2019/04/05
Source: xint.dtx 1.3e 2019/04/05 (doc 2019/04/05)
Author: Jean-Francois Burnol
Info: Expandable operations on big integers, decimals, fractions
License: LPPL 1.3e1.3e (2019/04/05)\xint(float)expr...\relax syntax. One can not use there the \xint(float)eval wrappers (anyhow they add overhead and can be replaced with the lower level syntax).xintexpr: \xintdefefunc, \xintdeffloatefunc, \xintdefiiefunc define functions which are not protected against expansion in the definition of other functions; refer to xint.pdf for the related explanations.
\xintdef(e)func, \xintNewExpr, \xintNewFunction is complex and to be considered still as work in progress as it has a number of shortcomings.inv(), ilog10(), sfloat(), behaviour of qfloat() slightly modified.\xintensuredummy, \xintrestorelettervar.The optional argument of \xintfloatexpr or \xintfloateval (it must be at start of braced argument) can be negative; it then means to trim (and round) from the output at float precision that many least significant digits.
1.3d (2019/01/06)xintexpr: the gcd() and lcm() functions formerly converted their arguments to integers via \xintNum. They now handle general input with no such modification.
xintexpr: former \xinteval, \xintieval, \xintiieval, and \xintfloateval renamed to \xintexpro, \xintiexpro, \xintiiexpro, and \xintfloatexpro.
xintexpr: the gcd() and lcm() multi-arguments functions have been refactored to handle general fractions. The dependency on xintgcd is removed.
xintexpr: three-way branching \xintifsgnexpr, \xintifsgnfloatexpr, \xintifsgniiexpr conditional macros.
xintexpr: \xintunassignexprfunc, \xintunassigniiexprfunc, \xintunassignfloatexprfunc to “undefine” functions.
xintexpr: \xintunassignvar really makes the (multi-letter) variable unknown (formerly, it only gave it value zero),
xintexpr: functions isone() and isint().
xintexpr: \xinteval, \xintieval, \xintiieval, and \xintfloateval as synonyms to \xinttheexpr...\relax etc…, but with the (comma-separated) expression as a usual braced macro argument.
\xintiiexpr was broken for a zero dividend and a one-digit divisor (e.g. 0//7) since 1.2p due to a bug in \xintiiDivMod for such arguments. The bug was signaled (thanks to Kpym for report) and fixed shortly after 1.3c release but I then completely forgot to upload a bugfix release to CTAN at that time, apologies for that.1.3c (2018/06/17)xintexpr: with \xintglobaldefstrue, \xintdefvar, \xintdeffunc, \xintNewExpr et al. make definitions with global scope.
xintexpr: qraw() for fast input of (very many) comma separated numbers (in suitable raw format).
xintexpr: the colon in the := part of the syntax for \xintdefvar and variants is now optional; and if present it may be an active character or have any (reasonable) catcode.
xintexpr: \xintdefvar, \xintdeffunc and their variants try to set the catcode of the semi-colon which delimits their arguments; of course this will not work if that catcode is already frozen.
\xintUniformDeviate is better documented and sourcexint.pdf is better hyperlinked and includes indices for the macros defined by each package.
1.3 release, it loaded xintgcd in contradiction to what the documentation says (hence also xintexpr loaded xintgcd automatically). There is no actual dependency so the loading is removed for now.1.3b (2018/05/18)All additions related to randomness are marked as work-in-progress. They require an engine providing the \(pdf)uniformdeviate primitive.
xintkernel: \xintUniformDeviate.
xint: \xintRandomDigits, \xintXRandomDigits, \xintiiRandRange, \xintiiRandRangeAtoB.
xintfrac: support macros (not public, mainly because internal format for floats is surely not final) for random() and qrand().
xintexpr: random(), qrand(), and randrange(A[, B]).
xintexpr: when a function foo() is declared via \xintdeffunc (et al.) to be parameter-less, it can be used as foo(); formerly foo(nil) syntax was required.
The usual provision of user manual “improvements”.
1.3a (2018/03/07)1.2o to add a deprecation mechanism; all deprecated macros have been removed at 1.3 so there was no reason to keep the code used for deprecating them.xintexpr: new conditionals ifone() and ifint().
xintfrac: \xintREZ is faster on inputs having one hundred digits or more.
Added to the user manual mention of macros such as \xintDivFloor, \xintMod, \xintModTrunc, which had been left out so far.
1.3: in particular recursive definitions became possible. But an oversight made these recursive functions quite inefficient (to remain polite.) This release fixes the problem.1.3 (2018/03/01)xintcore, xint, xintfrac: all macros deprecated at 1.2o got removed.
xintfrac: addition and subtraction of a/b and c/d now use the l.c.m. of the denominators. Similarly the macro supporting the modulo operator /: uses a l.c.m. for the denominator of the result.
xintexpr: the addition, subtraction, modulo /:, and the mod() and divmod() functions produce generally smaller denominators (see previous item).
xintexpr: formerly, the internal macros which are internally associated to user-declared functions were using comma separated parameter texts. They now do not use such commas (their meanings, which may again change in future, are written for information to the log under \xintverbosetrue).
xintexpr: user-defined functions may now be of a recursive nature. This was made possible by a refactoring of the \xintNewExpr mechanism. It became both leaner and more extensive than formerly.
xintfrac: new macros \xintPIrr and \xintDecToString. The latter is a backport of a polexpr 0.4 utility, and it is to be considered unstable.
xintexpr: new function preduce() associated with \xintPIrr.
1.2q (2018/02/06)3!4!5! or (1+2)3.1.2l of subtraction left an extra character in an inner macro causing breakage in some rare circumstances. This should not have escaped our test suite!1.2p (2017/12/05)xintgcd: \xintBezout{a}{b}’s output consists of {u}{v}{d} with u*a+v*b==d, with d the GCD. Formerly it was {a}{b}{u}{v}{d}, and with u*a-v*b==d.
xintgcd: \xintBezout{0}{0} expands to {0}{0}{0}. Formerly (since 1.2l) it raised InvalidOperation.
xintcore: \xintiiMod is now associated with floored division. The former meaning (associated with truncated division) is available as \xintiiModTrunc.
xintfrac: \xintMod is now associated with floored division. The former meaning is available as \xintModTrunc.
xintexpr: the // operator and its associated modulo 'mod' (or /:) now correspond to floored division, like the Python language //, %, and divmod(x, y). Formerly they had been associated to truncated division. This is breaking change for operands of opposite signs.
xinttools: \xintListWithSep, which had remained unchanged since its introduction at 1.04 (2013/04/25), was rewritten for increased speed.
xintexpr: \xintdefvar’s syntax is extended to allow simultaneous assignments. Examples: \xintdefvar x1, x2, x3 := 1, 3**10, 3**20; or \xintdefiivar A, B := B, A 'mod' B; for already defined variables A and B.
xintexpr: added divmod() to the built-in functions. It is associated with floored division, like the Python language divmod(). Related support macros added to xintcore, and xintfrac.
xintgcd: \xintBezout{6}{3} (for example) expanded to {6}{3}{-0}{-1}{3}, but the -0 should have been 0.
xintgcd: it still used macro \xintiAbs although the latter had been deprecated from xintcore.
xintexpr: in float expressions the // and /: (aka 'mod') operators did not round their operands to the float precision prior to computing with them, contrarily to other infix arithmetic operators and to the mod(f,g) function; thus, mod(f,g) and f 'mod' g were not completely equivalent.
various documentation fixes; in particular, the partial dependency of xintcfrac on xinttools had not been mentioned.
1.2o (2017/08/29)\xintAND, \xintOR, … and similar Boolean logic macros do not apply anymore \xintNum (or \xintRaw if xintfrac is loaded), to their arguments (often, from internal usage of \xintSgn), but only f-expand them (using e.g. \xintiiSgn). This is kept un-modified even if loading xintfrac.Deprecated macros raise an error but, generally, then expand as in former releases. They will all get removed at some future release.
xintcore: \xintiOpp, \xintiAbs, \xintiAdd, \xintiSub, \xintiMul, \xintiDivision, \xintiQuo, \xintiRem, \xintiDivRound, \xintiDivTrunc, \xintiMod, \xintiSqr, \xintiPow, and \xintiFac are deprecated. Only the ii-named variants get defined.
xintcore: \xintCmp and \xintSgn are deprecated from xintcore (which only defines \xintiiCmp and \xintiiSgn) as they actually belong to xintfrac.
xintcore: \xintiiFDg, resp. \xintiiLDg, are renamed \xintFDg, resp. \xintLDg. Former denominations are deprecated.
xint: \xintMON, \xintMMON, \xintiMax, \xintiMin, \xintiMaxof, \xintiMinof, \xintiSquareRoot, \xintiSqrt, \xintiSqrtR, \xintiBinomial, and \xintiPFactorial are deprecated. Only ii-named variants get defined.
xint: \xintEq, \xintGeq, \xintGt, \xintLt, \xintGtorEq, \xintLtorEq, \xintIsZero, \xintIsNotZero, \xintIsOne, \xintOdd, \xintEven, \xintifSgn, \xintifCmp, \xintifEq, \xintifGt, \xintifLt, \xintifZero, \xintifNotZero, \xintifOne, \xintifOdd, are deprecated. These macros belong to xintfrac. Package xint defines only the ii-named variants.
xint: \xintNeq was renamed to \xintNotEq which however is only provided by xintfrac. Package xint defines \xintiiNotEq, and \xintNeq is deprecated.
xint: \xintNot was renamed to \xintNOT, former denomination is deprecated. See also item about Boolean logic macros in the Incompatible Changes section.
1.2n (2017/08/06)xintbinhex has only xintkernel as dependency.
Macros of xintbinhex have been improved for speed and increased maximal sizes of allowable inputs.
1.2m (2017/07/31)xintbinhex: the length of the input is now limited. The maximum size depends on the macro and ranges from about 4000 to about 19900 digits.
xintbinhex: \xintCHexToBin is now the variant of \xintHexToBin which does not remove leading binary zeroes: N hex-digits give on output exactly 4N binary digits.
1.08 of 2013/06/07.) The new macros are faster but limited to a few thousand digits. The 1.08 routines could handle tens of thousands of digits, but not in a reasonable time.user manual: the Changes section wrongly stated at 1.2l that the macros of xintbinhex had been made robust against non terminated input such as \number\mathcode`\-. Unfortunately the author fell into the trap of believing his own documentation and he forgot to actually implement the change. Now done.
user manual: the PDF bookmarks were messed up.
xint, xintfrac: \xintGeq, \xintMax, \xintMin, suffered from some extra overhead. This was caused by use of some auxiliaries from the very early days which got redefined at some stage. This is fixed here with some additional efficiency improvements and pruning of old code.
1.2l (2017/07/26)\xintiiSumExpr, \xintiiPrdExpr (xint) and \xintSumExpr, \xintPrdExpr (xintfrac). They had not been formally deprecated, but had been left un-documented since 1.09d (2013/10/22).
internal macro \xint_gob_til_xint_relax removed.
the underscore character _ is accepted by the xintexpr parsers as a digit separator (the space character already could be used for improved readability of big numbers). It is not allowed as first character of a number, as it would then be mis-interpreted as the start of a possible variable name.
some refactoring in xintcore auxiliary routines and in \xintiiSub and \xintiiCmp for some small efficiency gains.
code comments in xintcore are better formatted, but remain sparse.
xintcore, xint, xintfrac, … : some macros were not robust against arguments whose expansion looks forward for some termination (e.g. \number\mathcode`\-), and particularly, most were fragile against inputs using non-terminated \numexpr (such as \xintiiAdd{\the\numexpr1}{2} or \xintRaw{\numexpr1}). This was not a bug per se, as the user manual did not claim such inputs were legal, but it was slightly inconvenient. Most macros (particularly those of xintfrac) have now been made robust against such inputs. Some macros from xintcore primarily destined to internal usage still accept only properly terminated arguments such as \the\mathcode`\-<space> or \the\numexpr1\relax.
The situation with expressions is unchanged: syntax such as \xintexpr \numexpr1+2\relax is illegal as the ending \relax token will get swallowed by the \numexpr; but it is needed by the xintexpr-ession parser, hence the parser will expand forward and presumably end with in an “illegal token” error, or provoke some low-level TeX error (N.B.: a closing brace } for example can not terminate an xintexpr-ession, the parser must find a \relax token at some point). Thus there must be in this example a second \relax.
experimental code for error conditions; there is no complete user interface yet, it is done in preparation for next major release and is completely unstable and undocumented.
xintbinhex: since 1.2 (2015/10/10), \xintHexToDec was broken due to an undefined macro (it was in xint.sty, but the module by itself is supposedly dependent only upon xintcore.sty).
xintgcd: macro \xintBezout produced partially wrong output if one of its two arguments was zero.
xintfrac: the manual said one could use directly \numexpr compatible expressions in arithmetic macros (without even a \numexpr encapsulation) if they were expressed with up to 8 tokens. There was a bug if these 8 tokens evaluated to zero. The bug has been fixed, and up to 9 tokens are now accepted. But it is simpler to use \the\numexpr prefix and not to worry about the token count… The ending \relax is now un-needed.
1.2k (2017/01/06)macro \xintFloat which rounds its input to a floating point number does not print anymore 10.0...0eN to signal an upwards rounding to the next power of ten. The mantissa has in all cases except the zero input exactly one digit before the decimal mark.
some floating point computations may differ in the least significant digits, due to a change in the rounding algorithm applied to macro arguments expressed as fractions and to an improvement in precision regarding half-integer powers in expressions. See next.
the initial rounding to the target precision P which is applied by the floating point macros from xintfrac to their arguments achieves the exact (aka correct) rounding even for inputs which are fractions with more than P+2 digits in their numerators and denominators (>1.) Hence the computed values depend only on the arguments as rational numbers and not upon their representatives. This is not relevant to expressions (xintexpr), because the \xintfloatexpr parser sees there / as an operator and does not (apart from special constructs) get to manipulate fractions as such.
\xintnewdummy is public interface to a 1.2e macro which serves to declare any given catcode 11 character as a dummy variable for expressions (xintexpr). This is useful for Unicode engines (the Latin letters being already all pre-declared as dummy variables.)
added \xintiSqrtR, there was only \xintiiSqrtR alongside \xintiSqrt and \xintiiSqrt (xint).
added non public \xintLastItem:f:csv to xinttools for faster last() function, and improved \xintNewExpr compatibility. Also \xintFirstItem:f:csv.
the 1.2f half-integer powers computed within \xintfloatexpr had a silly rounding to the target precision just before the final square-root extraction, thus possibly losing some precision. The 1.2k implementation keeps guard digits for this final square root extraction. As for integer exponents, it is guaranteed that the computed value differs from the exact one by less than 0.52 ulp (for inputs having at most \xinttheDigits digits.)
more regressions from 1.2i were fixed: \xintLen (xint, xintfrac) and \xintDouble (xintcore) had forgotten that their argument was allowed to be negative. A regression test suite is now in place and is being slowly expanded to cover more macros.
\xintiiSquareRoot{0} now produces {1}{1}, which fits better the general documented behaviour of this macro than 11.
1.2j (2016/12/22)xinttools and xintexpr:
slightly improves the speed of \xintTrim.
speed gains for the handlers of comma separated lists implementing Python-like slicing and item extraction. Relevant non (user) documented macros better documented in sourcexint.pdf.
significant documentations tweaks (inclusive of suppressing things!), and among them two beautiful hyperlinked tables with both horizontal and vertical rules which bring the documentation of the xintexpr syntax to a kind of awe-inspiring perfection… except that implementation of some math functions is still lacking.
1.2i regressions caused by undefined macros (\xintNthElt in certain branches and [list][N] item extraction in certain cases.) The test files existed but were not executed prior to release. Automation in progress.1.2i (2016/12/13)\xintDecSplit second argument must have no sign (former code replaced it with its absolute value, a sign now may cause an error.)\xintifTrue, \xintifTrueFalse, \xintQuo, \xintRem, \xintquo, \xintrem.xintkernel: \xintLength is faster. New macros:
\xintLastItem to fetch the last item from its argument,
\romannumeral\xintgobble for gobbling many (up to 531440) upstream braced items or tokens.
\romannumeral\xintreplicate which is copied over from the expl3 \prg_replicate:nn with some minor changes.
xinttools: general token list handling routines \xintKeep, \xintTrim and \xintNthElt are faster; but the novel \xintTrim can only remove up to a maximum of 531440 items.
Also, \xintFor partially improves on some issues which are reported upon in the documentation.
some old macros have been rewritten entirely or partially using techniques which xint started using in release 1.2:
xintcore: \xintDouble, \xintHalf, \xintInc, \xintDec, \xintiiLDg, \xintDSR (originally from xint), a novel \xintDSRr.
xint: \xintDSH, \xintDSx, \xintDecSplit, \xintiiE.
xintfrac: as a result of the above \xintTrunc, \xintRound and \xintXTrunc got faster. But the main improvement for them is with decimal inputs which formerly had not been treated separately from the general fraction case. Also, \xintXTrunc does not anymore create a dependency of xintfrac on xinttools.
the documentation has again been (slightly) re-organized; it has a new sub-section on the Miller-Rabin primality test, to illustrate some use of \xintNewFunction for recursive definitions.
the documentation has dropped the LaTeX “command” terminology (which had been used initially in 2013 for some forgotten reasons and should have been removed long ago) and uses only the more apt “macro”, as after all, all of xint is about expansion of macros (plus the use of \numexpr).
\xintDecSplitL and \xintDecSplitR from xint produced their output in a spurious brace pair (bug introduced in 1.2f).1.2h (2016/11/20)new macro \xintNewFunction in xintexpr which allows to extend the parser syntax with functions in situations where \xintdeffunc is not usable (typically, because dummy variables are used over a not yet determined range of values because it depends on the variables).
after three years of strict obedience to xint prefix, now \thexintexpr, \thexintiexpr, \thexintfloatexpr, and \thexintiiexpr are provided as synonyms to \xinttheexpr, etc…
the (cond)?{foo}{bar} operator from xintexpr mis-behaved in certain circumstances (such as an empty foo).
the xintexpr 1.2f binomial function (which uses \xintiiBinomial from xint.sty or \xintFloatBinomial from xintfrac.sty) deliberately raised an error for binomial(x,y) with y<0 or x<y. This was unfortunate, and it now simply evaluates to zero in such cases.
similarly the pfactorial function was very strict and pfactorial(x,y) deliberately raised an out-of-range error if not used with non-negative integers with x less than y. It now avoids doing that and allows negative arguments.
the add and mul from xintexpr, which work with dummy variables since 1.1, raised an error since 1.2c 2015/11/16 when the dummy variable was given an empty range (or list) of values, rather than producing respectively 0 and 1 as formerly.
1.2g (2016/03/19)inside expressions, list item selector [L][n] counts starting at zero, not at one. This is more coherent with [L][a:b] which was already exactly like in Python since its introduction. A function len(L) replaces earlier [L][0].
former iter keyword now called iterr. Indeed it matched with rrseq, the new iter (which was somehow missing from 1.1) is the one matching rseq. Allows to iterate more easily with a “list” variable.
in xintexpr.sty: list selectors [L][n] and [L][a:b] are more efficient: the earlier 1.1 routines did back and forth conversions from comma separated values to braced tokens, the 1.2g routines use macros from xinttools.sty handling directly the encountered lists of comma separated values.
in xinttools.sty: slight improvements in the efficiency of the \xintNthElt, \xintKeep, \xintTrim routines and new routines handling directly comma separated values. The latter are not included in the user manual (they are not \long, they don’t make efforts to preserve some braces, do not worry about spaces, all those worries being irrelevant to the use in expressions for list selectors).
a slight speed improvement to \xintFloatSqrt in its quest of correct rounding.
float multiplication and division handle more swiftly operands (non-fractional) with few digits, when the float precision is large.
the syntax of expressions is described in a devoted chapter of the documentation; an example shows how to implement (expandably) the Brent-Salamin algorithm for computation of Pi using iter in a float expression.
1.2f (2016/03/12)\xintFac macro but \xintiFac/\xintiiFac/\xintFloatFac.functions binomial, pfactorial and factorial in both integer and float versions.
macros \xintiiBinomial, \xintiiPFactorial (xint.sty) and \xintFloatBinomial, \xintFloatPFactorial (xintfrac.sty). Improvements to \xintFloatFac.
faster implementation and increased accuracy of float power macros. Half-integer exponents are now accepted inside float expressions.
faster implementation of both integral and float square root macros.
the float square root achieves correct (aka exact) rounding in arbitrary precision.
modified behaviour for the \xintPFloat macro, used by \xintthefloatexpr to prettify its output. It now opts for decimal notation if and only if scientific notation would use an exponent between -5 and 5 inclusive. The zero value is printed 0. with a dot.
the float macros for addition, subtraction, multiplication, division now first round their two operands to P, not P+2, significant places before doing the actual computation (P being the target precision). The same applies to the power macros and to the square root macro.
the documentation offers a more precise (and accurate) discussion of floating point issues.
various under-the-hood code improvements; the floatexpr operations are chained in a faster way, from skipping some unneeded parsing on results of earlier computations. The absence of a real inner data structure for floats (incorporating their precisions, for one) is however still a bit hair raising: currently the lengths of the mantissas of the operands are computed again by each float macro or expression operation.
(TeXperts only) the macros defined (internally) from \xintdeffunc et al. constructs do not incorporate an initial \romannumeral anymore.
renewed desperate efforts at improving the documentation by random shuffling of sections and well thought additions; cuts were considered and even performed.
squaring macro \xintSqr from xintfrac.sty was broken due to a misspelled sub-macro name. Dates back to 1.1 release of 2014/10/28 :-((.
1.2c’s fix to the subtraction bug from 1.2 introduced another bug, which in some cases could create leading zeroes in the output, or even worse. This could invalidate other routines using subtractions, like \xintiiSquareRoot.
the comparison operators were not recognized by \xintNewIIExpr and \xintdefiifunc constructs.
1.2e (2015/11/22)macro \xintunassignvar.
slight modifications of the logged messages in case of \xintverbosetrue.
a space in \xintdeffunc f(x)<space>:= expression ; is now accepted.
documentation enhancements: the Quick Sort section with its included code samples has been entirely re-written; the Commands of the xintexpr package section has been extended and reviewed entirely.
in xintfrac: the \xintFloatFac from release 1.2 parsed its argument only through \numexpr but it should have used \xintNum.
in xintexpr: release 1.2d had broken the recognition of sub-expressions immediately after variable names (with tacit multiplication).
in xintexpr: contrarily to what 1.2d documentation said, tacit multiplication was not yet always done with enhanced precedence. Now yes.
1.2d (2015/11/18)the function definitions done by \xintdeffunc et al., as well as the macro declarations by \xintNewExpr et al. now have only local scope.
tacit multiplication applies to more cases, for example (x+y)z, and always ties more than standard * infix operator, e.g. x/2y is like x/(2*y).
some documentation enhancements, particularly in the chapter on xintexpr.sty, and also in the code source comments.
1.2c had inadvertently broken the \xintiiDivRound macro.1.2c (2015/11/16)macros \xintdeffunc, \xintdefiifunc, \xintdeffloatfunc and boolean \ifxintverbose.
on-going code improvements and documentation enhancements, but stopped in order to issue this bugfix release.
1.2 introduced a bug in the subtraction (happened when 00000001 was found under certain circumstances at certain mod 8 locations).1.2b (2015/10/29)1.2 introduced a bug in the division macros, causing a crash when the divisor started with 99999999 (it was attempted to use with 1+99999999 a subroutine expecting only 8-digits numbers).1.2a (2015/10/19)added \xintKeepUnbraced, \xintTrimUnbraced (xinttools) and fixed documentation of \xintKeep and \xintTrim regarding brace stripping.
added \xintiiMaxof/\xintiiMinof (xint).
TeX hackers only: replaced all code uses of \romannumeral-`0 by the quicker \romannumeral`&&@ (^ being used as letter, had to find another character usable with catcode 7).
1.2 introduced a bad bug in the parsing of decimal numbers and as a result \xinttheexpr 0.01\relax expanded to 0 ! (sigh…)1.2 (2015/10/10)\xintAdd, \xintSub, \xintMul, \xintMax, \xintMin, \xintMaxof, \xintMinof are removed from package xint, and only exist in the versions from xintfrac. With only xintcore or xint loaded, one must use \xintiiAdd, \xintiiSub, …, or \xintiAdd, \xintiSub, etc…the basic arithmetic implemented in xintcore has been entirely rewritten. The mathematics remains the elementary school one, but the TeX implementation achieves higher speed (except, regarding addition/subtraction, for numbers up to about thirty digits), the gains becoming quite significant for numbers with hundreds of digits.
the inputs must have less than 19959 digits. But computations with thousands of digits take time.
a previously standing limitation of \xintexpr, \xintiiexpr, and of \xintfloatexpr to numbers of less than 5000 digits has been lifted.
a qint function is provided to help the parser gather huge integers in one-go, as an exception to its normal mode of operation which expands token by token.
\xintFloatFac macro for computing the factorials of integers as floating point numbers to a given precision. The ! postfix operator inside \xintfloatexpr maps to this new macro rather than to the exact factorial as used by \xintexpr and \xintiiexpr.
there is more flexibility in the parsing done by the macros from xintfrac on fractional input: the decimal parts of both the numerator and the denominator may arise from a separate expansion via \romannumeral-`0. Also the strict A/B[N] format is a bit relaxed: N may be anything understood by \numexpr (it could even be empty but that possibility has been removed by later 1.2f release.)
on the other hand an isolated dot . is not legal syntax anymore inside the expression parsers: there must be digits either before or after. It remains legal input for the macros of xintfrac.
added \ht, \dp, \wd, \fontcharht, etc… to the tokens recognized by the parsers and expanded by \number.
an obscure bug in package xintkernel has been fixed, regarding the sanitization of catcodes: under certain circumstances (which could not occur in a normal LaTeX context), unusual catcodes could end up being propagated to the external world.
an effort at randomly shuffling around various pieces of the documentation has been done.
1.1c (2015/09/12)bugfix regarding macro \xintAssign from xinttools which did not behave correctly in some circumstances (if there was a space before \to, in particular).
very minor code improvements, and correction of some issues regarding the source code formatting in sourcexint.pdf, and minor issues in Makefile.mk.
1.1b (2015/08/31)bugfix: some macros needed by the integer division routine from xintcore had been left in xint.sty since release 1.1. This for example broke the \xintGCD from xintgcd if package xint was not loaded.
Slight enhancements to the documentation, particularly in the Read this first section.
1.1a (2014/11/07)fixed a bug which prevented \xintNewExpr from producing correctly working macros from a comma separated replacement text.
\xintiiSqrtR for rounded integer square root; former \xintiiSqrt already produced truncated integer square root; corresponding function sqrtr added to \xintiiexpr..\relax syntax.
use of straight quotes in the documentation for better legibility.
added \xintiiIsOne, \xintiiifOne, \xintiiifCmp, \xintiiifEq, \xintiiifGt, \xintiiifLt, \xintiiifOdd, \xintiiCmp, \xintiiEq, \xintiiGt, \xintiiLt, \xintiiLtorEq, \xintiiGtorEq, \xintiiNeq, mainly for efficiency of \xintiiexpr.
for the same reason, added \xintiiGCD and \xintiiLCM.
added the previously mentioned ii macros, and some others from 1.1, to the user manual. But their main usage is internal to \xintiiexpr, to skip unnecessary overheads.
various typographical fixes throughout the documentation, and a bit of clean up of the code comments. Improved \Factors example of nested subs, rseq, iter in \xintiiexpr.
1.1 (2014/10/28)in \xintiiexpr, / does rounded division, rather than the Euclidean division (for positive arguments, this is truncated division). The // operator does truncated division,
the : operator for three-way branching is gone, replaced with ??,
1e(3+5) is now illegal. The number parser identifies e and E in the same way it does for the decimal mark, earlier versions treated e as E rather as infix operators of highest precedence,
the add and mul have a new syntax, old syntax is with `+` and `*` (left quotes mandatory), sum and prd are gone,
no more special treatment for encountered brace pairs {..} by the number scanner, a/b[N] notation can be used without use of braces (the N will end up as is in a \numexpr, it is not parsed by the \xintexpr-ession scanner),
in earlier releases, place holders for \xintNewExpr could either be denoted #1, #2, … or also $1, $2, … Only the usual # form is now accepted and the special cases previously treated via the second form are now managed via a protect(...) function.
xintfrac: \xintFloor and \xintCeil add a trailing /1[0] to their (integer) output. New \xintiFloor and \xintiCeil do not.
\xintnumexpr, \xintthenumexpr, \xintNewNumExpr: use \xintiexpr, \xinttheiexpr, \xintNewIExpr.\xintDivision, \xintQuo, \xintRem: use \xintiDivision, \xintiQuo, \xintiRem.
\xintMax, \xintMin, \xintAdd, \xintSub, \xintMul (xint): their usage without xintfrac is deprecated; use \xintiMax, \xintiMin, \xintiAdd, \xintiSub, \xintiMul.
the & and | as Boolean operators in xintexpr-essions are deprecated in favour of && and ||. The single letter operators might be assigned some other meaning in some later release (bitwise operations, perhaps). Do not use them.
new package xintcore has been split off xint. It contains the core arithmetic macros (it is loaded by LaTeX package bnumexpr),
neither xint nor xintfrac load xinttools. Only xintexpr does,
whenever some portion of code has been revised, often use has been made of the \xint_dothis and \xint_orthat pair of macros for expandably branching,
these tiny helpful macros, and a few others are in package xintkernel which contains also the catcode and loading order management code, initially inspired by code found in Heiko Oberdiek’s packages,
the source code, which was suppressed from xint.pdf in release 1.09n, is now compiled into a separate file sourcexint.pdf,
faster handling by \xintAdd, \xintSub, \xintMul, … of the case where one of the arguments is zero,
the \xintAdd and \xintSub macros from package xintfrac check if one of the denominators is a multiple of the other, and only if this is not the case do they multiply the denominators. But systematic reduction would be too costly,
this naturally will be also the case for the + and - operations in \xintexpr,
xint added \xintiiDivRound, \xintiiDivTrunc, \xintiiMod for rounded and truncated division of big integers (next to \xintiiQuo and \xintiiRem),
with xintfrac loaded, the \xintNum macro does \xintTTrunc (which is truncation to an integer, same as \xintiTrunc {0}),
added \xintMod to xintfrac for modulo operation with fractional numbers,
added \xintiFloor and \xintiCeil to xintfrac,
\xintiexpr, \xinttheiexpr admit an optional argument within brackets [d], they round the computation result (or results, if comma separated) to d digits after decimal mark, (the whole computation is done exactly, as in xintexpr),
\xintfloatexpr, \xintthefloatexpr similarly admit an optional argument which serves to keep only d digits of precision, getting rid of cumulated uncertainties in the last digits (the whole computation is done according to the precision set via \xintDigits),
\xinttheexpr and \xintthefloatexpr pretty-print if possible, the former removing unit denominator or [0] brackets, the latter avoiding scientific notation if decimal notation is practical,
the // does truncated division and /: is the associated modulo,
multi-character operators &&, ||, ==, <=, >=, !=, **,
multi-letter infix binary words 'and', 'or', 'xor', 'mod' (straight quotes mandatory),
functions even, odd,
\xintdefvar A3:=3.1415; for variable definitions (non expandable, naturally), usable in subsequent expressions; variable names may contain letters, digits, underscores. They should not start with a digit, the @ is reserved, and single lowercase and uppercase Latin letters are predefined to work as dummy variables (see next),
generation of comma separated lists a..b, a..[d]..b,
Python syntax-like list extractors [list][n:], [list][:n], [list][a:b] allowing negative indices, but no optional step argument, and [list][n] (n=0 for the number of items in the list),
functions first, last, reversed,
itemwise operations on comma separated lists a*[list], etc.., possible on both sides a*[list]^b, and obeying the same precedence rules as with numbers,
add and mul must use a dummy variable: add(x(x+1)(x-1), x=-10..10),
variable substitutions with subs: subs(subs(add(x^2+y^2,x=1..y),y=t),t=20),
sequence generation using seq with a dummy variable: seq(x^3, x=-10..10),
simple recursive lists with rseq, with @ given the last value, rseq(1;2@+1,i=1..10),
higher recursion with rrseq, @1, @2, @3, @4, and @@(n) for earlier values, up to n=K where K is the number of terms of the initial stretch rrseq(0,1;@1+@2,i=2..100),
iteration with iter which is like rrseq but outputs only the last K terms, where K was the number of initial terms,
inside seq, rseq, rrseq, iter, possibility to use omit, abort and break to control termination,
n++ potentially infinite index generation for seq, rseq, rrseq, and iter, it is advised to use abort or break(..) at some point,
the add, mul, seq, … are nestable,
\xintthecoords converts a comma separated list of an even number of items to the format expected by the TikZ coordinates syntax,
completely new version \xintNewExpr, protect function to handle external macros. The dollar sign $ for place holders is not accepted anymore, only the standard macro parameter #. Not all constructs are compatible with \xintNewExpr.
\xintZapFirstSpaces hence also \xintZapSpaces from package xinttools were buggy when used with an argument either empty or containing only space tokens.
\xintiiexpr did not strip leading zeroes, hence \xinttheiiexpr 001+1\relax did not obtain the expected result …
\xinttheexpr \xintiexpr 1.23\relax\relax should have produced 1, but it produced 1.23
the catcode of ; was not set at package launching time.
the \XINTinFloatPrd:csv macro name had a typo, hence prd was non-functional in \xintfloatexpr.
1.09n (2014/04/01)the user manual does not include by default the source code anymore: the \NoSourceCode toggle in file xint.tex has to be set to 0 before compilation to get source code inclusion (later release 1.1 made source code available as sourcexint.pdf).
bug fix (xinttools) in \XINT_nthelt_finish (this bug was introduced in 1.09i of 2013/12/18 and showed up when the index N was larger than the number of elements of the list).
1.09m (2014/02/26)new in xinttools: \xintKeep keeps the first N or last N elements of a list (sequence of braced items); \xintTrim cuts out either the first N or the last N elements from a list.
new in xintcfrac: \xintFGtoC finds the initial partial quotients common to two numbers or fractions f and g; \xintGGCFrac is a clone of \xintGCFrac which however does not assume that the coefficients of the generalized continued fraction are numeric quantities. Some other minor changes.
1.09kb (2014/02/13)bug fix (xintexpr): an aloof modification done by 1.09i to \xintNewExpr had resulted in a spurious trailing space present in the outputs of all macros created by \xintNewExpr, making nesting of such macros impossible.
bug fix (xinttools): \xintBreakFor and \xintBreakForAndDo were buggy when used in the last iteration of an \xintFor loop.
bug fix (xinttools): \xintSeq from 1.09k needed a \chardef which was missing from xinttools.sty, it was in xint.sty.
1.09k (2014/01/21)inside \xintexpr..\relax (and its variants) tacit multiplication is implied when a number or operand is followed directly with an opening parenthesis,
the " for denoting (arbitrarily big) hexadecimal numbers is recognized by \xintexpr and its variants (package xintbinhex is required); a fractional hexadecimal part introduced by a dot . is allowed.
re-organization of the first sections of the user manual.
bug fix (xinttools, xint, …): forgotten catcode check of " at loading time has been added.
1.09j (2014/01/09)(xint) the core division routines have been re-written for some (limited) efficiency gain, more pronounced for small divisors. As a result the computation of one thousand digits of π is close to three times faster than with earlier releases.
some various other small improvements, particularly in the power routines.
(xintfrac) a macro \xintXTrunc is designed to produce thousands or even tens of thousands of digits of the decimal expansion of a fraction. Although completely expandable it has its use limited to inside an \edef, \write, \message, . It can thus not be nested as argument to another package macro.
(xintexpr) the tacit multiplication done in \xintexpr..\relax on encountering a count register or variable, or a \numexpr, while scanning a (decimal) number, is extended to the case of a sub \xintexpr-ession.
\xintexpr can now be used in an \edef with no \xintthe prefix; it will execute completely the computation, and the error message about a missing \xintthe will be inhibited. Previously, in the absence of \xintthe, expansion could only be a full one (with \romannumeral-`0), not a complete one (with \edef). Note that this differs from the behavior of the non-expandable \numexpr: \the or \number (or \romannumeral) are needed not only to print but also to trigger the computation, whereas \xintthe is mandatory only for the printing step.
the default behavior of \xintAssign is changed, it now does not do any further expansion beyond the initial full-expansion which provided the list of items to be assigned to macros.
bug fix (xintfrac): 1.09i did an unexplainable change to \XINT_infloat_zero which broke the floating point routines for vanishing operands =:(((
bug fix: the 1.09i xint.ins file produced a buggy xint.tex file.
1.09i (2013/12/18)(xintexpr) \xintiiexpr is a variant of \xintexpr which is optimized to deal only with (long) integers, / does a euclidean quotient.
deprecated: \xintnumexpr, \xintthenumexpr, \xintNewNumExpr are renamed, respectively, \xintiexpr, \xinttheiexpr, \xintNewIExpr. The earlier denominations are kept but are to be removed at some point.
it is now possible within \xintexpr...\relax and its variants to use count, dimen, and skip registers or variables without explicit \the/\number: the parser inserts automatically \number and a tacit multiplication is implied when a register or variable immediately follows a number or fraction. Regarding dimensions and \number, see the further discussion in Dimensions.
(xintfrac) conditional \xintifOne; \xintifTrueFalse renamed to \xintifTrueAelseB; macros \xintTFrac (fractional part, mapped to function frac in \xintexpr-essions), \xintFloatE.
(xinttools) \xintAssign admits an optional argument to specify the expansion type to be used: [] (none, default), [o] (once), [oo] (twice), [f] (full), [e] (\edef),… to define the macros
xinttools defines \odef, \oodef, \fdef (if the names have already been assigned, it uses \xintoodef etc…). These tools are provided for the case one uses the package macros in a non-expandable context. \oodef expands twice the macro replacement text, and \fdef applies full expansion. They are useful in situations where one does not want a full \edef. \fdef appears to be faster than \oodef in almost all cases (with less than thousand digits in the result), and even faster than \edef for expanding the package macros when the result has a few dozens of digits. \oodef needs that expansion ends up in thousands of digits to become competitive with the other two.
some across the board slight efficiency improvement as a result of modifications of various types to fork macros and branching conditionals which are used internally.
bug fix (xint): \xintAND and \xintOR inserted a space token in some cases and did not expand as promised in two steps :-(( (bug dating back to 1.09a I think; this bug was without consequences when using & and | in \xintexpr-essions, it affected only the macro form).
bug fix (xintcfrac): \xintFtoCCv still ended fractions with the [0]’s which were supposed to have been removed since release 1.09b.
deprecated: \xintifTrueFalse, \xintifTrue; use \xintifTrueAelseB.
1.09h (2013/11/28)parts of the documentation have been re-written or re-organized, particularly the discussion of expansion issues and of input and output formats.
the expansion types of macro arguments are documented in the margin of the macro descriptions, with conventions mainly taken over from those in the LaTeX3 documentation.
a dependency of xinttools on xint (inside \xintSeq) has been removed.
(xintgcd) \xintTypesetEuclideAlgorithm and \xintTypesetBezoutAlgorithm have been slightly modified (regarding indentation).
(xint) macros xintiSum and xintiPrd are renamed to \xintiiSum and \xintiiPrd.
(xinttools) a count register used in 1.09g in the \xintFor loops for parsing purposes has been removed and replaced by use of a \numexpr.
the few uses of \loop have been replaced by \xintloop/\xintiloop.
all macros of xinttools for which it makes sense are now declared \long.
1.09g (2013/11/22)a package xinttools is detached from xint, to make tools such as \xintFor, \xintApplyUnbraced, and \xintiloop available without the xint overhead.
expandable nestable loops \xintloop and \xintiloop.
bugfix: \xintFor and \xintFor* do not modify anymore the value of \count 255.
1.09f (2013/11/04)(xint) \xintZapFirstSpaces, \xintZapLastSpaces, \xintZapSpaces, \xintZapSpacesB, for expandably stripping away leading and/or ending spaces.
\xintCSVtoList by default uses \xintZapSpacesB to strip away spaces around commas (or at the start and end of the comma separated list).
also the \xintFor loop will strip out all spaces around commas and at the start and the end of its list argument; and similarly for \xintForpair, \xintForthree, \xintForfour.
\xintFor et al. accept all macro parameters from #1 to #9.
for reasons of inner coherence some macros previously with one extra i in their names (e.g. \xintiMON) now have a doubled ii (\xintiiMON) to indicate that they skip the overhead of parsing their inputs via \xintNum. Macros with a single i such as \xintiAdd are those which maintain the non-xintfrac output format for big integers, but do parse their inputs via \xintNum (since release 1.09a). They too may have doubled-i variants for matters of programming optimization when working only with (big) integers and not fractions or decimal numbers.
1.09e (2013/10/29)(xint) \xintintegers, \xintdimensions, \xintrationals for infinite \xintFor loops, interrupted with \xintBreakFor and \xintBreakForAndDo.
\xintifForFirst, \xintifForLast for the \xintFor and \xintFor* loops,
the \xintFor and xintFor* loops are now \long, the replacement text and the items may contain explicit \par’s.
conditionals \xintifCmp, \xintifInt, \xintifOdd.
bug fix (xint): the \xintFor loop (not \xintFor*) did not correctly detect an empty list.
bug fix (xint): \xintiSqrt {0} crashed. :-((
the documentation has been enriched with various additional examples, such as the the quick sort algorithm illustrated or the various ways of computing prime numbers.
the documentation explains with more details various expansion related issues, particularly in relation to conditionals.
1.09d (2013/10/22)bug fix (xint): \xintFor* is modified to gracefully handle a space token (or more than one) located at the very end of its list argument (as the space before \do in \xintFor* #1 in {{a}{b}{c}<space>} \do {stuff}; spaces at other locations were already harmless). Furthermore this new version f-expands the un-braced list items. After \def\x{{1}{2}} and \def\y{{a}\x {b}{c}\x }, \y will appear to \xintFor* exactly as if it had been defined as \def\y{{a}{1}{2}{b}{c}{1}{2}}.
same bug fix for \xintApplyInline.
1.09c (2013/10/09)(xintexpr) added bool and togl to the \xintexpr syntax; also added \xintboolexpr and \xintifboolexpr.
added \xintNewNumExpr.
the factorial ! and branching ?, :, operators (in \xintexpr...\relax) have now less precedence than a function name located just before,
(xint) \xintFor is a new type of loop, whose replacement text inserts the comma separated values or list items via macro parameters, rather than encapsulated in macros; the loops are nestable up to four levels (nine levels since 1.09f) and their replacement texts are allowed to close groups as happens with the tabulation in alignments,
\xintForpair, \xintForthree, \xintForfour are experimental variants of \xintFor,
\xintApplyInline has been enhanced in order to be usable for generating rows (partially or completely) in an alignment,
command \xintSeq to generate (expandably) arithmetic sequences of (short) integers,
again various improvements and changes in the documentation.
1.09b (2013/10/03)various improvements in the documentation,
more economical catcode management and re-loading handling,
removal of all those [0]’s previously forcefully added at the end of fractions by various macros of xintcfrac,
\xintNthElt with a negative index returns from the tail of the list,
macro \xintPRaw to have something like what \xintFrac does in math mode; i.e. a \xintRaw which does not print the denominator if it is one.
1.09a (2013/09/24)(xintexpr) \xintexpr..\relax and \xintfloatexpr..\relax admit functions in their syntax, with comma separated values as arguments, among them reduce, sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm, max, min, sum, prd, add, mul, not, all, any, xor.
comparison (<, >, =) and logical (|, &) operators.
the command \xintthe which converts \xintexpressions into printable format (like \the with \numexpr) is more efficient, for example one can do \xintthe\x if \x was defined to be an \xintexpr..\relax:
\def\x{\xintexpr 3^57\relax}
\def\y{\xintexpr \x^(-2)\relax}
\def\z{\xintexpr \y-3^-114\relax}
\xintthe\z\xintnumexpr .. \relax (now renamed \xintiexpr) is \xintexpr round( .. ) \relax.
\xintNewExpr now works with the standard macro parameter character #.
both regular \xintexpr-essions and commands defined by \xintNewExpr will work with comma separated lists of expressions,
commands \xintFloor, \xintCeil, \xintMaxof, \xintMinof (package xintfrac), \xintGCDof, \xintLCM, \xintLCMof (package xintgcd), \xintifLt, \xintifGt, \xintifSgn, \xintANDof, …
The arithmetic macros from package xint now filter their operands via \xintNum which means that they may use directly count registers and \numexpr-essions without having to prefix them by \the. This is thus similar to the situation holding previously already when xintfrac was loaded.
a bug (xintfrac) introduced in 1.08b made \xintCmp crash when one of its arguments was zero. :-((
1.08b (2013/06/14)(xintexpr) Correction of a problem with spaces inside \xintexpr-essions.
(xintfrac) Additional improvements to the handling of floating point numbers.
section Use of count registers documenting how count registers may be directly used in arguments to the macros of xintfrac.
1.08a (2013/06/11)(xintfrac) Improved efficiency of the basic conversion from exact fractions to floating point numbers, with ensuing speed gains especially for the power function macros \xintFloatPow and \xintFloatPower,
Better management by \xintCmp, \xintMax, \xintMin and \xintGeq of inputs having big powers of ten in them.
Macros for floating point numbers added to the xintseries package.
1.08 (2013/06/07)(xint and xintfrac) Macros for extraction of square roots, for floating point numbers (\xintFloatSqrt), and integers (\xintiSqrt).
new package xintbinhex providing conversion routines to and from binary and hexadecimal bases.
1.07 (2013/05/25)The xintexpr package is a new core constituent (which loads automatically xintfrac and xint) and implements the expandable expanding parser
\xintexpr . . . \relax,and its variant
\xintfloatexpr . . . \relaxallowing on input formulas using the infix operators +, -, *, /, and ^, and arbitrary levels of parenthesizing. Within a float expression the operations are executed according to the current value set by \xintDigits. Within an \xintexpr-ession the binary operators are computed exactly.
To write the \xintexpr parser I benefited from the commented source of the l3fp parser; the \xintexpr parser has its own features and peculiarities. See its documentation.
The floating point precision D is set (this is a local assignment to a \mathchar variable) with \xintDigits := D; and queried with \xinttheDigits. It may be set to anything up to 32767.1 The macro incarnations of the binary operations admit an optional argument which will replace pointwise D; this argument may exceed the 32767 bound.
The xintfrac macros now accept numbers written in scientific notation, the \xintFloat command serves to output its argument with a given number D of significant figures. The value of D is either given as optional argument to \xintFloat or set with \xintDigits := D;. The default value is 16.
1.06b (2013/05/14)1.06 (2013/05/07)Some code improvements, particularly for macros of xint doing loops.
New utilities in xint for expandable manipulations of lists:
\xintNthElt, \xintCSVtoList, \xintRevWithBracesThe macros did only a double expansion of their arguments. They now fully expand them (using \romannumeral-`0). Furthermore, in the case of arguments constrained to obey the TeX bounds they will be inserted inside a \numexpr..\relax, hence completely expanded, one may use count registers, even infix arithmetic operations, etc…
1.05 (2013/05/01)Minor changes and additions to xintfrac and xintcfrac.
1.04 (2013/04/25)New component xintcfrac devoted to continued fractions.
xint: faster division.
xint: added expandable macros \xintListWithSep and \xintApply to handle token lists.
xintfrac: added \xintRound.
xintseries has a new implementation of \xintPowerSeries based on a Horner scheme, and new macro \xintRationalSeries. Both to help deal with the denominator buildup plague.
tex xint.dtx extracts style files (no need for a xint.ins).
Bug fix (xintfrac): \xintIrr {0} crashed.
1.03 (2013/04/14)New modules xintfrac (expandable operations on fractions) and xintseries (expandable partial sums with xint package).
Slightly improved division and faster multiplication (the best ordering of the arguments is chosen automatically).
Added illustration of Machin algorithm to the documentation.
1.0 (2013/03/28)Initial announcement:
The xint package implements with expandable TeX macros the basic arithmetic operations of addition, subtraction, multiplication and division, as applied to arbitrarily long numbers represented as chains of digits with an optional minus sign.
The xintgcd package provides implementations of the Euclidean algorithm and of its typesetting.
The packages may be used with Plain and with LaTeX.
but values higher than 100 or 200 will presumably give too slow evaluations.↩