Source:  xint.dtx (v1.1a, 2014/11/07)
Author:  Jean-Francois Burnol
Info:    Expandable operations on big integers, decimals, fractions
License: LPPL 1.3c or later

## 1.1a (2014/11/07)

• fixed a bug which prevented \xintNewExpr to produce correctly working macros from a comma separated replacement text.

• new \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 v1.1, 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)

bug fixes
• \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.

breaking changes
• in \xintiiexpr, / does rounded division, rather than the Euclidean division (for positive arguments, this is truncated division). The new // 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),

• although & and | are still available as Boolean operators the use of && and || is strongly recommended. The single letter operators might be assigned some other meaning in later releases (bitwise operations, perhaps). Do not use them.

There are many novelties, most to be found in package xintexpr. But first the other changes.

• new package xintcore has been split off xint. It contains the core arithmetic macros. It is loaded by 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,

• new macros \xintiiDivRound, \xintiiDivTrunc and \xintiiMod for rounded and truncated division of big integers (now in xintcore), alongside the earlier \xintiiQuo and \xintiiRem,

• with xintfrac loaded, the \xintNum macro does \xintTTrunc (which is truncation to an integer, same as \xintiTrunc {0}),

• new macro \xintMod in xintfrac for modulo operation with fractional numbers,

• \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, an 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. Not all constructs are compatible with \xintNewExpr.

## 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.

• 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 new 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 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.

• \xintnumexpr, \xintthenumexpr, \xintNewNumExpr are renamed, respectively, \xintiexpr, \xinttheiexpr, \xintNewIExpr. The earlier denominations are kept but 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) new conditional \xintifOne; \xintifTrueFalse renamed to \xintifTrueAelseB; new 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, particularly \oodef which expands twice the macro replacement text and is thus a faster alternative to \edef taking into account that the xint bundle macros expand already completely in only two steps. This can be significant when repeatedly making \def-initions expanding to hundreds of digits.

• 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.

## 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.

• new 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) new \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) new \xintintegers, \xintdimensions, \xintrationals for infinite \xintFor loops, interrupted with \xintBreakFor and \xintBreakForAndDo.

• new \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.

• new 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 (now \xintNewIExpr and \xintNewBoolExpr),

• 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,

• new 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,

• 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,

• new 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,

• new 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.

• new 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 . . . \relax

allowing 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 of \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)

• Minor code and documentation improvements. Everywhere in the source code, a more modern underscore has replaced the @ sign.

## 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, \xintRevWithBraces
• The 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.

• bug fix (xintfrac): \xintIrr {0} crashed.

• faster division routine in xint, new macros to deal expandably with token lists.

• \xintRound added.

• 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).

## 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.

1. but values higher than 100 or 200 will presumably give too slow evaluations.