitle: A Radical Approach to Computation with Real Numbers
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/superfri.bib

entry:
@Article{Gustafson:2016:RAC,
  author =       "John L. Gustafson",
  title =        "A Radical Approach to Computation with Real Numbers",
  journal =      j-SUPERFRI,
  volume =       "3",
  number =       "2",
  pages =        "38--53",
  month =        "????",
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.14529/jsfi160203",
  ISSN =         "2409-6008 (print), 2313-8734 (electronic)",
  ISSN-L =       "2313-8734",
  bibdate =      "Sat Nov 11 07:15:27 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/superfri.bib",
  URL =          "http://superfri.org/superfri/article/view/94",
  acknowledgement = ack-nhfb,
  fjournal =     "Supercomputing Frontiers and Innovations",
  journal-URL =  "http://superfri.org/superfri/issue/archive",
}


bibtype: misc
filename: fparith.bib
label: Gustafson:2016:TGD
author: John Gustafson and William Kahan
title: Transcription of ``The Great Debate'': John Gustafson vs. William Kahan on Unum Arithmetic
pages: 50
day: 12
month: jul
monthnumber: 7
year: 2016
bibdate: Thu Jan 18 06:36:55 2018
bibtimestamp: 2018.01.18 06:36:55 ???
acknowledgement: ack-nhfb
howpublished: Web document.
rawauthor: John Gustafson and William Kahan
rawtitle: Transcription of {``The} Great Debate'': {John Gustafson} vs. {William Kahan} on Unum Arithmetic
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib; https://www.math.utah.edu/pub/bibnet/authors/p/parlett-beresford-n.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

URL:
http://www.johngustafson.net/pdfs/DebateTranscription.pdf; https://www.youtube.com/watch?v=KEAKYDyUua4

entry:
@Misc{Gustafson:2016:TGD,
  author =       "John Gustafson and William Kahan",
  title =        "Transcription of {``The} Great Debate'': {John
                 Gustafson} vs. {William Kahan} on Unum Arithmetic",
  howpublished = "Web document.",
  pages =        "50",
  day =          "12",
  month =        jul,
  year =         "2016",
  bibdate =      "Thu Jan 18 06:36:55 2018",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/p/parlett-beresford-n.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.johngustafson.net/pdfs/DebateTranscription.pdf;
                 https://www.youtube.com/watch?v=KEAKYDyUua4",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Hopkins:2016:WMN
author: David Hopkins
title: Will my numbers add up correctly if I round them?
journal: j-MATH-GAZ
volume: 100
number: 549
pages: 396--409
month: nov
monthnumber: 11
year: 2016
CODEN: MAGAAS
DOI: https://doi.org/10.1017/mag.2016.104
ISSN: 0025-5572 (print), 2056-6328 (electronic) OR 00255572 (print), 20566328 (electronic)
ISSNL: 0025-5572 OR 00255572
bibdate: Thu Nov 17 10:32:54 MST 2016
bibtimestamp: 2016.11.17 10:32:54 MST
URL: https://www.cambridge.org/core/product/88F5753DFE9F0DDDEAD1F2552B0F8B22
fjournal: The Mathematical Gazette
keywords: correct rounding; floating-point arithmetic; rounding in fixed-point addition
acknowledgement: ack-nhfb
ajournal: Math. Gaz.
onlinedate: 17 October 2016
rawauthor: David Hopkins
rawtitle: Will my numbers add up correctly if {I} round them?
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib

note:
The probability that rounding after fixed-point summation of $n$ terms gives the same result as summation of rounded terms is given by $ p(n) = (2 / pi ) int _0^infty (sin (x) / x)^n + 1 d x$, and that function is always a rational number. Its values are $ p(n) = 1, 3 / 4, 2 / 3, 115 / 192, 11 / 20, 5887 / 11520, 151 / 315, 259723 / 573440, ldots $ for $ n = 1$ to $8$.

journalURL:
http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620

rawnote:
The probability that rounding after fixed-point summation of $n$ terms gives the same result as summation of rounded terms is given by $ p(n) = (2 / \pi) \int_0^\infty (\sin (x) / x)^{n + 1} \, d x$, and that function is always a rational number. Its values are $ p(n) = 1, 3 / 4, 2 / 3, 115 / 192, 11 / 20, 5887 / 11520, 151 / 315, 259723 / 573440, \ldots {}$ for $ n = 1$ to $8$.

entry:
@Article{Hopkins:2016:WMN,
  author =       "David Hopkins",
  title =        "Will my numbers add up correctly if {I} round them?",
  journal =      j-MATH-GAZ,
  volume =       "100",
  number =       "549",
  pages =        "396--409",
  month =        nov,
  year =         "2016",
  CODEN =        "MAGAAS",
  DOI =          "https://doi.org/10.1017/mag.2016.104",
  ISSN =         "0025-5572 (print), 2056-6328 (electronic)",
  ISSN-L =       "0025-5572",
  bibdate =      "Thu Nov 17 10:32:54 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib",
  note =         "The probability that rounding after fixed-point
                 summation of $n$ terms gives the same result as
                 summation of rounded terms is given by $ p(n) = (2 /
                 \pi) \int_0^\infty (\sin (x) / x)^{n + 1} \, d x$, and
                 that function is always a rational number. Its values
                 are $ p(n) = 1, 3 / 4, 2 / 3, 115 / 192, 11 / 20, 5887
                 / 11520, 151 / 315, 259723 / 573440, \ldots {}$ for $ n
                 = 1$ to $8$.",
  URL =          "https://www.cambridge.org/core/product/88F5753DFE9F0DDDEAD1F2552B0F8B22",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Gaz.",
  fjournal =     "The Mathematical Gazette",
  journal-URL =  "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
  keywords =     "correct rounding; floating-point arithmetic; rounding
                 in fixed-point addition",
  onlinedate =   "17 October 2016",
  remark-1 =     "See also \cite{Nelsen:1987:PSR} for an earlier
                 independent derivation of part of this work.",
  remark-2 =     "From a Maple computer-algebra session: \\
                 p := proc(n) return (2/Pi) * int((sin(x)/x)^(n+1), x =
                 0 .. infinity); end proc; \\
                 for k from 1 to 20 do printf({"}%2d\t%a\n{"}, k, p(k))
                 end do; \\
                 1 1 \\
                 2 3/4 \\
                 3 2/3 \\
                 4 115/192 \\
                 5 11/20 \\
                 6 5887/11520 \\
                 7 151/315 \\
                 8 259723/573440 \\
                 9 15619/36288 \\
                 10 381773117/928972800 \\
                 11 655177/1663200 \\
                 12 20646903199/54499737600 \\
                 13 27085381/74131200 \\
                 14 467168310097/1322526965760 \\
                 15 2330931341/6810804000 \\
                 16 75920439315929441/228532659683328000 \\
                 17 12157712239/37638881280 \\
                 18 5278968781483042969/16783438527143608320 \\
                 19 37307713155613/121645100408832 \\
                 20 9093099984535515162569/30370031620545576960000 \\
                 for k from 10 to 100 by 10 do printf({"}%2d\t%.4f\n{"},
                 k, evalf(p(k),20)) end do; \\
                 10 0.4110 \\
                 20 0.2994 \\
                 30 0.2470 \\
                 40 0.2150 \\
                 50 0.1929 \\
                 60 0.1765 \\
                 70 0.1637 \\
                 80 0.1533 \\
                 90 0.1446 \\
                 100 NaN \\
                 Maple evaluates p(100) as undefined / Pi.",
  remark-3 =     "Equivalent code in the Mathematica computer-algebra
                 system is \\
                 p = Function[n, (2 / Pi) * Integrate[(Sin[x] / x)^(n +
                 1), {x, 0, Infinity}]]; \\
                 For[k = 1, k <= 20, ++k, Print[k, {"} {"}, p[k]]]\\
                 \ldots{} \\
                 Mathematica finds p[100] = 0.137307\ldots{}",
  remark-4 =     "The Maxima computer-algebra system is unable to
                 evaluate p(n) := (2 / %pi) * int((sin(x) / x)^(n + 1),
                 x, 0, inf);.",
}


bibtype: article
filename: fparith.bib
label: Hormigo:2016:MIW
author: Javier Hormigo and Julio Villalba-Moreno
journal: j-IEEE-TRANS-VLSI-SYST
volume: 24
number: 6
pages: 2369--2377
month: jun
monthnumber: 6
year: 2016
CODEN: IEVSE9
DOI: https://doi.org/10.1109/TVLSI.2015.2502318
ISSN: 1063-8210 (print), 1557-9999 (electronic) OR 10638210 (print), 15579999 (electronic)
ISSNL: 1063-8210 OR 10638210
bibdate: Mon Feb 10 08:19:41 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 08:19:41 ???
URL: https://ieeexplore.ieee.org/document/7349231
fjournal: IEEE Transactions on Very Large Scale Integration (VLSI) Systems
keywords: HUB (Half-Unit-Biased) floating-point format
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/issues?punumber=92
rawauthor: Javier Hormigo and Julio Villalba-Moreno
title:
Measuring Improvement When Using HUB Formats to Implement Floating-Point Systems under Round-to-Nearest

rawtitle:
Measuring Improvement When Using {HUB} Formats to Implement Floating-Point Systems under Round-to-Nearest

entry:
@Article{Hormigo:2016:MIW,
  author =       "Javier Hormigo and Julio Villalba-Moreno",
  title =        "Measuring Improvement When Using {HUB} Formats to
                 Implement Floating-Point Systems under
                 Round-to-Nearest",
  journal =      j-IEEE-TRANS-VLSI-SYST,
  volume =       "24",
  number =       "6",
  pages =        "2369--2377",
  month =        jun,
  year =         "2016",
  CODEN =        "IEVSE9",
  DOI =          "https://doi.org/10.1109/TVLSI.2015.2502318",
  ISSN =         "1063-8210 (print), 1557-9999 (electronic)",
  ISSN-L =       "1063-8210",
  bibdate =      "Mon Feb 10 08:19:41 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://ieeexplore.ieee.org/document/7349231",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Very Large Scale Integration
                 (VLSI) Systems",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/issues?punumber=92",
  keywords =     "HUB (Half-Unit-Biased) floating-point format",
}


bibtype: article
filename: fparith.bib
label: Hormigo:2016:NFC
author: Javier Hormigo and Julio Villalba
title: New Formats for Computing with Real-Numbers under Round-to-Nearest
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 7
pages: 2158--2168
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2479623
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Wed Jun 15 06:49:34 MDT 2016
bibtimestamp: 2016.06.15 06:49:34 MDT
fjournal: IEEE Transactions on Computers -
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Javier Hormigo and Julio Villalba
rawtitle: New Formats for Computing with Real-Numbers under Round-to-Nearest
entry:
@Article{Hormigo:2016:NFC,
  author =       "Javier Hormigo and Julio Villalba",
  title =        "New Formats for Computing with Real-Numbers under
                 Round-to-Nearest",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "7",
  pages =        "2158--2168",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2479623",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Wed Jun 15 06:49:34 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers -",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Hsu:2016:TPE
author: J. Hsu
title: Three paths to exascale supercomputing
journal: j-IEEE-SPECTRUM
volume: 53
number: 1
pages: 14--15
month: jan
monthnumber: 1
year: 2016
CODEN: IEESAM
DOI: https://doi.org/10.1109/MSPEC.2016.7367447
ISSN: 0018-9235 (print), 1939-9340 (electronic) OR 00189235 (print), 19399340 (electronic)
ISSNL: 0018-9235 OR 00189235
bibdate: Sat Jan 18 07:02:09 2020
bibtimestamp: 2020.01.18 07:02:09 ???
fjournal: IEEE Spectrum
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6
rawauthor: J. Hsu
rawtitle: Three paths to exascale supercomputing
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeespectrum2010.bib; https://www.math.utah.edu/pub/tex/bib/super.bib

keywords:
exascale computers; exascale supercomputing; floating point arithmetic; floating-point operations; high-performance computing; parallel machines

entry:
@Article{Hsu:2016:TPE,
  author =       "J. Hsu",
  title =        "Three paths to exascale supercomputing",
  journal =      j-IEEE-SPECTRUM,
  volume =       "53",
  number =       "1",
  pages =        "14--15",
  month =        jan,
  year =         "2016",
  CODEN =        "IEESAM",
  DOI =          "https://doi.org/10.1109/MSPEC.2016.7367447",
  ISSN =         "0018-9235 (print), 1939-9340 (electronic)",
  ISSN-L =       "0018-9235",
  bibdate =      "Sat Jan 18 07:02:09 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeespectrum2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/super.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Spectrum",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6",
  keywords =     "exascale computers; exascale supercomputing; floating
                 point arithmetic; floating-point operations;
                 high-performance computing; parallel machines",
}


bibtype: mastersthesis
filename: fparith.bib
label: Hunhold:2016:UNF
author: Laslo Hunhold
type: Bachelorarbeit
school: Universitat zu Koln
address: Koln, Germany
pages: iv + 91
day: 8
month: nov
monthnumber: 11
year: 2016
bibdate: Fri Jan 06 19:16:38 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.01.06 19:16:38 ???
URL: http://frign.de/publications/2016-11-08-the_unum_number_format.pdf
keywords: interval arithmetic; Unum
acknowledgement: ack-nhfb
advisor: Angela Kunoth and Samuel Leweke
rawauthor: Laslo Hunhold
title:
The Unum Number Format: Mathematical Foundations, Implementation and Comparison to IEEE 754 Floating-Point Numbers

abstract:
This thesis examines a modern concept for machine numbers based on interval arithmetic called `Unums' and compares it to IEEE 754 floating-point arithmetic, evaluating possible uses of this format where floating-point numbers are inadequate. In the course of this examination, this thesis builds theoretical foundations for IEEE 754 floating-point numbers, interval arithmetic based on the projectively extended real numbers and Unums.

rawabstract:
This thesis examines a modern concept for machine numbers based on interval arithmetic called `Unums' and compares it to IEEE 754 floating-point arithmetic, evaluating possible uses of this format where floating-point numbers are inadequate. In the course of this examination, this thesis builds theoretical foundations for IEEE 754 floating-point numbers, interval arithmetic based on the projectively extended real numbers and Unums.

rawtitle:
The {Unum} Number Format: Mathematical Foundations, Implementation and Comparison to {IEEE 754} Floating-Point Numbers

entry:
@MastersThesis{Hunhold:2016:UNF,
  author =       "Laslo Hunhold",
  title =        "The {Unum} Number Format: Mathematical Foundations,
                 Implementation and Comparison to {IEEE 754}
                 Floating-Point Numbers",
  type =         "{Bachelorarbeit}",
  school =       "Universit{\"a}t zu K{\"o}ln",
  address =      "K{\"o}ln, Germany",
  pages =        "iv + 91",
  day =          "8",
  month =        nov,
  year =         "2016",
  bibdate =      "Fri Jan 06 19:16:38 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://frign.de/publications/2016-11-08-the_unum_number_format.pdf",
  abstract =     "This thesis examines a modern concept for machine
                 numbers based on interval arithmetic called `Unums' and
                 compares it to IEEE 754 floating-point arithmetic,
                 evaluating possible uses of this format where
                 floating-point numbers are inadequate. In the course of
                 this examination, this thesis builds theoretical
                 foundations for IEEE 754 floating-point numbers,
                 interval arithmetic based on the projectively extended
                 real numbers and Unums.",
  acknowledgement = ack-nhfb,
  advisor =      "Angela Kunoth and Samuel Leweke",
  keywords =     "interval arithmetic; Unum",
}


bibtype: article
filename: fparith.bib
label: Isupov:2016:AMC
author: Konstantin Isupov
title: An Algorithm for Magnitude Comparison in RNS based on Mixed-Radix Conversion II
journal: j-INT-J-COMP-APPL
volume: 141
pages: 1--4
month: may
monthnumber: 5
year: 2016
DOI: https://doi.org/10.5120/ijca2016909626
ISSN: 0975-8887 OR 09758887
ISSNL: 0975-8887 OR 09758887
bibdate: Fri Jan 24 09:27:51 MST 2025
bibtimestamp: 2025.01.24 09:27:51 MST
URL: https://www.ijcaonline.org/archives/volume141/number5/24777-2016909626/
fjournal: International Journal of Computer Applications
acknowledgement: ack-nhfb
ajournal: Intern. J. of Computer Applications
articleno: 5
journalURL: https://www.ijcaonline.org/
rawauthor: Konstantin Isupov
rawtitle: An Algorithm for Magnitude Comparison in {RNS} based on Mixed-Radix Conversion {II}
entry:
@Article{Isupov:2016:AMC,
  author =       "Konstantin Isupov",
  title =        "An Algorithm for Magnitude Comparison in {RNS} based
                 on Mixed-Radix Conversion {II}",
  journal =      j-INT-J-COMP-APPL,
  volume =       "141",
  number =       "??",
  pages =        "1--4",
  month =        may,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.5120/ijca2016909626",
  ISSN =         "0975-8887",
  ISSN-L =       "0975-8887",
  bibdate =      "Fri Jan 24 09:27:51 MST 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/intjcompappl.bib",
  URL =          "https://www.ijcaonline.org/archives/volume141/number5/24777-2016909626/",
  acknowledgement = ack-nhfb,
  ajournal =     "Intern. J. of Computer Applications",
  articleno =    "5",
  fjournal =     "International Journal of Computer Applications",
  journal-URL =  "https://www.ijcaonline.org/",
}


bibtype: inproceedings
filename: fparith.bib
label: Jaberipur:2016:FFC
author: Ghassem Jaberipur and Behrooz Parhami and Dariush Abedi
title: A Formulation of Fast Carry Chains Suitable for Efficient Implementation with Majority Elements
crossref: Montuschi:2016:ISC
pages: 8--15
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.14
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Ghassem Jaberipur and Behrooz Parhami and Dariush Abedi
rawtitle: A Formulation of Fast Carry Chains Suitable for Efficient Implementation with Majority Elements
entry:
@InProceedings{Jaberipur:2016:FFC,
  author =       "Ghassem Jaberipur and Behrooz Parhami and Dariush
                 Abedi",
  title =        "A Formulation of Fast Carry Chains Suitable for
                 Efficient Implementation with Majority Elements",
  crossref =     "Montuschi:2016:ISC",
  pages =        "8--15",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.14",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: misc
filename: fparith.bib
label: Jaeger:2016:OHQ
author: Andreas Jaeger
title: OpenLibm: A high quality system independent, portable, open source libm implementation
pages: 8--15
year: 2016
bibdate: Thu Oct 17 05:37:38 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.17 05:37:38 ???
URL: http://openlibm.org/
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Andreas Jaeger
rawtitle: {OpenLibm}: A high quality system independent, portable, open source libm implementation
abstract:
OpenLibm is an effort to have a high quality, portable, standalone C mathematical library (libm). It can be used standalone in applications and programming language implementations. The project was born out of a need to have a good libm for the Julia programming langage that worked consistently across compilers and operating systems, and in 32-bit and 64-bit environments.

rawabstract:
OpenLibm is an effort to have a high quality, portable, standalone C mathematical library (libm). It can be used standalone in applications and programming language implementations. The project was born out of a need to have a good libm for the Julia programming langage that worked consistently across compilers and operating systems, and in 32-bit and 64-bit environments.

entry:
@Misc{Jaeger:2016:OHQ,
  author =       "Andreas Jaeger",
  title =        "{OpenLibm}: A high quality system independent,
                 portable, open source libm implementation",
  howpublished = "Web site",
  year =         "2016",
  bibdate =      "Thu Oct 17 05:37:38 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://openlibm.org/",
  abstract =     "OpenLibm is an effort to have a high quality,
                 portable, standalone C mathematical library (libm). It
                 can be used standalone in applications and programming
                 language implementations. The project was born out of a
                 need to have a good libm for the Julia programming
                 langage that worked consistently across compilers and
                 operating systems, and in 32-bit and 64-bit
                 environments.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Jeannerod:2016:RIE
author: Claude-Pierre Jeannerod
title: A Radix-Independent Error Analysis of the Cornea--Harrison--Tang Method
journal: j-TOMS
volume: 42
number: 3
pages: 19:1--19:20
month: may
monthnumber: 5
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2824252
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon May 23 16:40:02 MDT 2016
bibtimestamp: 2016.05.23 16:40:02 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 19
journalURL: https://dl.acm.org/loi/toms
rawauthor: Claude-Pierre Jeannerod
rawtitle: A Radix-Independent Error Analysis of the {Cornea--Harrison--Tang} Method
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
Assuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea--Harrison--Tang method aims to evaluate expressions of the form $ a b + c d $ with high relative accuracy. In this article, we provide a rounding error analysis of this method, which unlike previous studies is not restricted to binary floating-point arithmetic but holds for any radix $ beta $. We show first that an asymptotically optimal bound on the relative error of this method is $ 2 beta u + 2 u^2 / beta - 2 u^2 = 2 u + 2 / beta u^2 + O (u^3) $, where $ u = 1 / 2 beta ^1 - p $ is the unit roundoff in radix $ beta $ and precision $p$. Then we show that the possibility of removing the $ O (u^2)$ term from this bound is governed by the radix parity and the tie-breaking strategy used for rounding: if $ beta $ is odd or rounding is to nearest even, then the simpler bound $ 2 u$ is obtained, while if $ beta $ is even and rounding is to nearest away, then there exist floating-point inputs $a$, $b$, $c$, $d$ that lead to a relative error larger than $ 2 u + 2 / beta u^2 - 4 u^3$. All these results hold provided underflows and overflows do not occur and under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats.

rawabstract:
Assuming floating-point arithmetic with a fused multiply-add operation and rounding to nearest, the Cornea--Harrison--Tang method aims to evaluate expressions of the form $ a b + c d $ with high relative accuracy. In this article, we provide a rounding error analysis of this method, which unlike previous studies is not restricted to binary floating-point arithmetic but holds for any radix $ \beta $. We show first that an asymptotically optimal bound on the relative error of this method is $ 2 \beta u + 2 u^2 / \beta - 2 u^2 = 2 u + 2 / \beta u^2 + O (u^3) $, where $ u = 1 / 2 \beta^{1 - p} $ is the unit roundoff in radix $ \beta $ and precision $p$. Then we show that the possibility of removing the $ O (u^2)$ term from this bound is governed by the radix parity and the tie-breaking strategy used for rounding: if $ \beta $ is odd or rounding is to nearest even, then the simpler bound $ 2 u$ is obtained, while if $ \beta $ is even and rounding is to nearest away, then there exist floating-point inputs $a$, $b$, $c$, $d$ that lead to a relative error larger than $ 2 u + 2 / \beta u^2 - 4 u^3$. All these results hold provided underflows and overflows do not occur and under some mild assumptions on $p$ satisfied by IEEE 754-2008 formats.

entry:
@Article{Jeannerod:2016:RIE,
  author =       "Claude-Pierre Jeannerod",
  title =        "A Radix-Independent Error Analysis of the
                 {Cornea--Harrison--Tang} Method",
  journal =      j-TOMS,
  volume =       "42",
  number =       "3",
  pages =        "19:1--19:20",
  month =        may,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2824252",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon May 23 16:40:02 MDT 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "Assuming floating-point arithmetic with a fused
                 multiply-add operation and rounding to nearest, the
                 Cornea--Harrison--Tang method aims to evaluate
                 expressions of the form $ a b + c d $ with high
                 relative accuracy. In this article, we provide a
                 rounding error analysis of this method, which unlike
                 previous studies is not restricted to binary
                 floating-point arithmetic but holds for any radix $
                 \beta $. We show first that an asymptotically optimal
                 bound on the relative error of this method is $ 2 \beta
                 u + 2 u^2 / \beta - 2 u^2 = 2 u + 2 / \beta u^2 + O
                 (u^3) $, where $ u = 1 / 2 \beta^{1 - p} $ is the unit
                 roundoff in radix $ \beta $ and precision $p$. Then we
                 show that the possibility of removing the $ O (u^2)$
                 term from this bound is governed by the radix parity
                 and the tie-breaking strategy used for rounding: if $
                 \beta $ is odd or rounding is to nearest even, then the
                 simpler bound $ 2 u$ is obtained, while if $ \beta $ is
                 even and rounding is to nearest away, then there exist
                 floating-point inputs $a$, $b$, $c$, $d$ that lead to a
                 relative error larger than $ 2 u + 2 / \beta u^2 - 4
                 u^3$. All these results hold provided underflows and
                 overflows do not occur and under some mild assumptions
                 on $p$ satisfied by IEEE 754-2008 formats.",
  acknowledgement = ack-nhfb,
  articleno =    "19",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: fparith.bib
label: Jeannerod:2016:SEB
author: Claude-Pierre Jeannerod and Nicolas Louvet and Jean-Michel Muller and Antoine Plet
title: Sharp error bounds for complex floating-point inversion
journal: j-NUMER-ALGORITHMS
volume: 73
number: 3
pages: 735--760
month: nov
monthnumber: 11
year: 2016
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-016-0115-x
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Mar 1 09:12:13 MST 2017
bibtimestamp: 2017.03.01 09:12:13 MST
URL: http://link.springer.com/article/10.1007/s11075-016-0115-x
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Claude-Pierre Jeannerod and Nicolas Louvet and Jean-Michel Muller and Antoine Plet
rawtitle: Sharp error bounds for complex floating-point inversion
bibsource:
http://link.springer.com/journal/11075/73/3; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Jeannerod:2016:SEB,
  author =       "Claude-Pierre Jeannerod and Nicolas Louvet and
                 Jean-Michel Muller and Antoine Plet",
  title =        "Sharp error bounds for complex floating-point
                 inversion",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "73",
  number =       "3",
  pages =        "735--760",
  month =        nov,
  year =         "2016",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-016-0115-x",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Mar 1 09:12:13 MST 2017",
  bibsource =    "http://link.springer.com/journal/11075/73/3;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-016-0115-x",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: fparith.bib
label: Jiang:2016:ARB
author: Honglan Jiang and Jie Han and Fei Qiao and Fabrizio Lombardi
title: Approximate Radix-8 Booth Multipliers for Low-Power and High-Performance Operation
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 8
pages: 2638--2644
month: aug
monthnumber: 8
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2493547
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 11 18:58:52 MDT 2016
bibtimestamp: 2016.07.11 18:58:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Honglan Jiang and Jie Han and Fei Qiao and Fabrizio Lombardi
rawtitle: Approximate Radix-8 {Booth} Multipliers for Low-Power and High-Performance Operation
entry:
@Article{Jiang:2016:ARB,
  author =       "Honglan Jiang and Jie Han and Fei Qiao and Fabrizio
                 Lombardi",
  title =        "Approximate Radix-8 {Booth} Multipliers for Low-Power
                 and High-Performance Operation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "8",
  pages =        "2638--2644",
  month =        aug,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2493547",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 11 18:58:52 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Joldes:2016:AAE
author: Mioara Joldes and Olivier Marty and Jean-Michel Muller and Valentina Popescu
title: Arithmetic Algorithms for Extended Precision Using Floating-Point Expansions
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 4
pages: 1197--1210
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2441714
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Mar 19 07:47:06 MDT 2016
bibtimestamp: 2016.03.19 07:47:06 MDT
fjournal: IEEE Transactions on Computers -
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Mioara Joldes and Olivier Marty and Jean-Michel Muller and Valentina Popescu
rawtitle: Arithmetic Algorithms for Extended Precision Using Floating-Point Expansions
entry:
@Article{Joldes:2016:AAE,
  author =       "Mioara Joldes and Olivier Marty and Jean-Michel Muller
                 and Valentina Popescu",
  title =        "Arithmetic Algorithms for Extended Precision Using
                 Floating-Point Expansions",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "4",
  pages =        "1197--1210",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2441714",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Mar 19 07:47:06 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers -",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Kadric:2016:APF
author: Edin Kadric and Paul Gurniak and Andre DeHon
title: Accurate Parallel Floating-Point Accumulation
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 11
pages: 3224--3238
month: nov
monthnumber: 11
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2016.2532874
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Oct 11 05:14:24 MDT 2016
bibtimestamp: 2016.10.11 05:14:24 MDT
fjournal: IEEE Transactions on Computers
keywords: accurate floating-point summation; floating-point arithmetic; rounding errors
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Edin Kadric and Paul Gurniak and Andr{\'e} DeHon
rawtitle: Accurate Parallel Floating-Point Accumulation
entry:
@Article{Kadric:2016:APF,
  author =       "Edin Kadric and Paul Gurniak and Andr{\'e} DeHon",
  title =        "Accurate Parallel Floating-Point Accumulation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "11",
  pages =        "3224--3238",
  month =        nov,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2016.2532874",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Oct 11 05:14:24 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "accurate floating-point summation; floating-point
                 arithmetic; rounding errors",
}


bibtype: book
filename: fparith.bib
label: Kneusel:2016:NC
author: Ronald T. Kneusel
title: Numbers and Computers
publisher: pub-SV
address: pub-SV:adr
pages: xi + 231
year: 2016
ISBN: 3-319-35940-1 (softcover), 3-319-17260-3 (e-book) OR 3319359401 (softcover), 3319172603 (ebook)
LCCN: QA241 .K54 2016
bibdate: Tue Aug 22 05:53:26 MDT 2017
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.08.22 05:53:26 MDT
subject: Number theory; Numerals; Numeration; Computer science; Mathematics
acknowledgement: ack-nhfb
rawauthor: Ronald T. Kneusel
rawtitle: Numbers and Computers
ISBN13:
978-3-319-35940-3 (softcover), 978-3-319-17260-6 (e-book) OR 9783319359403 (softcover), 9783319172606 (ebook)

abstract:
This is a book about numbers and how those numbers are represented in and operated on by computers. It is crucial that developers understand this area because the numerical operations allowed by computers, and the limitations of those operations, especially in the area of floating point math, affect virtually everything people try to do with computers. This book aims to fill this gap by exploring, in sufficient but not overwhelming detail, just what it is that computers do with numbers. Divided into two parts, the first deals with standard representations of integers and floating point numbers, while the second details several other number representations. Each chapter ends with exercises to review the key points. Topics covered include interval arithmetic, fixed-point numbers, floating point numbers, big integers and rational arithmetic. This book is for anyone who develops software including software engineering, scientists, computer science students, engineering students and anyone who programs for fun.

TOC:
Number Systems Integers Floating Point Big Integers and Rational Arithmetic Fixed-Point Numbers Decimal Floating Point Interval Arithmetic

rawabstract:
This is a book about numbers and how those numbers are represented in and operated on by computers. It is crucial that developers understand this area because the numerical operations allowed by computers, and the limitations of those operations, especially in the area of floating point math, affect virtually everything people try to do with computers. This book aims to fill this gap by exploring, in sufficient but not overwhelming detail, just what it is that computers do with numbers. Divided into two parts, the first deals with standard representations of integers and floating point numbers, while the second details several other number representations. Each chapter ends with exercises to review the key points. Topics covered include interval arithmetic, fixed-point numbers, floating point numbers, big integers and rational arithmetic. This book is for anyone who develops software including software engineering, scientists, computer science students, engineering students and anyone who programs for fun.

rawTOC:
Number Systems \\ Integers \\ Floating Point \\ Big Integers and Rational Arithmetic \\ Fixed-Point Numbers \\ Decimal Floating Point \\ Interval Arithmetic

entry:
@Book{Kneusel:2016:NC,
  author =       "Ronald T. Kneusel",
  title =        "Numbers and Computers",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xi + 231",
  year =         "2016",
  ISBN =         "3-319-35940-1 (softcover), 3-319-17260-3 (e-book)",
  ISBN-13 =      "978-3-319-35940-3 (softcover), 978-3-319-17260-6
                 (e-book)",
  LCCN =         "QA241 .K54 2016",
  bibdate =      "Tue Aug 22 05:53:26 MDT 2017",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This is a book about numbers and how those numbers are
                 represented in and operated on by computers. It is
                 crucial that developers understand this area because
                 the numerical operations allowed by computers, and the
                 limitations of those operations, especially in the area
                 of floating point math, affect virtually everything
                 people try to do with computers. This book aims to fill
                 this gap by exploring, in sufficient but not
                 overwhelming detail, just what it is that computers do
                 with numbers. Divided into two parts, the first deals
                 with standard representations of integers and floating
                 point numbers, while the second details several other
                 number representations. Each chapter ends with
                 exercises to review the key points. Topics covered
                 include interval arithmetic, fixed-point numbers,
                 floating point numbers, big integers and rational
                 arithmetic. This book is for anyone who develops
                 software including software engineering, scientists,
                 computer science students, engineering students and
                 anyone who programs for fun.",
  acknowledgement = ack-nhfb,
  subject =      "Number theory; Numerals; Numeration; Computer science;
                 Mathematics",
  tableofcontents = "Number Systems \\
                 Integers \\
                 Floating Point \\
                 Big Integers and Rational Arithmetic \\
                 Fixed-Point Numbers \\
                 Decimal Floating Point \\
                 Interval Arithmetic",
}


bibtype: book
filename: fparith.bib
label: Kumm:2016:MCM
author: Martin Kumm
title: Multiple Constant Multiplication Optimizations for Field Programmable Gate Arrays
publisher: pub-SV
address: pub-SV:adr
pages: xxxiii + 206 + 47
year: 2016
DOI: https://doi.org/10.1007/978-3-658-13323-8
ISBN: 3-658-13322-8 (print), 3-658-13323-6 (e-book) OR 3658133228 (print), 3658133236 (ebook)
LCCN: TK7895.G36
bibdate: Fri Dec 8 13:11:57 MST 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.08 13:11:57 MST
URL: https://link.springer.com/book/10.1007/978-3-658-13323-8
acknowledgement: ack-nhfb
rawauthor: Martin Kumm
rawtitle: Multiple Constant Multiplication Optimizations for Field Programmable Gate Arrays
ISBN13:
978-3-658-13322-1 (print), 978-3-658-13323-8 (e-book) OR 9783658133221 (print), 9783658133238 (ebook)

abstract:
This work covers field programmable gate array (FPGA)-specific optimizations of circuits computing the multiplication of a variable by several constants, commonly denoted as multiple constant multiplication (MCM). These optimizations focus on low resource usage but high performance. They comprise the use of fast carry-chains in adder-based constant multiplications including ternary (3-input) adders as well as the integration of look-up table-based constant multipliers and embedded multipliers to get the optimal mapping to modern FPGAs. The proposed methods can be used for the efficient implementation of digital filters, discrete transforms and many other circuits in the domain of digital signal processing, communication and image processing.

subject:
Engineering; Computer hardware; Applied mathematics; Engineering mathematics; Electrical engineering; Ingenierie; Mathematiques de l'ingenieur; Genie electrique; engineering.; electrical engineering.; Electrical engineering.; Engineering.; Engineering mathematics.

TOC:
Front Matter / i--xxxiii Introduction / 1--7 Background / 9--35 The Pipelined Multiple Constant Multiplication Problem Front Matter / 37--37 Optimal Pipelining of Precomputed Adder Graphs / 39--51 The Reduced Pipelined Adder Graph Algorithm / 53--85 Optimally Solving MCM Related Problems Using Integer Linear Programming / 87--111 A Heuristic for the Constant Matrix Multiplication Problem / 113--124 FPGA Specific MCM Optimizations Front Matter / 125--125 Combining Adder Graphs with LUT-Based Constant Multipliers / 127--137 Optimization of Hybrid Adder Graphs Containing Embedded Multipliers / 139--152 Floating Point Multiple Constant Multiplication / 153--162 Optimization of Adder Graphs with Ternary (3-Input) Adders / 163--175 Conclusion and Future Work / 177--179 Back Matter / 181--206

rawabstract:
This work covers field programmable gate array (FPGA)-specific optimizations of circuits computing the multiplication of a variable by several constants, commonly denoted as multiple constant multiplication (MCM). These optimizations focus on low resource usage but high performance. They comprise the use of fast carry-chains in adder-based constant multiplications including ternary (3-input) adders as well as the integration of look-up table-based constant multipliers and embedded multipliers to get the optimal mapping to modern FPGAs. The proposed methods can be used for the efficient implementation of digital filters, discrete transforms and many other circuits in the domain of digital signal processing, communication and image processing.

rawTOC:
Front Matter / i--xxxiii \\ Introduction / 1--7 \\ Background / 9--35 \\ The Pipelined Multiple Constant Multiplication Problem \\ Front Matter / 37--37 \\ Optimal Pipelining of Precomputed Adder Graphs / 39--51 \\ The Reduced Pipelined Adder Graph Algorithm / 53--85 \\ Optimally Solving MCM Related Problems Using Integer Linear Programming / 87--111 \\ A Heuristic for the Constant Matrix Multiplication Problem / 113--124 \\ FPGA Specific MCM Optimizations \\ Front Matter / 125--125 \\ Combining Adder Graphs with LUT-Based Constant Multipliers / 127--137 \\ Optimization of Hybrid Adder Graphs Containing Embedded Multipliers / 139--152 \\ Floating Point Multiple Constant Multiplication / 153--162 \\ Optimization of Adder Graphs with Ternary (3-Input) Adders / 163--175 \\ Conclusion and Future Work / 177--179 \\ Back Matter / 181--206

entry:
@Book{Kumm:2016:MCM,
  author =       "Martin Kumm",
  title =        "Multiple Constant Multiplication Optimizations for
                 Field Programmable Gate Arrays",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xxxiii + 206 + 47",
  year =         "2016",
  DOI =          "https://doi.org/10.1007/978-3-658-13323-8",
  ISBN =         "3-658-13322-8 (print), 3-658-13323-6 (e-book)",
  ISBN-13 =      "978-3-658-13322-1 (print), 978-3-658-13323-8
                 (e-book)",
  LCCN =         "TK7895.G36",
  bibdate =      "Fri Dec 8 13:11:57 MST 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://link.springer.com/book/10.1007/978-3-658-13323-8",
  abstract =     "This work covers field programmable gate array
                 (FPGA)-specific optimizations of circuits computing the
                 multiplication of a variable by several constants,
                 commonly denoted as multiple constant multiplication
                 (MCM). These optimizations focus on low resource usage
                 but high performance. They comprise the use of fast
                 carry-chains in adder-based constant multiplications
                 including ternary (3-input) adders as well as the
                 integration of look-up table-based constant multipliers
                 and embedded multipliers to get the optimal mapping to
                 modern FPGAs. The proposed methods can be used for the
                 efficient implementation of digital filters, discrete
                 transforms and many other circuits in the domain of
                 digital signal processing, communication and image
                 processing.",
  acknowledgement = ack-nhfb,
  subject =      "Engineering; Computer hardware; Applied mathematics;
                 Engineering mathematics; Electrical engineering;
                 Ing{\'e}nierie; Math{\'e}matiques de l'ing{\'e}nieur;
                 G{\'e}nie {\'e}lectrique; engineering.; electrical
                 engineering.; Electrical engineering.; Engineering.;
                 Engineering mathematics.",
  tableofcontents = "Front Matter / i--xxxiii \\
                 Introduction / 1--7 \\
                 Background / 9--35 \\
                 The Pipelined Multiple Constant Multiplication Problem
                 \\
                 Front Matter / 37--37 \\
                 Optimal Pipelining of Precomputed Adder Graphs / 39--51
                 \\
                 The Reduced Pipelined Adder Graph Algorithm / 53--85
                 \\
                 Optimally Solving MCM Related Problems Using Integer
                 Linear Programming / 87--111 \\
                 A Heuristic for the Constant Matrix Multiplication
                 Problem / 113--124 \\
                 FPGA Specific MCM Optimizations \\
                 Front Matter / 125--125 \\
                 Combining Adder Graphs with LUT-Based Constant
                 Multipliers / 127--137 \\
                 Optimization of Hybrid Adder Graphs Containing Embedded
                 Multipliers / 139--152 \\
                 Floating Point Multiple Constant Multiplication /
                 153--162 \\
                 Optimization of Adder Graphs with Ternary (3-Input)
                 Adders / 163--175 \\
                 Conclusion and Future Work / 177--179 \\
                 Back Matter / 181--206",
}


bibtype: inproceedings
filename: fparith.bib
label: Langhammer:2016:SPN
author: Martin Langhammer and Bogdan Pasca
title: Single Precision Natural Logarithm Architecture for Hard Floating-Point and DSP-Enabled FPGAs
crossref: Montuschi:2016:ISC
pages: 164--171
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.20
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Martin Langhammer and Bogdan Pasca
rawtitle: Single Precision Natural Logarithm Architecture for Hard Floating-Point and {DSP}-Enabled {FPGAs}
entry:
@InProceedings{Langhammer:2016:SPN,
  author =       "Martin Langhammer and Bogdan Pasca",
  title =        "Single Precision Natural Logarithm Architecture for
                 Hard Floating-Point and {DSP}-Enabled {FPGAs}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "164--171",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.20",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: inproceedings
filename: fparith.bib
label: Langlois:2016:RNR
author: Philippe Langlois and Rafife Nheili and Christophe Denis
title: Recovering Numerical Reproducibility in Hydrodynamic Simulations
crossref: Montuschi:2016:ISC
pages: 63--70
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.27
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Philippe Langlois and Rafife Nheili and Christophe Denis
rawtitle: Recovering Numerical Reproducibility in Hydrodynamic Simulations
entry:
@InProceedings{Langlois:2016:RNR,
  author =       "Philippe Langlois and Rafife Nheili and Christophe
                 Denis",
  title =        "Recovering Numerical Reproducibility in Hydrodynamic
                 Simulations",
  crossref =     "Montuschi:2016:ISC",
  pages =        "63--70",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.27",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: techreport
filename: fparith.bib
label: Lee:2016:VBM
author: Wonyeol Lee and Rahul Sharma and Alex Aiken
title: Verifying Bit-Manipulations of Floating-Point
type: Report
institution: Stanford University
address: Stanford, CA, USA
pages: 15
day: 15
month: apr
monthnumber: 4
year: 2016
bibdate: Mon May 23 16:07:51 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.05.23 16:07:51 ???
URL: http://theory.stanford.edu/~aiken/publications/papers/pldi16b.pdf
acknowledgement: ack-nhfb
rawauthor: Wonyeol Lee and Rahul Sharma and Alex Aiken
rawtitle: Verifying Bit-Manipulations of Floating-Point
remark:
To appear in ACM PLDI'2016 conference proceedings, usually published as an issue of booktitle ACM SIGPLAN Notices.

entry:
@TechReport{Lee:2016:VBM,
  author =       "Wonyeol Lee and Rahul Sharma and Alex Aiken",
  title =        "Verifying Bit-Manipulations of Floating-Point",
  type =         "Report",
  institution =  "Stanford University",
  address =      "Stanford, CA, USA",
  pages =        "15",
  day =          "15",
  month =        apr,
  year =         "2016",
  bibdate =      "Mon May 23 16:07:51 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://theory.stanford.edu/~aiken/publications/papers/pldi16b.pdf",
  acknowledgement = ack-nhfb,
  remark =       "To appear in ACM PLDI'2016 conference proceedings,
                 usually published as an issue of \booktitle{ACM SIGPLAN
                 Notices}.",
}


bibtype: inproceedings
filename: fparith.bib
label: Lefevre:2016:CRA
author: Vincent Lefevre
title: Correctly Rounded Arbitrary-Precision Floating-Point Summation
crossref: Montuschi:2016:ISC
pages: 71--78
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.9
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: accurate summation; ARITH 2016; ARITH-23; correct rounding; floating-point arithmetic
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\`e}vre
rawtitle: Correctly Rounded Arbitrary-Precision Floating-Point Summation
entry:
@InProceedings{Lefevre:2016:CRA,
  author =       "Vincent Lef{\`e}vre",
  title =        "Correctly Rounded Arbitrary-Precision Floating-Point
                 Summation",
  crossref =     "Montuschi:2016:ISC",
  pages =        "71--78",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.9",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "accurate summation; ARITH 2016; ARITH-23; correct
                 rounding; floating-point arithmetic",
}


bibtype: techreport
filename: fparith.bib
label: Lefevre:2016:OBB
author: Vincent Lefevre and Paul Zimmermann
title: Optimized Binary64 and Binary128 Arithmetic with GNU MPFR
type: Report
number: hal-01502326
pages: 10
month: apr
monthnumber: 4
year: 2016
bibdate: Wed Apr 05 06:51:37 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/gnu.bib
bibtimestamp: 2017.04.05 06:51:37 ???
note: To appear in IEEE ARITH'2016 proceedings London, UK (24--26 July, 2016).
URL: https://hal.inria.fr/hal-01502326
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\`e}vre and Paul Zimmermann
rawnote: To appear in IEEE ARITH'2016 proceedings London, UK (24--26 July, 2016).
rawtitle: Optimized {Binary64} and {Binary128} Arithmetic with {GNU MPFR}
institution:
Inria Grenoble --- Rhone-Alpes, LIP --- Laboratoire de l'Informatique du Parallelisme and Inria Nancy --- Grand Est, LORIA --- ALGO --- Department of Algorithms, Computation, Image and Geometry

abstract:
We describe algorithms used to optimize the GNU MPFR library when the operands fit into one or two words. On modern processors, a correctly rounded addition of two quadruple precision numbers is now performed in 22 cycles, a subtraction in 24 cycles, a multiplication in 32 cycles, a division in 64 cycles, and a square root in 69 cycles. We also introduce a new faithful rounding mode, which enables even faster computations. Those optimizations will be available in version 4 of MPFR.

rawabstract:
We describe algorithms used to optimize the GNU MPFR library when the operands fit into one or two words. On modern processors, a correctly rounded addition of two quadruple precision numbers is now performed in 22 cycles, a subtraction in 24 cycles, a multiplication in 32 cycles, a division in 64 cycles, and a square root in 69 cycles. We also introduce a new faithful rounding mode, which enables even faster computations. Those optimizations will be available in version 4 of MPFR.

entry:
@TechReport{Lefevre:2016:OBB,
  author =       "Vincent Lef{\`e}vre and Paul Zimmermann",
  title =        "Optimized {Binary64} and {Binary128} Arithmetic with
                 {GNU MPFR}",
  type =         "Report",
  number =       "hal-01502326",
  institution =  "Inria Grenoble --- Rh{\^o}ne-Alpes, LIP ---
                 Laboratoire de l'Informatique du Parall{\'e}lisme and
                 Inria Nancy --- Grand Est, LORIA --- ALGO ---
                 Department of Algorithms, Computation, Image and
                 Geometry",
  pages =        "10",
  month =        apr,
  year =         "2016",
  bibdate =      "Wed Apr 05 06:51:37 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/gnu.bib",
  note =         "To appear in IEEE ARITH'2016 proceedings London, UK
                 (24--26 July, 2016).",
  URL =          "https://hal.inria.fr/hal-01502326",
  abstract =     "We describe algorithms used to optimize the GNU MPFR
                 library when the operands fit into one or two words. On
                 modern processors, a correctly rounded addition of two
                 quadruple precision numbers is now performed in 22
                 cycles, a subtraction in 24 cycles, a multiplication in
                 32 cycles, a division in 64 cycles, and a square root
                 in 69 cycles. We also introduce a new faithful rounding
                 mode, which enables even faster computations. Those
                 optimizations will be available in version 4 of MPFR.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: LeMaire:2016:CFP
author: Julien Le Maire and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
title: Computing floating-point logarithms with fixed-point operations
crossref: Montuschi:2016:ISC
pages: 156--163
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.24
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Julien {Le Maire} and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
rawtitle: Computing floating-point logarithms with fixed-point operations
entry:
@InProceedings{LeMaire:2016:CFP,
  author =       "Julien {Le Maire} and Nicolas Brunie and Florent de
                 Dinechin and Jean-Michel Muller",
  title =        "Computing floating-point logarithms with fixed-point
                 operations",
  crossref =     "Montuschi:2016:ISC",
  pages =        "156--163",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.24",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: inproceedings
filename: fparith.bib
label: Li:2016:SDT
author: Ang Li and Shuaiwen Leon Song and Mark Wijtvliet and Akash Kumar and Henk Corporaal
title: SFU-Driven Transparent Approximation Acceleration on GPUs
publisher: pub-ACM
address: pub-ACM:adr
pages: 1--14
month: jun
monthnumber: 6
year: 2016
DOI: https://doi.org/10.1145/2925426.2926255
ISBN: 1-4503-4361-9 OR 1450343619
ISBN13: 978-1-4503-4361-9 OR 9781450343619
LCCN: QA76.88
bibdate: Mon Sep 11 07:17:26 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.09.11 07:17:26 ???
acknowledgement: ack-nhfb
rawauthor: Ang Li and Shuaiwen Leon Song and Mark Wijtvliet and Akash Kumar and Henk Corporaal
rawtitle: {SFU}-Driven Transparent Approximation Acceleration on {GPUs}
booktitle:
ICS'16: Proceedings of the 2016 International Conference on Supercomputing, Istanbul Turkey, June 1--3, 2016

abstract:
Approximate computing, the technique that sacrifices certain amount of accuracy in exchange for substantial performance boost or power reduction, is one of the most promising solutions to enable power control and performance scaling towards exascale. Although most existing approximation designs target the emerging data-intensive applications that are comparatively more error-tolerable, there is still high demand for the acceleration of traditional scientific applications (e.g., weather and nuclear simulation), which often comprise intensive transcendental function calls and are very sensitive to accuracy loss. To address this challenge, we focus on a very important but long ignored approximation unit on today's commercial GPUs the special-function unit (SFU), and clarify its unique role in performance acceleration of accuracy-sensitive applications in the context of approximate computing. To better understand its features, we conduct a thorough empirical analysis on three generations of NVIDIA GPU architectures to evaluate all the single-precision and double-precision numeric transcendental functions that can be accelerated by SFUs, in terms of their performance, accuracy and power consumption. Based on the insights from the evaluation, we propose a transparent, tractable and portable design framework for SFU-driven approximate acceleration on GPUs. Our design is software-based and requires no hardware or application modifications. Experimental results on three NVIDIA GPU platforms demonstrate that our proposed framework can provide fine-grained tuning for performance and accuracy trade-offs, thus facilitating applications to achieve the maximum performance under certain accuracy constraints.

rawabstract:
Approximate computing, the technique that sacrifices certain amount of accuracy in exchange for substantial performance boost or power reduction, is one of the most promising solutions to enable power control and performance scaling towards exascale. Although most existing approximation designs target the emerging data-intensive applications that are comparatively more error-tolerable, there is still high demand for the acceleration of traditional scientific applications (e.g., weather and nuclear simulation), which often comprise intensive transcendental function calls and are very sensitive to accuracy loss. To address this challenge, we focus on a very important but long ignored approximation unit on today's commercial GPUs the special-function unit (SFU), and clarify its unique role in performance acceleration of accuracy-sensitive applications in the context of approximate computing. To better understand its features, we conduct a thorough empirical analysis on three generations of NVIDIA GPU architectures to evaluate all the single-precision and double-precision numeric transcendental functions that can be accelerated by SFUs, in terms of their performance, accuracy and power consumption. Based on the insights from the evaluation, we propose a transparent, tractable and portable design framework for SFU-driven approximate acceleration on GPUs. Our design is software-based and requires no hardware or application modifications. Experimental results on three NVIDIA GPU platforms demonstrate that our proposed framework can provide fine-grained tuning for performance and accuracy trade-offs, thus facilitating applications to achieve the maximum performance under certain accuracy constraints.

rawbooktitle:
{ICS'16}: Proceedings of the {2016 International Conference on Supercomputing, Istanbul Turkey, June 1--3, 2016}

entry:
@InProceedings{Li:2016:SDT,
  author =       "Ang Li and Shuaiwen Leon Song and Mark Wijtvliet and
                 Akash Kumar and Henk Corporaal",
  booktitle =    "{ICS'16}: Proceedings of the {2016 International
                 Conference on Supercomputing, Istanbul Turkey, June
                 1--3, 2016}",
  title =        "{SFU}-Driven Transparent Approximation Acceleration on
                 {GPUs}",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  pages =        "1--14",
  month =        jun,
  year =         "2016",
  DOI =          "https://doi.org/10.1145/2925426.2926255",
  ISBN =         "1-4503-4361-9",
  ISBN-13 =      "978-1-4503-4361-9",
  LCCN =         "QA76.88",
  bibdate =      "Mon Sep 11 07:17:26 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Approximate computing, the technique that sacrifices
                 certain amount of accuracy in exchange for substantial
                 performance boost or power reduction, is one of the
                 most promising solutions to enable power control and
                 performance scaling towards exascale. Although most
                 existing approximation designs target the emerging
                 data-intensive applications that are comparatively more
                 error-tolerable, there is still high demand for the
                 acceleration of traditional scientific applications
                 (e.g., weather and nuclear simulation), which often
                 comprise intensive transcendental function calls and
                 are very sensitive to accuracy loss. To address this
                 challenge, we focus on a very important but long
                 ignored approximation unit on today's commercial GPUs
                 the special-function unit (SFU), and clarify its unique
                 role in performance acceleration of accuracy-sensitive
                 applications in the context of approximate computing.
                 To better understand its features, we conduct a
                 thorough empirical analysis on three generations of
                 NVIDIA GPU architectures to evaluate all the
                 single-precision and double-precision numeric
                 transcendental functions that can be accelerated by
                 SFUs, in terms of their performance, accuracy and power
                 consumption. Based on the insights from the evaluation,
                 we propose a transparent, tractable and portable design
                 framework for SFU-driven approximate acceleration on
                 GPUs. Our design is software-based and requires no
                 hardware or application modifications. Experimental
                 results on three NVIDIA GPU platforms demonstrate that
                 our proposed framework can provide fine-grained tuning
                 for performance and accuracy trade-offs, thus
                 facilitating applications to achieve the maximum
                 performance under certain accuracy constraints.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Lichtenau:2016:QPF
author: Cedric Lichtenau and Steven Carlough and Silvia Melitta Mueller
title: Quad Precision Floating Point on the IBM z13
crossref: Montuschi:2016:ISC
pages: 87--94
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.26
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Cedric Lichtenau and Steven Carlough and Silvia Melitta Mueller
rawtitle: Quad Precision Floating Point on the {IBM z13}
entry:
@InProceedings{Lichtenau:2016:QPF,
  author =       "Cedric Lichtenau and Steven Carlough and Silvia
                 Melitta Mueller",
  title =        "Quad Precision Floating Point on the {IBM z13}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "87--94",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.26",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Liu:2016:DAI
author: W. Liu and L. Chen and C. Wang and M. O'Neill and F. Lombardi
title: Design and Analysis of Inexact Floating-Point Adders
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 1
pages: 308--314
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2417549
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Dec 15 09:36:24 MST 2015
bibtimestamp: 2015.12.15 09:36:24 MST
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: W. Liu and L. Chen and C. Wang and M. O'Neill and F. Lombardi
rawtitle: Design and Analysis of Inexact Floating-Point Adders
entry:
@Article{Liu:2016:DAI,
  author =       "W. Liu and L. Chen and C. Wang and M. O'Neill and F.
                 Lombardi",
  title =        "Design and Analysis of Inexact Floating-Point Adders",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "1",
  pages =        "308--314",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2417549",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Dec 15 09:36:24 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Martin-Dorel:2016:PTB
author: Erik Martin-Dorel and Guillaume Melquiond
title: Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq
journal: j-J-AUTOM-REASON
volume: 57
number: 3
pages: 187--217
month: oct
monthnumber: 10
year: 2016
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-015-9350-4
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Fri Sep 2 06:39:36 MDT 2016
bibtimestamp: 2016.09.02 06:39:36 MDT
URL: http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4
fjournal: Journal of Automated Reasoning
acknowledgement: ack-nhfb
ajournal: J. Autom. Reason.
journalURL: http://link.springer.com/journal/10817
rawauthor: {\'E}rik Martin-Dorel and Guillaume Melquiond
rawtitle: Proving Tight Bounds on Univariate Expressions with Elementary Functions in {Coq}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib

entry:
@Article{Martin-Dorel:2016:PTB,
  author =       "{\'E}rik Martin-Dorel and Guillaume Melquiond",
  title =        "Proving Tight Bounds on Univariate Expressions with
                 Elementary Functions in {Coq}",
  journal =      j-J-AUTOM-REASON,
  volume =       "57",
  number =       "3",
  pages =        "187--217",
  month =        oct,
  year =         "2016",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-015-9350-4",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Fri Sep 2 06:39:36 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jautomreason.bib",
  URL =          "http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Autom. Reason.",
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
}


bibtype: article
filename: fparith.bib
label: Mascarenhas:2016:FPN
author: W. F. Mascarenhas
title: Floating point numbers are real numbers
journal: arxiv.org
pages: 57
month: may
monthnumber: 5
year: 2016
bibdate: Tue May 31 07:07:46 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.05.31 07:07:46 ???
URL: http://arxiv.org/abs/1605.09202
keywords: Mathematics - Numerical Analysis
acknowledgement: ack-nhfb
rawauthor: W. F. Mascarenhas
rawtitle: Floating point numbers are real numbers
abstract:
Floating-point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which continuous mathematics leads to sharp, simple and new results about the evaluation of sums, square roots and dot products in floating point arithmetic.

rawabstract:
Floating-point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which continuous mathematics leads to sharp, simple and new results about the evaluation of sums, square roots and dot products in floating point arithmetic.

entry:
@Article{Mascarenhas:2016:FPN,
  author =       "W. F. Mascarenhas",
  title =        "Floating point numbers are real numbers",
  journal =      "arxiv.org",
  pages =        "57",
  month =        may,
  year =         "2016",
  bibdate =      "Tue May 31 07:07:46 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1605.09202",
  abstract =     "Floating-point arithmetic allows us to use a finite
                 machine, the digital computer, to reach conclusions
                 about models based on continuous mathematics. In this
                 article we work in the other direction, that is, we
                 present examples in which continuous mathematics leads
                 to sharp, simple and new results about the evaluation
                 of sums, square roots and dot products in floating
                 point arithmetic.",
  acknowledgement = ack-nhfb,
  archiveprefix = "arXiv",
  eprint =       "1605.09202",
  keywords =     "Mathematics - Numerical Analysis",
  primaryclass = "math.NA",
}


bibtype: inproceedings
filename: fparith.bib
label: Meloni:2016:RDR
author: Nicolas Meloni and M. Anwar Hasan
title: Random Digit Representation of Integers
crossref: Montuschi:2016:ISC
pages: 118--125
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.11
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23; modular exponentiation
acknowledgement: ack-nhfb
rawauthor: Nicolas M{\'e}loni and M. Anwar Hasan
rawtitle: Random Digit Representation of Integers
entry:
@InProceedings{Meloni:2016:RDR,
  author =       "Nicolas M{\'e}loni and M. Anwar Hasan",
  title =        "Random Digit Representation of Integers",
  crossref =     "Montuschi:2016:ISC",
  pages =        "118--125",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.11",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23; modular exponentiation",
}


bibtype: article
filename: fparith.bib
label: Montuschi:2016:MCA
author: Paolo Montuschi and Jean-Michel Muller
title: Modern Computer Arithmetic
journal: j-IEEE-CGA
volume: 49
number: 9
pages: 12--12
month: sep
monthnumber: 9
year: 2016
CODEN: ICGADZ
DOI: https://doi.org/10.1109/MC.2016.277
ISSN: 0272-1716 (print), 1558-1756 (electronic) OR 02721716 (print), 15581756 (electronic)
ISSNL: 0272-1716 OR 02721716
bibdate: Tue Oct 25 06:11:19 MDT 2016
bibtimestamp: 2016.10.25 06:11:19 MDT
URL: https://www.computer.org/csdl/mags/co/2016/09/mco2016090012.html
fjournal: IEEE Computer Graphics and Applications
acknowledgement: ack-nhfb
journalURL: http://www.computer.org/portal/web/csdl/magazines/cga
rawauthor: Paolo Montuschi and Jean-Michel Muller
rawtitle: Modern Computer Arithmetic
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeecga.bib

remark:
This one-page editorial draws attention to three papers on computer arithmetic in booktitle IEEE Transactions on Computers cite Demmel:2015:PRS,Joldes:2016:AAE,Chen:2016:DAR..

entry:
@Article{Montuschi:2016:MCA,
  author =       "Paolo Montuschi and Jean-Michel Muller",
  title =        "Modern Computer Arithmetic",
  journal =      j-IEEE-CGA,
  volume =       "49",
  number =       "9",
  pages =        "12--12",
  month =        sep,
  year =         "2016",
  CODEN =        "ICGADZ",
  DOI =          "https://doi.org/10.1109/MC.2016.277",
  ISSN =         "0272-1716 (print), 1558-1756 (electronic)",
  ISSN-L =       "0272-1716",
  bibdate =      "Tue Oct 25 06:11:19 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeecga.bib",
  URL =          "https://www.computer.org/csdl/mags/co/2016/09/mco2016090012.html",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Computer Graphics and Applications",
  journal-URL =  "http://www.computer.org/portal/web/csdl/magazines/cga",
  remark =       "This one-page editorial draws attention to three
                 papers on computer arithmetic in \booktitle{IEEE
                 Transactions on Computers}
                 \cite{Demmel:2015:PRS,Joldes:2016:AAE,Chen:2016:DAR}..",
}


bibtype: inproceedings
filename: fparith.bib
label: Morancho:2016:UAF
author: Enric Morancho
booktitle: 2016 Euromicro Conference on Digital System Design (DSD)
title: Unum: Adaptive Floating-Point Arithmetic
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 651--656
year: 2016
DOI: https://doi.org/10.1109/DSD.2016.39
bibdate: Fri Dec 15 07:38:18 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.15 07:38:18 ???
acknowledgement: ack-nhfb
rawauthor: Enric Morancho
rawbooktitle: {2016 Euromicro Conference on Digital System Design (DSD)}
rawtitle: Unum: Adaptive Floating-Point Arithmetic
entry:
@InProceedings{Morancho:2016:UAF,
  author =       "Enric Morancho",
  booktitle =    "{2016 Euromicro Conference on Digital System Design
                 (DSD)}",
  title =        "Unum: Adaptive Floating-Point Arithmetic",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "651--656",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/DSD.2016.39",
  bibdate =      "Fri Dec 15 07:38:18 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Moroz:2016:FCI
title: Fast calculation of inverse square root with the use of magic constant --- analytical approach
journal: arXiv.org
pages: 1--23
day: 14
month: mar
monthnumber: 3
year: 2016
DOI: https://doi.org/10.48550/arXiv.1603.04483
bibdate: Wed Dec 20 07:34:12 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2023.12.20 07:34:12 ???
URL: https://arxiv.org/pdf/1603.04483.pdf
acknowledgement: ack-nhfb
rawtitle: Fast calculation of inverse square root with the use of magic constant --- analytical approach
author:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cieslinski

abstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

rawabstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

rawauthor:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cie{\'s}li{\'n}ski

entry:
@Article{Moroz:2016:FCI,
  author =       "Leonid V. Moroz and Cezary J. Walczyk and Andriy
                 Hrynchyshyn and Vijay Holimath and Jan L.
                 Cie{\'s}li{\'n}ski",
  title =        "Fast calculation of inverse square root with the use
                 of magic constant --- analytical approach",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--23",
  day =          "14",
  month =        mar,
  year =         "2016",
  DOI =          "https://doi.org/10.48550/arXiv.1603.04483",
  bibdate =      "Wed Dec 20 07:34:12 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/pdf/1603.04483.pdf",
  abstract =     "We present a mathematical analysis of transformations
                 used in fast calculation of inverse square root for
                 single-precision floating-point numbers. Optimal values
                 of the so called magic constants are derived in a
                 systematic way, minimizing either absolute or relative
                 errors at subsequent stages of the discussed
                 algorithm.",
  acknowledgement = ack-nhfb,
}


bibtype: inproceedings
filename: fparith.bib
label: Mukunoki:2016:RPF
author: Daichi Mukunoki and Toshiyuki Imamura
booktitle: 2016 IEEE International Conference on Cluster Computing (CLUSTER)
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 144--145
year: 2016
DOI: https://doi.org/10.1109/CLUSTER.2016.77
bibdate: Wed Oct 1 06:40:18 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.01 06:40:18 ???
acknowledgement: ack-nhfb
rawauthor: Daichi Mukunoki and Toshiyuki Imamura
rawbooktitle: {2016 IEEE International Conference on Cluster Computing (CLUSTER)}
title:
Reduced-Precision Floating-Point Formats on GPUs for High Performance and Energy Efficient Computation

keywords:
Computers; Conferences; Containers; Graphics processing units; Kernel; Linear algebra; Sparse matrices

rawtitle:
Reduced-Precision Floating-Point Formats on {GPUs} for High Performance and Energy Efficient Computation

entry:
@InProceedings{Mukunoki:2016:RPF,
  author =       "Daichi Mukunoki and Toshiyuki Imamura",
  booktitle =    "{2016 IEEE International Conference on Cluster
                 Computing (CLUSTER)}",
  title =        "Reduced-Precision Floating-Point Formats on {GPUs} for
                 High Performance and Energy Efficient Computation",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "144--145",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/CLUSTER.2016.77",
  bibdate =      "Wed Oct 1 06:40:18 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Computers; Conferences; Containers; Graphics
                 processing units; Kernel; Linear algebra; Sparse
                 matrices",
}


bibtype: inproceedings
filename: fparith.bib
label: Muller:2016:NMA
author: Jean-Michel Muller and Valentina Popescu and Ping Tak Peter Tang
title: A New Multiplication Algorithm for Extended Precision Using Floating-Point Expansions
crossref: Montuschi:2016:ISC
pages: 39--46
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.18
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Jean-Michel Muller and Valentina Popescu and Ping Tak Peter Tang
rawtitle: A New Multiplication Algorithm for Extended Precision Using Floating-Point Expansions
entry:
@InProceedings{Muller:2016:NMA,
  author =       "Jean-Michel Muller and Valentina Popescu and Ping Tak
                 Peter Tang",
  title =        "A New Multiplication Algorithm for Extended Precision
                 Using Floating-Point Expansions",
  crossref =     "Montuschi:2016:ISC",
  pages =        "39--46",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.18",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: misc
filename: fparith.bib
label: Munshi:2016:OCS
author: Aaftab Munshi and Lee Howes and Bartosz Sochacki and Khronos OpenCL Working Group
title: The OpenCL C Specification Version: 2.0 Document Revision: 33
pages: 205
day: 13
month: apr
monthnumber: 4
year: 2016
bibdate: Mon Apr 16 14:05:49 2018
bibtimestamp: 2018.04.16 14:05:49 ???
URL: https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf
acknowledgement: ack-nhfb
howpublished: Web document.
rawauthor: Aaftab Munshi and Lee Howes and Bartosz Sochacki and {Khronos OpenCL Working Group}
rawtitle: The {OpenCL} {C} Specification Version: 2.0 Document Revision: 33
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib

remark:
Section 6.1.3.2 Math Functions, pages 74ff, defines a function repertoire extended beyond that of ISO C, including tt acospi, tt asinpi, tt atanpi, tt atan2pi, tt cospi, tt sinpi, tt tanpi, tt cospi, tt fract, tt lgamma_r, tt mad (approximation to tt a * b + c), tt minmag, tt pown, tt rootn, tt sincos, tt sinpi, and tt tanpi.

entry:
@Misc{Munshi:2016:OCS,
  author =       "Aaftab Munshi and Lee Howes and Bartosz Sochacki and
                 {Khronos OpenCL Working Group}",
  title =        "The {OpenCL} {C} Specification Version: 2.0 Document
                 Revision: 33",
  howpublished = "Web document.",
  pages =        "205",
  day =          "13",
  month =        apr,
  year =         "2016",
  bibdate =      "Mon Apr 16 14:05:49 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/pvm.bib",
  URL =          "https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf",
  acknowledgement = ack-nhfb,
  remark =       "Section 6.1.3.2 Math Functions, pages 74ff, defines a
                 function repertoire extended beyond that of ISO C,
                 including {\tt acospi}, {\tt asinpi}, {\tt atanpi},
                 {\tt atan2pi}, {\tt cospi}, {\tt sinpi}, {\tt tanpi},
                 {\tt cospi}, {\tt fract}, {\tt lgamma\_r}, {\tt mad}
                 (approximation to {\tt a * b + c}), {\tt minmag}, {\tt
                 pown}, {\tt rootn}, {\tt sincos}, {\tt sinpi}, and {\tt
                 tanpi}.",
}


bibtype: article
filename: fparith.bib
label: Nannarelli:2016:PPS
author: Alberto Nannarelli
title: Performance / Power Space Exploration for Binary64 Division Units
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 5
pages: 1671--1677
month: may
monthnumber: 5
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2448097
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue May 31 08:56:47 MDT 2016
bibtimestamp: 2016.05.31 08:56:47 MDT
fjournal: IEEE Transactions on Computers -
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Alberto Nannarelli
rawtitle: Performance\slash Power Space Exploration for {Binary64} Division Units
entry:
@Article{Nannarelli:2016:PPS,
  author =       "Alberto Nannarelli",
  title =        "Performance\slash Power Space Exploration for
                 {Binary64} Division Units",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "5",
  pages =        "1671--1677",
  month =        may,
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2448097",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue May 31 08:56:47 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Computers -",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Notzli:2016:LVP
author: Andres Notzli and Fraser Brown
title: LifeJacket: Verifying precise floating-point optimizations in LLVM
journal: arxiv.org
pages: ??--??
day: 30
month: mar
monthnumber: 3
year: 2016
bibdate: Sat Apr 2 06:26:03 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.04.02 06:26:03 ???
URL: http://arxiv.org/abs/1603.09290
acknowledgement: ack-nhfb
rawauthor: Andres N{\"o}tzli and Fraser Brown
rawtitle: {LifeJacket}: Verifying precise floating-point optimizations in {LLVM}
abstract:
Optimizing floating-point arithmetic is vital because it is ubiquitous, costly, and used in compute-heavy workloads. Implementing precise optimizations correctly, however, is difficult, since developers must account for all the esoteric properties of floating-point arithmetic to ensure that their transformations do not alter the output of a program. Manual reasoning is error prone and stifles incorporation of new optimizations. We present an approach to automate reasoning about floating-point optimizations using satisfiability modulo theories (SMT) solvers. We implement the approach in LifeJacket, a system for automatically verifying precise floating-point optimizations for the LLVM assembly language. We have used LifeJacket to verify 43 LLVM optimizations and to discover eight incorrect ones, including three previously unreported problems. LifeJacket is an open source extension of the Alive system for optimization verification.

rawabstract:
Optimizing floating-point arithmetic is vital because it is ubiquitous, costly, and used in compute-heavy workloads. Implementing precise optimizations correctly, however, is difficult, since developers must account for all the esoteric properties of floating-point arithmetic to ensure that their transformations do not alter the output of a program. Manual reasoning is error prone and stifles incorporation of new optimizations. We present an approach to automate reasoning about floating-point optimizations using satisfiability modulo theories (SMT) solvers. We implement the approach in LifeJacket, a system for automatically verifying precise floating-point optimizations for the LLVM assembly language. We have used LifeJacket to verify 43 LLVM optimizations and to discover eight incorrect ones, including three previously unreported problems. LifeJacket is an open source extension of the Alive system for optimization verification.

entry:
@Article{Notzli:2016:LVP,
  author =       "Andres N{\"o}tzli and Fraser Brown",
  title =        "{LifeJacket}: Verifying precise floating-point
                 optimizations in {LLVM}",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "30",
  month =        mar,
  year =         "2016",
  bibdate =      "Sat Apr 2 06:26:03 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://arxiv.org/abs/1603.09290",
  abstract =     "Optimizing floating-point arithmetic is vital because
                 it is ubiquitous, costly, and used in compute-heavy
                 workloads. Implementing precise optimizations
                 correctly, however, is difficult, since developers must
                 account for all the esoteric properties of
                 floating-point arithmetic to ensure that their
                 transformations do not alter the output of a program.
                 Manual reasoning is error prone and stifles
                 incorporation of new optimizations. We present an
                 approach to automate reasoning about floating-point
                 optimizations using satisfiability modulo theories
                 (SMT) solvers. We implement the approach in LifeJacket,
                 a system for automatically verifying precise
                 floating-point optimizations for the LLVM assembly
                 language. We have used LifeJacket to verify 43 LLVM
                 optimizations and to discover eight incorrect ones,
                 including three previously unreported problems.
                 LifeJacket is an open source extension of the Alive
                 system for optimization verification.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Ozaki:2016:EFT
author: Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi Oishi
title: Error-free transformation of matrix multiplication with a posteriori validation
journal: j-NUM-LIN-ALG-APPL
volume: 23
number: 5
pages: 931--946
month: oct
monthnumber: 10
year: 2016
CODEN: NLAAEM
DOI: https://doi.org/10.1002/nla.2061
ISSN: 1070-5325 (print), 1099-1506 (electronic) OR 10705325 (print), 10991506 (electronic)
ISSNL: 1070-5325 OR 10705325
bibdate: Thu Sep 29 08:59:10 MDT 2016
bibtimestamp: 2016.09.29 08:59:10 MDT
fjournal: Numerical Linear Algebra with Applications
acknowledgement: ack-nhfb
journalURL: http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-1506
rawauthor: Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi Oishi
rawtitle: Error-free transformation of matrix multiplication with a posteriori validation
bibsource:
http://www.interscience.wiley.com/jpages/1070-5325; http://www3.interscience.wiley.com/journalfinder.html; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numlinaa.bib

keywords:
accurate computing; accurate floating-point summation; floating-point arithmetic; interval arithmetic; matrix multiplication

entry:
@Article{Ozaki:2016:EFT,
  author =       "Katsuhisa Ozaki and Takeshi Ogita and Shin'ichi
                 Oishi",
  title =        "Error-free transformation of matrix multiplication
                 with a posteriori validation",
  journal =      j-NUM-LIN-ALG-APPL,
  volume =       "23",
  number =       "5",
  pages =        "931--946",
  month =        oct,
  year =         "2016",
  CODEN =        "NLAAEM",
  DOI =          "https://doi.org/10.1002/nla.2061",
  ISSN =         "1070-5325 (print), 1099-1506 (electronic)",
  ISSN-L =       "1070-5325",
  bibdate =      "Thu Sep 29 08:59:10 MDT 2016",
  bibsource =    "http://www.interscience.wiley.com/jpages/1070-5325;
                 http://www3.interscience.wiley.com/journalfinder.html;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/numlinaa.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Linear Algebra with Applications",
  journal-URL =  "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-1506",
  keywords =     "accurate computing; accurate floating-point summation;
                 floating-point arithmetic; interval arithmetic; matrix
                 multiplication",
}


bibtype: article
filename: fparith.bib
label: Ozaki:2016:SFP
author: Katsuhisa Ozaki and Florian Bunger and Takeshi Ogita
title: Simple floating-point filters for the two-dimensional orientation problem
journal: j-BIT-NUM-MATH
volume: 56
number: 2
pages: 729--749
month: jun
monthnumber: 6
year: 2016
CODEN: BITTEL, NBITAB
DOI: https://doi.org/10.1007/s10543-015-0574-9
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Mon May 30 06:11:11 MDT 2016
bibtimestamp: 2016.05.30 06:11:11 MDT
URL: http://link.springer.com/article/10.1007/s10543-015-0574-9
fjournal: BIT Numerical Mathematics
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Katsuhisa Ozaki and Florian B{\"u}nger and Takeshi Ogita
rawtitle: Simple floating-point filters for the two-dimensional orientation problem
bibsource:
http://link.springer.com/journal/10543/56/2; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Ozaki:2016:SFP,
  author =       "Katsuhisa Ozaki and Florian B{\"u}nger and Takeshi
                 Ogita",
  title =        "Simple floating-point filters for the two-dimensional
                 orientation problem",
  journal =      j-BIT-NUM-MATH,
  volume =       "56",
  number =       "2",
  pages =        "729--749",
  month =        jun,
  year =         "2016",
  CODEN =        "BITTEL, NBITAB",
  DOI =          "https://doi.org/10.1007/s10543-015-0574-9",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Mon May 30 06:11:11 MDT 2016",
  bibsource =    "http://link.springer.com/journal/10543/56/2;
                 https://www.math.utah.edu/pub/tex/bib/bit.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/article/10.1007/s10543-015-0574-9",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
}


bibtype: inproceedings
filename: fparith.bib
label: Parfieniuk:2016:VQM
author: Marek Parfieniuk and Sang Yoon Park
booktitle: 2016 14th IEEE International New Circuits and Systems Conference (NEWCAS)
title: A versatile quaternion multiplier based on sparse-iteration 4D CORDIC
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--4
year: 2016
DOI: https://doi.org/10.1109/NEWCAS.2016.7604788
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
keywords: Hardware; Logic gates; Quaternions; Switches; Table lookup; Transforms; Two dimensional displays
acknowledgement: ack-nhfb
rawauthor: Marek Parfieniuk and Sang Yoon Park
rawbooktitle: {2016 14th IEEE International New Circuits and Systems Conference (NEWCAS)}
rawtitle: A versatile quaternion multiplier based on sparse-iteration {4D} {CORDIC}
entry:
@InProceedings{Parfieniuk:2016:VQM,
  author =       "Marek Parfieniuk and Sang Yoon Park",
  booktitle =    "{2016 14th IEEE International New Circuits and Systems
                 Conference (NEWCAS)}",
  title =        "A versatile quaternion multiplier based on
                 sparse-iteration {4D} {CORDIC}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--4",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/NEWCAS.2016.7604788",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Hardware; Logic gates; Quaternions; Switches; Table
                 lookup; Transforms; Two dimensional displays",
}


bibtype: article
filename: fparith.bib
label: Paulk:2016:IFP
author: Mark Paulk and Lori Cameron
title: IEEE Floating Point Standard
journal: j-COMPUTER
volume: 49
number: 6
pages: 10--10
month: jun
monthnumber: 6
year: 2016
CODEN: CPTRB4
ISSN: 0018-9162 (print), 1558-0814 (electronic) OR 00189162 (print), 15580814 (electronic)
ISSNL: 0018-9162 OR 00189162
bibdate: Tue Jun 21 15:00:51 MDT 2016
bibtimestamp: 2016.06.21 15:00:51 MDT
URL: http://csdl.computer.org/csdl/mags/co/2016/06/mco2016060010.html
fjournal: Computer
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2
rawauthor: Mark Paulk and Lori Cameron
rawtitle: {IEEE} Floating Point Standard
bibsource:
https://www.math.utah.edu/pub/tex/bib/computer2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Paulk:2016:IFP,
  author =       "Mark Paulk and Lori Cameron",
  title =        "{IEEE} Floating Point Standard",
  journal =      j-COMPUTER,
  volume =       "49",
  number =       "6",
  pages =        "10--10",
  month =        jun,
  year =         "2016",
  CODEN =        "CPTRB4",
  ISSN =         "0018-9162 (print), 1558-0814 (electronic)",
  ISSN-L =       "0018-9162",
  bibdate =      "Tue Jun 21 15:00:51 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computer2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://csdl.computer.org/csdl/mags/co/2016/06/mco2016060010.html",
  abstract-URL = "http://csdl.computer.org/csdl/mags/co/2016/06/mco2016060010-abs.html",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=2",
}


bibtype: article
filename: fparith.bib
label: Petrovsky:2016:CLF
author: Nick Petrovsky and Andrew Stankevich and Alexander Petrovsky
journal: Multidimensional Syst. Signal Process.
volume: 27
number: 3
pages: 667--695
year: 2016
DOI: https://doi.org/10.1007/s11045-015-0323-x
ISSN: 0923-6082 OR 09236082
bibdate: Mon Oct 27 10:32:44 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2025.10.27 10:32:44 ???
fjournal: Multidimensional Systems and Signal Processing
keywords: 94A08,94A12
ZMnumber: 1441.94022
acknowledgement: ack-nhfb
rawauthor: Nick Petrovsky and Andrew Stankevich and Alexander Petrovsky
title:
CORDIC-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

rawtitle:
{CORDIC}-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

entry:
@Article{Petrovsky:2016:CLF,
  author =       "Nick Petrovsky and Andrew Stankevich and Alexander
                 Petrovsky",
  title =        "{CORDIC}-lifting factorization of paraunitary filter
                 banks based on the quaternionic multipliers for
                 lossless image coding",
  journal =      "Multidimensional Syst. Signal Process.",
  volume =       "27",
  number =       "3",
  pages =        "667--695",
  year =         "2016",
  DOI =          "https://doi.org/10.1007/s11045-015-0323-x",
  ISSN =         "0923-6082",
  bibdate =      "Mon Oct 27 10:32:44 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  ZMnumber =     "1441.94022",
  acknowledgement = ack-nhfb,
  fjournal =     "Multidimensional Systems and Signal Processing",
  keywords =     "94A08,94A12",
  ZBmath =       "6758814",
}


bibtype: article
filename: fparith.bib
label: Phatak:2016:NDA
author: Dhananjay S. Phatak and Steven D. Houston
title: New distributed algorithms for fast sign detection in residue number systems (RNS)
journal: j-J-PAR-DIST-COMP
volume: 97
pages: 78--95
month: nov
monthnumber: 11
year: 2016
CODEN: JPDCER
ISSN: 0743-7315 (print), 1096-0848 (electronic) OR 07437315 (print), 10960848 (electronic)
ISSNL: 0743-7315 OR 07437315
bibdate: Thu Aug 25 18:20:29 MDT 2016
bibtimestamp: 2016.08.25 18:20:29 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0743731516300703
fjournal: Journal of Parallel and Distributed Computing
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/07437315/
rawauthor: Dhananjay S. Phatak and Steven D. Houston
rawtitle: New distributed algorithms for fast sign detection in residue number systems ({RNS})
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jpardistcomp.bib

entry:
@Article{Phatak:2016:NDA,
  author =       "Dhananjay S. Phatak and Steven D. Houston",
  title =        "New distributed algorithms for fast sign detection in
                 residue number systems ({RNS})",
  journal =      j-J-PAR-DIST-COMP,
  volume =       "97",
  number =       "??",
  pages =        "78--95",
  month =        nov,
  year =         "2016",
  CODEN =        "JPDCER",
  ISSN =         "0743-7315 (print), 1096-0848 (electronic)",
  ISSN-L =       "0743-7315",
  bibdate =      "Thu Aug 25 18:20:29 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jpardistcomp.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0743731516300703",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Parallel and Distributed Computing",
  journal-URL =  "http://www.sciencedirect.com/science/journal/07437315/",
}


bibtype: inproceedings
filename: fparith.bib
label: Rane:2016:SPF
author: Ashey Rane and Calvin Lin and Mohit Tiwari
editor: T. Holz and S. Savage
booktitle: Proceedings of the 25th USENIX Security 2016, August 10--12, 2016, Austin, TX
title: Secure, precise, and fast floating-point operations on x86 processors
publisher: pub-USENIX
address: pub-USENIX:adr
pages: 71--86
year: 2016
bibdate: Mon Aug 26 17:02:50 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2024.08.26 17:02:50 ???
URL: https://www.usenix.org/conference/usenixsecurity16/technical-sessions/presentation/rane
acknowledgement: ack-nhfb
rawauthor: Ashey Rane and Calvin Lin and Mohit Tiwari
rawbooktitle: Proceedings of the {25th USENIX Security 2016, August 10--12, 2016, Austin, TX}
raweditor: T. Holz and S. Savage
rawtitle: Secure, precise, and fast floating-point operations on x86 processors
abstract:
Floating-point computations introduce several side channels. This paper describes the first solution that closes these side channels while preserving the precision of non-secure executions. Our solution exploits microarchitectural features of the x86 architecture along with novel compilation techniques to provide low overhead.par Because of the details of x86 execution, the evaluation of floating-point side channel defenses is quite involved, but we show that our solution is secure, precise, and fast. Our solution closes more side channels than any prior solution. Despite the added security, our solution does not compromise on the precision of the floating-point operations. Finally, for a set of microkernels, our solution is an order of magnitude more efficient than the previous solution.

remark:
ISBN-13 978-1-931971-32-4 from publisher Web site BibTeX entry is incorrect: it is for an unrelated 2005 conference paper. Library catalog searches fail to find this proceedings volume.

rawabstract:
Floating-point computations introduce several side channels. This paper describes the first solution that closes these side channels while preserving the precision of non-secure executions. Our solution exploits microarchitectural features of the x86 architecture along with novel compilation techniques to provide low overhead.\par Because of the details of x86 execution, the evaluation of floating-point side channel defenses is quite involved, but we show that our solution is secure, precise, and fast. Our solution closes more side channels than any prior solution. Despite the added security, our solution does not compromise on the precision of the floating-point operations. Finally, for a set of microkernels, our solution is an order of magnitude more efficient than the previous solution.

entry:
@InProceedings{Rane:2016:SPF,
  author =       "Ashey Rane and Calvin Lin and Mohit Tiwari",
  editor =       "T. Holz and S. Savage",
  booktitle =    "Proceedings of the {25th USENIX Security 2016, August
                 10--12, 2016, Austin, TX}",
  title =        "Secure, precise, and fast floating-point operations on
                 x86 processors",
  publisher =    pub-USENIX,
  address =      pub-USENIX:adr,
  pages =        "71--86",
  year =         "2016",
  bibdate =      "Mon Aug 26 17:02:50 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://www.usenix.org/conference/usenixsecurity16/technical-sessions/presentation/rane",
  abstract =     "Floating-point computations introduce several side
                 channels. This paper describes the first solution that
                 closes these side channels while preserving the
                 precision of non-secure executions. Our solution
                 exploits microarchitectural features of the x86
                 architecture along with novel compilation techniques to
                 provide low overhead.\par

                 Because of the details of x86 execution, the evaluation
                 of floating-point side channel defenses is quite
                 involved, but we show that our solution is secure,
                 precise, and fast. Our solution closes more side
                 channels than any prior solution. Despite the added
                 security, our solution does not compromise on the
                 precision of the floating-point operations. Finally,
                 for a set of microkernels, our solution is an order of
                 magnitude more efficient than the previous solution.",
  acknowledgement = ack-nhfb,
  remark =       "ISBN-13 978-1-931971-32-4 from publisher Web site
                 BibTeX entry is incorrect: it is for an unrelated 2005
                 conference paper. Library catalog searches fail to find
                 this proceedings volume.",
  xxISBN =       "1-931971-32-3",
  xxISBN-13 =    "978-1-931971-32-4",
}


bibtype: article
filename: fparith.bib
label: Rashidi:2016:HSH
author: Bahram Rashidi and Sayed Masoud Sayedi and Reza Rezaeian Farashahi
title: High-speed hardware architecture of scalar multiplication for binary elliptic curve cryptosystems
journal: j-MICROELECT-J
volume: 52
pages: 49--65
month: jun
monthnumber: 6
year: 2016
CODEN: MICEB9
DOI: https://doi.org/10.1016/j.mejo.2016.03.006
ISSN: 0026-2692 (print), 1879-2391 (electronic) OR 00262692 (print), 18792391 (electronic)
ISSNL: 0026-2692 OR 00262692
bibdate: Fri Jun 24 16:29:16 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2022.06.24 16:29:16 ???
fjournal: Microelectronics Journal
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00262692
rawauthor: Bahram Rashidi and Sayed Masoud Sayedi and Reza Rezaeian Farashahi
rawtitle: High-speed hardware architecture of scalar multiplication for binary elliptic curve cryptosystems
entry:
@Article{Rashidi:2016:HSH,
  author =       "Bahram Rashidi and Sayed Masoud Sayedi and Reza
                 Rezaeian Farashahi",
  title =        "High-speed hardware architecture of scalar
                 multiplication for binary elliptic curve
                 cryptosystems",
  journal =      j-MICROELECT-J,
  volume =       "52",
  pages =        "49--65",
  month =        jun,
  year =         "2016",
  CODEN =        "MICEB9",
  DOI =          "https://doi.org/10.1016/j.mejo.2016.03.006",
  ISSN =         "0026-2692 (print), 1879-2391 (electronic)",
  ISSN-L =       "0026-2692",
  bibdate =      "Fri Jun 24 16:29:16 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Microelectronics Journal",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00262692",
}


bibtype: inproceedings
filename: fparith.bib
label: Revy:2016:ADF
author: Guillaume Revy
title: Automated Design of Floating-Point Logarithm Functions on Integer Processors
crossref: Montuschi:2016:ISC
pages: 172--180
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.28
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Guillaume Revy
rawtitle: Automated Design of Floating-Point Logarithm Functions on Integer Processors
entry:
@InProceedings{Revy:2016:ADF,
  author =       "Guillaume Revy",
  title =        "Automated Design of Floating-Point Logarithm Functions
                 on Integer Processors",
  crossref =     "Montuschi:2016:ISC",
  pages =        "172--180",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.28",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Roux:2016:FPR
author: Pierre Roux
journal: j-J-AUTOM-REASON
volume: 57
number: 2
pages: 135--156
month: aug
monthnumber: 8
year: 2016
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-015-9339-z
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Fri Sep 2 06:39:36 MDT 2016
bibtimestamp: 2016.09.02 06:39:36 MDT
URL: http://link.springer.com/accesspage/article/10.1007/s10817-015-9339-z
fjournal: Journal of Automated Reasoning
keywords: floating-point arithmetic; rounding error
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10817
rawauthor: Pierre Roux
title:
Formal Proofs of Rounding Error Bounds: With Application to an Automatic Positive Definiteness Check

rawtitle:
Formal Proofs of Rounding Error Bounds: With Application to an Automatic Positive Definiteness Check

entry:
@Article{Roux:2016:FPR,
  author =       "Pierre Roux",
  title =        "Formal Proofs of Rounding Error Bounds: With
                 Application to an Automatic Positive Definiteness
                 Check",
  journal =      j-J-AUTOM-REASON,
  volume =       "57",
  number =       "2",
  pages =        "135--156",
  month =        aug,
  year =         "2016",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-015-9339-z",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Fri Sep 2 06:39:36 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jautomreason.bib",
  URL =          "http://link.springer.com/accesspage/article/10.1007/s10817-015-9339-z",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
  keywords =     "floating-point arithmetic; rounding error",
}


bibtype: inproceedings
filename: fparith.bib
label: Rubio-Gonzalez:2016:FPP
booktitle: Proceedings of the 38th International Conference on Software Engineering
title: Floating-point precision tuning using blame analysis
publisher: pub-ACM
address: pub-ACM:adr
pages: 1074--1085
year: 2016
bibdate: Sat Jun 04 17:19:46 2016
bibtimestamp: 2016.06.04 17:19:46 ???
acknowledgement: ack-nhfb
rawbooktitle: Proceedings of the 38th International Conference on Software Engineering
rawtitle: Floating-point precision tuning using blame analysis
author:
Cindy Rubio-Gonzalez and Cuong Nguyen and Benjamin Mehne and Koushik Sen and James Demmel and William Kahan and Costin Iancu and Wim Lavrijsen and David H. Bailey and David Hough

rawauthor:
Cindy Rubio-Gonz{\'a}lez and Cuong Nguyen and Benjamin Mehne and Koushik Sen and James Demmel and William Kahan and Costin Iancu and Wim Lavrijsen and David H. Bailey and David Hough

entry:
@InProceedings{Rubio-Gonzalez:2016:FPP,
  author =       "Cindy Rubio-Gonz{\'a}lez and Cuong Nguyen and Benjamin
                 Mehne and Koushik Sen and James Demmel and William
                 Kahan and Costin Iancu and Wim Lavrijsen and David H.
                 Bailey and David Hough",
  booktitle =    "Proceedings of the 38th International Conference on
                 Software Engineering",
  title =        "Floating-point precision tuning using blame analysis",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  pages =        "1074--1085",
  year =         "2016",
  bibdate =      "Sat Jun 04 17:19:46 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/k/kahan-william-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Rump:2016:DUR
author: Siegfried M. Rump and Marko Lange
title: On the definition of unit roundoff
journal: j-BIT-NUM-MATH
volume: 56
number: 1
pages: 309--317
month: mar
monthnumber: 3
year: 2016
CODEN: BITTEL, NBITAB
DOI: https://doi.org/10.1007/s10543-015-0554-0
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Mon May 30 06:11:10 MDT 2016
bibtimestamp: 2016.05.30 06:11:10 MDT
URL: http://link.springer.com/article/10.1007/s10543-015-0554-0
fjournal: BIT Numerical Mathematics
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Siegfried M. Rump and Marko Lange
rawtitle: On the definition of unit roundoff
bibsource:
http://link.springer.com/journal/10543/56/1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Rump:2016:DUR,
  author =       "Siegfried M. Rump and Marko Lange",
  title =        "On the definition of unit roundoff",
  journal =      j-BIT-NUM-MATH,
  volume =       "56",
  number =       "1",
  pages =        "309--317",
  month =        mar,
  year =         "2016",
  CODEN =        "BITTEL, NBITAB",
  DOI =          "https://doi.org/10.1007/s10543-015-0554-0",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Mon May 30 06:11:10 MDT 2016",
  bibsource =    "http://link.springer.com/journal/10543/56/1;
                 https://www.math.utah.edu/pub/tex/bib/bit.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/article/10.1007/s10543-015-0554-0",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
}


bibtype: article
filename: fparith.bib
label: Rump:2016:IAF
author: Siegfried M. Rump and Takeshi Ogita and Yusuke Morikura and Shin'ichi Oishi
title: Interval arithmetic with fixed rounding mode
journal: j-NONLINEAR-THEORY-APPL
volume: 7
number: 3
pages: 362--373
year: 2016
DOI: https://doi.org/10.1587/nolta.7.362
ISSN: 2185-4106 OR 21854106
ISSNL: 2185-4106 OR 21854106
bibdate: Wed Jan 19 10:32:15 2022
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2022.01.19 10:32:15 ???
fjournal: Nonlinear Theory and Its Applications, IEICE
acknowledgement: ack-nhfb
journalURL: https://www.jstage.jst.go.jp/browse/nolta/
rawauthor: Siegfried M. Rump and Takeshi Ogita and Yusuke Morikura and Shin'ichi Oishi
rawtitle: Interval arithmetic with fixed rounding mode
abstract:
We discuss several methods to simulate interval arithmetic operations using floating-point operations with fixed rounding mode. In particular we present formulas using only rounding to nearest and using only chop rounding (towards zero). The latter was the default and only rounding on GPU (Graphics Processing Unit) and cell processors, which in turn are very fast and therefore attractive in scientific computations.

rawabstract:
We discuss several methods to simulate interval arithmetic operations using floating-point operations with fixed rounding mode. In particular we present formulas using only rounding to nearest and using only chop rounding (towards zero). The latter was the default and only rounding on GPU (Graphics Processing Unit) and cell processors, which in turn are very fast and therefore attractive in scientific computations.

entry:
@Article{Rump:2016:IAF,
  author =       "Siegfried M. Rump and Takeshi Ogita and Yusuke
                 Morikura and Shin'ichi Oishi",
  title =        "Interval arithmetic with fixed rounding mode",
  journal =      j-NONLINEAR-THEORY-APPL,
  volume =       "7",
  number =       "3",
  pages =        "362--373",
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1587/nolta.7.362",
  ISSN =         "2185-4106",
  ISSN-L =       "2185-4106",
  bibdate =      "Wed Jan 19 10:32:15 2022",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We discuss several methods to simulate interval
                 arithmetic operations using floating-point operations
                 with fixed rounding mode. In particular we present
                 formulas using only rounding to nearest and using only
                 chop rounding (towards zero). The latter was the
                 default and only rounding on GPU (Graphics Processing
                 Unit) and cell processors, which in turn are very fast
                 and therefore attractive in scientific computations.",
  acknowledgement = ack-nhfb,
  fjournal =     "Nonlinear Theory and Its Applications, {IEICE}",
  journal-URL =  "https://www.jstage.jst.go.jp/browse/nolta/",
}


bibtype: article
filename: fparith.bib
label: Rump:2016:IEB
author: Siegfried M. Rump and Florian Bunger and Claude-Pierre Jeannerod
title: Improved error bounds for floating-point products and Horner's scheme
journal: j-BIT-NUM-MATH
volume: 56
number: 1
pages: 293--307
month: mar
monthnumber: 3
year: 2016
CODEN: BITTEL, NBITAB
DOI: https://doi.org/10.1007/s10543-015-0555-z
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Mon May 30 06:11:10 MDT 2016
bibtimestamp: 2016.05.30 06:11:10 MDT
URL: http://link.springer.com/article/10.1007/s10543-015-0555-z
fjournal: BIT Numerical Mathematics
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Siegfried M. Rump and Florian B{\"u}nger and Claude-Pierre Jeannerod
rawtitle: Improved error bounds for floating-point products and {Horner}'s scheme
bibsource:
http://link.springer.com/journal/10543/56/1; https://www.math.utah.edu/pub/tex/bib/bit.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Rump:2016:IEB,
  author =       "Siegfried M. Rump and Florian B{\"u}nger and
                 Claude-Pierre Jeannerod",
  title =        "Improved error bounds for floating-point products and
                 {Horner}'s scheme",
  journal =      j-BIT-NUM-MATH,
  volume =       "56",
  number =       "1",
  pages =        "293--307",
  month =        mar,
  year =         "2016",
  CODEN =        "BITTEL, NBITAB",
  DOI =          "https://doi.org/10.1007/s10543-015-0555-z",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Mon May 30 06:11:10 MDT 2016",
  bibsource =    "http://link.springer.com/journal/10543/56/1;
                 https://www.math.utah.edu/pub/tex/bib/bit.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer.com/article/10.1007/s10543-015-0555-z",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
}


bibtype: article
filename: fparith.bib
label: Sayed:2016:WCR
author: Wafaa S. Sayed and Hossam A. H. Fahmy
title: What are the Correct Results for the Special Values of the Operands of the Power Operation?
journal: j-TOMS
volume: 42
number: 2
pages: 14:1--14:17
month: jun
monthnumber: 6
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2809783
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri Jun 3 18:52:21 MDT 2016
bibtimestamp: 2016.06.03 18:52:21 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 14
journalURL: https://dl.acm.org/loi/toms
rawauthor: Wafaa S. Sayed and Hossam A. H. Fahmy
rawtitle: What are the Correct Results for the Special Values of the Operands of the Power Operation?
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = X$^Y$. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.

rawabstract:
Language standards such as C99 and C11, as well as the IEEE Standard for Floating-Point Arithmetic 754 (IEEE Std 754-2008) specify the expected behavior of binary and decimal floating-point arithmetic in computer-programming environments and the handling of special values and exception conditions. Many researchers focus on verifying the compliance of implementations for binary and decimal floating-point operations with these standards. In this article, we are concerned with the special values of the operands of the power function Z = X$^Y$. We study how the standards define the correct results for this operation, propose a mathematically justified definition for the correct results of the power function on the occurrence of these special values as its operands, test how different software implementations for the power function deal with these special values, and classify the behavior of different programming languages from the viewpoint of how much they conform to the standards and our proposed mathematical definition. We present inconsistencies between the implementations and the standards, and discuss incompatibilities between different versions of the same software.

entry:
@Article{Sayed:2016:WCR,
  author =       "Wafaa S. Sayed and Hossam A. H. Fahmy",
  title =        "What are the Correct Results for the Special Values of
                 the Operands of the Power Operation?",
  journal =      j-TOMS,
  volume =       "42",
  number =       "2",
  pages =        "14:1--14:17",
  month =        jun,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2809783",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri Jun 3 18:52:21 MDT 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "Language standards such as C99 and C11, as well as the
                 IEEE Standard for Floating-Point Arithmetic 754 (IEEE
                 Std 754-2008) specify the expected behavior of binary
                 and decimal floating-point arithmetic in
                 computer-programming environments and the handling of
                 special values and exception conditions. Many
                 researchers focus on verifying the compliance of
                 implementations for binary and decimal floating-point
                 operations with these standards. In this article, we
                 are concerned with the special values of the operands
                 of the power function Z = X$^Y$. We study how the
                 standards define the correct results for this
                 operation, propose a mathematically justified
                 definition for the correct results of the power
                 function on the occurrence of these special values as
                 its operands, test how different software
                 implementations for the power function deal with these
                 special values, and classify the behavior of different
                 programming languages from the viewpoint of how much
                 they conform to the standards and our proposed
                 mathematical definition. We present inconsistencies
                 between the implementations and the standards, and
                 discuss incompatibilities between different versions of
                 the same software.",
  acknowledgement = ack-nhfb,
  articleno =    "14",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: inproceedings
filename: fparith.bib
label: Schaffner:2016:APT
author: Michael Schaffner and Michael Gautschi and Frank K. Gurkaynak and Luca Benini
title: Accuracy and Performance Trade-Offs of Logarithmic Number Units in Multi-Core Clusters
crossref: Montuschi:2016:ISC
pages: 95--103
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.10
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: Michael Schaffner and Michael Gautschi and Frank K. G{\"u}rkaynak and Luca Benini
rawtitle: Accuracy and Performance Trade-Offs of Logarithmic Number Units in Multi-Core Clusters
entry:
@InProceedings{Schaffner:2016:APT,
  author =       "Michael Schaffner and Michael Gautschi and Frank K.
                 G{\"u}rkaynak and Luca Benini",
  title =        "Accuracy and Performance Trade-Offs of Logarithmic
                 Number Units in Multi-Core Clusters",
  crossref =     "Montuschi:2016:ISC",
  pages =        "95--103",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.10",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: article
filename: fparith.bib
label: Schkufza:2016:SPO
author: Eric Schkufza and Rahul Sharma and Alex Aiken
title: Stochastic program optimization
journal: j-CACM
volume: 59
number: 2
pages: 114--122
month: feb
monthnumber: 2
year: 2016
CODEN: CACMA2
DOI: https://doi.org/10.1145/2863701
ISSN: 0001-0782 (print), 1557-7317 (electronic) OR 00010782 (print), 15577317 (electronic)
ISSNL: 0001-0782 OR 00010782
bibdate: Tue Feb 16 15:45:15 MST 2016
bibtimestamp: 2016.02.16 15:45:15 MST
URL: http://cacm.acm.org/magazines/2016/2/197428/fulltext
fjournal: Communications of the ACM
keywords: Hacker's delight; Henry S. Warren; Montgomery multiplication; SAXPY
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/cacm
rawauthor: Eric Schkufza and Rahul Sharma and Alex Aiken
rawtitle: Stochastic program optimization
bibsource:
http://www.acm.org/pubs/contents/journals/cacm/; https://www.math.utah.edu/pub/tex/bib/cacm2010.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

abstract:
The optimization of short sequences of loop-free, fixed-point assembly code sequences is an important problem in high-performance computing. However, the competing constraints of transformation correctness and performance improvement often force even special purpose compilers to produce sub-optimal code. We show that by encoding these constraints as terms in a cost function, and using a Markov Chain Monte Carlo sampler to rapidly explore the space of all possible code sequences, we are able to generate aggressively optimized versions of a given target code sequence. Beginning from binaries compiled by 11vm --O0, we are able to produce provably correct code sequences that either match or outperform the code produced by qcc --O3, icc --O3, and in some cases expert handwritten assembly.

rawabstract:
The optimization of short sequences of loop-free, fixed-point assembly code sequences is an important problem in high-performance computing. However, the competing constraints of transformation correctness and performance improvement often force even special purpose compilers to produce sub-optimal code. We show that by encoding these constraints as terms in a cost function, and using a Markov Chain Monte Carlo sampler to rapidly explore the space of all possible code sequences, we are able to generate aggressively optimized versions of a given target code sequence. Beginning from binaries compiled by 11vm --O0, we are able to produce provably correct code sequences that either match or outperform the code produced by qcc --O3, icc --O3, and in some cases expert handwritten assembly.

entry:
@Article{Schkufza:2016:SPO,
  author =       "Eric Schkufza and Rahul Sharma and Alex Aiken",
  title =        "Stochastic program optimization",
  journal =      j-CACM,
  volume =       "59",
  number =       "2",
  pages =        "114--122",
  month =        feb,
  year =         "2016",
  CODEN =        "CACMA2",
  DOI =          "https://doi.org/10.1145/2863701",
  ISSN =         "0001-0782 (print), 1557-7317 (electronic)",
  ISSN-L =       "0001-0782",
  bibdate =      "Tue Feb 16 15:45:15 MST 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/cacm/;
                 https://www.math.utah.edu/pub/tex/bib/cacm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://cacm.acm.org/magazines/2016/2/197428/fulltext",
  abstract =     "The optimization of short sequences of loop-free,
                 fixed-point assembly code sequences is an important
                 problem in high-performance computing. However, the
                 competing constraints of transformation correctness and
                 performance improvement often force even special
                 purpose compilers to produce sub-optimal code. We show
                 that by encoding these constraints as terms in a cost
                 function, and using a Markov Chain Monte Carlo sampler
                 to rapidly explore the space of all possible code
                 sequences, we are able to generate aggressively
                 optimized versions of a given target code sequence.
                 Beginning from binaries compiled by 11vm --O0, we are
                 able to produce provably correct code sequences that
                 either match or outperform the code produced by qcc
                 --O3, icc --O3, and in some cases expert handwritten
                 assembly.",
  acknowledgement = ack-nhfb,
  fjournal =     "Communications of the ACM",
  journal-URL =  "https://dl.acm.org/loi/cacm",
  keywords =     "Hacker's delight; Henry S. Warren; Montgomery
                 multiplication; SAXPY",
}


bibtype: article
filename: fparith.bib
label: Seo:2016:HMR
author: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
title: Hybrid Montgomery Reduction
journal: j-TECS
volume: 15
number: 3
pages: 58:1--58:??
month: jul
monthnumber: 7
year: 2016
DOI: https://doi.org/10.1145/2890502
ISSN: 1539-9087 (print), 1558-3465 (electronic) OR 15399087 (print), 15583465 (electronic)
ISSNL: 1539-9087 OR 15399087
bibdate: Thu Jul 21 17:18:13 MDT 2016
bibtimestamp: 2016.07.21 17:18:13 MDT
fjournal: ACM Transactions on Embedded Computing Systems
acknowledgement: ack-nhfb
articleno: 58
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J840
rawauthor: Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and Jongseok Choi and Howon Kim
rawtitle: Hybrid {Montgomery} Reduction
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tecs.bib

abstract:
In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of $n$-limb Montgomery reduction from $ theta (n^2 + n)$ to asymptotic complexity $ theta (7 n^2 / 8 + n)$. Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11%.

rawabstract:
In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of $n$-limb Montgomery reduction from $ \theta (n^2 + n)$ to asymptotic complexity $ \theta (7 n^2 / 8 + n)$. Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11\%.

entry:
@Article{Seo:2016:HMR,
  author =       "Hwajeong Seo and Zhe Liu and Yasuyuki Nogami and
                 Jongseok Choi and Howon Kim",
  title =        "Hybrid {Montgomery} Reduction",
  journal =      j-TECS,
  volume =       "15",
  number =       "3",
  pages =        "58:1--58:??",
  month =        jul,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2890502",
  ISSN =         "1539-9087 (print), 1558-3465 (electronic)",
  ISSN-L =       "1539-9087",
  bibdate =      "Thu Jul 21 17:18:13 MDT 2016",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/tecs.bib",
  abstract =     "In this article, we present a hybrid method to improve
                 the performance of the Montgomery reduction by taking
                 advantage of the Karatsuba technique. We divide the
                 Montgomery reduction into two sub-parts, including one
                 for the conventional Montgomery reduction and the other
                 one for Karatsuba-aided multiplication. This approach
                 reduces the multiplication complexity of $n$-limb
                 Montgomery reduction from $ \theta (n^2 + n)$ to
                 asymptotic complexity $ \theta (7 n^2 / 8 + n)$. Our
                 practical implementation results over an 8-bit
                 microcontroller also show performance enhancements by
                 11\%.",
  acknowledgement = ack-nhfb,
  articleno =    "58",
  fjournal =     "ACM Transactions on Embedded Computing Systems",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J840",
}


bibtype: article
filename: fparith.bib
label: Sohn:2016:FFP
author: Jongwook Sohn and Earl E. Swartzlander, Jr.
title: A Fused Floating-Point Four-Term Dot Product Unit
journal: j-IEEE-TRANS-CIRCUITS-SYST-1
volume: 63
number: 3
pages: 370--378
month: mar
monthnumber: 3
year: 2016
DOI: https://doi.org/10.1109/TCSI.2016.2525042
ISSN: 1549-8328 (print), 1558-0806 (electronic) OR 15498328 (print), 15580806 (electronic)
ISSNL: 1549-8328 OR 15498328
bibdate: Mon Feb 10 09:06:33 2020
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2020.02.10 09:06:33 ???
URL: https://ieeexplore.ieee.org/document/7416176
fjournal: IEEE Transactions on Circuits and Systems I: Regular Papers
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8919
rawauthor: Jongwook Sohn and Earl E. {Swartzlander, Jr.}
rawtitle: A Fused Floating-Point Four-Term Dot Product Unit
entry:
@Article{Sohn:2016:FFP,
  author =       "Jongwook Sohn and Earl E. {Swartzlander, Jr.}",
  title =        "A Fused Floating-Point Four-Term Dot Product Unit",
  journal =      j-IEEE-TRANS-CIRCUITS-SYST-1,
  volume =       "63",
  number =       "3",
  pages =        "370--378",
  month =        mar,
  year =         "2016",
  DOI =          "https://doi.org/10.1109/TCSI.2016.2525042",
  ISSN =         "1549-8328 (print), 1558-0806 (electronic)",
  ISSN-L =       "1549-8328",
  bibdate =      "Mon Feb 10 09:06:33 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://ieeexplore.ieee.org/document/7416176",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Circuits and Systems I: Regular
                 Papers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=8919",
}


bibtype: article
filename: fparith.bib
label: Tada:2016:ESG
author: Jubee Tada and Maiki Hosokawa and Ryusuke Egawa and Hiroaki Kobayashi
title: Effects of Stacking Granularity on $3$-D Stacked Floating-point Fused Multiply-Add Units
journal: j-COMP-ARCH-NEWS
volume: 44
number: 4
pages: 62--67
month: sep
monthnumber: 9
year: 2016
CODEN: CANED2
DOI: https://doi.org/10.1145/3039902.3039914
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Jan 12 18:43:44 MST 2017
bibtimestamp: 2017.01.12 18:43:44 MST
fjournal: ACM SIGARCH Computer Architecture News
remark: HEART '16 conference proceedings.
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/sigarch
rawauthor: Jubee Tada and Maiki Hosokawa and Ryusuke Egawa and Hiroaki Kobayashi
rawtitle: Effects of Stacking Granularity on {$3$-D} Stacked Floating-point Fused Multiply-Add Units
bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/sigarch.bib

abstract:
Three-dimensional stacked integrated circuits (3D-SICs) have been expected to overcome the limitations of conventional two-dimensional (2-D) implemented circuits. Since a stacking strategy affects the performance and the power consumption of 3D-SICs, this paper examines two stacking strategies for designing the 3-D stacked floating-point fused multiply-add (FP-FMA) module which contains four FP-FMA units. Experimental results show that a coarse-grain stacking strategy is suitable for reducing critical path delay of the 3-D stacked FP-FMA module. On the other hand, a fine-grain stacking strategy is suitable for reducing power consumption. The 3-D stacked FP-FMA module which is designed based on a fine-grain stacking strategy achieves an 8.4% critical path delay reduction and an 18% average power reduction compared with the 2-D implementation.

rawabstract:
Three-dimensional stacked integrated circuits (3D-SICs) have been expected to overcome the limitations of conventional two-dimensional (2-D) implemented circuits. Since a stacking strategy affects the performance and the power consumption of 3D-SICs, this paper examines two stacking strategies for designing the 3-D stacked floating-point fused multiply-add (FP-FMA) module which contains four FP-FMA units. Experimental results show that a coarse-grain stacking strategy is suitable for reducing critical path delay of the 3-D stacked FP-FMA module. On the other hand, a fine-grain stacking strategy is suitable for reducing power consumption. The 3-D stacked FP-FMA module which is designed based on a fine-grain stacking strategy achieves an 8.4\% critical path delay reduction and an 18\% average power reduction compared with the 2-D implementation.

entry:
@Article{Tada:2016:ESG,
  author =       "Jubee Tada and Maiki Hosokawa and Ryusuke Egawa and
                 Hiroaki Kobayashi",
  title =        "Effects of Stacking Granularity on {$3$-D} Stacked
                 Floating-point Fused Multiply-Add Units",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "44",
  number =       "4",
  pages =        "62--67",
  month =        sep,
  year =         "2016",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/3039902.3039914",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Jan 12 18:43:44 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigarch.bib",
  abstract =     "Three-dimensional stacked integrated circuits
                 (3D-SICs) have been expected to overcome the
                 limitations of conventional two-dimensional (2-D)
                 implemented circuits. Since a stacking strategy affects
                 the performance and the power consumption of 3D-SICs,
                 this paper examines two stacking strategies for
                 designing the 3-D stacked floating-point fused
                 multiply-add (FP-FMA) module which contains four FP-FMA
                 units. Experimental results show that a coarse-grain
                 stacking strategy is suitable for reducing critical
                 path delay of the 3-D stacked FP-FMA module. On the
                 other hand, a fine-grain stacking strategy is suitable
                 for reducing power consumption. The 3-D stacked FP-FMA
                 module which is designed based on a fine-grain stacking
                 strategy achieves an 8.4\% critical path delay
                 reduction and an 18\% average power reduction compared
                 with the 2-D implementation.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "https://dl.acm.org/loi/sigarch",
  remark =       "HEART '16 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Tay:2016:NIM
author: T. Fatt Tay and C. Chang
title: A Non-Iterative Multiple Residue Digit Error Detection and Correction Algorithm in RRNS
journal: j-IEEE-TRANS-COMPUT
volume: 65
number: 2
pages: 396--408
year: 2016
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2015.2435773
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Tue Jan 19 07:06:51 MST 2016
bibtimestamp: 2016.01.19 07:06:51 MST
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: T. Fatt Tay and C. Chang
rawtitle: A Non-Iterative Multiple Residue Digit Error Detection and Correction Algorithm in {RRNS}
entry:
@Article{Tay:2016:NIM,
  author =       "T. Fatt Tay and C. Chang",
  title =        "A Non-Iterative Multiple Residue Digit Error Detection
                 and Correction Algorithm in {RRNS}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "65",
  number =       "2",
  pages =        "396--408",
  month =        "????",
  year =         "2016",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2015.2435773",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Tue Jan 19 07:06:51 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: fparith.bib
label: Ugurdag:2016:ECC
author: H. Fatih Ugurdag and Anil Bayram and Vecdi Emre Levent and Sezer Goren
title: Efficient Combinational Circuits for Division by Small Integer Constants
crossref: Montuschi:2016:ISC
pages: 1--7
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.23
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23
acknowledgement: ack-nhfb
rawauthor: H. Fatih Ugurdag and Anil Bayram and Vecdi Emre Levent and Sezer G{\"o}ren
rawtitle: Efficient Combinational Circuits for Division by Small Integer Constants
entry:
@InProceedings{Ugurdag:2016:ECC,
  author =       "H. Fatih Ugurdag and Anil Bayram and Vecdi Emre Levent
                 and Sezer G{\"o}ren",
  title =        "Efficient Combinational Circuits for Division by Small
                 Integer Constants",
  crossref =     "Montuschi:2016:ISC",
  pages =        "1--7",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.23",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23",
}


bibtype: inproceedings
filename: fparith.bib
label: vanderHoeven:2016:ESL
author: Joris van der Hoeven and Gregoire Lecerf
title: Evaluating Straight-Line Programs over Balls
crossref: Montuschi:2016:ISC
pages: 142--149
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.12
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23; interval arithmetic
acknowledgement: ack-nhfb
rawauthor: Joris van der Hoeven and Gr{\'e}goire Lecerf
rawtitle: Evaluating Straight-Line Programs over Balls
entry:
@InProceedings{vanderHoeven:2016:ESL,
  author =       "Joris van der Hoeven and Gr{\'e}goire Lecerf",
  title =        "Evaluating Straight-Line Programs over Balls",
  crossref =     "Montuschi:2016:ISC",
  pages =        "142--149",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.12",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23; interval arithmetic",
}


bibtype: article
filename: fparith.bib
label: vanderHoeven:2016:MSA
author: Joris van der Hoeven and Gregoire Lecerf and Guillaume Quintin
title: Modular SIMD arithmetic in Mathemagix
journal: j-TOMS
volume: 43
number: 1
pages: 5:1--5:37
month: aug
monthnumber: 8
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2876503
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Nov 22 17:45:25 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2016.11.22 17:45:25 MST
URL: http://dl.acm.org/citation.cfm?id=2876503
fjournal: ACM Transactions on Mathematical Software
acknowledgement: ack-nhfb
articleno: 5
journalURL: http://dl.acm.org/pub.cfm?id=J782
rawauthor: Joris van der Hoeven and Gr{\'e}goire Lecerf and Guillaume Quintin
rawtitle: Modular {SIMD} arithmetic in {Mathemagix}
abstract:
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular integer arithmetic and present detailed vectorized counterparts. We also describe several applications, such as fast modular Fourier transforms and multiplication of integer polynomials and matrices. The vectorized algorithms have been implemented in C++ inside the free computer algebra and analysis system Mathemagix. The performance of our implementation is illustrated by various benchmarks.

rawabstract:
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular integer arithmetic and present detailed vectorized counterparts. We also describe several applications, such as fast modular Fourier transforms and multiplication of integer polynomials and matrices. The vectorized algorithms have been implemented in C++ inside the free computer algebra and analysis system Mathemagix. The performance of our implementation is illustrated by various benchmarks.

entry:
@Article{vanderHoeven:2016:MSA,
  author =       "Joris van der Hoeven and Gr{\'e}goire Lecerf and
                 Guillaume Quintin",
  title =        "Modular {SIMD} arithmetic in {Mathemagix}",
  journal =      j-TOMS,
  volume =       "43",
  number =       "1",
  pages =        "5:1--5:37",
  month =        aug,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2876503",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Nov 22 17:45:25 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "http://dl.acm.org/citation.cfm?id=2876503",
  abstract =     "Modular integer arithmetic occurs in many algorithms
                 for computer algebra, cryptography, and error
                 correcting codes. Although recent microprocessors
                 typically offer a wide range of highly optimized
                 arithmetic functions, modular integer operations still
                 require dedicated implementations. In this article, we
                 survey existing algorithms for modular integer
                 arithmetic and present detailed vectorized
                 counterparts. We also describe several applications,
                 such as fast modular Fourier transforms and
                 multiplication of integer polynomials and matrices. The
                 vectorized algorithms have been implemented in C++
                 inside the free computer algebra and analysis system
                 Mathemagix. The performance of our implementation is
                 illustrated by various benchmarks.",
  acknowledgement = ack-nhfb,
  articleno =    "5",
  fjournal =     "ACM Transactions on Mathematical Software",
  journal-URL =  "http://dl.acm.org/pub.cfm?id=J782",
}


bibtype: inproceedings
filename: fparith.bib
label: Villalba-Moreno:2016:DRF
author: Julio Villalba-Moreno
title: Digit Recurrence Floating-Point Division under HUB Format
crossref: Montuschi:2016:ISC
pages: 79--86
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.17
bibdate: Fri Dec 16 15:17:20 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH 2016; ARITH-23; HUB (Half-Unit-Biased) floating-point format
acknowledgement: ack-nhfb
rawauthor: Julio Villalba-Moreno
rawtitle: Digit Recurrence Floating-Point Division under {HUB} Format
entry:
@InProceedings{Villalba-Moreno:2016:DRF,
  author =       "Julio Villalba-Moreno",
  title =        "Digit Recurrence Floating-Point Division under {HUB}
                 Format",
  crossref =     "Montuschi:2016:ISC",
  pages =        "79--86",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.17",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2016; ARITH-23; HUB (Half-Unit-Biased)
                 floating-point format",
}


bibtype: inproceedings
filename: fparith.bib
label: Wang:2016:DFP
author: Ran Wang and Daming Zou and Xinrui He and Yingfei Xiong and Lu Zhang and Gang Huang
title: Detecting and fixing precision-specific operations for measuring floating-point errors
publisher: pub-ACM
address: pub-ACM:adr
pages: 79--86
year: 2016
DOI: https://doi.org/10.1145/2950290.2950355
bibdate: Thu Oct 17 06:42:14 2019
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2019.10.17 06:42:14 ???
acknowledgement: ack-nhfb
rawauthor: Ran Wang and Daming Zou and Xinrui He and Yingfei Xiong and Lu Zhang and Gang Huang
rawtitle: Detecting and fixing precision-specific operations for measuring floating-point errors
booktitle:
Proceedings of the 2016 24th ACM SIGSOFT International Symposium on Foundations of Software Engineering --- FSE 2016

abstract:
The accuracy of the floating-point calculation is critical to many applications and different methods have been proposed around floating-point accuracies, such as detecting the errors in the program, verifying the accuracy of the program, and optimizing the program to produce more accurate results. These approaches need a specification of the program to understand the ideal calculation performed by the program, which is usually approached by interpreting the program in a precision-unspecific way.par However, many operations programmed in existing code are inherently precision-specific, which cannot be easily interpreted in a precision-unspecific way. In fact, the semantics used in existing approaches usually fail to interpret precision-specific operations correctly.par In this paper, we present a systematic study on precision-specific operations. First, we propose a detection approach to detect precision-specific operations. Second, we propose a fixing approach to enable the tuning of precisions under the presence of precision-specific operations. Third, we studied the precision-specific operations in the GNU C standard math library based on our detection and fixing approaches. Our results show that (1) a significant number of code fragments in the standard C math library are precision-specific operations, and some large inaccuracies reported in existing studies are false positives or potential false positives due to precision-specific operations; (2) our detection approach has high precision and recall; (3) our fixing approach can lead to overall more accurate result.

rawabstract:
The accuracy of the floating-point calculation is critical to many applications and different methods have been proposed around floating-point accuracies, such as detecting the errors in the program, verifying the accuracy of the program, and optimizing the program to produce more accurate results. These approaches need a specification of the program to understand the ideal calculation performed by the program, which is usually approached by interpreting the program in a precision-unspecific way.\par However, many operations programmed in existing code are inherently precision-specific, which cannot be easily interpreted in a precision-unspecific way. In fact, the semantics used in existing approaches usually fail to interpret precision-specific operations correctly.\par In this paper, we present a systematic study on precision-specific operations. First, we propose a detection approach to detect precision-specific operations. Second, we propose a fixing approach to enable the tuning of precisions under the presence of precision-specific operations. Third, we studied the precision-specific operations in the GNU C standard math library based on our detection and fixing approaches. Our results show that (1) a significant number of code fragments in the standard C math library are precision-specific operations, and some large inaccuracies reported in existing studies are false positives or potential false positives due to precision-specific operations; (2) our detection approach has high precision and recall; (3) our fixing approach can lead to overall more accurate result.

rawbooktitle:
{Proceedings of the 2016 24th ACM SIGSOFT International Symposium on Foundations of Software Engineering --- FSE 2016}

entry:
@InProceedings{Wang:2016:DFP,
  author =       "Ran Wang and Daming Zou and Xinrui He and Yingfei
                 Xiong and Lu Zhang and Gang Huang",
  booktitle =    "{Proceedings of the 2016 24th ACM SIGSOFT
                 International Symposium on Foundations of Software
                 Engineering --- FSE 2016}",
  title =        "Detecting and fixing precision-specific operations for
                 measuring floating-point errors",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  year =         "2016",
  DOI =          "https://doi.org/10.1145/2950290.2950355",
  bibdate =      "Thu Oct 17 06:42:14 2019",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The accuracy of the floating-point calculation is
                 critical to many applications and different methods
                 have been proposed around floating-point accuracies,
                 such as detecting the errors in the program, verifying
                 the accuracy of the program, and optimizing the program
                 to produce more accurate results. These approaches need
                 a specification of the program to understand the ideal
                 calculation performed by the program, which is usually
                 approached by interpreting the program in a
                 precision-unspecific way.\par

                 However, many operations programmed in existing code
                 are inherently precision-specific, which cannot be
                 easily interpreted in a precision-unspecific way. In
                 fact, the semantics used in existing approaches usually
                 fail to interpret precision-specific operations
                 correctly.\par

                 In this paper, we present a systematic study on
                 precision-specific operations. First, we propose a
                 detection approach to detect precision-specific
                 operations. Second, we propose a fixing approach to
                 enable the tuning of precisions under the presence of
                 precision-specific operations. Third, we studied the
                 precision-specific operations in the GNU C standard
                 math library based on our detection and fixing
                 approaches. Our results show that (1) a significant
                 number of code fragments in the standard C math library
                 are precision-specific operations, and some large
                 inaccuracies reported in existing studies are false
                 positives or potential false positives due to
                 precision-specific operations; (2) our detection
                 approach has high precision and recall; (3) our fixing
                 approach can lead to overall more accurate result.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: fparith.bib
label: Wilson:2016:UAA
author: David Wilson and Greg Stitt
journal: j-TRETS
volume: 9
number: 3
pages: 21:1--21:??
month: jul
monthnumber: 7
year: 2016
DOI: https://doi.org/10.1145/2809432
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Thu Jul 14 16:35:43 MDT 2016
bibtimestamp: 2016.07.14 16:35:43 MDT
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
articleno: 21
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: David Wilson and Greg Stitt
title:
The Unified Accumulator Architecture: a Configurable, Portable, and Extensible Floating-Point Accumulator

abstract:
Applications accelerated by field-programmable gate arrays (FPGAs) often require pipelined floating-point accumulators with a variety of different trade-offs. Although previous work has introduced numerous floating-point accumulation architectures, few cores are available for public use, which forces designers to use fixed-point implementations or vendor-provided cores that are not portable and are often not optimized for the desired set of trade-offs. In this article, we combine and extend previous floating-point accumulator architectures into a configurable, open-source core, referred to as the unified accumulator architecture (UAA), which enables designers to choose between different trade-offs for different applications. UAA is portable across FPGAs and allows designers to specialize the underlying adder core to take advantage of device-specific optimizations. By providing an extensible, open-source implementation, we hope for the research community to extend the provided core with new architectures and optimizations.

rawabstract:
Applications accelerated by field-programmable gate arrays (FPGAs) often require pipelined floating-point accumulators with a variety of different trade-offs. Although previous work has introduced numerous floating-point accumulation architectures, few cores are available for public use, which forces designers to use fixed-point implementations or vendor-provided cores that are not portable and are often not optimized for the desired set of trade-offs. In this article, we combine and extend previous floating-point accumulator architectures into a configurable, open-source core, referred to as the unified accumulator architecture (UAA), which enables designers to choose between different trade-offs for different applications. UAA is portable across FPGAs and allows designers to specialize the underlying adder core to take advantage of device-specific optimizations. By providing an extensible, open-source implementation, we hope for the research community to extend the provided core with new architectures and optimizations.

rawtitle:
The Unified Accumulator Architecture: a Configurable, Portable, and Extensible Floating-Point Accumulator

entry:
@Article{Wilson:2016:UAA,
  author =       "David Wilson and Greg Stitt",
  title =        "The Unified Accumulator Architecture: a Configurable,
                 Portable, and Extensible Floating-Point Accumulator",
  journal =      j-TRETS,
  volume =       "9",
  number =       "3",
  pages =        "21:1--21:??",
  month =        jul,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2809432",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Thu Jul 14 16:35:43 MDT 2016",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/trets.bib",
  abstract =     "Applications accelerated by field-programmable gate
                 arrays (FPGAs) often require pipelined floating-point
                 accumulators with a variety of different trade-offs.
                 Although previous work has introduced numerous
                 floating-point accumulation architectures, few cores
                 are available for public use, which forces designers to
                 use fixed-point implementations or vendor-provided
                 cores that are not portable and are often not optimized
                 for the desired set of trade-offs. In this article, we
                 combine and extend previous floating-point accumulator
                 architectures into a configurable, open-source core,
                 referred to as the unified accumulator architecture
                 (UAA), which enables designers to choose between
                 different trade-offs for different applications. UAA is
                 portable across FPGAs and allows designers to
                 specialize the underlying adder core to take advantage
                 of device-specific optimizations. By providing an
                 extensible, open-source implementation, we hope for the
                 research community to extend the provided core with new
                 architectures and optimizations.",
  acknowledgement = ack-nhfb,
  articleno =    "21",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: article
filename: fparith.bib
label: Zhou:2016:PUH
author: Yuanyuan Zhou
title: Programming Uncertain $<$T$>$ hings
journal: j-COMP-ARCH-NEWS
volume: 44
number: 2
pages: 1--2
month: may
monthnumber: 5
year: 2016
CODEN: CANED2
DOI: https://doi.org/10.1145/2980024.2872416
ISSN: 0163-5964 (print), 1943-5851 (electronic) OR 01635964 (print), 19435851 (electronic)
ISSNL: 0163-5964 OR 01635964
bibdate: Thu Jan 12 18:43:42 MST 2017
bibtimestamp: 2017.01.12 18:43:42 MST
fjournal: ACM SIGARCH Computer Architecture News
remark: ASPLOS'16 conference proceedings.
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/sigarch
rawauthor: Yuanyuan Zhou
rawtitle: Programming Uncertain {$<$T$>$ hings}
abstract:
Innovation flourishes with good abstractions. For instance, codification of the IEEE Floating Point standard in 1985 was critical to the subsequent success of scientific computing. Programming languages currently lack appropriate abstractions for uncertain data. Applications already use estimates from sensors, machine learning, big data, humans, and approximate algorithms, but most programming languages do not help developers address correctness, programmability, and optimization problems due to estimates. To address these problems, we propose a new programming abstraction called Uncertain. We encourage the community to develop and use abstractions for estimates.

rawabstract:
Innovation flourishes with good abstractions. For instance, codification of the IEEE Floating Point standard in 1985 was critical to the subsequent success of scientific computing. Programming languages currently lack appropriate abstractions for uncertain data. Applications already use estimates from sensors, machine learning, big data, humans, and approximate algorithms, but most programming languages do not help developers address correctness, programmability, and optimization problems due to estimates. To address these problems, we propose a new programming abstraction called Uncertain. We encourage the community to develop and use abstractions for estimates.

entry:
@Article{Zhou:2016:PUH,
  author =       "Yuanyuan Zhou",
  title =        "Programming Uncertain {$<$T$>$ hings}",
  journal =      j-COMP-ARCH-NEWS,
  volume =       "44",
  number =       "2",
  pages =        "1--2",
  month =        may,
  year =         "2016",
  CODEN =        "CANED2",
  DOI =          "https://doi.org/10.1145/2980024.2872416",
  ISSN =         "0163-5964 (print), 1943-5851 (electronic)",
  ISSN-L =       "0163-5964",
  bibdate =      "Thu Jan 12 18:43:42 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigarch.bib",
  abstract =     "Innovation flourishes with good abstractions. For
                 instance, codification of the IEEE Floating Point
                 standard in 1985 was critical to the subsequent success
                 of scientific computing. Programming languages
                 currently lack appropriate abstractions for uncertain
                 data. Applications already use estimates from sensors,
                 machine learning, big data, humans, and approximate
                 algorithms, but most programming languages do not help
                 developers address correctness, programmability, and
                 optimization problems due to estimates. To address
                 these problems, we propose a new programming
                 abstraction called Uncertain. We encourage the
                 community to develop and use abstractions for
                 estimates.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM SIGARCH Computer Architecture News",
  journal-URL =  "https://dl.acm.org/loi/sigarch",
  remark =       "ASPLOS'16 conference proceedings.",
}


bibtype: article
filename: fparith.bib
label: Aliasgari:2017:SCH
author: Mehrdad Aliasgari and Marina Blanton and Fattaneh Bayatbabolghani
journal: j-INT-J-INFO-SEC
volume: 16
number: 6
pages: 577--601
month: nov
monthnumber: 11
year: 2017
DOI: https://doi.org/10.1007/s10207-016-0350-0
ISSN: 1615-5262 (print), 1615-5270 (electronic) OR 16155262 (print), 16155270 (electronic)
ISSNL: 1615-5262 OR 16155262
bibdate: Tue Jan 23 16:01:49 MST 2018
bibtimestamp: 2018.01.23 16:01:49 MST
URL: http://link.springer.com/article/10.1007/s10207-016-0350-0
fjournal: International Journal of Information Security
keywords: Floating point; Gaussian mixture models; Hidden Markov models; Secure computation
acknowledgement: ack-nhfb
journalURL: https://link.springer.com/journal/10207
rawauthor: Mehrdad Aliasgari and Marina Blanton and Fattaneh Bayatbabolghani
title:
Secure computation of hidden Markov models and secure floating-point arithmetic in the malicious model

bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/intjinfosec.bib

rawtitle:
Secure computation of hidden {Markov} models and secure floating-point arithmetic in the malicious model

entry:
@Article{Aliasgari:2017:SCH,
  author =       "Mehrdad Aliasgari and Marina Blanton and Fattaneh
                 Bayatbabolghani",
  title =        "Secure computation of hidden {Markov} models and
                 secure floating-point arithmetic in the malicious
                 model",
  journal =      j-INT-J-INFO-SEC,
  volume =       "16",
  number =       "6",
  pages =        "577--601",
  month =        nov,
  year =         "2017",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10207-016-0350-0",
  ISSN =         "1615-5262 (print), 1615-5270 (electronic)",
  ISSN-L =       "1615-5262",
  bibdate =      "Tue Jan 23 16:01:49 MST 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/intjinfosec.bib",
  URL =          "http://link.springer.com/article/10.1007/s10207-016-0350-0",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Information Security",
  journal-URL =  "https://link.springer.com/journal/10207",
  keywords =     "Floating point; Gaussian mixture models; Hidden Markov
                 models; Secure computation",
}


bibtype: article
filename: fparith.bib
label: Anderson:2017:EMF
author: Andrew Anderson and Servesh Muralidharan and David Gregg
title: Efficient Multibyte Floating Point Data Formats Using Vectorization
journal: j-IEEE-TRANS-COMPUT
volume: 66
number: 12
pages: 2081--2096
year: 2017
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2017.2716355
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Nov 10 08:32:25 MST 2017
bibtimestamp: 2017.11.10 08:32:25 MST
URL: https://ieeexplore.ieee.org/document/7950938/
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Andrew Anderson and Servesh Muralidharan and David Gregg
rawtitle: Efficient Multibyte Floating Point Data Formats Using Vectorization
entry:
@Article{Anderson:2017:EMF,
  author =       "Andrew Anderson and Servesh Muralidharan and David
                 Gregg",
  title =        "Efficient Multibyte Floating Point Data Formats Using
                 Vectorization",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "66",
  number =       "12",
  pages =        "2081--2096",
  month =        "????",
  year =         "2017",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2017.2716355",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Nov 10 08:32:25 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  URL =          "https://ieeexplore.ieee.org/document/7950938/",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: fparith.bib
label: Angerd:2017:FAC
author: Alexandra Angerd and Erik Sintorn and Per Stenstrom
journal: j-TACO
volume: 14
number: 4
pages: 46:1--46:??
month: dec
monthnumber: 12
year: 2017
DOI: https://doi.org/10.1145/3151032
ISSN: 1544-3566 (print), 1544-3973 (electronic) OR 15443566 (print), 15443973 (electronic)
ISSNL: 1544-3566 OR 15443566
bibdate: Fri Dec 22 18:25:55 MST 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/taco.bib
bibtimestamp: 2017.12.22 18:25:55 MST
fjournal: ACM Transactions on Architecture and Code Optimization (TACO)
acknowledgement: ack-nhfb
articleno: 46
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J924
rawauthor: Alexandra Angerd and Erik Sintorn and Per Stenstr{\"o}m
title:
A Framework for Automated and Controlled Floating-Point Accuracy Reduction in Graphics Applications on GPUs

abstract:
Reducing the precision of floating-point values can improve performance and/or reduce energy expenditure in computer graphics, among other, applications. However, reducing the precision level of floating-point values in a controlled fashion needs support both at the compiler and at the microarchitecture level. At the compiler level, a method is needed to automate the reduction of precision of each floating-point value. At the microarchitecture level, a lower precision of each floating-point register can allow more floating-point values to be packed into a register file. This, however, calls for new register file organizations. This article proposes an automated precision-selection method and a novel GPU register file organization that can store floating-point register values at arbitrary precisions densely. The automated precision-selection method uses a data-driven approach for setting the precision level of floating-point values, given a quality threshold and a representative set of input data. By allowing a small, but acceptable, degradation in output quality, our method can remove a significant amount of the bits needed to represent floating-point values in the investigated kernels (between 28% and 60%). Our proposed register file organization exploits these lower-precision floating-point values by packing several of them into the same physical register. This reduces the register pressure per thread by up to 48%, and by 27% on average, for a negligible output-quality degradation. This can enable GPUs to keep up to twice as many threads in flight simultaneously.

rawabstract:
Reducing the precision of floating-point values can improve performance and/or reduce energy expenditure in computer graphics, among other, applications. However, reducing the precision level of floating-point values in a controlled fashion needs support both at the compiler and at the microarchitecture level. At the compiler level, a method is needed to automate the reduction of precision of each floating-point value. At the microarchitecture level, a lower precision of each floating-point register can allow more floating-point values to be packed into a register file. This, however, calls for new register file organizations. This article proposes an automated precision-selection method and a novel GPU register file organization that can store floating-point register values at arbitrary precisions densely. The automated precision-selection method uses a data-driven approach for setting the precision level of floating-point values, given a quality threshold and a representative set of input data. By allowing a small, but acceptable, degradation in output quality, our method can remove a significant amount of the bits needed to represent floating-point values in the investigated kernels (between 28\% and 60\%). Our proposed register file organization exploits these lower-precision floating-point values by packing several of them into the same physical register. This reduces the register pressure per thread by up to 48\%, and by 27\% on average, for a negligible output-quality degradation. This can enable GPUs to keep up to twice as many threads in flight simultaneously.

rawtitle:
A Framework for Automated and Controlled Floating-Point Accuracy Reduction in Graphics Applications on {GPUs}

entry:
@Article{Angerd:2017:FAC,
  author =       "Alexandra Angerd and Erik Sintorn and Per
                 Stenstr{\"o}m",
  title =        "A Framework for Automated and Controlled
                 Floating-Point Accuracy Reduction in Graphics
                 Applications on {GPUs}",
  journal =      j-TACO,
  volume =       "14",
  number =       "4",
  pages =        "46:1--46:??",
  month =        dec,
  year =         "2017",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/3151032",
  ISSN =         "1544-3566 (print), 1544-3973 (electronic)",
  ISSN-L =       "1544-3566",
  bibdate =      "Fri Dec 22 18:25:55 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/taco.bib",
  abstract =     "Reducing the precision of floating-point values can
                 improve performance and/or reduce energy expenditure in
                 computer graphics, among other, applications. However,
                 reducing the precision level of floating-point values
                 in a controlled fashion needs support both at the
                 compiler and at the microarchitecture level. At the
                 compiler level, a method is needed to automate the
                 reduction of precision of each floating-point value. At
                 the microarchitecture level, a lower precision of each
                 floating-point register can allow more floating-point
                 values to be packed into a register file. This,
                 however, calls for new register file organizations.
                 This article proposes an automated precision-selection
                 method and a novel GPU register file organization that
                 can store floating-point register values at arbitrary
                 precisions densely. The automated precision-selection
                 method uses a data-driven approach for setting the
                 precision level of floating-point values, given a
                 quality threshold and a representative set of input
                 data. By allowing a small, but acceptable, degradation
                 in output quality, our method can remove a significant
                 amount of the bits needed to represent floating-point
                 values in the investigated kernels (between 28\% and
                 60\%). Our proposed register file organization exploits
                 these lower-precision floating-point values by packing
                 several of them into the same physical register. This
                 reduces the register pressure per thread by up to 48\%,
                 and by 27\% on average, for a negligible output-quality
                 degradation. This can enable GPUs to keep up to twice
                 as many threads in flight simultaneously.",
  acknowledgement = ack-nhfb,
  articleno =    "46",
  fjournal =     "ACM Transactions on Architecture and Code Optimization
                 (TACO)",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J924",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:AI
author: Anonymous
title: Author index
crossref: Burgess:2017:ISC
pages: 196--196
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.41
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Presents an index of the authors whose articles are published in the conference proceedings record.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Presents an index of the authors whose articles are published in the conference proceedings record.
rawauthor: Anonymous
rawtitle: Author index
entry:
@InProceedings{Anonymous:2017:AI,
  author =       "Anonymous",
  title =        "Author index",
  crossref =     "Burgess:2017:ISC",
  pages =        "196--196",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.41",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Presents an index of the authors whose articles are
                 published in the conference proceedings record.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:C
author: Anonymous
title: Committees
crossref: Burgess:2017:ISC
pages: x--x
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.5
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Provides a listing of current committee members and society officers.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Provides a listing of current committee members and society officers.
rawauthor: Anonymous
rawtitle: Committees
entry:
@InProceedings{Anonymous:2017:C,
  author =       "Anonymous",
  title =        "Committees",
  crossref =     "Burgess:2017:ISC",
  pages =        "x--x",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.5",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Provides a listing of current committee members and
                 society officers.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:CN
author: Anonymous
title: [Copyright notice]
crossref: Burgess:2017:ISC
pages: iv--iv
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.3
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Presents the copyright information for the conference. May include reprint permission information.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Presents the copyright information for the conference. May include reprint permission information.
rawauthor: Anonymous
rawtitle: [{Copyright} notice]
entry:
@InProceedings{Anonymous:2017:CN,
  author =       "Anonymous",
  title =        "[{Copyright} notice]",
  crossref =     "Burgess:2017:ISC",
  pages =        "iv--iv",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.3",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Presents the copyright information for the conference.
                 May include reprint permission information.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:F
author: Anonymous
title: Foreword
crossref: Burgess:2017:ISC
pages: viii--ix
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.4
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawauthor: Anonymous
rawtitle: Foreword
abstract:
Presents the introductory welcome message from the conference proceedings. May include the conference officers' congratulations to all involved with the conference event and publication of the proceedings record.

rawabstract:
Presents the introductory welcome message from the conference proceedings. May include the conference officers' congratulations to all involved with the conference event and publication of the proceedings record.

entry:
@InProceedings{Anonymous:2017:F,
  author =       "Anonymous",
  title =        "Foreword",
  crossref =     "Burgess:2017:ISC",
  pages =        "viii--ix",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.4",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Presents the introductory welcome message from the
                 conference proceedings. May include the conference
                 officers' congratulations to all involved with the
                 conference event and publication of the proceedings
                 record.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:FC
author: Anonymous
title: [Front cover]
crossref: Burgess:2017:ISC
pages: c1--c1
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.43
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Presents the front cover or splash screen of the proceedings record.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Presents the front cover or splash screen of the proceedings record.
rawauthor: Anonymous
rawtitle: [{Front} cover]
entry:
@InProceedings{Anonymous:2017:FC,
  author =       "Anonymous",
  title =        "[{Front} cover]",
  crossref =     "Burgess:2017:ISC",
  pages =        "c1--c1",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.43",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Presents the front cover or splash screen of the
                 proceedings record.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:PCM
author: Anonymous
title: Program Committee Members
crossref: Burgess:2017:ISC
pages: xi--xi
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.6
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Provides a listing of current committee members and society officers.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Provides a listing of current committee members and society officers.
rawauthor: Anonymous
rawtitle: Program Committee Members
entry:
@InProceedings{Anonymous:2017:PCM,
  author =       "Anonymous",
  title =        "Program Committee Members",
  crossref =     "Burgess:2017:ISC",
  pages =        "xi--xi",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.6",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Provides a listing of current committee members and
                 society officers.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:PI
author: Anonymous
title: [Publisher's information]
crossref: Burgess:2017:ISC
pages: 198--198
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.42
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Provides a listing of current committee members and society officers.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Provides a listing of current committee members and society officers.
rawauthor: Anonymous
rawtitle: [{Publisher}'s information]
entry:
@InProceedings{Anonymous:2017:PI,
  author =       "Anonymous",
  title =        "[{Publisher}'s information]",
  crossref =     "Burgess:2017:ISC",
  pages =        "198--198",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.42",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Provides a listing of current committee members and
                 society officers.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:SC
author: Anonymous
title: Steering Committee
crossref: Burgess:2017:ISC
pages: xii--xii
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.7
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Provides a listing of current committee members and society officers.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Provides a listing of current committee members and society officers.
rawauthor: Anonymous
rawtitle: Steering Committee
entry:
@InProceedings{Anonymous:2017:SC,
  author =       "Anonymous",
  title =        "Steering Committee",
  crossref =     "Burgess:2017:ISC",
  pages =        "xii--xii",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.7",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Provides a listing of current committee members and
                 society officers.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:TC
author: Anonymous
title: Table of contents
crossref: Burgess:2017:ISC
pages: v--vii
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.8
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
acknowledgement: ack-nhfb
rawauthor: Anonymous
rawtitle: Table of contents
abstract:
The following topics are dealt with: multiprecision arithmetic; computer arithmetic; DSP; floating-point error analysis; reproducible arithmetic; FPGA; matrix computations; and cryptography.

keywords:
ARITH 2017; ARITH-24; computer arithmetic; cryptography; digital signal processing chips; DSP; error analysis; field programmable gate arrays; floating point arithmetic; floating-point error analysis; FPGA; matrix algebra; matrix computations; multiprecision arithmetic; reproducible arithmetic

rawabstract:
The following topics are dealt with: multiprecision arithmetic; computer arithmetic; DSP; floating-point error analysis; reproducible arithmetic; FPGA; matrix computations; and cryptography.

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  author =       "Anonymous",
  title =        "Table of contents",
  crossref =     "Burgess:2017:ISC",
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  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The following topics are dealt with: multiprecision
                 arithmetic; computer arithmetic; DSP; floating-point
                 error analysis; reproducible arithmetic; FPGA; matrix
                 computations; and cryptography.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24; computer arithmetic;
                 cryptography; digital signal processing chips; DSP;
                 error analysis; field programmable gate arrays;
                 floating point arithmetic; floating-point error
                 analysis; FPGA; matrix algebra; matrix computations;
                 multiprecision arithmetic; reproducible arithmetic",
}


bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:TP
author: Anonymous
title: [Title page i]
crossref: Burgess:2017:ISC
pages: i--i
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.1
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Presents the title page of the proceedings record.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Presents the title page of the proceedings record.
rawauthor: Anonymous
rawtitle: [{Title} page i]
entry:
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  author =       "Anonymous",
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  crossref =     "Burgess:2017:ISC",
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  bibdate =      "Fri Nov 17 09:10:14 2017",
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  abstract =     "Presents the title page of the proceedings record.",
  acknowledgement = ack-nhfb,
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bibtype: inproceedings
filename: fparith.bib
label: Anonymous:2017:TPI
author: Anonymous
title: [Title page iii]
crossref: Burgess:2017:ISC
pages: iii--iii
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1109/ARITH.2017.2
ISSN: 1063-6889 OR 10636889
bibdate: Fri Nov 17 09:10:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/fparith.bib
bibtimestamp: 2017.11.17 09:10:14 ???
abstract: Presents the title page of the proceedings record.
keywords: ARITH 2017; ARITH-24
acknowledgement: ack-nhfb
rawabstract: Presents the title page of the proceedings record.
rawauthor: Anonymous
rawtitle: [{Title} page iii]
entry:
@InProceedings{Anonymous:2017:TPI,
  author =       "Anonymous",
  title =        "[{Title} page iii]",
  crossref =     "Burgess:2017:ISC",
  pages =        "iii--iii",
  month =        jul,
  year =         "2017",
  DOI =          "https://doi.org/10.1109/ARITH.2017.2",
  ISSN =         "1063-6889",
  bibdate =      "Fri Nov 17 09:10:14 2017",
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  abstract =     "Presents the title page of the proceedings record.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH 2017; ARITH-24",
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bibtype: article
filename: fparith.bib
label: Aurentz:2017:CCS
author: Jared L. Aurentz and Lloyd N. Trefethen
title: Chopping a Chebyshev Series
journal: j-TOMS
volume: 43
number: 4
pages: 33:1--33:21
month: mar
monthnumber: 3
year: 2017
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2998442
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65G50 (65Y04)
MRnumber: 3638570
bibdate: Fri Mar 24 08:51:05 MDT 2017
bibtimestamp: 2017.03.24 08:51:05 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 33
journalURL: https://dl.acm.org/loi/toms
rawauthor: Jared L. Aurentz and Lloyd N. Trefethen
rawtitle: Chopping a {Chebyshev} Series
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
Chebfun and related software projects for numerical computing with functions are based on the idea that at each step of a computation, a function $ f(x) $ defined on an interval $ [a, b] $ is ``rounded'' to a prescribed precision by constructing a Chebyshev series and chopping it at an appropriate point. Designing a chopping algorithm with the right properties proves to be a surprisingly complex and interesting problem. We describe the chopping algorithm introduced in Chebfun Version 5.3 in 2015 after many years of discussion and the considerations that led to this design.

rawabstract:
Chebfun and related software projects for numerical computing with functions are based on the idea that at each step of a computation, a function $ f(x) $ defined on an interval $ [a, b] $ is ``rounded'' to a prescribed precision by constructing a Chebyshev series and chopping it at an appropriate point. Designing a chopping algorithm with the right properties proves to be a surprisingly complex and interesting problem. We describe the chopping algorithm introduced in Chebfun Version 5.3 in 2015 after many years of discussion and the considerations that led to this design.

entry:
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  author =       "Jared L. Aurentz and Lloyd N. Trefethen",
  title =        "Chopping a {Chebyshev} Series",
  journal =      j-TOMS,
  volume =       "43",
  number =       "4",
  pages =        "33:1--33:21",
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  DOI =          "https://doi.org/10.1145/2998442",
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  MRclass =      "65G50 (65Y04)",
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  bibdate =      "Fri Mar 24 08:51:05 MDT 2017",
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                 https://www.math.utah.edu/pub/bibnet/authors/t/trefethen-lloyd-n.bib;
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                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "Chebfun and related software projects for numerical
                 computing with functions are based on the idea that at
                 each step of a computation, a function $ f(x) $ defined
                 on an interval $ [a, b] $ is ``rounded'' to a
                 prescribed precision by constructing a Chebyshev series
                 and chopping it at an appropriate point. Designing a
                 chopping algorithm with the right properties proves to
                 be a surprisingly complex and interesting problem. We
                 describe the chopping algorithm introduced in Chebfun
                 Version 5.3 in 2015 after many years of discussion and
                 the considerations that led to this design.",
  acknowledgement = ack-nhfb,
  articleno =    "33",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: fparith.bib
label: B:2017:GML
author: Sudeepa K. B. and Ganesh Aithal
journal: j-J-COMPUT-SCI
volume: 21
pages: 379--386
month: jul
monthnumber: 7
year: 2017
DOI: https://doi.org/10.1016/j.jocs.2016.10.006
ISSN: 1877-7503 (print), 1877-7511 (electronic) OR 18777503 (print), 18777511 (electronic)
ISSNL: 1877-7503 OR 18777503
MRclass: 94A60
MRnumber: 3685019
bibdate: Tue Sep 19 13:54:25 MDT 2023
bibtimestamp: 2023.09.19 13:54:25 MDT
URL: https://www.sciencedirect.com/science/article/pii/S1877750316302071
fjournal: Journal of Computational Science
acknowledgement: ack-nhfb
ajournal: J. Comput. Sci.
journalURL: https://www.sciencedirect.com/journal/journal-of-computational-science
rawauthor: Sudeepa K. B. and Ganesh Aithal
title:
Generation of maximum length non-binary key sequence and its application for stream cipher based on residue number system

bibsource:
https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcomputsci.bib

rawtitle:
Generation of maximum length non-binary key sequence and its application for stream cipher based on residue number system

entry:
@Article{B:2017:GML,
  author =       "Sudeepa K. B. and Ganesh Aithal",
  title =        "Generation of maximum length non-binary key sequence
                 and its application for stream cipher based on residue
                 number system",
  journal =      j-J-COMPUT-SCI,
  volume =       "21",
  pages =        "379--386",
  month =        jul,
  year =         "2017",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1016/j.jocs.2016.10.006",
  ISSN =         "1877-7503 (print), 1877-7511 (electronic)",
  ISSN-L =       "1877-7503",
  MRclass =      "94A60",
  MRnumber =     "3685019",
  bibdate =      "Tue Sep 19 13:54:25 MDT 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputsci.bib",
  URL =          "https://www.sciencedirect.com/science/article/pii/S1877750316302071",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Comput. Sci.",
  fjournal =     "Journal of Computational Science",
  journal-URL =  "https://www.sciencedirect.com/journal/journal-of-computational-science",
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bibtype: book
filename: fparith.bib
label: Beebe:2017:MFC
author: Nelson H. F. Beebe
publisher: pub-SV
address: pub-SV:adr
pages: xxxvi + 1114
year: 2017
DOI: https://doi.org/10.1007/978-3-319-64110-2
ISBN: 3-319-64109-3 (hardcover), 3-319-64110-7 (e-book) OR 3319641093 (hardcover), 3319641107 (ebook)
LCCN: QA75.5-76.95
bibdate: Sat Jul 15 19:34:43 MDT 2017
bibtimestamp: 2017.07.15 19:34:43 MDT
URL: http://www.springer.com/us/book/9783319641096
acknowledgement: ack-nhfb
rawauthor: Nelson H. F. Beebe
ORCID: Beebe, Nelson H. F./0000-0001-7281-4263
title:
The Mathematical-Function Computation Handbook: Programming Using the MathCW Portable Software Library

ISBN13:
978-3-319-64109-6 (hardcover), 978-3-319-64110-2 (e-book) OR 9783319641096 (hardcover), 9783319641102 (ebook)

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib; https://www.math.utah.edu/pub/tex/bib/axiom.bib; https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/mupad.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/redbooks.bib; https://www.math.utah.edu/pub/tex/bib/utah-math-dept-books.bib

TOC:
List of figures / xxv List of tables / xxxi Quick start / xxxv 1: Introduction / 1 1.1: Programming conventions / 2 1.2: Naming conventions / 4 1.3: Library contributions and coverage / 5 1.4: Summary / 6 2: Iterative solutions and other tools / 7 2.1: Polynomials and Taylor series / 7 2.2: First-order Taylor series approximation / 8 2.3: Second-order Taylor series approximation / 9 2.4: Another second-order Taylor series approximation / 9 2.5: Convergence of second-order methods / 10 2.6: Taylor series for elementary functions / 10 2.7: Continued fractions / 12 2.8: Summation of continued fractions / 17 2.9: Asymptotic expansions / 19 2.10: Series inversion / 20 2.11: Summary / 22 3: Polynomial approximations / 23 3.1: Computation of odd series / 23 3.2: Computation of even series / 25 3.3: Computation of general series / 25 3.4: Limitations of Cody / Waite polynomials / 28 3.5: Polynomial fits with Maple / 32 3.6: Polynomial fits with Mathematica / 33 3.7: Exact polynomial coefficients / 42 3.8: Cody / Waite rational polynomials / 43 3.9: Chebyshev polynomial economization / 43 3.10: Evaluating Chebyshev polynomials / 48 3.11: Error compensation in Chebyshev fits / 50 3.12: Improving Chebyshev fits / 51 3.13: Chebyshev fits in rational form / 52 3.14: Chebyshev fits with Mathematica / 56 3.15: Chebyshev fits for function representation / 57 3.16: Extending the library / 57 3.17: Summary and further reading / 58 4: Implementation issues / 61 4.1: Error magnification / 61 4.2: Machine representation and machine epsilon / 62 4.3: IEEE 754 arithmetic / 63 4.4: Evaluation order in C / 64 4.5: The tt volatile type qualifier / 65 4.6: Rounding in floating-point arithmetic / 66 4.7: Signed zero / 69 4.8: Floating-point zero divide / 70 4.9: Floating-point overflow / 71 4.10: Integer overflow / 72 4.11: Floating-point underflow / 77 4.12: Subnormal numbers / 78 4.13: Floating-point inexact operation / 79 4.14: Floating-point invalid operation / 79 4.15: Remarks on NaN tests / 80 4.16: Ulps --- units in the last place / 81 4.17: Fused multiply-add / 85 4.18: Fused multiply-add and polynomials / 88 4.19: Significance loss / 89 4.20: Error handling and reporting / 89 4.21: Interpreting error codes / 93 4.22: C99 changes to error reporting / 94 4.23: Error reporting with threads / 95 4.24: Comments on error reporting / 95 4.25: Testing function implementations / 96 4.26: Extended data types on Hewlett--Packard HP-UX IA-64 / 100 4.27: Extensions for decimal arithmetic / 101 4.28: Further reading / 103 4.29: Summary / 104 5: The floating-point environment / 105 5.1: IEEE 754 and programming languages / 105 5.2: IEEE 754 and the mathcw library / 106 5.3: Exceptions and traps / 106 5.4: Access to exception flags and rounding control / 107 5.5: The environment access pragma / 110 5.6: Implementation of exception-flag and rounding-control access / 110 5.7: Using exception flags: simple cases / 112 5.8: Using rounding control / 115 5.9: Additional exception flag access / 116 5.10: Using exception flags: complex case / 120 5.11: Access to precision control / 123 5.12: Using precision control / 126 5.13: Summary / 127 6: Converting floating-point values to integers / 129 6.1: Integer conversion in programming languages / 129 6.2: Programming issues for conversions to integers / 130 6.3: Hardware out-of-range conversions / 131 6.4: Rounding modes and integer conversions / 132 6.5: Extracting integral and fractional parts / 132 6.6: Truncation functions / 135 6.7: Ceiling and floor functions / 136 6.8: Floating-point rounding functions with fixed rounding / 137 6.9: Floating-point rounding functions: current rounding / 138 6.10: Floating-point rounding functions without em inexact exception / 139 6.11: Integer rounding functions with fixed rounding / 140 6.12: Integer rounding functions with current rounding / 142 6.13: Remainder / 143 6.14: Why the remainder functions are hard / 144 6.15: Computing tt fmod / 146 6.16: Computing tt remainder / 148 6.17: Computing tt remquo / 150 6.18: Computing one remainder from the other / 152 6.19: Computing the remainder in nonbinary bases / 155 6.20: Summary / 156 7: Random numbers / 157 7.1: Guidelines for random-number software / 157 7.2: Creating generator seeds / 158 7.3: Random floating-point values / 160 7.4: Random integers from floating-point generator / 165 7.5: Random integers from an integer generator / 166 7.6: Random integers in ascending order / 168 7.7: How random numbers are generated / 169 7.8: Removing generator bias / 178 7.9: Improving a poor random number generator / 178 7.10: Why long periods matter / 179 7.11: Inversive congruential generators / 180 7.12: Inversive congruential generators, revisited / 189 7.13: Distributions of random numbers / 189 7.14: Other distributions / 195 7.15: Testing random-number generators / 196 7.16: Applications of random numbers / 202 7.17: The textsf mathcw random number routines / 208 7.18: Summary, advice, and further reading / 214 8: Roots / 215 8.1: Square root / 215 8.2: Hypotenuse and vector norms / 222 8.3: Hypotenuse by iteration / 227 8.4: Reciprocal square root / 233 8.5: Cube root / 237 8.6: Roots in hardware / 240 8.7: Summary / 242 9: Argument reduction / 243 9.1: Simple argument reduction / 243 9.2: Exact argument reduction / 250 9.3: Implementing exact argument reduction / 253 9.4: Testing argument reduction / 265 9.5: Retrospective on argument reduction / 265 10: Exponential and logarithm / 267 10.1: Exponential functions / 267 10.2: Exponential near zero / 273 10.3: Logarithm functions / 282 10.4: Logarithm near one / 290 10.5: Exponential and logarithm in hardware / 292 10.6: Compound interest and annuities / 294 10.7: Summary / 298 11: Trigonometric functions / 299 11.1: Sine and cosine properties / 299 11.2: Tangent properties / 302 11.3: Argument conventions and units / 304 11.4: Computing the cosine and sine / 306 11.5: Computing the tangent / 310 11.6: Trigonometric functions in degrees / 313 11.7: Trigonometric functions in units of $ pi $ / 315 11.8: Computing the cosine and sine together / 320 11.9: Inverse sine and cosine / 323 11.10: Inverse tangent / 331 11.11: Inverse tangent, take two / 336 11.12: Trigonometric functions in hardware / 338 11.13: Testing trigonometric functions / 339 11.14: Retrospective on trigonometric functions / 340 12: Hyperbolic functions / 341 12.1: Hyperbolic functions / 341 12.2: Improving the hyperbolic functions / 345 12.3: Computing the hyperbolic functions together / 348 12.4: Inverse hyperbolic functions / 348 12.5: Hyperbolic functions in hardware / 350 12.6: Summary / 352 13: Pair-precision arithmetic / 353 13.1: Limitations of pair-precision arithmetic / 354 13.2: Design of the pair-precision software interface / 355 13.3: Pair-precision initialization / 356 13.4: Pair-precision evaluation / 357 13.5: Pair-precision high part / 357 13.6: Pair-precision low part / 357 13.7: Pair-precision copy / 357 13.8: Pair-precision negation / 358 13.9: Pair-precision absolute value / 358 13.10: Pair-precision sum / 358 13.11: Splitting numbers into pair sums / 359 13.12: Premature overflow in splitting / 362 13.13: Pair-precision addition / 365 13.14: Pair-precision subtraction / 367 13.15: Pair-precision comparison / 368 13.16: Pair-precision multiplication / 368 13.17: Pair-precision division / 371 13.18: Pair-precision square root / 373 13.19: Pair-precision cube root / 377 13.20: Accuracy of pair-precision arithmetic / 379 13.21: Pair-precision vector sum / 384 13.22: Exact vector sums / 385 13.23: Pair-precision dot product / 385 13.24: Pair-precision product sum / 386 13.25: Pair-precision decimal arithmetic / 387 13.26: Fused multiply-add with pair precision / 388 13.27: Higher intermediate precision and the FMA / 393 13.28: Fused multiply-add without pair precision / 395 13.29: Fused multiply-add with multiple precision / 401 13.30: Fused multiply-add, Boldo/penalty exhyphenpenalty Melquiond style / 403 13.31: Error correction in fused multiply-add / 406 13.32: Retrospective on pair-precision arithmetic / 407 14: Power function / 411 14.1: Why the power function is hard to compute / 411 14.2: Special cases for the power function / 412 14.3: Integer powers / 414 14.4: Integer powers, revisited / 420 14.5: Outline of the power-function algorithm / 421 14.6: Finding $a$ and $p$ / 423 14.7: Table searching / 424 14.8: Computing $log _n(g/a)$ / 426 14.9: Accuracy required for $log _n(g/a)$ / 429 14.10: Exact products / 430 14.11: Computing $w$, $w_1$ and $w_2$ / 433 14.12: Computing $n^w_2$ / 437 14.13: The choice of $q$ / 438 14.14: Testing the power function / 438 14.15: Retrospective on the power function / 440 15: Complex arithmetic primitives / 441 15.1: Support macros and type definitions / 442 15.2: Complex absolute value / 443 15.3: Complex addition / 445 15.4: Complex argument / 445 15.5: Complex conjugate / 446 15.6: Complex conjugation symmetry / 446 15.7: Complex conversion / 448 15.8: Complex copy / 448 15.9: Complex division: C99 style / 449 15.10: Complex division: Smith style / 451 15.11: Complex division: Stewart style / 452 15.12: Complex division: Priest style / 453 15.13: Complex division: avoiding subtraction loss / 455 15.14: Complex imaginary part / 456 15.15: Complex multiplication / 456 15.16: Complex multiplication: error analysis / 458 15.17: Complex negation / 459 15.18: Complex projection / 460 15.19: Complex real part / 460 15.20: Complex subtraction / 461 15.21: Complex infinity test / 462 15.22: Complex NaN test / 462 15.23: Summary / 463 16: Quadratic equations / 465 16.1: Solving quadratic equations / 465 16.2: Root sensitivity / 471 16.3: Testing a quadratic-equation solver / 472 16.4: Summary / 474 17: Elementary functions in complex arithmetic / 475 17.1: Research on complex elementary functions / 475 17.2: Principal values / 476 17.3: Branch cuts / 476 17.4: Software problems with negative zeros / 478 17.5: Complex elementary function tree / 479 17.6: Series for complex functions / 479 17.7: Complex square root / 480 17.8: Complex cube root / 485 17.9: Complex exponential / 487 17.10: Complex exponential near zero / 492 17.11: Complex logarithm / 495 17.12: Complex logarithm near one / 497 17.13: Complex power / 500 17.14: Complex trigonometric functions / 502 17.15: Complex inverse trigonometric functions / 504 17.16: Complex hyperbolic functions / 509 17.17: Complex inverse hyperbolic functions / 514 17.18: Summary / 520 18: The Greek functions: gamma, psi, and zeta / 521 18.1: Gamma and log-gamma functions / 521 18.2: The tt psi and tt psiln functions / 536 18.3: Polygamma functions / 547 18.4: Incomplete gamma functions / 560 18.5: A Swiss diversion: Bernoulli and Euler / 568 18.6: An Italian excursion: Fibonacci numbers / 575 18.7: A German gem: the Riemann zeta function / 579 18.8: Further reading / 590 18.9: Summary / 591 19: Error and probability functions / 593 19.1: Error functions / 593 19.2: Scaled complementary error function / 598 19.3: Inverse error functions / 600 19.4: Normal distribution functions and inverses / 610 19.5: Summary / 617 20: Elliptic integral functions / 619 20.1: The arithmetic-geometric mean / 619 20.2: Elliptic integral functions of the first kind / 624 20.3: Elliptic integral functions of the second kind / 627 20.4: Elliptic integral functions of the third kind / 630 20.5: Computing $K(m)$ and $K'(m)$ / 631 20.6: Computing $E(m)$ and $E'(m)$ / 637 20.7: Historical algorithms for elliptic integrals / 643 20.8: Auxiliary functions for elliptic integrals / 645 20.9: Computing the elliptic auxiliary functions / 648 20.10: Historical elliptic functions / 650 20.11: Elliptic functions in software / 652 20.12: Applications of elliptic auxiliary functions / 653 20.13: Elementary functions from elliptic auxiliary functions / 654 20.14: Computing elementary functions via $R_C(x,y)$ / 655 20.15: Jacobian elliptic functions / 657 20.16: Inverses of Jacobian elliptic functions / 664 20.17: The modulus and the nome / 668 20.18: Jacobian theta functions / 673 20.19: Logarithmic derivatives of the Jacobian theta functions / 675 20.20: Neville theta functions / 678 20.21: Jacobian Eta, Theta, and Zeta functions / 679 20.22: Weierstrass elliptic functions / 682 20.23: Weierstrass functions by duplication / 689 20.24: Complete elliptic functions, revisited / 690 20.25: Summary / 691 21: Bessel functions / 693 21.1: Cylindrical Bessel functions / 694 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 21.4: Experiments with recurrences for $J_0(x)$ / 705 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 21.6: Computing $J_n(x)$ / 710 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 21.8: Computing $Y_n(x)$ / 715 21.9: Improving Bessel code near zeros / 716 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 21.14: Properties of spherical Bessel functions / 731 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 21.17: Modified spherical Bessel functions / 743 21.18: Software for Bessel-function sequences / 755 21.19: Retrospective on Bessel functions / 761 22: Testing the library / 763 22.1: Testing tt tgamma and tt lgamma / 765 22.2: Testing tt psi and tt psiln / 768 22.3: Testing tt erf and tt erfc / 768 22.4: Testing cylindrical Bessel functions / 769 22.5: Testing exponent/penalty exhyphenpenalty significand manipulation / 769 22.6: Testing inline assembly code / 769 22.7: Testing with Maple / 770 22.8: Testing floating-point arithmetic / 773 22.9: The Berkeley Elementary Functions Test Suite / 774 22.10: The AT&T floating-point test package / 775 22.11: The Antwerp test suite / 776 22.12: Summary / 776 23: Pair-precision elementary functions / 777 23.1: Pair-precision integer power / 777 23.2: Pair-precision machine epsilon / 779 23.3: Pair-precision exponential / 780 23.4: Pair-precision logarithm / 787 23.5: Pair-precision logarithm near one / 793 23.6: Pair-precision exponential near zero / 793 23.7: Pair-precision base-$n$ exponentials / 795 23.8: Pair-precision trigonometric functions / 796 23.9: Pair-precision inverse trigonometric functions / 801 23.10: Pair-precision hyperbolic functions / 804 23.11: Pair-precision inverse hyperbolic functions / 808 23.12: Summary / 808 24: Accuracy of the Cody / Waite algorithms / 811 25: Improving upon the Cody / Waite algorithms / 823 25.1: The Bell Labs libraries / 823 25.2: The Cephes library / 823 25.3: The Sun libraries / 824 25.4: Mathematical functions on EPIC / 824 25.5: The GNU libraries / 825 25.6: The French libraries / 825 25.7: The NIST effort / 826 25.8: Commercial mathematical libraries / 826 25.9: Mathematical libraries for decimal arithmetic / 826 25.10: Mathematical library research publications / 826 25.11: Books on computing mathematical functions / 827 25.12: Summary / 828 26: Floating-point output / 829 26.1: Output character string design issues / 830 26.2: Exact output conversion / 831 26.3: Hexadecimal floating-point output / 832 26.4: Octal floating-point output / 850 26.5: Binary floating-point output / 851 26.6: Decimal floating-point output / 851 26.7: Accuracy of output conversion / 865 26.8: Output conversion to a general base / 865 26.9: Output conversion of Infinity / 866 26.10: Output conversion of NaN / 866 26.11: Number-to-string conversion / 867 26.12: The tt printf family / 867 26.13: Summary / 878 27: Floating-point input / 879 27.1: Binary floating-point input / 879 27.2: Octal floating-point input / 894 27.3: Hexadecimal floating-point input / 895 27.4: Decimal floating-point input / 895 27.5: Based-number input / 899 27.6: General floating-point input / 900 27.7: The tt scanf family / 901 27.8: Summary / 910 A: Ada interface / 911 A.1: Building the Ada interface / 911 A.2: Programming the Ada interface / 912 A.3: Using the Ada interface / 915 B: C# interface / 917 B.1: C# on the CLI virtual machine / 917 B.2: Building the C# interface / 918 B.3: Programming the C# interface / 920 B.4: Using the C# interface / 922 C: C++ interface / 923 C.1: Building the C++ interface / 923 C.2: Programming the C++ interface / 924 C.3: Using the C++ interface / 925 D: Decimal arithmetic / 927 D.1: Why we need decimal floating-point arithmetic / 927 D.2: Decimal floating-point arithmetic design issues / 928 D.3: How decimal and binary arithmetic differ / 931 D.4: Initialization of decimal floating-point storage / 935 D.5: The tt <decfloat.h> header file / 936 D.6: Rounding in decimal arithmetic / 936 D.7: Exact scaling in decimal arithmetic / 937 E: Errata in the Cody / Waite book / 939 F: Fortran interface / 941 F.1: Building the Fortran interface / 943 F.2: Programming the Fortran interface / 944 F.3: Using the Fortran interface / 945 H: Historical floating-point architectures / 947 H.1: CDC family / 949 H.2: Cray family / 952 H.3: DEC PDP-10 / 953 H.4: DEC PDP-11 and VAX / 956 H.5: General Electric 600 series / 958 H.6: IBM family / 959 H.7: Lawrence Livermore S-1 Mark IIA / 965 H.8: Unusual floating-point systems / 966 H.9: Historical retrospective / 967 I: Integer arithmetic / 969 I.1: Memory addressing and integers / 971 I.2: Representations of signed integers / 971 I.3: Parity testing / 975 I.4: Sign testing / 975 I.5: Arithmetic exceptions / 975 I.6: Notations for binary numbers / 977 I.7: Summary / 978 J: Java interface / 979 J.1: Building the Java interface / 979 J.2: Programming the Java MathCW class / 980 J.3: Programming the Java C interface / 982 J.4: Using the Java interface / 985 L: Letter notation / 987 P: Pascal interface / 989 P.1: Building the Pascal interface / 989 P.2: Programming the Pascal MathCW module / 990 P.3: Using the Pascal module interface / 993 P.4: Pascal and numeric programming / 994 Bibliography / 995 Author/editor index / 1039 Function and macro index / 1049 Subject index / 1065 Colophon / 1115

rawtitle:
The Mathematical-Function Computation Handbook: Programming Using the {MathCW} Portable Software Library

rawTOC:
List of figures / xxv \\ List of tables / xxxi \\ Quick start / xxxv \\ 1: Introduction / 1 \\ 1.1: Programming conventions / 2 \\ 1.2: Naming conventions / 4 \\ 1.3: Library contributions and coverage / 5 \\ 1.4: Summary / 6 \\ 2: Iterative solutions and other tools / 7 \\ 2.1: Polynomials and Taylor series / 7 \\ 2.2: First-order Taylor series approximation / 8 \\ 2.3: Second-order Taylor series approximation / 9 \\ 2.4: Another second-order Taylor series approximation / 9 \\ 2.5: Convergence of second-order methods / 10 \\ 2.6: Taylor series for elementary functions / 10 \\ 2.7: Continued fractions / 12 \\ 2.8: Summation of continued fractions / 17 \\ 2.9: Asymptotic expansions / 19 \\ 2.10: Series inversion / 20 \\ 2.11: Summary / 22 \\ 3: Polynomial approximations / 23 \\ 3.1: Computation of odd series / 23 \\ 3.2: Computation of even series / 25 \\ 3.3: Computation of general series / 25 \\ 3.4: Limitations of Cody\slash Waite polynomials / 28 \\ 3.5: Polynomial fits with Maple / 32 \\ 3.6: Polynomial fits with Mathematica / 33 \\ 3.7: Exact polynomial coefficients / 42 \\ 3.8: Cody\slash Waite rational polynomials / 43 \\ 3.9: Chebyshev polynomial economization / 43 \\ 3.10: Evaluating Chebyshev polynomials / 48 \\ 3.11: Error compensation in Chebyshev fits / 50 \\ 3.12: Improving Chebyshev fits / 51 \\ 3.13: Chebyshev fits in rational form / 52 \\ 3.14: Chebyshev fits with Mathematica / 56 \\ 3.15: Chebyshev fits for function representation / 57 \\ 3.16: Extending the library / 57 \\ 3.17: Summary and further reading / 58 \\ 4: Implementation issues / 61 \\ 4.1: Error magnification / 61 \\ 4.2: Machine representation and machine epsilon / 62 \\ 4.3: IEEE 754 arithmetic / 63 \\ 4.4: Evaluation order in C / 64 \\ 4.5: The {\tt volatile} type qualifier / 65 \\ 4.6: Rounding in floating-point arithmetic / 66 \\ 4.7: Signed zero / 69 \\ 4.8: Floating-point zero divide / 70 \\ 4.9: Floating-point overflow / 71 \\ 4.10: Integer overflow / 72 \\ 4.11: Floating-point underflow / 77 \\ 4.12: Subnormal numbers / 78 \\ 4.13: Floating-point inexact operation / 79 \\ 4.14: Floating-point invalid operation / 79 \\ 4.15: Remarks on NaN tests / 80 \\ 4.16: Ulps --- units in the last place / 81 \\ 4.17: Fused multiply-add / 85 \\ 4.18: Fused multiply-add and polynomials / 88 \\ 4.19: Significance loss / 89 \\ 4.20: Error handling and reporting / 89 \\ 4.21: Interpreting error codes / 93 \\ 4.22: C99 changes to error reporting / 94 \\ 4.23: Error reporting with threads / 95 \\ 4.24: Comments on error reporting / 95 \\ 4.25: Testing function implementations / 96 \\ 4.26: Extended data types on Hewlett--Packard HP-UX IA-64 / 100 \\ 4.27: Extensions for decimal arithmetic / 101 \\ 4.28: Further reading / 103 \\ 4.29: Summary / 104 \\ 5: The floating-point environment / 105 \\ 5.1: IEEE 754 and programming languages / 105 \\ 5.2: IEEE 754 and the mathcw library / 106 \\ 5.3: Exceptions and traps / 106 \\ 5.4: Access to exception flags and rounding control / 107 \\ 5.5: The environment access pragma / 110 \\ 5.6: Implementation of exception-flag and rounding-control access / 110 \\ 5.7: Using exception flags: simple cases / 112 \\ 5.8: Using rounding control / 115 \\ 5.9: Additional exception flag access / 116 \\ 5.10: Using exception flags: complex case / 120 \\ 5.11: Access to precision control / 123 \\ 5.12: Using precision control / 126 \\ 5.13: Summary / 127 \\ 6: Converting floating-point values to integers / 129 \\ 6.1: Integer conversion in programming languages / 129 \\ 6.2: Programming issues for conversions to integers / 130 \\ 6.3: Hardware out-of-range conversions / 131 \\ 6.4: Rounding modes and integer conversions / 132 \\ 6.5: Extracting integral and fractional parts / 132 \\ 6.6: Truncation functions / 135 \\ 6.7: Ceiling and floor functions / 136 \\ 6.8: Floating-point rounding functions with fixed rounding / 137 \\ 6.9: Floating-point rounding functions: current rounding / 138 \\ 6.10: Floating-point rounding functions without {\em inexact\/} exception / 139 \\ 6.11: Integer rounding functions with fixed rounding / 140 \\ 6.12: Integer rounding functions with current rounding / 142 \\ 6.13: Remainder / 143 \\ 6.14: Why the remainder functions are hard / 144 \\ 6.15: Computing {\tt fmod} / 146 \\ 6.16: Computing {\tt remainder} / 148 \\ 6.17: Computing {\tt remquo} / 150 \\ 6.18: Computing one remainder from the other / 152 \\ 6.19: Computing the remainder in nonbinary bases / 155 \\ 6.20: Summary / 156 \\ 7: Random numbers / 157 \\ 7.1: Guidelines for random-number software / 157 \\ 7.2: Creating generator seeds / 158 \\ 7.3: Random floating-point values / 160 \\ 7.4: Random integers from floating-point generator / 165 \\ 7.5: Random integers from an integer generator / 166 \\ 7.6: Random integers in ascending order / 168 \\ 7.7: How random numbers are generated / 169 \\ 7.8: Removing generator bias / 178 \\ 7.9: Improving a poor random number generator / 178 \\ 7.10: Why long periods matter / 179 \\ 7.11: Inversive congruential generators / 180 \\ 7.12: Inversive congruential generators, revisited / 189 \\ 7.13: Distributions of random numbers / 189 \\ 7.14: Other distributions / 195 \\ 7.15: Testing random-number generators / 196 \\ 7.16: Applications of random numbers / 202 \\ 7.17: The \textsf {mathcw} random number routines / 208 \\ 7.18: Summary, advice, and further reading / 214 \\ 8: Roots / 215 \\ 8.1: Square root / 215 \\ 8.2: Hypotenuse and vector norms / 222 \\ 8.3: Hypotenuse by iteration / 227 \\ 8.4: Reciprocal square root / 233 \\ 8.5: Cube root / 237 \\ 8.6: Roots in hardware / 240 \\ 8.7: Summary / 242 \\ 9: Argument reduction / 243 \\ 9.1: Simple argument reduction / 243 \\ 9.2: Exact argument reduction / 250 \\ 9.3: Implementing exact argument reduction / 253 \\ 9.4: Testing argument reduction / 265 \\ 9.5: Retrospective on argument reduction / 265 \\ 10: Exponential and logarithm / 267 \\ 10.1: Exponential functions / 267 \\ 10.2: Exponential near zero / 273 \\ 10.3: Logarithm functions / 282 \\ 10.4: Logarithm near one / 290 \\ 10.5: Exponential and logarithm in hardware / 292 \\ 10.6: Compound interest and annuities / 294 \\ 10.7: Summary / 298 \\ 11: Trigonometric functions / 299 \\ 11.1: Sine and cosine properties / 299 \\ 11.2: Tangent properties / 302 \\ 11.3: Argument conventions and units / 304 \\ 11.4: Computing the cosine and sine / 306 \\ 11.5: Computing the tangent / 310 \\ 11.6: Trigonometric functions in degrees / 313 \\ 11.7: Trigonometric functions in units of $ \pi $ / 315 \\ 11.8: Computing the cosine and sine together / 320 \\ 11.9: Inverse sine and cosine / 323 \\ 11.10: Inverse tangent / 331 \\ 11.11: Inverse tangent, take two / 336 \\ 11.12: Trigonometric functions in hardware / 338 \\ 11.13: Testing trigonometric functions / 339 \\ 11.14: Retrospective on trigonometric functions / 340 \\ 12: Hyperbolic functions / 341 \\ 12.1: Hyperbolic functions / 341 \\ 12.2: Improving the hyperbolic functions / 345 \\ 12.3: Computing the hyperbolic functions together / 348 \\ 12.4: Inverse hyperbolic functions / 348 \\ 12.5: Hyperbolic functions in hardware / 350 \\ 12.6: Summary / 352 \\ 13: Pair-precision arithmetic / 353 \\ 13.1: Limitations of pair-precision arithmetic / 354 \\ 13.2: Design of the pair-precision software interface / 355 \\ 13.3: Pair-precision initialization / 356 \\ 13.4: Pair-precision evaluation / 357 \\ 13.5: Pair-precision high part / 357 \\ 13.6: Pair-precision low part / 357 \\ 13.7: Pair-precision copy / 357 \\ 13.8: Pair-precision negation / 358 \\ 13.9: Pair-precision absolute value / 358 \\ 13.10: Pair-precision sum / 358 \\ 13.11: Splitting numbers into pair sums / 359 \\ 13.12: Premature overflow in splitting / 362 \\ 13.13: Pair-precision addition / 365 \\ 13.14: Pair-precision subtraction / 367 \\ 13.15: Pair-precision comparison / 368 \\ 13.16: Pair-precision multiplication / 368 \\ 13.17: Pair-precision division / 371 \\ 13.18: Pair-precision square root / 373 \\ 13.19: Pair-precision cube root / 377 \\ 13.20: Accuracy of pair-precision arithmetic / 379 \\ 13.21: Pair-precision vector sum / 384 \\ 13.22: Exact vector sums / 385 \\ 13.23: Pair-precision dot product / 385 \\ 13.24: Pair-precision product sum / 386 \\ 13.25: Pair-precision decimal arithmetic / 387 \\ 13.26: Fused multiply-add with pair precision / 388 \\ 13.27: Higher intermediate precision and the FMA / 393 \\ 13.28: Fused multiply-add without pair precision / 395 \\ 13.29: Fused multiply-add with multiple precision / 401 \\ 13.30: Fused multiply-add, Boldo/\penalty \exhyphenpenalty Melquiond style / 403 \\ 13.31: Error correction in fused multiply-add / 406 \\ 13.32: Retrospective on pair-precision arithmetic / 407 \\ 14: Power function / 411 \\ 14.1: Why the power function is hard to compute / 411 \\ 14.2: Special cases for the power function / 412 \\ 14.3: Integer powers / 414 \\ 14.4: Integer powers, revisited / 420 \\ 14.5: Outline of the power-function algorithm / 421 \\ 14.6: Finding $a$ and $p$ / 423 \\ 14.7: Table searching / 424 \\ 14.8: Computing $\log_n(g/a)$ / 426 \\ 14.9: Accuracy required for $\log_n(g/a)$ / 429 \\ 14.10: Exact products / 430 \\ 14.11: Computing $w$, $w_1$ and $w_2$ / 433 \\ 14.12: Computing $n^{w_2}$ / 437 \\ 14.13: The choice of $q$ / 438 \\ 14.14: Testing the power function / 438 \\ 14.15: Retrospective on the power function / 440 \\ 15: Complex arithmetic primitives / 441 \\ 15.1: Support macros and type definitions / 442 \\ 15.2: Complex absolute value / 443 \\ 15.3: Complex addition / 445 \\ 15.4: Complex argument / 445 \\ 15.5: Complex conjugate / 446 \\ 15.6: Complex conjugation symmetry / 446 \\ 15.7: Complex conversion / 448 \\ 15.8: Complex copy / 448 \\ 15.9: Complex division: C99 style / 449 \\ 15.10: Complex division: Smith style / 451 \\ 15.11: Complex division: Stewart style / 452 \\ 15.12: Complex division: Priest style / 453 \\ 15.13: Complex division: avoiding subtraction loss / 455 \\ 15.14: Complex imaginary part / 456 \\ 15.15: Complex multiplication / 456 \\ 15.16: Complex multiplication: error analysis / 458 \\ 15.17: Complex negation / 459 \\ 15.18: Complex projection / 460 \\ 15.19: Complex real part / 460 \\ 15.20: Complex subtraction / 461 \\ 15.21: Complex infinity test / 462 \\ 15.22: Complex NaN test / 462 \\ 15.23: Summary / 463 \\ 16: Quadratic equations / 465 \\ 16.1: Solving quadratic equations / 465 \\ 16.2: Root sensitivity / 471 \\ 16.3: Testing a quadratic-equation solver / 472 \\ 16.4: Summary / 474 \\ 17: Elementary functions in complex arithmetic / 475 \\ 17.1: Research on complex elementary functions / 475 \\ 17.2: Principal values / 476 \\ 17.3: Branch cuts / 476 \\ 17.4: Software problems with negative zeros / 478 \\ 17.5: Complex elementary function tree / 479 \\ 17.6: Series for complex functions / 479 \\ 17.7: Complex square root / 480 \\ 17.8: Complex cube root / 485 \\ 17.9: Complex exponential / 487 \\ 17.10: Complex exponential near zero / 492 \\ 17.11: Complex logarithm / 495 \\ 17.12: Complex logarithm near one / 497 \\ 17.13: Complex power / 500 \\ 17.14: Complex trigonometric functions / 502 \\ 17.15: Complex inverse trigonometric functions / 504 \\ 17.16: Complex hyperbolic functions / 509 \\ 17.17: Complex inverse hyperbolic functions / 514 \\ 17.18: Summary / 520 \\ 18: The Greek functions: gamma, psi, and zeta / 521 \\ 18.1: Gamma and log-gamma functions / 521 \\ 18.2: The {\tt psi} and {\tt psiln} functions / 536 \\ 18.3: Polygamma functions / 547 \\ 18.4: Incomplete gamma functions / 560 \\ 18.5: A Swiss diversion: Bernoulli and Euler / 568 \\ 18.6: An Italian excursion: Fibonacci numbers / 575 \\ 18.7: A German gem: the Riemann zeta function / 579 \\ 18.8: Further reading / 590 \\ 18.9: Summary / 591 \\ 19: Error and probability functions / 593 \\ 19.1: Error functions / 593 \\ 19.2: Scaled complementary error function / 598 \\ 19.3: Inverse error functions / 600 \\ 19.4: Normal distribution functions and inverses / 610 \\ 19.5: Summary / 617 \\ 20: Elliptic integral functions / 619 \\ 20.1: The arithmetic-geometric mean / 619 \\ 20.2: Elliptic integral functions of the first kind / 624 \\ 20.3: Elliptic integral functions of the second kind / 627 \\ 20.4: Elliptic integral functions of the third kind / 630 \\ 20.5: Computing $K(m)$ and $K'(m)$ / 631 \\ 20.6: Computing $E(m)$ and $E'(m)$ / 637 \\ 20.7: Historical algorithms for elliptic integrals / 643 \\ 20.8: Auxiliary functions for elliptic integrals / 645 \\ 20.9: Computing the elliptic auxiliary functions / 648 \\ 20.10: Historical elliptic functions / 650 \\ 20.11: Elliptic functions in software / 652 \\ 20.12: Applications of elliptic auxiliary functions / 653 \\ 20.13: Elementary functions from elliptic auxiliary functions / 654 \\ 20.14: Computing elementary functions via $R_C(x,y)$ / 655 \\ 20.15: Jacobian elliptic functions / 657 \\ 20.16: Inverses of Jacobian elliptic functions / 664 \\ 20.17: The modulus and the nome / 668 \\ 20.18: Jacobian theta functions / 673 \\ 20.19: Logarithmic derivatives of the Jacobian theta functions / 675 \\ 20.20: Neville theta functions / 678 \\ 20.21: Jacobian Eta, Theta, and Zeta functions / 679 \\ 20.22: Weierstrass elliptic functions / 682 \\ 20.23: Weierstrass functions by duplication / 689 \\ 20.24: Complete elliptic functions, revisited / 690 \\ 20.25: Summary / 691 \\ 21: Bessel functions / 693 \\ 21.1: Cylindrical Bessel functions / 694 \\ 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 \\ 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 \\ 21.4: Experiments with recurrences for $J_0(x)$ / 705 \\ 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 \\ 21.6: Computing $J_n(x)$ / 710 \\ 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 \\ 21.8: Computing $Y_n(x)$ / 715 \\ 21.9: Improving Bessel code near zeros / 716 \\ 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 \\ 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 \\ 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 \\ 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 \\ 21.14: Properties of spherical Bessel functions / 731 \\ 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 \\ 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 \\ 21.17: Modified spherical Bessel functions / 743 \\ 21.18: Software for Bessel-function sequences / 755 \\ 21.19: Retrospective on Bessel functions / 761 \\ 22: Testing the library / 763 \\ 22.1: Testing {\tt tgamma} and {\tt lgamma} / 765 \\ 22.2: Testing {\tt psi} and {\tt psiln} / 768 \\ 22.3: Testing {\tt erf} and {\tt erfc} / 768 \\ 22.4: Testing cylindrical Bessel functions / 769 \\ 22.5: Testing exponent/\penalty \exhyphenpenalty significand manipulation / 769 \\ 22.6: Testing inline assembly code / 769 \\ 22.7: Testing with Maple / 770 \\ 22.8: Testing floating-point arithmetic / 773 \\ 22.9: The Berkeley Elementary Functions Test Suite / 774 \\ 22.10: The AT\&T floating-point test package / 775 \\ 22.11: The Antwerp test suite / 776 \\ 22.12: Summary / 776 \\ 23: Pair-precision elementary functions / 777 \\ 23.1: Pair-precision integer power / 777 \\ 23.2: Pair-precision machine epsilon / 779 \\ 23.3: Pair-precision exponential / 780 \\ 23.4: Pair-precision logarithm / 787 \\ 23.5: Pair-precision logarithm near one / 793 \\ 23.6: Pair-precision exponential near zero / 793 \\ 23.7: Pair-precision base-$n$ exponentials / 795 \\ 23.8: Pair-precision trigonometric functions / 796 \\ 23.9: Pair-precision inverse trigonometric functions / 801 \\ 23.10: Pair-precision hyperbolic functions / 804 \\ 23.11: Pair-precision inverse hyperbolic functions / 808 \\ 23.12: Summary / 808 \\ 24: Accuracy of the Cody\slash Waite algorithms / 811 \\ 25: Improving upon the Cody\slash Waite algorithms / 823 \\ 25.1: The Bell Labs libraries / 823 \\ 25.2: The {Cephes} library / 823 \\ 25.3: The {Sun} libraries / 824 \\ 25.4: Mathematical functions on EPIC / 824 \\ 25.5: The GNU libraries / 825 \\ 25.6: The French libraries / 825 \\ 25.7: The NIST effort / 826 \\ 25.8: Commercial mathematical libraries / 826 \\ 25.9: Mathematical libraries for decimal arithmetic / 826 \\ 25.10: Mathematical library research publications / 826 \\ 25.11: Books on computing mathematical functions / 827 \\ 25.12: Summary / 828 \\ 26: Floating-point output / 829 \\ 26.1: Output character string design issues / 830 \\ 26.2: Exact output conversion / 831 \\ 26.3: Hexadecimal floating-point output / 832 \\ 26.4: Octal floating-point output / 850 \\ 26.5: Binary floating-point output / 851 \\ 26.6: Decimal floating-point output / 851 \\ 26.7: Accuracy of output conversion / 865 \\ 26.8: Output conversion to a general base / 865 \\ 26.9: Output conversion of Infinity / 866 \\ 26.10: Output conversion of NaN / 866 \\ 26.11: Number-to-string conversion / 867 \\ 26.12: The {\tt printf} family / 867 \\ 26.13: Summary / 878 \\ 27: Floating-point input / 879 \\ 27.1: Binary floating-point input / 879 \\ 27.2: Octal floating-point input / 894 \\ 27.3: Hexadecimal floating-point input / 895 \\ 27.4: Decimal floating-point input / 895 \\ 27.5: Based-number input / 899 \\ 27.6: General floating-point input / 900 \\ 27.7: The {\tt scanf} family / 901 \\ 27.8: Summary / 910 \\ A: Ada interface / 911 \\ A.1: Building the Ada interface / 911 \\ A.2: Programming the Ada interface / 912 \\ A.3: Using the Ada interface / 915 \\ B: C\# interface / 917 \\ B.1: C\# on the CLI virtual machine / 917 \\ B.2: Building the C\# interface / 918 \\ B.3: Programming the C\# interface / 920 \\ B.4: Using the C\# interface / 922 \\ C: C++ interface / 923 \\ C.1: Building the C++ interface / 923 \\ C.2: Programming the C++ interface / 924 \\ C.3: Using the C++ interface / 925 \\ D: Decimal arithmetic / 927 \\ D.1: Why we need decimal floating-point arithmetic / 927 \\ D.2: Decimal floating-point arithmetic design issues / 928 \\ D.3: How decimal and binary arithmetic differ / 931 \\ D.4: Initialization of decimal floating-point storage / 935 \\ D.5: The {\tt <decfloat.h>} header file / 936 \\ D.6: Rounding in decimal arithmetic / 936 \\ D.7: Exact scaling in decimal arithmetic / 937 \\ E: Errata in the Cody\slash Waite book / 939 \\ F: Fortran interface / 941 \\ F.1: Building the Fortran interface / 943 \\ F.2: Programming the Fortran interface / 944 \\ F.3: Using the Fortran interface / 945 \\ H: Historical floating-point architectures / 947 \\ H.1: CDC family / 949 \\ H.2: Cray family / 952 \\ H.3: DEC PDP-10 / 953 \\ H.4: DEC PDP-11 and VAX / 956 \\ H.5: General Electric 600 series / 958 \\ H.6: IBM family / 959 \\ H.7: Lawrence Livermore S-1 Mark IIA / 965 \\ H.8: Unusual floating-point systems / 966 \\ H.9: Historical retrospective / 967 \\ I: Integer arithmetic / 969 \\ I.1: Memory addressing and integers / 971 \\ I.2: Representations of signed integers / 971 \\ I.3: Parity testing / 975 \\ I.4: Sign testing / 975 \\ I.5: Arithmetic exceptions / 975 \\ I.6: Notations for binary numbers / 977 \\ I.7: Summary / 978 \\ J: Java interface / 979 \\ J.1: Building the Java interface / 979 \\ J.2: Programming the Java MathCW class / 980 \\ J.3: Programming the Java C interface / 982 \\ J.4: Using the Java interface / 985 \\ L: Letter notation / 987 \\ P: Pascal interface / 989 \\ P.1: Building the Pascal interface / 989 \\ P.2: Programming the Pascal MathCW module / 990 \\ P.3: Using the Pascal module interface / 993 \\ P.4: Pascal and numeric programming / 994 \\ Bibliography / 995 \\ Author/editor index / 1039 \\ Function and macro index / 1049 \\ Subject index / 1065 \\ Colophon / 1115

entry:
@Book{Beebe:2017:MFC,
  author =       "Nelson H. F. Beebe",
  title =        "The Mathematical-Function Computation Handbook:
                 Programming Using the {MathCW} Portable Software
                 Library",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xxxvi + 1114",
  year =         "2017",
  DOI =          "https://doi.org/10.1007/978-3-319-64110-2",
  ISBN =         "3-319-64109-3 (hardcover), 3-319-64110-7 (e-book)",
  ISBN-13 =      "978-3-319-64109-6 (hardcover), 978-3-319-64110-2
                 (e-book)",
  LCCN =         "QA75.5-76.95",
  bibdate =      "Sat Jul 15 19:34:43 MDT 2017",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/b/beebe-nelson-h-f.bib;
                 https://www.math.utah.edu/pub/tex/bib/axiom.bib;
                 https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/maple-extract.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/mupad.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/redbooks.bib;
                 https://www.math.utah.edu/pub/tex/bib/utah-math-dept-books.bib",
  URL =          "http://www.springer.com/us/book/9783319641096",
  acknowledgement = ack-nhfb,
  ORCID-numbers = "Beebe, Nelson H. F./0000-0001-7281-4263",
  tableofcontents = "List of figures / xxv \\
                 List of tables / xxxi \\
                 Quick start / xxxv \\
                 1: Introduction / 1 \\
                 1.1: Programming conventions / 2 \\
                 1.2: Naming conventions / 4 \\
                 1.3: Library contributions and coverage / 5 \\
                 1.4: Summary / 6 \\
                 2: Iterative solutions and other tools / 7 \\
                 2.1: Polynomials and Taylor series / 7 \\
                 2.2: First-order Taylor series approximation / 8 \\
                 2.3: Second-order Taylor series approximation / 9 \\
                 2.4: Another second-order Taylor series approximation /
                 9 \\
                 2.5: Convergence of second-order methods / 10 \\
                 2.6: Taylor series for elementary functions / 10 \\
                 2.7: Continued fractions / 12 \\
                 2.8: Summation of continued fractions / 17 \\
                 2.9: Asymptotic expansions / 19 \\
                 2.10: Series inversion / 20 \\
                 2.11: Summary / 22 \\
                 3: Polynomial approximations / 23 \\
                 3.1: Computation of odd series / 23 \\
                 3.2: Computation of even series / 25 \\
                 3.3: Computation of general series / 25 \\
                 3.4: Limitations of Cody\slash Waite polynomials / 28
                 \\
                 3.5: Polynomial fits with Maple / 32 \\
                 3.6: Polynomial fits with Mathematica / 33 \\
                 3.7: Exact polynomial coefficients / 42 \\
                 3.8: Cody\slash Waite rational polynomials / 43 \\
                 3.9: Chebyshev polynomial economization / 43 \\
                 3.10: Evaluating Chebyshev polynomials / 48 \\
                 3.11: Error compensation in Chebyshev fits / 50 \\
                 3.12: Improving Chebyshev fits / 51 \\
                 3.13: Chebyshev fits in rational form / 52 \\
                 3.14: Chebyshev fits with Mathematica / 56 \\
                 3.15: Chebyshev fits for function representation / 57
                 \\
                 3.16: Extending the library / 57 \\
                 3.17: Summary and further reading / 58 \\
                 4: Implementation issues / 61 \\
                 4.1: Error magnification / 61 \\
                 4.2: Machine representation and machine epsilon / 62
                 \\
                 4.3: IEEE 754 arithmetic / 63 \\
                 4.4: Evaluation order in C / 64 \\
                 4.5: The {\tt volatile} type qualifier / 65 \\
                 4.6: Rounding in floating-point arithmetic / 66 \\
                 4.7: Signed zero / 69 \\
                 4.8: Floating-point zero divide / 70 \\
                 4.9: Floating-point overflow / 71 \\
                 4.10: Integer overflow / 72 \\
                 4.11: Floating-point underflow / 77 \\
                 4.12: Subnormal numbers / 78 \\
                 4.13: Floating-point inexact operation / 79 \\
                 4.14: Floating-point invalid operation / 79 \\
                 4.15: Remarks on NaN tests / 80 \\
                 4.16: Ulps --- units in the last place / 81 \\
                 4.17: Fused multiply-add / 85 \\
                 4.18: Fused multiply-add and polynomials / 88 \\
                 4.19: Significance loss / 89 \\
                 4.20: Error handling and reporting / 89 \\
                 4.21: Interpreting error codes / 93 \\
                 4.22: C99 changes to error reporting / 94 \\
                 4.23: Error reporting with threads / 95 \\
                 4.24: Comments on error reporting / 95 \\
                 4.25: Testing function implementations / 96 \\
                 4.26: Extended data types on Hewlett--Packard HP-UX
                 IA-64 / 100 \\
                 4.27: Extensions for decimal arithmetic / 101 \\
                 4.28: Further reading / 103 \\
                 4.29: Summary / 104 \\
                 5: The floating-point environment / 105 \\
                 5.1: IEEE 754 and programming languages / 105 \\
                 5.2: IEEE 754 and the mathcw library / 106 \\
                 5.3: Exceptions and traps / 106 \\
                 5.4: Access to exception flags and rounding control /
                 107 \\
                 5.5: The environment access pragma / 110 \\
                 5.6: Implementation of exception-flag and
                 rounding-control access / 110 \\
                 5.7: Using exception flags: simple cases / 112 \\
                 5.8: Using rounding control / 115 \\
                 5.9: Additional exception flag access / 116 \\
                 5.10: Using exception flags: complex case / 120 \\
                 5.11: Access to precision control / 123 \\
                 5.12: Using precision control / 126 \\
                 5.13: Summary / 127 \\
                 6: Converting floating-point values to integers / 129
                 \\
                 6.1: Integer conversion in programming languages / 129
                 \\
                 6.2: Programming issues for conversions to integers /
                 130 \\
                 6.3: Hardware out-of-range conversions / 131 \\
                 6.4: Rounding modes and integer conversions / 132 \\
                 6.5: Extracting integral and fractional parts / 132 \\
                 6.6: Truncation functions / 135 \\
                 6.7: Ceiling and floor functions / 136 \\
                 6.8: Floating-point rounding functions with fixed
                 rounding / 137 \\
                 6.9: Floating-point rounding functions: current
                 rounding / 138 \\
                 6.10: Floating-point rounding functions without {\em
                 inexact\/} exception / 139 \\
                 6.11: Integer rounding functions with fixed rounding /
                 140 \\
                 6.12: Integer rounding functions with current rounding
                 / 142 \\
                 6.13: Remainder / 143 \\
                 6.14: Why the remainder functions are hard / 144 \\
                 6.15: Computing {\tt fmod} / 146 \\
                 6.16: Computing {\tt remainder} / 148 \\
                 6.17: Computing {\tt remquo} / 150 \\
                 6.18: Computing one remainder from the other / 152 \\
                 6.19: Computing the remainder in nonbinary bases / 155
                 \\
                 6.20: Summary / 156 \\
                 7: Random numbers / 157 \\
                 7.1: Guidelines for random-number software / 157 \\
                 7.2: Creating generator seeds / 158 \\
                 7.3: Random floating-point values / 160 \\
                 7.4: Random integers from floating-point generator /
                 165 \\
                 7.5: Random integers from an integer generator / 166
                 \\
                 7.6: Random integers in ascending order / 168 \\
                 7.7: How random numbers are generated / 169 \\
                 7.8: Removing generator bias / 178 \\
                 7.9: Improving a poor random number generator / 178 \\
                 7.10: Why long periods matter / 179 \\
                 7.11: Inversive congruential generators / 180 \\
                 7.12: Inversive congruential generators, revisited /
                 189 \\
                 7.13: Distributions of random numbers / 189 \\
                 7.14: Other distributions / 195 \\
                 7.15: Testing random-number generators / 196 \\
                 7.16: Applications of random numbers / 202 \\
                 7.17: The \textsf {mathcw} random number routines / 208
                 \\
                 7.18: Summary, advice, and further reading / 214 \\
                 8: Roots / 215 \\
                 8.1: Square root / 215 \\
                 8.2: Hypotenuse and vector norms / 222 \\
                 8.3: Hypotenuse by iteration / 227 \\
                 8.4: Reciprocal square root / 233 \\
                 8.5: Cube root / 237 \\
                 8.6: Roots in hardware / 240 \\
                 8.7: Summary / 242 \\
                 9: Argument reduction / 243 \\
                 9.1: Simple argument reduction / 243 \\
                 9.2: Exact argument reduction / 250 \\
                 9.3: Implementing exact argument reduction / 253 \\
                 9.4: Testing argument reduction / 265 \\
                 9.5: Retrospective on argument reduction / 265 \\
                 10: Exponential and logarithm / 267 \\
                 10.1: Exponential functions / 267 \\
                 10.2: Exponential near zero / 273 \\
                 10.3: Logarithm functions / 282 \\
                 10.4: Logarithm near one / 290 \\
                 10.5: Exponential and logarithm in hardware / 292 \\
                 10.6: Compound interest and annuities / 294 \\
                 10.7: Summary / 298 \\
                 11: Trigonometric functions / 299 \\
                 11.1: Sine and cosine properties / 299 \\
                 11.2: Tangent properties / 302 \\
                 11.3: Argument conventions and units / 304 \\
                 11.4: Computing the cosine and sine / 306 \\
                 11.5: Computing the tangent / 310 \\
                 11.6: Trigonometric functions in degrees / 313 \\
                 11.7: Trigonometric functions in units of $ \pi $ / 315
                 \\
                 11.8: Computing the cosine and sine together / 320 \\
                 11.9: Inverse sine and cosine / 323 \\
                 11.10: Inverse tangent / 331 \\
                 11.11: Inverse tangent, take two / 336 \\
                 11.12: Trigonometric functions in hardware / 338 \\
                 11.13: Testing trigonometric functions / 339 \\
                 11.14: Retrospective on trigonometric functions / 340
                 \\
                 12: Hyperbolic functions / 341 \\
                 12.1: Hyperbolic functions / 341 \\
                 12.2: Improving the hyperbolic functions / 345 \\
                 12.3: Computing the hyperbolic functions together / 348
                 \\
                 12.4: Inverse hyperbolic functions / 348 \\
                 12.5: Hyperbolic functions in hardware / 350 \\
                 12.6: Summary / 352 \\
                 13: Pair-precision arithmetic / 353 \\
                 13.1: Limitations of pair-precision arithmetic / 354
                 \\
                 13.2: Design of the pair-precision software interface /
                 355 \\
                 13.3: Pair-precision initialization / 356 \\
                 13.4: Pair-precision evaluation / 357 \\
                 13.5: Pair-precision high part / 357 \\
                 13.6: Pair-precision low part / 357 \\
                 13.7: Pair-precision copy / 357 \\
                 13.8: Pair-precision negation / 358 \\
                 13.9: Pair-precision absolute value / 358 \\
                 13.10: Pair-precision sum / 358 \\
                 13.11: Splitting numbers into pair sums / 359 \\
                 13.12: Premature overflow in splitting / 362 \\
                 13.13: Pair-precision addition / 365 \\
                 13.14: Pair-precision subtraction / 367 \\
                 13.15: Pair-precision comparison / 368 \\
                 13.16: Pair-precision multiplication / 368 \\
                 13.17: Pair-precision division / 371 \\
                 13.18: Pair-precision square root / 373 \\
                 13.19: Pair-precision cube root / 377 \\
                 13.20: Accuracy of pair-precision arithmetic / 379 \\
                 13.21: Pair-precision vector sum / 384 \\
                 13.22: Exact vector sums / 385 \\
                 13.23: Pair-precision dot product / 385 \\
                 13.24: Pair-precision product sum / 386 \\
                 13.25: Pair-precision decimal arithmetic / 387 \\
                 13.26: Fused multiply-add with pair precision / 388 \\
                 13.27: Higher intermediate precision and the FMA / 393
                 \\
                 13.28: Fused multiply-add without pair precision / 395
                 \\
                 13.29: Fused multiply-add with multiple precision / 401
                 \\
                 13.30: Fused multiply-add, Boldo/\penalty
                 \exhyphenpenalty Melquiond style / 403 \\
                 13.31: Error correction in fused multiply-add / 406 \\
                 13.32: Retrospective on pair-precision arithmetic / 407
                 \\
                 14: Power function / 411 \\
                 14.1: Why the power function is hard to compute / 411
                 \\
                 14.2: Special cases for the power function / 412 \\
                 14.3: Integer powers / 414 \\
                 14.4: Integer powers, revisited / 420 \\
                 14.5: Outline of the power-function algorithm / 421 \\
                 14.6: Finding $a$ and $p$ / 423 \\
                 14.7: Table searching / 424 \\
                 14.8: Computing $\log_n(g/a)$ / 426 \\
                 14.9: Accuracy required for $\log_n(g/a)$ / 429 \\
                 14.10: Exact products / 430 \\
                 14.11: Computing $w$, $w_1$ and $w_2$ / 433 \\
                 14.12: Computing $n^{w_2}$ / 437 \\
                 14.13: The choice of $q$ / 438 \\
                 14.14: Testing the power function / 438 \\
                 14.15: Retrospective on the power function / 440 \\
                 15: Complex arithmetic primitives / 441 \\
                 15.1: Support macros and type definitions / 442 \\
                 15.2: Complex absolute value / 443 \\
                 15.3: Complex addition / 445 \\
                 15.4: Complex argument / 445 \\
                 15.5: Complex conjugate / 446 \\
                 15.6: Complex conjugation symmetry / 446 \\
                 15.7: Complex conversion / 448 \\
                 15.8: Complex copy / 448 \\
                 15.9: Complex division: C99 style / 449 \\
                 15.10: Complex division: Smith style / 451 \\
                 15.11: Complex division: Stewart style / 452 \\
                 15.12: Complex division: Priest style / 453 \\
                 15.13: Complex division: avoiding subtraction loss /
                 455 \\
                 15.14: Complex imaginary part / 456 \\
                 15.15: Complex multiplication / 456 \\
                 15.16: Complex multiplication: error analysis / 458 \\
                 15.17: Complex negation / 459 \\
                 15.18: Complex projection / 460 \\
                 15.19: Complex real part / 460 \\
                 15.20: Complex subtraction / 461 \\
                 15.21: Complex infinity test / 462 \\
                 15.22: Complex NaN test / 462 \\
                 15.23: Summary / 463 \\
                 16: Quadratic equations / 465 \\
                 16.1: Solving quadratic equations / 465 \\
                 16.2: Root sensitivity / 471 \\
                 16.3: Testing a quadratic-equation solver / 472 \\
                 16.4: Summary / 474 \\
                 17: Elementary functions in complex arithmetic / 475
                 \\
                 17.1: Research on complex elementary functions / 475
                 \\
                 17.2: Principal values / 476 \\
                 17.3: Branch cuts / 476 \\
                 17.4: Software problems with negative zeros / 478 \\
                 17.5: Complex elementary function tree / 479 \\
                 17.6: Series for complex functions / 479 \\
                 17.7: Complex square root / 480 \\
                 17.8: Complex cube root / 485 \\
                 17.9: Complex exponential / 487 \\
                 17.10: Complex exponential near zero / 492 \\
                 17.11: Complex logarithm / 495 \\
                 17.12: Complex logarithm near one / 497 \\
                 17.13: Complex power / 500 \\
                 17.14: Complex trigonometric functions / 502 \\
                 17.15: Complex inverse trigonometric functions / 504
                 \\
                 17.16: Complex hyperbolic functions / 509 \\
                 17.17: Complex inverse hyperbolic functions / 514 \\
                 17.18: Summary / 520 \\
                 18: The Greek functions: gamma, psi, and zeta / 521 \\
                 18.1: Gamma and log-gamma functions / 521 \\
                 18.2: The {\tt psi} and {\tt psiln} functions / 536 \\
                 18.3: Polygamma functions / 547 \\
                 18.4: Incomplete gamma functions / 560 \\
                 18.5: A Swiss diversion: Bernoulli and Euler / 568 \\
                 18.6: An Italian excursion: Fibonacci numbers / 575 \\
                 18.7: A German gem: the Riemann zeta function / 579 \\
                 18.8: Further reading / 590 \\
                 18.9: Summary / 591 \\
                 19: Error and probability functions / 593 \\
                 19.1: Error functions / 593 \\
                 19.2: Scaled complementary error function / 598 \\
                 19.3: Inverse error functions / 600 \\
                 19.4: Normal distribution functions and inverses / 610
                 \\
                 19.5: Summary / 617 \\
                 20: Elliptic integral functions / 619 \\
                 20.1: The arithmetic-geometric mean / 619 \\
                 20.2: Elliptic integral functions of the first kind /
                 624 \\
                 20.3: Elliptic integral functions of the second kind /
                 627 \\
                 20.4: Elliptic integral functions of the third kind /
                 630 \\
                 20.5: Computing $K(m)$ and $K'(m)$ / 631 \\
                 20.6: Computing $E(m)$ and $E'(m)$ / 637 \\
                 20.7: Historical algorithms for elliptic integrals /
                 643 \\
                 20.8: Auxiliary functions for elliptic integrals / 645
                 \\
                 20.9: Computing the elliptic auxiliary functions / 648
                 \\
                 20.10: Historical elliptic functions / 650 \\
                 20.11: Elliptic functions in software / 652 \\
                 20.12: Applications of elliptic auxiliary functions /
                 653 \\
                 20.13: Elementary functions from elliptic auxiliary
                 functions / 654 \\
                 20.14: Computing elementary functions via $R_C(x,y)$ /
                 655 \\
                 20.15: Jacobian elliptic functions / 657 \\
                 20.16: Inverses of Jacobian elliptic functions / 664
                 \\
                 20.17: The modulus and the nome / 668 \\
                 20.18: Jacobian theta functions / 673 \\
                 20.19: Logarithmic derivatives of the Jacobian theta
                 functions / 675 \\
                 20.20: Neville theta functions / 678 \\
                 20.21: Jacobian Eta, Theta, and Zeta functions / 679
                 \\
                 20.22: Weierstrass elliptic functions / 682 \\
                 20.23: Weierstrass functions by duplication / 689 \\
                 20.24: Complete elliptic functions, revisited / 690 \\
                 20.25: Summary / 691 \\
                 21: Bessel functions / 693 \\
                 21.1: Cylindrical Bessel functions / 694 \\
                 21.2: Behavior of $J_n(x)$ and $Y_n(x)$ / 695 \\
                 21.3: Properties of $J_n(z)$ and $Y_n(z)$ / 697 \\
                 21.4: Experiments with recurrences for $J_0(x)$ / 705
                 \\
                 21.5: Computing $J_0(x)$ and $J_1(x)$ / 707 \\
                 21.6: Computing $J_n(x)$ / 710 \\
                 21.7: Computing $Y_0(x)$ and $Y_1(x)$ / 713 \\
                 21.8: Computing $Y_n(x)$ / 715 \\
                 21.9: Improving Bessel code near zeros / 716 \\
                 21.10: Properties of $I_n(z)$ and $K_n(z)$ / 718 \\
                 21.11: Computing $I_0(x)$ and $I_1(x)$ / 724 \\
                 21.12: Computing $K_0(x)$ and $K_1(x)$ / 726 \\
                 21.13: Computing $I_n(x)$ and $K_n(x)$ / 728 \\
                 21.14: Properties of spherical Bessel functions / 731
                 \\
                 21.15: Computing $j_n(x)$ and $y_n(x)$ / 735 \\
                 21.16: Improving $j_1(x)$ and $y_1(x)$ / 740 \\
                 21.17: Modified spherical Bessel functions / 743 \\
                 21.18: Software for Bessel-function sequences / 755 \\
                 21.19: Retrospective on Bessel functions / 761 \\
                 22: Testing the library / 763 \\
                 22.1: Testing {\tt tgamma} and {\tt lgamma} / 765 \\
                 22.2: Testing {\tt psi} and {\tt psiln} / 768 \\
                 22.3: Testing {\tt erf} and {\tt erfc} / 768 \\
                 22.4: Testing cylindrical Bessel functions / 769 \\
                 22.5: Testing exponent/\penalty \exhyphenpenalty
                 significand manipulation / 769 \\
                 22.6: Testing inline assembly code / 769 \\
                 22.7: Testing with Maple / 770 \\
                 22.8: Testing floating-point arithmetic / 773 \\
                 22.9: The Berkeley Elementary Functions Test Suite /
                 774 \\
                 22.10: The AT\&T floating-point test package / 775 \\
                 22.11: The Antwerp test suite / 776 \\
                 22.12: Summary / 776 \\
                 23: Pair-precision elementary functions / 777 \\
                 23.1: Pair-precision integer power / 777 \\
                 23.2: Pair-precision machine epsilon / 779 \\
                 23.3: Pair-precision exponential / 780 \\
                 23.4: Pair-precision logarithm / 787 \\
                 23.5: Pair-precision logarithm near one / 793 \\
                 23.6: Pair-precision exponential near zero / 793 \\
                 23.7: Pair-precision base-$n$ exponentials / 795 \\
                 23.8: Pair-precision trigonometric functions / 796 \\
                 23.9: Pair-precision inverse trigonometric functions /
                 801 \\
                 23.10: Pair-precision hyperbolic functions / 804 \\
                 23.11: Pair-precision inverse hyperbolic functions /
                 808 \\
                 23.12: Summary / 808 \\
                 24: Accuracy of the Cody\slash Waite algorithms / 811
                 \\
                 25: Improving upon the Cody\slash Waite algorithms /
                 823 \\
                 25.1: The Bell Labs libraries / 823 \\
                 25.2: The {Cephes} library / 823 \\
                 25.3: The {Sun} libraries / 824 \\
                 25.4: Mathematical functions on EPIC / 824 \\
                 25.5: The GNU libraries / 825 \\
                 25.6: The French libraries / 825 \\
                 25.7: The NIST effort / 826 \\
                 25.8: Commercial mathematical libraries / 826 \\
                 25.9: Mathematical libraries for decimal arithmetic /
                 826 \\
                 25.10: Mathematical library research publications / 826
                 \\
                 25.11: Books on computing mathematical functions / 827
                 \\
                 25.12: Summary / 828 \\
                 26: Floating-point output / 829 \\
                 26.1: Output character string design issues / 830 \\
                 26.2: Exact output conversion / 831 \\
                 26.3: Hexadecimal floating-point output / 832 \\
                 26.4: Octal floating-point output / 850 \\
                 26.5: Binary floating-point output / 851 \\
                 26.6: Decimal floating-point output / 851 \\
                 26.7: Accuracy of output conversion / 865 \\
                 26.8: Output conversion to a general base / 865 \\
                 26.9: Output conversion of Infinity / 866 \\
                 26.10: Output conversion of NaN / 866 \\
                 26.11: Number-to-string conversion / 867 \\
                 26.12: The {\tt printf} family / 867 \\
                 26.13: Summary / 878 \\
                 27: Floating-point input / 879 \\
                 27.1: Binary floating-point input / 879 \\
                 27.2: Octal floating-point input / 894 \\
                 27.3: Hexadecimal floating-point input / 895 \\
                 27.4: Decimal floating-point input / 895 \\
                 27.5: Based-number input / 899 \\
                 27.6: General floating-point input / 900 \\
                 27.7: The {\tt scanf} family / 901 \\
                 27.8: Summary / 910 \\
                 A: Ada interface / 911 \\
                 A.1: Building the Ada interface / 911 \\
                 A.2: Programming the Ada interface / 912 \\
                 A.3: Using the Ada interface / 915 \\
                 B: C\# interface / 917 \\
                 B.1: C\# on the CLI virtual machine / 917 \\
                 B.2: Building the C\# interface / 918 \\
                 B.3: Programming the C\# interface / 920 \\
                 B.4: Using the C\# interface / 922 \\
                 C: C++ interface / 923 \\
                 C.1: Building the C++ interface / 923 \\
                 C.2: Programming the C++ interface / 924 \\
                 C.3: Using the C++ interface / 925 \\
                 D: Decimal arithmetic / 927 \\
                 D.1: Why we need decimal floating-point arithmetic /
                 927 \\
                 D.2: Decimal floating-point arithmetic design issues /
                 928 \\
                 D.3: How decimal and binary arithmetic differ / 931 \\
                 D.4: Initialization of decimal floating-point storage /
                 935 \\
                 D.5: The {\tt <decfloat.h>} header file / 936 \\
                 D.6: Rounding in decimal arithmetic / 936 \\
                 D.7: Exact scaling in decimal arithmetic / 937 \\
                 E: Errata in the Cody\slash Waite book / 939 \\
                 F: Fortran interface / 941 \\
                 F.1: Building the Fortran interface / 943 \\
                 F.2: Programming the Fortran interface / 944 \\
                 F.3: Using the Fortran interface / 945 \\
                 H: Historical floating-point architectures / 947 \\
                 H.1: CDC family / 949 \\
                 H.2: Cray family / 952 \\
                 H.3: DEC PDP-10 / 953 \\
                 H.4: DEC PDP-11 and VAX / 956 \\
                 H.5: General Electric 600 series / 958 \\
                 H.6: IBM family / 959 \\
                 H.7: Lawrence Livermore S-1 Mark IIA / 965 \\
                 H.8: Unusual floating-point systems / 966 \\
                 H.9: Historical retrospective / 967 \\
                 I: Integer arithmetic / 969 \\
                 I.1: Memory addressing and integers / 971 \\
                 I.2: Representations of signed integers / 971 \\
                 I.3: Parity testing / 975 \\
                 I.4: Sign testing / 975 \\
                 I.5: Arithmetic exceptions / 975 \\
                 I.6: Notations for binary numbers / 977 \\
                 I.7: Summary / 978 \\
                 J: Java interface / 979 \\
                 J.1: Building the Java interface / 979 \\
                 J.2: Programming the Java MathCW class / 980 \\
                 J.3: Programming the Java C interface / 982 \\
                 J.4: Using the Java interface / 985 \\
                 L: Letter notation / 987 \\
                 P: Pascal interface / 989 \\
                 P.1: Building the Pascal interface / 989 \\
                 P.2: Programming the Pascal MathCW module / 990 \\
                 P.3: Using the Pascal module interface / 993 \\
                 P.4: Pascal and numeric programming / 994 \\
                 Bibliography / 995 \\
                 Author/editor index / 1039 \\
                 Function and macro index / 1049 \\
                 Subject index / 1065 \\
                 Colophon / 1115",
}


