diff options
19 files changed, 6969 insertions, 6123 deletions
diff --git a/Master/texmf-dist/doc/generic/xint/CHANGES.html b/Master/texmf-dist/doc/generic/xint/CHANGES.html index 0df6a146b0c..d7f74b2e15d 100644 --- a/Master/texmf-dist/doc/generic/xint/CHANGES.html +++ b/Master/texmf-dist/doc/generic/xint/CHANGES.html @@ -4,7 +4,7 @@ <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta http-equiv="Content-Style-Type" content="text/css" /> <meta name="generator" content="pandoc" /> - <meta name="author" content="xint v1.1c" /> + <meta name="author" content="xint v1.2" /> <title>CHANGE LOG</title> <style type="text/css">code{white-space: pre;}</style> <style type="text/css"> @@ -17,15 +17,16 @@ <body> <div id="header"> <h1 class="title">CHANGE LOG</h1> -<h2 class="author">xint v1.1c</h2> -<h3 class="date">2015/09/12</h3> +<h2 class="author">xint v1.2</h2> +<h3 class="date">2015/10/10</h3> </div> <div id="TOC"> <ul> +<li><a href="#section"><code>1.2 (2015/10/10)</code></a></li> <li><a href="#c-20150912"><code>1.1c (2015/09/12)</code></a></li> <li><a href="#b-20150831"><code>1.1b (2015/08/31)</code></a></li> <li><a href="#a-20141107"><code>1.1a (2014/11/07)</code></a></li> -<li><a href="#section"><code>1.1 (2014/10/28)</code></a></li> +<li><a href="#section-1"><code>1.1 (2014/10/28)</code></a></li> <li><a href="#n-20140401"><code>1.09n (2014/04/01)</code></a></li> <li><a href="#m-20140226"><code>1.09m (2014/02/26)</code></a></li> <li><a href="#kb-20140213"><code>1.09kb (2014/02/13)</code></a></li> @@ -42,20 +43,34 @@ <li><a href="#a-20130924"><code>1.09a (2013/09/24)</code></a></li> <li><a href="#b-20130614"><code>1.08b (2013/06/14)</code></a></li> <li><a href="#a-20130611"><code>1.08a (2013/06/11)</code></a></li> -<li><a href="#section-1"><code>1.08 (2013/06/07)</code></a></li> -<li><a href="#section-2"><code>1.07 (2013/05/25)</code></a></li> +<li><a href="#section-2"><code>1.08 (2013/06/07)</code></a></li> +<li><a href="#section-3"><code>1.07 (2013/05/25)</code></a></li> <li><a href="#b-20130514"><code>1.06b (2013/05/14)</code></a></li> -<li><a href="#section-3"><code>1.06 (2013/05/07)</code></a></li> -<li><a href="#section-4"><code>1.05 (2013/05/01)</code></a></li> -<li><a href="#section-5"><code>1.04 (2013/04/25)</code></a></li> -<li><a href="#section-6"><code>1.03 (2013/04/14)</code></a></li> -<li><a href="#section-7"><code>1.0 (2013/03/28)</code></a></li> +<li><a href="#section-4"><code>1.06 (2013/05/07)</code></a></li> +<li><a href="#section-5"><code>1.05 (2013/05/01)</code></a></li> +<li><a href="#section-6"><code>1.04 (2013/04/25)</code></a></li> +<li><a href="#section-7"><code>1.03 (2013/04/14)</code></a></li> +<li><a href="#section-8"><code>1.0 (2013/03/28)</code></a></li> </ul> </div> -<pre><code>Source: xint.dtx v1.1c 2015/09/12 (doc 2015/09/17) +<pre><code>Source: xint.dtx v1.2 2015/10/10 (doc 2015/10/10) Author: Jean-Francois Burnol Info: Expandable operations on big integers, decimals, fractions License: LPPL 1.3c</code></pre> +<h2 id="section"><code>1.2 (2015/10/10)</code></h2> +<ul> +<li><p>the basic arithmetic implemented in <strong>xintcore</strong> has been entirely rewritten. The mathematics remains the elementary school one, but the <code>TeX</code> 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.</p></li> +<li><p>the inputs must have less than 19959 digits. But computations with thousands of digits take time.</p></li> +<li><p>a previously standing limitation of <code>\xintexpr</code>, <code>\xintiiexpr</code>, and of <code>\xintfloatexpr</code> to numbers of less than 5000 digits has been lifted.</p></li> +<li><p>a <em>qint</em> 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.</p></li> +<li><p>new <code>\xintFloatFac</code> macro for computing the factorials of integers as floating point numbers to a given precision. The <code>!</code> postfix operator inside <code>\xintfloatexpr</code> maps to this new macro rather than to the exact factorial as used by <code>\xintexpr</code> and <code>xintiiexpr</code>.</p></li> +<li><p>the macros <code>\xintAdd</code>, <code>\xintSub</code>, ..., now require package <strong>xintfrac</strong>. With only <strong>xintcore</strong> or <strong>xint</strong> loaded, one <em>must</em> use <code>\xintiiAdd</code>, <code>\xintiiSub</code>, ..., or <code>\xintiAdd</code>, <code>\xintiSub</code>, etc...</p></li> +<li><p>there is more flexibility in the parsing done by the macros from <strong>xintfrac</strong> on fractional input: the decimal parts of both the numerator and the denominator may arise from a separate expansion via <code>\romannumeral-`0</code>. Also the strict <code>A/B[N]</code> format is a bit relaxed: <code>N</code> may be empty or anything understood by <code>\numexpr</code>.</p></li> +<li><p>on the other hand an isolated dot <code>.</code> 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 <strong>xintfrac</strong>.</p></li> +<li><p>added <code>\ht</code>, <code>\dp</code>, <code>\wd</code>, <code>\fontcharht</code>, etc... to the tokens recognized by the parsers and expanded by <code>\number</code>.</p></li> +<li><p>an obscure bug in package <strong>xintkernel</strong> has been fixed, regarding the sanitization of catcodes: under certain circumstances (which could not occur in a normal <code>LaTeX</code> context), unusual catcodes could end up being propagated to the external world.</p></li> +<li><p>an effort at randomly shuffling around various pieces of the documentation has been done.</p></li> +</ul> <h2 id="c-20150912"><code>1.1c (2015/09/12)</code></h2> <ul> <li><p>bugfix regarding macro <code>\xintAssign</code> from <strong>xinttools</strong> which did not behave correctly in some circumstances (if there was a space before <code>\to</code>, in particular).</p></li> @@ -76,7 +91,7 @@ License: LPPL 1.3c</code></pre> <li><p>added the previously mentioned <code>ii</code> macros, and some others from <code>v1.1</code>, to the user manual. But their main usage is internal to <code>\xintiiexpr</code>, to skip unnecessary overheads.</p></li> <li><p>various typographical fixes throughout the documentation, and a bit of clean up of the code comments. Improved <code>\Factors</code> example of nested <code>subs</code>, <code>rseq</code>, <code>iter</code> in <code>\xintiiexpr</code>.</p></li> </ul> -<h2 id="section"><code>1.1 (2014/10/28)</code></h2> +<h2 id="section-1"><code>1.1 (2014/10/28)</code></h2> <dl> <dt>bug fixes</dt> <dd><ul> @@ -177,7 +192,7 @@ License: LPPL 1.3c</code></pre> <li><p>it is now possible within <code>\xintexpr...\relax</code> and its variants to use count, dimen, and skip registers or variables without explicit <code>\the/\number</code>: the parser inserts automatically <code>\number</code> and a tacit multiplication is implied when a register or variable immediately follows a number or fraction. Regarding dimensions and <code>\number</code>, see the further discussion in <em>Dimensions</em>.</p></li> <li><p>(<strong>xintfrac</strong>) new conditional <code>\xintifOne</code>; <code>\xintifTrueFalse</code> renamed to <code>\xintifTrueAelseB</code>; new macros <code>\xintTFrac</code> (<code>fractional part</code>, mapped to function <code>frac</code> in <code>\xintexpr</code>-essions), <code>\xintFloatE</code>.</p></li> <li><p>(<strong>xinttools</strong>) <code>\xintAssign</code> admits an optional argument to specify the expansion type to be used: <code>[]</code> (none, default), <code>[o]</code> (once), <code>[oo]</code> (twice), <code>[f]</code> (full), <code>[e]</code> (<code>\edef</code>),... to define the macros</p></li> -<li><p><strong>xinttools</strong> defines <code>\odef</code>, <code>\oodef</code>, <code>\fdef</code> (if the names have already been assigned, it uses <code>\xintoodef</code> etc...). These tools are provided for the case one uses the package macros in a non-expandable context, particularly <code>\oodef</code> which expands twice the macro replacement text and is thus a faster alternative to <code>\edef</code> taking into account that the <strong>xint</strong> bundle macros expand already completely in only two steps. This can be significant when repeatedly making <code>\def</code>-initions expanding to hundreds of digits.</p></li> +<li><p><strong>xinttools</strong> defines <code>\odef</code>, <code>\oodef</code>, <code>\fdef</code> (if the names have already been assigned, it uses <code>\xintoodef</code> etc...). These tools are provided for the case one uses the package macros in a non-expandable context. <code>\oodef</code> expands twice the macro replacement text, and <code>\fdef</code> applies full expansion. They are useful in situations where one does not want a full <code>\edef</code>. <code>\fdef</code> appears to be faster than <code>\oodef</code> in almost all cases (with less than thousand digits in the result), and even faster than <code>\edef</code> for expanding the package macros when the result has a few dozens of digits. <code>\oodef</code> needs that expansion ends up in thousands of digits to become competitive with the other two.</p></li> <li><p>some across the board slight efficiency improvement as a result of modifications of various types to <em>fork macros</em> and <em>branching conditionals</em> which are used internally.</p></li> <li><p>bug fix (<strong>xint</strong>): <code>\xintAND</code> and <code>\xintOR</code> inserted a space token in some cases and did not expand as promised in two steps <code>:-((</code> (bug dating back to <code>1.09a</code> I think; this bug was without consequences when using <code>&</code> and <code>|</code> in <code>\xintexpr-essions</code>, it affected only the macro form).</p></li> <li><p>bug fix (<strong>xintcfrac</strong>): <code>\xintFtoCCv</code> still ended fractions with the <code>[0]</code>'s which were supposed to have been removed since release <code>1.09b</code>.</p></li> @@ -270,12 +285,12 @@ License: LPPL 1.3c</code></pre> <li><p>Better management by <code>\xintCmp</code>, <code>\xintMax</code>, <code>\xintMin</code> and <code>\xintGeq</code> of inputs having big powers of ten in them.</p></li> <li><p>Macros for floating point numbers added to the <strong>xintseries</strong> package.</p></li> </ul> -<h2 id="section-1"><code>1.08 (2013/06/07)</code></h2> +<h2 id="section-2"><code>1.08 (2013/06/07)</code></h2> <ul> <li><p>(<strong>xint</strong> and <strong>xintfrac</strong>) Macros for extraction of square roots, for floating point numbers (<code>\xintFloatSqrt</code>), and integers (<code>\xintiSqrt</code>).</p></li> <li><p>New package <strong>xintbinhex</strong> providing <em>conversion routines</em> to and from binary and hexadecimal bases.</p></li> </ul> -<h2 id="section-2"><code>1.07 (2013/05/25)</code></h2> +<h2 id="section-3"><code>1.07 (2013/05/25)</code></h2> <ul> <li><p>The <strong>xintexpr</strong> package is a new core constituent (which loads automatically <strong>xintfrac</strong> and <strong>xint</strong>) and implements the expandable expanding parser</p> <pre><code>\xintexpr . . . \relax,</code></pre> @@ -290,16 +305,16 @@ License: LPPL 1.3c</code></pre> <ul> <li>Minor code and documentation improvements. Everywhere in the source code, a more modern underscore has replaced the @ sign.</li> </ul> -<h2 id="section-3"><code>1.06 (2013/05/07)</code></h2> +<h2 id="section-4"><code>1.06 (2013/05/07)</code></h2> <ul> <li><p>Some code improvements, particularly for macros of <strong>xint</strong> doing loops.</p></li> <li><p>New utilities in <strong>xint</strong> for expandable manipulations of lists:</p> <pre><code>\xintNthElt, \xintCSVtoList, \xintRevWithBraces</code></pre></li> <li><p>The macros did only a double expansion of their arguments. They now fully expand them (using <code>\romannumeral-`0</code>). Furthermore, in the case of arguments constrained to obey the TeX bounds they will be inserted inside a <code>\numexpr..\relax</code>, hence completely expanded, one may use count registers, even infix arithmetic operations, etc...</p></li> </ul> -<h2 id="section-4"><code>1.05 (2013/05/01)</code></h2> +<h2 id="section-5"><code>1.05 (2013/05/01)</code></h2> <p>Minor changes and additions to <strong>xintfrac</strong> and <strong>xintcfrac</strong>.</p> -<h2 id="section-5"><code>1.04 (2013/04/25)</code></h2> +<h2 id="section-6"><code>1.04 (2013/04/25)</code></h2> <ul> <li><p>New component <strong>xintcfrac</strong> devoted to continued fractions.</p></li> <li><p>bug fix (<strong>xintfrac</strong>): <code>\xintIrr {0}</code> crashed.</p></li> @@ -308,13 +323,13 @@ License: LPPL 1.3c</code></pre> <li><p><strong>xintseries</strong> has a new implementation of <code>\xintPowerSeries</code> based on a Horner scheme, and new macro <code>\xintRationalSeries</code>. Both to help deal with the <em>denominator buildup</em> plague.</p></li> <li><p><code>tex xint.dtx</code> extracts style files (no need for a <code>xint.ins</code>).</p></li> </ul> -<h2 id="section-6"><code>1.03 (2013/04/14)</code></h2> +<h2 id="section-7"><code>1.03 (2013/04/14)</code></h2> <ul> <li><p>new modules <strong>xintfrac</strong> (expandable operations on fractions) and <strong>xintseries</strong> (expandable partial sums with xint package).</p></li> <li><p>slightly improved division and faster multiplication (the best ordering of the arguments is chosen automatically).</p></li> <li><p>added illustration of Machin algorithm to the documentation.</p></li> </ul> -<h2 id="section-7"><code>1.0 (2013/03/28)</code></h2> +<h2 id="section-8"><code>1.0 (2013/03/28)</code></h2> <p>Initial announcement:</p> <blockquote> <p>The <strong>xint</strong> 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.</p> diff --git a/Master/texmf-dist/doc/generic/xint/CHANGES.pdf b/Master/texmf-dist/doc/generic/xint/CHANGES.pdf Binary files differindex e53c366fc0b..3528d49a696 100644 --- a/Master/texmf-dist/doc/generic/xint/CHANGES.pdf +++ b/Master/texmf-dist/doc/generic/xint/CHANGES.pdf diff --git a/Master/texmf-dist/doc/generic/xint/README b/Master/texmf-dist/doc/generic/xint/README index 3aff524f028..81b28347704 100644 --- a/Master/texmf-dist/doc/generic/xint/README +++ b/Master/texmf-dist/doc/generic/xint/README @@ -1,4 +1,4 @@ - Source: xint.dtx v1.1c 2015/09/12 (doc 2015/09/17) + Source: xint.dtx v1.2 2015/10/10 (doc 2015/10/10) Author: Jean-Francois Burnol Info: Expandable operations on big integers, decimals, fractions License: LPPL 1.3c @@ -12,6 +12,33 @@ as sourcexint.pdf. +AIM + + +The basic aim is provide _expandable_ computations on big integers, and +also big fractions. For example + + \xinttheexpr reduce(37189719/183618963+11390170/17310720)^17\relax + +will evaluate exactly the fraction (the result has 462 characters +including the fraction slash). One can also work with dummy variables. +For example + + \xinttheexpr mul(add(x(x+1)(x+2), x=y..y+15), y=171286,98762,9296)\relax + +evaluates to 15979066346135829902328007959448563667099190784. + +It is possible to use the package with Plain as well as with LaTeX. + +Sub-units xintcore, xint and xintfrac provide the underlying macros, and +xintexpr loads all of them and provides expandable parsers allowing +computations such as the above (and more). A more light-weight package +bnumexpr (LaTeX only) loads only xintcore and provides a parser which +handles only big integers, the four operations, the power operation and +the factorial (v1.2). + + + USAGE @@ -42,7 +69,7 @@ their standard catcodes. xintcore.sty and xinttools.sty both import xintkernel.sty which has the catcode handler and package identifier and defines a few utilities such -as \oodef or \xint_dothis/\xint_orthat. +as \oodef, \fdef, or \xint_dothis/\xint_orthat. diff --git a/Master/texmf-dist/doc/generic/xint/README.html b/Master/texmf-dist/doc/generic/xint/README.html index 2a14f30f696..2dd4044f9c3 100644 --- a/Master/texmf-dist/doc/generic/xint/README.html +++ b/Master/texmf-dist/doc/generic/xint/README.html @@ -4,7 +4,7 @@ <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta http-equiv="Content-Style-Type" content="text/css" /> <meta name="generator" content="pandoc" /> - <meta name="author" content="xint v1.1c" /> + <meta name="author" content="xint v1.2" /> <title>README</title> <style type="text/css">code{white-space: pre;}</style> <style type="text/css"> @@ -17,11 +17,12 @@ <body> <div id="header"> <h1 class="title">README</h1> -<h2 class="author">xint v1.1c</h2> -<h3 class="date">2015/09/12</h3> +<h2 class="author">xint v1.2</h2> +<h3 class="date">2015/10/10</h3> </div> <div id="TOC"> <ul> +<li><a href="#aim">Aim</a></li> <li><a href="#usage">Usage</a><ul> <li><a href="#with-latex">With LaTeX</a></li> <li><a href="#with-tex">With TeX</a></li> @@ -35,13 +36,21 @@ <li><a href="#license">License</a></li> </ul> </div> -<pre><code>Source: xint.dtx v1.1c 2015/09/12 (doc 2015/09/17) +<pre><code>Source: xint.dtx v1.2 2015/10/10 (doc 2015/10/10) Author: Jean-Francois Burnol Info: Expandable operations on big integers, decimals, fractions License: LPPL 1.3c</code></pre> <p>This <code>README</code> is also available as <code>README.pdf</code> and <code>README.html</code>.</p> <p>Change log is to be found in <code>CHANGES.pdf</code> or <code>CHANGES.html</code>.</p> <p>The user manual is <code>xint.pdf</code>, and the commented source code is available as <code>sourcexint.pdf</code>.</p> +<h1 id="aim">Aim</h1> +<p>The basic aim is provide <em>expandable</em> computations on big integers, and also big fractions. For example</p> +<pre><code>\xinttheexpr reduce(37189719/183618963+11390170/17310720)^17\relax</code></pre> +<p>will evaluate exactly the fraction (the result has 462 characters including the fraction slash). One can also work with dummy variables. For example</p> +<pre><code>\xinttheexpr mul(add(x(x+1)(x+2), x=y..y+15), y=171286,98762,9296)\relax</code></pre> +<p>evaluates to <code>15979066346135829902328007959448563667099190784</code>.</p> +<p>It is possible to use the package with Plain as well as with LaTeX.</p> +<p>Sub-units <code>xintcore</code>, <code>xint</code> and <code>xintfrac</code> provide the underlying macros, and <code>xintexpr</code> loads all of them and provides expandable parsers allowing computations such as the above (and more). A more light-weight package <a href="http://www.ctan.org/pkg/bnumexpr">bnumexpr</a> (LaTeX only) loads only <code>xintcore</code> and provides a parser which handles only big integers, the four operations, the power operation and the factorial (v1.2).</p> <h1 id="usage">Usage</h1> <h2 id="with-latex">With LaTeX</h2> <pre><code>\usepackage{xint} % expandable arithmetic with big integers @@ -52,7 +61,7 @@ License: LPPL 1.3c</code></pre> <p>One does for example:</p> <pre><code>\input xintexpr.sty</code></pre> <p>Again, all dependencies are handled automatically. The packages may be loaded in any catcode context such that letters, digits, <code>\</code> and <code>%</code> have their standard catcodes.</p> -<p><code>xintcore.sty</code> and <code>xinttools.sty</code> both import <code>xintkernel.sty</code> which has the catcode handler and package identifier and defines a few utilities such as <code>\oodef</code> or <code>\xint_dothis/\xint_orthat</code>.</p> +<p><code>xintcore.sty</code> and <code>xinttools.sty</code> both import <code>xintkernel.sty</code> which has the catcode handler and package identifier and defines a few utilities such as <code>\oodef</code>, <code>\fdef</code>, or <code>\xint_dothis/\xint_orthat</code>.</p> <h1 id="installation">Installation</h1> <h2 id="method-a-using-the-package-manager-of-your-tex-distribution">Method A: using the package manager of your TeX distribution</h2> <p><code>xint</code> is included in <a href="http://tug.org/texlive/">TeXLive</a> (hence also <a href="http://tug.org/mactex/">MacTeX</a>) and <a href="http://www.miktex.org/">MikTeX</a>.</p> diff --git a/Master/texmf-dist/doc/generic/xint/README.pdf b/Master/texmf-dist/doc/generic/xint/README.pdf Binary files differindex 6c187cd4ac3..6f19f2b2bad 100644 --- a/Master/texmf-dist/doc/generic/xint/README.pdf +++ b/Master/texmf-dist/doc/generic/xint/README.pdf diff --git a/Master/texmf-dist/doc/generic/xint/sourcexint.pdf b/Master/texmf-dist/doc/generic/xint/sourcexint.pdf Binary files differindex 5a0b24ed2ea..2f37c53c925 100644 --- a/Master/texmf-dist/doc/generic/xint/sourcexint.pdf +++ b/Master/texmf-dist/doc/generic/xint/sourcexint.pdf diff --git a/Master/texmf-dist/doc/generic/xint/xint.pdf b/Master/texmf-dist/doc/generic/xint/xint.pdf Binary files differindex 57951acbcc7..01ddc80bf62 100644 --- a/Master/texmf-dist/doc/generic/xint/xint.pdf +++ b/Master/texmf-dist/doc/generic/xint/xint.pdf diff --git a/Master/texmf-dist/source/generic/xint/xint.dtx b/Master/texmf-dist/source/generic/xint/xint.dtx index 3a3a4bc6fe3..553962e7b0a 100644 --- a/Master/texmf-dist/source/generic/xint/xint.dtx +++ b/Master/texmf-dist/source/generic/xint/xint.dtx @@ -1,25 +1,25 @@ % -*- coding: iso-latin-1; time-stamp-format: "%02d-%02m-%:y at %02H:%02M:%02S %Z" -*- % N.B.: this dtx file does NOT use \DocInput, only docstrip. The user manual -% latex source is NOT prefixed with %'s +% latex source is NOT prefixed with percent characters. %<*dtx> -\def\xintdtxtimestamp {Time-stamp: <17-09-2015 at 11:09:20 CEST>} +\def\xintdtxtimestamp {Time-stamp: <10-10-2015 at 23:42:39 CEST>} %</dtx> %<*drv> %% --------------------------------------------------------------- -\def\xintdocdate {2015/09/17} -\def\xintbndldate{2015/09/12} -\def\xintbndlversion {1.1c} +\def\xintdocdate {2015/10/10} +\def\xintbndldate{2015/10/10} +\def\xintbndlversion {1.2} %</drv> %<*dtx> \iffalse % meta-comment %</dtx> %<readme>% README %<changes>% CHANGE LOG -%<readme|changes>% xint v1.1c -%<readme|changes>% 2015/09/12 +%<readme|changes>% xint v1.2 +%<readme|changes>% 2015/10/10 %<*readme|changes> - Source: xint.dtx v1.1c 2015/09/12 (doc 2015/09/17) + Source: xint.dtx v1.2 2015/10/10 (doc 2015/10/10) Author: Jean-Francois Burnol Info: Expandable operations on big integers, decimals, fractions License: LPPL 1.3c @@ -27,7 +27,7 @@ %</readme|changes> %<*!readme&!changes&!dohtmlsh&!dopdfsh&!makefile> %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %<xintkernel>%% xintkernel: Paraphernalia for the xint packages %<xinttools>%% xinttools: Expandable and non-expandable utilities @@ -49,6 +49,34 @@ Change log is to be found in `CHANGES.pdf` or `CHANGES.html`. The user manual is `xint.pdf`, and the commented source code is available as `sourcexint.pdf`. + +Aim +=== + +The basic aim is provide *expandable* computations on big integers, +and also big fractions. For example + + \xinttheexpr reduce(37189719/183618963+11390170/17310720)^17\relax + +will evaluate exactly the fraction (the result has 462 characters +including the fraction slash). One can also work with dummy +variables. For example + + \xinttheexpr mul(add(x(x+1)(x+2), x=y..y+15), y=171286,98762,9296)\relax + +evaluates to `15979066346135829902328007959448563667099190784`. + +It is possible to use the package with Plain as well as with LaTeX. + +Sub-units `xintcore`, `xint` and `xintfrac` provide the underlying +macros, and `xintexpr` loads all of them and provides expandable +parsers allowing computations such as the above (and more). A more +light-weight package [bnumexpr](http://www.ctan.org/pkg/bnumexpr) +(LaTeX only) loads only `xintcore` and provides a parser which +handles only big integers, the four operations, the power operation +and the factorial (v1.2). + + Usage ===== @@ -78,7 +106,8 @@ be loaded in any catcode context such that letters, digits, `\` and `xintcore.sty` and `xinttools.sty` both import `xintkernel.sty` which has the catcode handler and package identifier and defines a -few utilities such as `\oodef` or `\xint_dothis/\xint_orthat`. +few utilities such as `\oodef`, `\fdef`, or `\xint_dothis/\xint_orthat`. + Installation ============ @@ -179,6 +208,7 @@ Finishing the installation in a TDS hierarchy: Depending on the destination, it may then be necessary to refresh a filename database. + License ======= @@ -207,6 +237,58 @@ pandoctpl.latex, doHTMLs.sh, doPDFs.sh, xint.dvi, xint.pdf, Makefile.mk.</div> %</readme>-------------------------------------------------------- %<*changes>------------------------------------------------------- +`1.2 (2015/10/10)` +---- + + - 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. + + - new `\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`. + + - the macros `\xintAdd`, `\xintSub`, ..., now require package + **xintfrac**. With only **xintcore** or **xint** loaded, one _must_ + use `\xintiiAdd`, `\xintiiSub`, ..., or `\xintiAdd`, `\xintiSub`, + etc... + + - 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 empty or anything understood by `\numexpr`. + + - 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)` ---- @@ -557,13 +639,15 @@ breaking changes 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. + 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 @@ -2373,6 +2457,8 @@ pdfpagemode=UseOutlines} \frenchspacing \renewcommand\familydefault\sfdefault +\def\liiibigint{\href{http://latex-project.org/svnroot/experimental/trunk/l3trial/l3bigint}{l3bigint}} + \begin{document}\thispagestyle{empty}% \ttzfamily already done \pdfbookmark[1]{Title page}{TOP} % \makeatletter % @ n'est plus actif dans dtx 1.1, ouf! @@ -2617,37 +2703,57 @@ This section provides recommended reading on first discovering the package. \subsection{The packages of the \xintname bundle} + \begin{framed} The \xintcorename and \xintname packages provide macros dedicated to \emph{expandable} computations on numbers exceeding the \TeX{} (and \eTeX{}) limit of \dtt{\number"7FFFFFFF} (\emph{i.e.} on numbers of $10$ digits or more.) +\medskip + With package \xintfracname also decimal numbers (with a dot \dtt{.} as decimal mark), numbers in scientific notation (with a lowercase \dtt{e}), and even fractions (with a forward slash \dtt{/}) are acceptable inputs. +\medskip + Package \xintexprname handles expressions written with the standard infix notations, thus providing a more convenient - interface. + interface. \begin{everbatim*} -\xinttheexpr (2981279/2662176-317127/17129791)^3\relax\par -\noindent\xintthefloatexpr 1.23456789123456789^123456789\relax{} -(<- notice the exponent size)\par -\end{everbatim*} +\xinttheexpr (2981.279/.2662176e2-3.17127e2/3.129791)^3\relax +\end{everbatim*}\newline +(the |A/B[n]| notation on output means $(A/B)\times 10^n$), or also: +\begin{everbatim*} +\xintthefloatexpr 1.23456789123456789^123456789\relax +\end{everbatim*} (<- notice the size of this exponent). + +\smallskip -\xintexprname is able to do computations with dummy variables, here an example: + Furthermore \xintexprname is also able since release |1.1| of |2014/10/28| to + do computations with dummy variables, as in this example: \begin{everbatim*} -\xinttheexpr reduce(add(1/x, x=1..100))\relax\par +\xinttheexpr seq(1+reduce(add(mul((x-i+1)/i,i=1..j),j=1..floor(x/2))), + x=10..20, 31, 51)\relax \end{everbatim*} -The reasonable range of use of the package arithmetics is with numbers of less -than $\boxed{100}$ digits (integer+decimal parts combined), as it allows to do -hundreds, even thousands of such computations with an acceptable time impact. -When developing in 2013 the basic macros still in \xintcorename, the author's -priority was the speed of operations for numbers in that range. + The reasonable range of use of the package arithmetics is with numbers of + less than \emph{one hundred or perhaps two hundred digits.} Release |1.2| + has significantly improved the speed of the basic operations for numbers + with more than $50$ digits, the speed gains getting better for bigger + numbers. Although numbers up to about \dtt{19950} digits are acceptable + inputs, the package is not at his peak efficiency when confronted with such + really big numbers having thousands of digits.\footnotemark \end{framed} +\footnotetext{The maximal handled size for inputs to multiplication is + \dtt{19959} digits. This limit is observed with the current default values + of some parameters of the tex executable (input save stack size at 5000, + maximal expansion depth at 10000). Nesting of macros will reduce it and it + is best to restrain numbers to at most \dtt{19900} digits. The output, as + naturally is the case with multiplication, may exceed the bound.} + The \eTeX{} extensions (dating back to 1999) must be enabled; this is the case by default in modern distributions, except for the |tex| executable itself which has to be the pure \textsc{D.~Knuth} software with no additions. The @@ -2659,11 +2765,6 @@ All components may be loaded with \LaTeX{} |\usepackage| or |\RequirePackage| or, for any other format based on \TeX{}, directly via \string\input{}, e.g. |\input xint.sty\relax|. There are no package options. -% -% {\makeatother\footnote{\csa{empty}, \csa{space}, \csa{z@}, -% \csa{@ne}, and \csa{m@ne} should have the same meaning as in Plain and -% \LaTeX.}} -% Each package automatically loads those not already loaded it depends on (but in a few rare cases there are some extra dependencies, for example the |gcd| function in \xintexprname expressions requires explicit loading of package @@ -2777,7 +2878,12 @@ Additional derived parsers: bool, togl, add, mul, seq, subs, rseq, rrseq, iter}. And |"| may serve for hexadecimal input (uppercase only; package \xintbinhexname required). - See also \autoref{ssec:syntax}, as well as \autoref{sec:expr11}. + |1.2| has added \dtt{qint, qfrac, qfloat} to tell the parser to skip its usual + token by token expansion when gathering the digits of a number. + + See \autoref{ssec:syntax} for the complete syntax, as well as + \autoref{sec:expr11} which focused on the extensions brought with |xintexpr + 1.1|. \end{framed} Here is an example of a computation: @@ -2818,18 +2924,10 @@ it will naturally not be able to digest a number in scientific notation or a fraction. Fixed point decimal numbers however can be understood by \TeX{} in the context of manipulation of dimensions. -The constraint of expandability exerces its spell on the programmer as a -challenge to raise, and has its rewards: this is my only excuse for pretending -that computing expandably with fractions of dozens of digits has any use. The -initial release of \xintname (|2013/03/28|) was quickly followed by -\xintfracname which handles exactly fractions, decimal numbers, scientific -numbers, hence in a derived way floating point numbers. A bit later -(|2013/05/25|) \xintexprname implemented an expandable parser of expressions -with the usual infix notations for the basic operations. -The core big integer routines (now in \xintcorename) have not been much -changed since (although my knowledge of \TeX{} programming increased a lot), -the effort of development going mainly in \xintexprname which was extended -first in september 2013, then substantially in october 2014. +% The constraint of expandability exerces its spell on the programmer as a +% challenge to raise, and has its rewards: this is my only excuse for pretending +% that computing expandably with fractions of dozens of digits has any use. + % The implementation is also maximally complicated as many functionalities of % the \TeX{} macro language can't be used in the source code of \xintname by the @@ -2844,11 +2942,11 @@ first in september 2013, then substantially in october 2014. % gains could be achieved via a complete rewrite of all basic macros of the % package. -The underlying macros to which |\xinttheexpr ...\relax| maps operations -are thus provided by packages \xintcorename, \xintname (for long) integers and -\xintfracname (for fractions, decimal numbers, scientific numbers). They are -nestable. For example to do |21+32*43|, the syntax would be (only -\xintcorename needed): +The underlying macros to which |\xinttheexpr ...\relax| and the other parsers +map the infix operations are provided by packages \xintcorename, \xintname (for +integers) and \xintfracname (for fractions, decimal numbers, and scientific +numbers). They are nestable. For example to do something like |21+32*43|, the +syntax would be (only \xintcorename is needed): \begin{everbatim*} \xintiiAdd{21}{\xintiiMul{32}{43}}\par \noindent\xintiiMul{21283978192739181739}{\xintiiSub {130938109831081320}{29810810281}} @@ -2862,11 +2960,11 @@ Needless to say this quickly becomes a bit painful. One more example (needs \end{everbatim*} This shows that package \xintfracname knows natively how to handle fractions -|A/B| (notice that |*|, |+| and |-| contrarily to |/| are not generally -accepted in the arguments to the \xintfracname macros; but see -\autoref{sec:inputs} and \autoref{sec:useofcount}) and that it has -a command \csbxint{Irr} to reduce to smallest terms (in an -|\xintexpr..\relax| this would be the |reduce| function). +|A/B|. Notice that |*|, |+| and |-| contrarily to the |/| which is treated as +a special optional delimiter are not accepted in the arguments to the +\xintfracname macros (see \autoref{sec:inputs} and \autoref{sec:useofcount} +for some exceptions). There is \csbxint{Irr} to reduce to smallest terms (in +an |\xintexpr..\relax| this would be the |reduce| function). Again, all computations done by |\xinttheexpr..\relax| are completely exact. Thus, very quickly very big numbers are created (and computation times @@ -2879,8 +2977,8 @@ assignment to |\xintDigits|): (<- notice the size of the power of ten: this surely largely exceeds your pocket calculator abilities). -It is also possible -to do some computer algebra like evaluations (only numerically though): +It is also possible to do some (expandable...) computer algebra like +evaluations (only numerically though): \begin{everbatim*} \xinttheiiexpr add(i^5, i=100..200)\relax\par \noindent\xinttheexpr add(x/(x+1), x = 1000..[3]..1020)\relax @@ -2895,31 +2993,31 @@ Make sure to read \autoref{ssec:userinterface}, \autoref{sec:expr11} and \subsection {Changes} -On |TeXLive| (and presumably |MikTeX|), issue in a console |texdoc --list -xint| to get access to the documentation files, among them |CHANGES.html| -provides the detailed cumulative change log since the initial release. +The initial \xintname (|2013/03/28|) was followed by \xintfracname +(|2013/04/14|) which handled exactly fractions and decimal numbers. Later came +\xintexprname (|2013/05/25|) and at the same time \xintfracname got extended +to handle floating point numbers. Later, \xinttoolsname was detached +(|2013/11/22|). The main focus of development during late 2013 and early 2014 +was kept on \xintexprname. One year later it got a significant upgrade with +|1.1| of |2014/10/28|. The core integer routines remained essentially +unmodified during all this time (apart from a slight improvement of division +early 2014) until their complete rewrite with release +|1.2| from |2015/10/10|. -It is also available on \href{http://ctan.org}{CTAN} via -\href{http://mirrors.ctan.org/macros/generic/xint/CHANGES.html}{this link}. -Or, running |etex xint.dtx| in a working repertory will extract a |CHANGES.md| -file with Markdown syntax. - -The most recent major changes came with release |1.1| from |2014/10/28|. - -Since, |xint| saw only minor modifications such as enhancements to the -documentation and a few bug fixes (|1.1c 2015/09/12| fixed a bug with -\csbxint{Assign}, |1.1b 2015/08/31| transferred some macros which had been -left in |xint.sty| and should have been part of |xintcore.sty|, and earlier -|1.1a 2014/11/07| had corrected a problem with \csbxint{NewExpr}). - -|1.1 2014/10/28| brought many additions, most of them to package -\xintexprname, such as for example the evaluation of expressions with dummy -variables, possibly iteratively, and with allowed nesting. See -\autoref{sec:expr11} for a description of these new functionalities. - -Apart from that, |1.1| brought also two main other changes: +\begin{description} +\item [|1.2 (2015/10/10):|] complete rewrite of the core arithmetic routines. + The efficiency for numbers with less than $20$ or $30$ digits is slightly + compromised (for addition/subtraction) but it is increased for bigger + numbers. For multiplication and division the gains are there for almost all + sizes, and become quite noticeable for numbers with hundreds of digits. The + allowable inputs are constrained to have less than about $19950$ digits + ($19968$ for addition, $19959$ for multiplication). +\item [|1.1 (2014/10/28):|] many extensions to package \xintexprname, such as + the evaluation of expressions with dummy variables, possibly iteratively, + with allowed nesting. See \autoref{sec:expr11} for a description of + these new functionalities. Also worthy of attention: \begin{enumerate} -\item |\xintiiexpr...\relax| now associates |/| with the \emph{rounded} +\item |\xintiiexpr...\relax| associates |/| with the \emph{rounded} division (the |//| operator being provided for the \emph{truncated} division) to be in synchrony with the habits of |\numexpr|, \item the \xintfracname macro \csbxint{Add} (corresponding to |+| in @@ -2928,13 +3026,24 @@ Apart from that, |1.1| brought also two main other changes: smallest terms, or systematically computing the |LCM| of the denominators would be too costly (I think). \end{enumerate} -Also worth mentioning is that \xintname does not load \xinttoolsname -anymore (only \xintexprname does) and that the core arithmetic macros have -been moved to a new package \xintcorename which is loaded automatically by -\xintname. +\xintname does not load \xinttoolsname +anymore (only \xintexprname does) and the core arithmetic macros are +moved to a new package \xintcorename (loaded automatically by +\xintname, itself loaded by \xintfracname, itself loaded by \xintexprname). Package \href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr} (which is \LaTeX{} only) now also loads only \xintcorename. +\end{description} + +There is a file |CHANGES.html| (also |CHANGES.pdf|) which provides the +detailed cumulative change log since the initial release. To access it, issue +on the command line |texdoc --list xint| (this works |TeXLive| and there is +probably an equivalent in |MikTeX|). + +It is also available on \href{http://ctan.org}{CTAN} via +\href{http://mirrors.ctan.org/macros/generic/xint/CHANGES.html}{this link}. +Or, running |etex xint.dtx| in a working repertory will extract a |CHANGES.md| +file with Markdown syntax. \subsection{Installation instructions} \label{ssec:install} @@ -2979,95 +3088,70 @@ If you have |xint.dtx|, no internet access and can not use the Makefile method: |etex xint.dtx| extracts all files and among them the |README| as a file with name |README.md|. Further help and options will be found therein. -\subsection{FAQ} - -\def\liiibigint{\href{http://latex-project.org/svnroot/experimental/trunk/l3trial/l3bigint}{l3bigint}} +% \subsection{FAQ} -% pour accéder à l'historique des commits: -% https://github.com/latex3/latex3/tree/master/l3trial/l3bigint +% % pour accéder à l'historique des commits: +% % https://github.com/latex3/latex3/tree/master/l3trial/l3bigint -We are honoured to present here this interview with the author, dating back to -late March 2014. -\begin{description} -\item[Will \xintexprname implement \texttt{exp}, \texttt{log}, - \texttt{cos}, \texttt{sin} \dots at some point?] -I guess so. - -\item[\xintseriesname already provides generic tools.] Right, although the - casual user of the \xintname bundle will not quite know how to do variable - reduction expandably in order to use some series or Pad\'e approximants. - Besides I wrote the code at the beginning of the project and perhaps I could - do it better now (I have not looked at it for a while). Anyhow, generic things - do not help much if one wants to optimize. - -\item[Optimizing? isn't \TeX's macro expansion mechanism intrinsically slow?] - Intensive use of \csa{numexpr} and some token manipulation algorithms - exploiting to the best I could \TeX{} macros with parameters grant \xintname - a significant speed up in expandable arithmetic on big integers compared to - previously available implementations. You can do some comparisons with - multiplication on numbers with $100$ digits or division of one of $100$ - digits by another of $50$ digits, for example. However expandability is - antagonist of speed, and I agree it is not very exciting to optimize slow - things. And I was disappointed last year to realize the slowness of \TeX's - mouth when it has to keep hundreds of tokens in cheek to mix them later with - new aliments. -% https://github.com/latex3/latex3/commit/f46e22cb772ee34aeb2fb200f7907ed3e6192bac -% -\footnote{\label{fn:l3bigint2015}\textbf{2015/09/15:} the recent - (experimental, and partially still in progress) new version of \liiibigint{} - by Bruno \textsc{Le Floch} works with comparable speed as \xintname for - numbers with less than (roughly) $100$ digits (the division was not - available for testing a.t.t.o.w.), and its multiplication becomes then much - better: about $5x$ faster for numbers with $400$ digits and $20x$ faster for - numbers with $1000$ digits. Bruno succeeded into implementing expandably - Karatsuba multiplication and he achieves - sub-quadratic growth in the computation time, whereas the xint integer - multiplication time drifts from quadratic to worse than cubic: from $50$ to - $100$ digits the time factor increases by about $4.6$ for \xintname and - about $3.1$ for \liiibigint, in the former case |xint| is the faster, in - the latter it is |l3bigint| the faster. As an illustration squaring a $8000$ - digits number on my computer is $470x$ faster with the new \liiibigint{} - compared to |xint| ($5.5s$ vz $43m$). As \liiibigint{} is still in progress, - this may well evolve further (currently it seems to be limited to numbers up - to $8192$ digits). The addition currently starts getting faster than the one - from |xint| with about $400$ digits (2013 |xint dev| had a variant of - addition which I think would be faster than the new one from \liiibigint{} - in the whole range up to circa $1000$ digits; but it did not make it into - the release as it was a bit slower than the adopted one up to - $50$ digits; and adopting it would have sped up multiplication only by a - factor of $2$ anyhow, and would not have solved the steep shape of the curve - of computation times.)} -% -Believe me, I try not to think too much about the fact that the whole -enterprise is made irrelevant by Lua\LaTeX's ability to access external -libraries. - -\item[Well, why isn't this \texttt{log} etc\dots thing done yet?] -I have to decide on the maximal precision to achieve: $24$, $32$, $48$, -$64$,\dots ; to settle that I would need to implement some initial versions and -benchmark them. - -\item[Fair enough. That's the common lot. So why not yet?] I am a bit - overworked. It is also an opportunity to think over - the basic underlying mathematics, and will need devoted thinking for some not - insignificant amount of time. So far I didn't find the time, or rather I found - out good means to waste it sillily. I also anticipate that originality could - very - well not pay off at all, so small is the window for the precision. - -\item[Any chance this could be done in time for TL2014?] No, - sorry.\newline Release |1.09m| of |[2014/02/26]| was the end of a cycle, and - this |1.09n| of |[2014/04/01]| is only for a bug fix and inclusion of this - |FAQ| in the documentation. - -\footnotesize - -\item[and in time for TL2015?] ... (indistinct mumbles, something like - \emph{too tired}, \emph{I need a life}, \emph{get yourself a calculator}, - \emph{we'll see}\dots) -\end{description} +% We are honoured to present here this interview with the author, dating back to +% late March 2014. +% \begin{description} +% \item[Will \xintexprname implement \texttt{exp}, \texttt{log}, +% \texttt{cos}, \texttt{sin} \dots at some point?] +% I guess so. + +% \item[\xintseriesname already provides generic tools.] Right, although the +% casual user of the \xintname bundle will not quite know how to do variable +% reduction expandably in order to use some series or Pad\'e approximants. +% Besides I wrote the code at the beginning of the project and perhaps I could +% do it better now (I have not looked at it for a while). Anyhow, generic things +% do not help much if one wants to optimize. + +% \item[Optimizing? isn't \TeX's macro expansion mechanism intrinsically slow?] +% Intensive use of \csa{numexpr} and some token manipulation algorithms +% exploiting to the best I could \TeX{} macros with parameters grant \xintname +% a significant speed up in expandable arithmetic on big integers compared to +% previously available implementations. You can do some comparisons with +% multiplication on numbers with $100$ digits or division of one of $100$ +% digits by another of $50$ digits, for example. However expandability is +% antagonist of speed, and I agree it is not very exciting to optimize slow +% things. And I was disappointed last year to realize the slowness of \TeX's +% mouth when it has to keep hundreds of tokens in cheek to mix them later with +% new aliments. +% % https://github.com/latex3/latex3/commit/f46e22cb772ee34aeb2fb200f7907ed3e6192bac +% % +% +% Believe me, I try not to think too much about the fact that the whole +% enterprise is made irrelevant by Lua\LaTeX's ability to access external +% libraries. + +% \item[Well, why isn't this \texttt{log} etc\dots thing done yet?] +% I have to decide on the maximal precision to achieve: $24$, $32$, $48$, +% $64$,\dots ; to settle that I would need to implement some initial versions and +% benchmark them. + +% \item[Fair enough. That's the common lot. So why not yet?] I am a bit +% overworked. It is also an opportunity to think over +% the basic underlying mathematics, and will need devoted thinking for some not +% insignificant amount of time. So far I didn't find the time, or rather I found +% out good means to waste it sillily. I also anticipate that originality could +% very +% well not pay off at all, so small is the window for the precision. + +% \item[Any chance this could be done in time for TL2014?] No, +% sorry.\newline Release |1.09m| of |[2014/02/26]| was the end of a cycle, and +% this |1.09n| of |[2014/04/01]| is only for a bug fix and inclusion of this +% |FAQ| in the documentation. + +% \footnotesize + +% \item[and in time for TL2015?] ... (indistinct mumbles, something like +% \emph{too tired}, \emph{I need a life}, \emph{get yourself a calculator}, +% \emph{we'll see}\dots) +% \end{description} \section{Introduction via examples} +\label{sec:examples} The main goal is to allow expandable computations with integers and fractions of arbitrary sizes. @@ -3108,188 +3192,6 @@ which can be used to achieve this splitting accross lines, and does work in inline math mode (however it doesn't allow to separate digits by groups of three, for example).\par -\subsection{User interface} -\label{ssec:userinterface} - -The user interface for executing operations on numbers is via macros such as -\csbxint{Add} or \csbxint{Mul} which have two arguments, or via expressions -\csbxint{expr}|..\relax| which use infix notations such as |+|, |-|, |*|, |/| -and |^| (or |**|) for the basic operations, and recognize functions of one or -more comma separated arguments (such as |max|, or |round|, or |sqrt|), -parentheses, logic operators of conjunction |&&|, disjunction \verb+||+, as -well as two-way |?| and three-way |??| conditionals and more. A few examples: -% -\begin{everbatim*} -\begin{enumerate}[nosep] - \item \xintiiAdd {2719873981798137981381789317981279}{13819093809180120910390190} - \item \xintiiMul {2719873981798137981381789317981279}{13819093809180120910390190} - \item \xintthefloatexpr (19317/21913+2198/9291)^3\relax - \item \xintDigits:=64;\xintthefloatexpr (19317/21913+2198/9291)^3\relax -% Let's compute the inner sum exactly, not as a float, before raising to third power: - \item \xintDigits:=16;\xintthefloatexpr \xintexpr 19317/21913+2198/9291\relax^3\relax -\end{enumerate} -\end{everbatim*} -In \csbxint{expr}|..\relax| the contents are expanded completely from left to -right until the ending |\relax| is found and swallowed, and spaces and even -(to some extent) catcodes do not matter. Algebraic operations are done -\emph{exactly}. - -The \csbxint{floatexpr} variant is for operations which are done using the -precision set via the |\xintDigits:=N;| assignment (default is with -significands of \dtt{16} digits). - -For all |\xintexpr| variants, prefixing with |\xintthe| allows to print -the result or use it in other contexts. Shortcuts \csbxint{theexpr}, -\csbxint{thefloatexpr}, \csbxint{theiiexpr}, \dots\ are available. - -The \csbxint{iiexpr} variant is only for big integers, it does not know -fractions. - -There are some important differences of syntax between |\numexpr| and -|\xintiiexpr| and variants: -\begin{itemize} -\item Contrarily to |\numexpr|, the |\xintiiexpr| parser will stop expanding - only after having encountered (and swallowed) a \emph{mandatory} |\relax| - token. -\item In particular, spaces between digits (and not only around infix - operators or parentheses) do not stop |\xintiiexpr|, contrarily to the - situation with |numexpr|: |\the\numexpr 7 + 3 5\relax| expands (in one step) - to \dtt{\detokenize\expandafter{\the\numexpr 7 + 3 5\relax}\unskip}, whereas - |\xintthe\xintiiexpr 7 + 3 5\relax| expands (in two steps) to - \dtt{\detokenize\expandafter\expandafter\expandafter {\xintthe\xintiiexpr 7 - + 3 5\relax}}. - \item Also worth mentioning is the fact that |\numexpr -(1)\relax| is - illegal. But this is perfectly legal and with the expected result in - |\xintiiexpr...\relax|. - \item Inside an |\edef|, expressions |\xintiiexpr...\relax| get fully - evaluated, but need the prefix |\xintthe| to get printed or used as - arguments to some macros, whereas expansion of |\numexpr| in an |\edef| - occurs only if prefixed with |\the| or |\number| (or |\romannumeral|, or - the expression is included in a bigger - |\numexpr| which will be the one to have to be prefixed\dots .) -\end{itemize} - -For macros such as \csbxint{Add} or -\csbxint{Mul} the arguments are each subjected to the process of \fexpan sion: -repeated expansion of the first token until finding something unexpandable (or -being stopped by a space token). - -Conversely this process of \fexpan sion always provokes the complete expansion -of the package macros and |\xintexpr..\relax| also will expand completely -under \fexpan sion, but to a private format; the \csbxint{the} prefix allows the -computation result either to be passed as argument to one of the package -macros,\footnote{the \csa{xintthe} prefix \fexpan ds the \csa{xintexpr}-ession - then unlocks it from its private format; it should not be used for - sub-expressions inside a bigger one as it is more efficient for the - expression parser to keep the result in the private format.} or also end up on -the printed page (or in an auxiliary file). -To recapitulate, all macros dealing with computations -\begin{enumerate} -\item \emph{expand completely under the sole process of repeated expansion of - the first token, (and two expansions suffice)},\footnote{see in - \autoref{sec:expansions} for more details.} - -\item \emph{apply this \fexpan sion to each one of their arguments.} -\end{enumerate} -Hence they can be nested one within the other up to arbitrary -depths. Conditional evaluations either within the macro arguments themselves, or -with branches defined in terms of these macros are made possible via macros such -as as \csbxint{ifSgn} or \csbxint{ifCmp}. - -\begin{framed} - There is no notion of \emph{declaration of a variable} to \xintname, - \xintfracname, or \xintexprname. - The user employs the |\def|, |\edef|, or - |\newcommand| (in \LaTeX) as usual, for example: \IMPORTANT -% -\begin{everbatim*} -\def\x{1729728} \def\y{352827927} \edef\z{\xintMul {\x}{\y}} -\meaning\z -\end{everbatim*}\ (see below for the |A/B[N]| output format; with |\xintiiMul| -in place of |\xintMul| there would not be the strange looking |/1[0]|.) - -As a faster alternative to |\edef| (when hundreds of digits are involved), the -package provides |\oodef| which only expands twice its argument. This provokes -full expansion of the \xintname \fexpan dable macros (nested to possibly many -levels), inclusive of |\xintexpr| and variants. -\end{framed} - -\begingroup % pour \z, \zz -The \xintexprname package has a private internal -representation for the evaluated computation result. With -% -\begin{everbatim*} -\oodef\z {\xintexpr 3.141^18\relax} -\end{everbatim*} -% -the macro |\z| is already fully evaluated (two expansions were applied, and this -is enough), and can be reused in other |\xintexpr|-essions, such as for example -% -\begin{everbatim*} -\edef\zz {\xintexpr \z+1/\z\relax} - % (using short macro names such as \z and \zz is not too recommended in real - % life, some may have already definitions; I did it all in a group). -\end{everbatim*} -% -But to print it, or to use it as argument to one of the package macros, -it must be prefixed by |\xintthe| (a synonym for |\xintthe\xintexpr| is -\csbxint{theexpr}). Application of this |\xintthe| prefix outputs the -value in the \xintfracname semi-private internal format -|A/B[N]|,\footnote{there is also the notion of \csbxint{floatexpr}, for - which the output format after the action of \csa{xintthe} is a number in - floating point scientific notation.} representing the fraction -$(A/B)\times 10^N$. The |\zz| above produces a somewhat large output: -\begin{everbatim*} -\printnumber{\xintthe\zz }${}\approx{}$\xintFloat{\xintthe\zz} -\end{everbatim*} -\endgroup % pour \z, \zz - - \begin{framed} - By default, computations done by the macros of \xintfracname or within - |\xintexpr..\relax| are exact. Inputs containing decimal points or - scientific parts do not make the package switch to a `floating-point' mode. - The inputs, however long, are converted into exact internal representations. -% - % Floating point evaluations are done via special macros containing - % `Float' in their names, or inside |\xintfloatexpr|-essions. - - Manipulating exactly big fractions quickly leads to \dots bigger fractions. - There is a command \csbxint{Irr} (or the function |reduce| in an expression) - to reduce to smallest terms, but it has to be explicitely requested. Prior - to release |1.1| addition and subtraction blindly multiplied denominators; - they now check if one is a multiple of the other.\IMPORTANT\ But systematic - reduction of the result to its smallest terms would be too - costly.\def\everbatimindent{0pt } -\begin{everbatim*} -\xinttheexpr 27/25+46/50\relax\ is a bit simpler than \xinttheexpr (27*50+25*46)/(25*50)\relax, -but less so than \xinttheexpr reduce(27/25+46/50)\relax. And \xinttheexpr 3/75+4/50+2/100\relax\ -looks weird, but systematically reducing fractions would be too costly. -\end{everbatim*} - \end{framed} - -% -The |A/B[N]| shape is the output format of most \xintfracname macros, it -benefits from accelerated parsing when used on input, compared to the normal -user syntax which has no |[N]| part. An example of valid user input for a -fraction is -% -\leftedline{|-123.45602e78/+765.987e-123|} -% -where both the decimal parts, the scientific exponent parts, and the whole -denominator are optional components. The corresponding semi-private form in this -case would be -% -\leftedline{\xintRaw{-123.45602e78/+765.987e-123}} -% -The forward slash |/| is simply a delimiter to separate numerator and -denominator, in order to allow inputs having such denominators. - -Reduction to the irreducible form of the output must be asked for explicitely -via the \csbxint{Irr} macro or the |reduce| function within -|\xintexpr..\relax|. Elementary operations on fractions do very little of the -simplifications which could be obvious to (some) human beings. - - \subsection{Randomly chosen examples} Here are some examples of use of the package macros. The first one uses only @@ -3307,10 +3209,46 @@ Some inputs are simplified by the use of the \xintexprname package. |\xintTrunc {1500}{1234/56789}\dots|: \dtt{\printnumber {\xintTrunc {1500}{1234/56789}}\dots } -\item {$0.99^{-100}$ with 200 digits after the decimal point:}\\ -|\xinttheexpr trunc(.99^-100,200)\relax\dots|: -\dtt{\printnumber{\xinttheexpr trunc(.99^-100,200)\relax}\dots } +\item {$0.99^{-100}$ with 200 (+1) digits after the decimal point.}\\ + |\xinttheiexpr [201] .99^-100\relax|: + \dtt{\printnumber{\xinttheiexpr [201] .99^-100\relax}}\\ + Notice that this is rounded, hence we asked |\xinttheiexpr| for one + additional digit. To get a truncated result with 200 digits after the decimal + mark, we should have issued + |\xinttheexpr trunc(.99^-100,200)\relax|, rather. + +\begin{snugframed} + The fraction |0.99^-100|'s denominator is first evaluated \emph{exactly} + (\emph{i.e.} the integer |99^100| is evaluated exactly and then used to + divide the suitable power of ten to get the requested digits); for + some longer inputs, such as for example |0.7123045678952^-243|, the + exact evaluation before truncation would be costly, and it is more efficient + to use floating point numbers: +% +\leftedline{|\xintDigits:=20; + \np{\xintthefloatexpr .7123045678952^-243\relax}|}% +% +\leftedline{\xintDigits:=20;\dtt{\np{\xintthefloatexpr .7123045678952^-243\relax }}} +% +\xintDigits:=16;% +% +Side note: the exponent |-243| didn't have to be put inside parentheses, +contrarily to what happens with some professional computational +software. |;-)| +% 6.342,022,117,488,416,127,3 10^35 +% maple n'aime pas ^-243 il veut les parenthèses, bon et il donne, en Digits +% = 24: 0.634202211748841612732270 10^36 +\end{snugframed} + +\item {$200!$:}\\ +|\xinttheiiexpr 200!\relax|: +\dtt{\printnumber{\xinttheiiexpr 200!\relax}} +\item {$2000!$ as a float. As \xintexprname does not handle |exp/log| so far, + the computation is done internally without the Stirling formula, + by repeated multiplications truncated suitably:}\\ + |\xintDigits:=50;|\newline |\xintthefloatexpr 2000!\relax|: + {\xintDigits:=50;\dtt{\printnumber{\xintthefloatexpr 2000!\relax}}} \item Just to show off (again), let's print 300 digits (after the decimal point) of the decimal expansion of $0.7^{-25}$:% @@ -3330,28 +3268,11 @@ Some inputs are simplified by the use of the \xintexprname package. This computation is with \csbxint{theexpr} from package \xintexprname, which allows to use standard infix notations and function names to access the package macros, such as here |trunc| which corresponds to the \xintfracname macro -\csbxint{Trunc}. - -\begin{snugframed} - The fraction |0.7^-25| is first evaluated \emph{exactly}; for some more - complex inputs, such as |0.7123045678952^-243|, the exact evaluation before - truncation would be rather costly, and one would rather use floating point - numbers: -% -\leftedline{|\xintDigits:=20; - \np{\xintthefloatexpr .7123045678952^-243\relax}|}% -% -\leftedline{\xintDigits:=20;\dtt{\np{\xintthefloatexpr .7123045678952^-243\relax }}} -% -Side note: the exponent |-243| didn't have to be put inside parentheses, -contrarily to what happens with some professional computational -software. |;-)| -% 6.342,022,117,488,416,127,3 10^35 -% maple n'aime pas ^-243 il veut les parenthèses, bon et il donne, en Digits -% = 24: 0.634202211748841612732270 10^36 -\end{snugframed} - -\xintDigits:=16; +\csbxint{Trunc}. Regarding this computation, please keep in mind that +\csbxint{theexpr} computes \emph{exactly} the result before truncating. As +powers with fractions lead quickly to very big ones, it is good to know that +\xintexprname also provides \csbxint{thefloatexpr} which does computations +with floating point numbers. \item Computation of a Bezout identity with |7^200-3^200| and |2^200-1|: (with \xintgcdname)\par @@ -3510,51 +3431,57 @@ digits. This is not so many, let us print them here: examples in \autoref{sec:expr11}. \end{itemize} Almost all of the computational results interspersed throughout the -documentation are not hard-coded in the source of the document. They are -the result of evaluation of the package macros, and were selected to not -impact too much the compilation time of this documentation. -Nevertheless, there are so many computations done that compilation time -is significantly increased compared to a \LaTeX\ run on a typical -document of about the same size. - -\section{The \xintname bundle} +documentation are not hard-coded in the source file of this document but are +obtained via the expansion of the package macros during the \TeX{} +run.% +% +\footnote{The CPU of my computer hates me for all those re-compilations + after changing a single letter in the \LaTeX{} source, which require each + time to do all the zillions of evaluations contained in this document\dots} +% +% on examples which were selected to not impact too much the compilation time of +% this documentation. -\subsection{General overview} +% Nevertheless, there are so many computations done that compilation time +% is significantly increased compared to a \LaTeX\ run on a typical +% document of about the same size. -The main characteristics are: -\begin{enumerate} -\item exact algebra on arbitrarily big numbers, integers as well as fractions, -\item floating point variants with user-chosen precision, -\item implemented via macros compatible with expansion-only - context. -\end{enumerate} +\section{The \xintname bundle} -`Arbitrarily big' means with less than - |2^31-1|\dtt{=\number"7FFFFFFF} digits, as most of the macros will - have to compute the length of the inputs and these lengths must be treatable - as \TeX{} integers, which are at most \dtt{\number "7FFFFFFF} - in absolute value. - This is a distant irrelevant upper bound, as no such thing can fit - in \TeX's memory! The \emph{time} taken up by the expansion only - mechanisms is a much more stringent constraint. +\subsection{Characteristics} \begin{framed} - The reasonable range is for operations with numbers of up to about - (integer+decimal part) \dtt{100} digits. + The main characteristics are: + \begin{enumerate} + \item exact algebra on arbitrarily big numbers, integers as well as + fractions, + \item floating point variants with user-chosen precision, + \item implemented via macros compatible with expansion-only context, + \item and with a parser of infix operations implementing features such as + dummy variables, and coming in various incarnations depending on the kind + of computation desired: purely on integers, on integers and fractions, or + on floating point numbers. + \end{enumerate} + + `Arbitrarily big' currently means with less than about \dtt{19950} digits: the + maximal% + \MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{Changed!} + number of digits for addition is at \dtt{19968} digits, + and it is \dtt{19959} for multiplication. \end{framed} -As just recalled, ten-digits integers starting with a $3$ already exceed the -\TeX{} bound; and \TeX{} does not have a native processing of -floating point numbers (multiplication by a decimal number of a dimension -register is allowed --- this is used for example by the -\href{http://mirror.ctan.org/graphics/pgf/base}{pgf} basic math -engine.) +Integers with only $10$ digits and starting with a $3$ already exceed the +\TeX{} bound; and \TeX{} does not have a native processing of floating point +numbers (multiplication by a decimal number of a dimension register is allowed +--- this is used for example by the +\href{http://mirror.ctan.org/graphics/pgf/base}{pgf} basic math engine.) \TeX{} elementary operations on numbers are done via the non-expandable -\emph{advance, multiply, \emph{and} divide} assignments. This was changed with -\eTeX{}'s |\numexpr| which does expandable computations using standard infix -notations with \TeX{} integers. But \eTeX{} did not modify the \TeX{} bound on -acceptable integers, and did not add floating point support. +\emph{\char92advance, \char92multiply, \emph{and} \char92divide} assignments. +This was changed with \eTeX{}'s |\numexpr| which does expandable computations +using standard infix notations with \TeX{} integers. But \eTeX{} did not +modify the \TeX{} bound on acceptable integers, and did not add floating point +support. The \href{http://www.ctan.org/pkg/bigintcalc}{bigintcalc} package by \textsc{Heiko Oberdiek} provided expandable operations (using some of |\numexpr| @@ -3581,81 +3508,73 @@ ${}^{\text{,\,}}$% The \LaTeX3 project has implemented expandably floating-point computations with 16 significant figures (\href{http://www.ctan.org/pkg/l3kernel}{l3fp}), including -special functions such as exp, log, sine and cosine.% -% -\footnote{at the time of writing (2014/10/28) the +special functions such as exp, log, sine and cosine.\footnote{at the time of writing (2014/10/28) the \href{http://www.ctan.org/pkg/l3kernel}{l3fp} (exactly represented) floating point numbers have their exponents limited to $\pm$\dtt{9999}.} % -There is -also \liiibigint, which is part of the experimental trunk of the -\href{http://latex-project.org}{\LaTeX3 Project}. Like -\href{http://www.ctan.org/pkg/bigintcalc}{bigintcalc} and \xintname it -provides macros for big integer arithmetics. All three implementations of the -basic arithmetic macros can be mapped to easier infix notations via the -services of the \href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr} -package. -% -\footnote{2015/09/15: the currently available \liiibigint{} is not compatible - with the current latest \href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr} - release (1.1b 2014/10/28).} - -The \xintname package can be used for $24$, $40$, etc\dots{} significant figures -but one rather quickly (not much beyond $100$ figures) hits against a -`wall' created by the constraint of expandability: currently, multiplying out -two one-hundred digits numbers takes circa $80$ or $90$ times longer than for -two ten-digits numbers, which is reasonable, but multiplying out two -one-thousand digits numbers takes more than $500$ times longer than for two one -hundred-digits numbers. This shows that the algorithm is drifting from quadratic -to cubic in that range. On my laptop multiplication of two $1000$-digits numbers -takes some seconds, so it can not be done routinely in a -document.% -% -\footnote{2013/06/07: without entering into too much technical details, the - source of this `wall' is that when dealing with two long operands, when one - wants to pick some digits from the second one, one has to jump above all - digits constituting the first one, which can not be stored away: - expandability forbids assignments to memory storage. One may envision some - sophisticated schemes, dealing with this problem in less naive ways, trying - to move big chunks of data higher up in the input stream and come back to it - later, etc...; but each `better' algorithm adds overhead for the smaller - inputs. For example, I have another version of addition which is twice - faster on inputs with 500 digits or more, but it is slightly less efficient - for 50 digits or less. This `wall' dissuaded me to look into implementing - `intelligent' multiplication which would be sub-quadratic in a model where - storing and retrieving from memory would not cost much.} -\footnote{\textbf{2015/09/15:} the new \liiibigint{} has brilliantly - successfully implemented Karatsuba multiplication, solving the issue faced - by xint in 2013 (the method is based on a clever preparation of the inputs - into a form which allows precisely a ``sophisticated scheme'' as above on - where to locate the data to be manipulated and it has I presume the cost, as - described above, of a slight overhead for small numbers -- but I did not - test that point): the speed gain compared to xint for multiplication of - numbers of about $1000$ digits is about $20x$; for numbers of $100$ digits - \liiibigint{} is only a bit faster than \xintcorename; for small numbers - \xintcorename is probably (not tested) a bit faster. See also - \autoref{fn:l3bigint2015}.} - -The conclusion perhaps could be that it is in the end lucky that the speed gains -brought by \xintname for expandable operations on big numbers do open some -non-empty range of applicability in terms of the number of kept digits for -routine floating point operations. - -The second conclusion, somewhat depressing after all the hard work, is that if -one really wants to do computations with \emph{hundreds} of digits, one should -drop the expandability requirement. Indeed, as clearly demonstrated long ago -by the \href{http://www.ctan.org/pkg/pi}{pi computing file} by \textsc{D. - Roegel} one can program \TeX{} to compute with many digits at a much higher -speed than what \xintname achieves: but, direct access to memory storage in -one form or another seems a necessity for this kind of speed and one has to -renounce at the complete expandability.\footnote{2015/09/15: the latest - developments on the \liiibigint{} side do not really modify this conclusion, - because the computations remain extremely slow compared to what one can do - in other programming structures, and also slow in comparison to what one - could do non-expandably in \TeX{} (although I know to this day no available - macro package which has done this successfully for big integer arithmetics; - perhaps I am too optimistic with respect to what one can achieve without the - burden of expandability?). + +More directly related to the \xintname bundle, there is the promising new +version of the \liiibigint{} package. It was still in development a.t.t.o.w +(2015/10/09, no division yet) and is part of the experimental trunk of the +\href{http://latex-project.org}{\LaTeX3 Project}. It is devoted to expandable +computations on big integers with an associated expression parser. Its author +(Bruno \textsc{Le Floch}) succeeded brilliantly into implementing expandably +the Karatsuba multiplication algorithm and he achieves \emph{sub-quadratic + growth for the computation time}. This shows up very clearly with numbers +having more than one thousand digits (up to the maximum which a.t.t.o.w was at +$8192$ digits). + +I report here briefly on a quick comparison, although as \liiibigint{} is work +in progress, the reported results could well have to be modified soon. The +test was on a comparison of |\bigint_eval:n {#1*#2}| from the \liiibigint{} as +available in September 2015, on one hand, and on the other hand +|\xinttheiiexpr #1*#2\relax| from \xintexprname 1.2 (rather than directly +|\xintiiMul|, to be fairer to the parsing time induced by use of +|\bigint_eval:n|) and the computations were done with +|#1=#2=9999888877999988877...repeated...|. I observed: +\begin{itemize} +\item \csbxint{iiexpr}'s multiplication appears slightly faster (about |1.5x| + or |2x| to give an average order of magnitude) up to about + $900$ digits, +\item at $1000$ digits, \liiibigint{} runs between |15%| and |20%| faster, +\item then its sub-quadratic growth shows up, and at $8000$ digits I observed + it to be about |7.6x| faster (I tried on two computers and on my laptop the + ratio was more like |8.5x--9x|). Its computation time increased from $1000$ + digits to $8000$ digits by a factor smaller than |30|, whereas for + \csbxint{iiexpr} it was a factor only slightly inferior to |200| (|225| on + my laptop) ... + Karatsuba multiplication brilliantly pays off ! +\item One observes the transition at the powers of two for the \liiibigint{} + algorithm, for example I observed \liiibigint{} to be |3.5x| faster at + $4000$ digits but only |3.1x| faster at $5000$ digits. +\end{itemize} + +Once one accepts a small overhead, one can on the basis of the lengths decide +for the best algorithm to use, and it is tempting viewing the above to imagine +that some mixed approach could combine the best of both. But again all this is +a bit premature as both packages may still evolve further. + +Anyhow, all this being said, even the superior multiplication implementation +from \liiibigint{} takes of the order of seconds on my laptop for a single +multiplication of two $5000$-digits numbers. Hence it is not possible to do +routinely such computations in a document. I have long been thinking that +without the expandability constraint much higher speeds could be achieved, but +perhaps I have not given enough thought to sustain that optimistic +stance.\footnote{The \href{http://www.ctan.org/pkg/apnum}{apnum} package + implements non-expandably arbitrary precision arithmetic operations.} + +I remain of the opinion that if one really wants to do computations with +\emph{thousands} of digits, one should drop the expandability requirement. +Indeed, as clearly demonstrated long ago by the +\href{http://www.ctan.org/pkg/pi}{pi computing file} by \textsc{D. Roegel} one +can program \TeX{} to compute with many digits at a much higher speed than +what \xintname achieves: but, direct access to memory storage in one form or +another seems a necessity for this kind of speed and one has to renounce at +the complete expandability.% +% +\footnote{2015/09/15: as I said the latest developments on the \liiibigint{} + side do not really modify this conclusion, because the computations remain + extremely slow compared to what one can do in other programming structures. Another remark one could do is that it would be tremendously easier to enhance \eTeX{} than it is to embark into writing hundreds of lines of sometimes very clever \TeX{} macro programming.} @@ -3712,9 +3631,16 @@ for \csa{numexpr} has ten digits). The present package is the result of this initial questioning. -% \begin{framed}\centering -% \xintname requires the \eTeX{} extensions. -% \end{framed} +For the record, \xintname 1.2 also got its impulse from a fast ``reversing'' +macro, which I wrote after my interest got awakened again as a result of +correspondance with Bruno \textsc{Le Floch}: this new reverse uses a \TeX nique +which \emph{requires} the tokens to be digits. I wrote a routine which works +(expandably) in quasi-linear time, but a less fancy |O(N^2)| variant which I +developed concurrently proved to be faster all the way up to perhaps $7000$ +digits, thus I dropped the quasi-linear one. The less fancy variant has the +advantage that \xintname can handle numbers with more than $19900$ digits (but +not much more than $19950$). This is with the current common values of the input +save stack and maximal expansion depth: $5000$ and $10000$ respectively. \subsection{Expansion matters} \label{sec:expansions} @@ -3893,47 +3819,195 @@ such expandable macros: creates the |\AplusBC| macro doing the above and expanding in two expansion steps. -\subsection{Efficiency; floating point macros} - -The size of the manipulated numbers is limited by two -factors:\footnote{there is an intrinsic limit of - \dtt{\number"7FFFFFFF} on the number of digits, but it is - irrelevant, in view of the other limiting factors.} (1.)~\emph{the -available memory as configured in the |tex| executable}, -(2.)~\emph{the \emph{time} necessary to fully expand the computations - themselves}. -I discovered progressively, during the first few weeks of developing the -package, that the most limiting factor is the second one, the time -needed for multiplication, division (even more for powers). It -explodes with increasing input sizes long before the computations could -get limited by constraints on \TeX's available memory: -computations with $100$ digits are still reasonably fast, but the -situation then deteriorates swiftly and multiplication with $1000$ -digits takes some seconds.% -% -\footnote{Perhaps some faster routines could emerge from an approach - which, while maintaining expandability would renounce at \fexpan - dability (without impacting the input save stack). There is one such - routine \csbxint{XTrunc} which is able to write to a file (or inside - an \csa{edef}) tens of thousands of digits of a (reasonably-sized) - fraction.} -% -\footnote{\textbf{2015/09/15}: faster routines for numbers with more than - (about) $100$ digits (for multiplication) are part of a new (experimental) - release \liiibigint{}. It succeeds in implementing Karatsuba multiplication - (division yet to be done) and this drastically improves the dependency of - time upon size. See \autoref{fn:l3bigint2015}. Some technical limitations - of \TeX{} (which can be lifted from recompiling the binary with changed - settings) currently limit the range to up to $8192$ digits. Nevertheless - multiplication with $5000$ digits takes about the same time as what - \xintcorename achieves for $1000$ digits, \emph{i.e.} some seconds on my - computer. Although the usable range of \liiibigint{} thus goes well into the - hundreds of digits, which improves greatly on the \xintcorename - implementation of multiplication (up to some overhead for numbers of a few - dozen digits), here also computation times become dissuasive far before the - \TeX{} main memory would be saturated.} - -To address this issue, floating +\subsection{User interface} +\label{ssec:userinterface} + +The user interface for executing operations on numbers is via macros such as +\csbxint{Add} or \csbxint{Mul} which have two arguments, or via expressions +\csbxint{expr}|..\relax| which use infix notations such as |+|, |-|, |*|, |/| +and |^| (or |**|) for the basic operations, and recognize functions of one or +more comma separated arguments (such as |max|, or |round|, or |sqrt|), +parentheses, logic operators of conjunction |&&|, disjunction \verb+||+, as +well as two-way |?| and three-way |??| conditionals and more. A few examples: +% +\begin{everbatim*} +\begin{enumerate}[nosep] + \item \xintiiAdd {2719873981798137981381789317981279}{13819093809180120910390190} + \item \xintiiMul {2719873981798137981381789317981279}{13819093809180120910390190} + \item \xintthefloatexpr (19317/21913+2198/9291)^3\relax + \item \xintDigits:=64;\xintthefloatexpr (19317/21913+2198/9291)^3\relax +% Let's compute the inner sum exactly, not as a float, before raising to third power: + \item \xintDigits:=16;\xintthefloatexpr \xintexpr 19317/21913+2198/9291\relax^3\relax +\end{enumerate} +\end{everbatim*} +In \csbxint{expr}|..\relax| the contents are expanded completely from left to +right until the ending |\relax| is found and swallowed, and spaces and even +(to some extent) catcodes do not matter. Algebraic operations are done +\emph{exactly}. + +The \csbxint{floatexpr} variant is for operations which are done using the +precision set via the |\xintDigits:=N;| assignment (default is with +significands of \dtt{16} digits). + +For all |\xintexpr| variants, prefixing with |\xintthe| allows to print +the result or use it in other contexts. Shortcuts \csbxint{theexpr}, +\csbxint{thefloatexpr}, \csbxint{theiiexpr}, \dots\ are available. + +The \csbxint{iiexpr} variant is only for big integers, it does not know +fractions. + +There are some important differences of syntax between |\numexpr| and +|\xintiiexpr| and variants: +\begin{itemize} +\item Contrarily to |\numexpr|, the |\xintiiexpr| parser will stop expanding + only after having encountered (and swallowed) a \emph{mandatory} |\relax| + token. +\item In particular, spaces between digits (and not only around infix + operators or parentheses) do not stop |\xintiiexpr|, contrarily to the + situation with |numexpr|: |\the\numexpr 7 + 3 5\relax| expands (in one step) + to \dtt{\detokenize\expandafter{\the\numexpr 7 + 3 5\relax}\unskip}, whereas + |\xintthe\xintiiexpr 7 + 3 5\relax| expands (in two steps) to + \dtt{\detokenize\expandafter\expandafter\expandafter {\xintthe\xintiiexpr 7 + + 3 5\relax}}. + \item Also worth mentioning is the fact that |\numexpr -(1)\relax| is + illegal. But this is perfectly legal and with the expected result in + |\xintiiexpr...\relax|. + \item Inside an |\edef|, expressions |\xintiiexpr...\relax| get fully + evaluated, but need the prefix |\xintthe| to get printed or used as + arguments to some macros, whereas expansion of |\numexpr| in an |\edef| + occurs only if prefixed with |\the| or |\number| (or |\romannumeral|, or + the expression is included in a bigger + |\numexpr| which will be the one to have to be prefixed\dots .) +\end{itemize} + +For macros such as \csbxint{Add} or +\csbxint{Mul} the arguments are each subjected to the process of \fexpan sion: +repeated expansion of the first token until finding something unexpandable (or +being stopped by a space token). + +Conversely this process of \fexpan sion always provokes the complete expansion +of the package macros and |\xintexpr..\relax| also will expand completely +under \fexpan sion, but to a private format; the \csbxint{the} prefix allows the +computation result either to be passed as argument to one of the package +macros,\footnote{the \csa{xintthe} prefix \fexpan ds the \csa{xintexpr}-ession + then unlocks it from its private format; it should not be used for + sub-expressions inside a bigger one as it is more efficient for the + expression parser to keep the result in the private format.} or also end up on +the printed page (or in an auxiliary file). +To recapitulate, all macros dealing with computations +\begin{enumerate} +\item \emph{expand completely under the sole process of repeated expansion of + the first token, (and two expansions suffice)},\footnote{see in + \autoref{sec:expansions} for more details.} + +\item \emph{apply this \fexpan sion to each one of their arguments.} +\end{enumerate} +Hence they can be nested one within the other up to arbitrary +depths. Conditional evaluations either within the macro arguments themselves, or +with branches defined in terms of these macros are made possible via macros such +as as \csbxint{ifSgn} or \csbxint{ifCmp}. + +\begin{framed} + There is no notion of \emph{declaration of a variable} to \xintname, + \xintfracname, or \xintexprname. + The user employs the |\def|, |\edef|, or + |\newcommand| (in \LaTeX) as usual, for example: \IMPORTANT +% +\begin{everbatim*} +\def\x{1729728} \def\y{352827927} \edef\z{\xintMul {\x}{\y}} +\meaning\z +\end{everbatim*}\ (see below for the |A/B[N]| output format; with |\xintiiMul| +in place of |\xintMul| there would not be the strange looking |/1[0]|.) + +The package provides |\oodef| which only expands twice its argument. This +provokes full expansion of the \xintname macros (nested to possibly many +levels), inclusive of |\xintexpr| and variants. However, it is typically slower +then |\edef| (and quite slower for small things) when the expansion ends up +consisting of less than about one thousand digits. The second utility next to +|\oodef| is |\fdef| which applies full expansion upfront and appears to be +competitive with and even faster than |\edef| already in the case of expansion +leading to a few dozen digits. +\end{framed} + +\begingroup % pour \z, \zz +The \xintexprname package has a private internal +representation for the evaluated computation result. With +% +\begin{everbatim*} +\edef\z {\xintexpr 3.141^18\relax} +\end{everbatim*} +% +the macro |\z| is already fully evaluated (two expansions were applied, and this +is enough), and can be reused in other |\xintexpr|-essions, such as for example +% +\begin{everbatim*} +\edef\zz {\xintexpr \z+1/\z\relax} + % (using short macro names such as \z and \zz is not too recommended in real + % life, some may have already definitions; I did it all in a group). +\end{everbatim*} +% +But to print it, or to use it as argument to one of the package macros, +it must be prefixed by |\xintthe| (a synonym for |\xintthe\xintexpr| is +\csbxint{theexpr}). Application of this |\xintthe| prefix outputs the +value in the \xintfracname semi-private internal format +|A/B[N]|,\footnote{there is also the notion of \csbxint{floatexpr}, for + which the output format after the action of \csa{xintthe} is a number in + floating point scientific notation.} representing the fraction +$(A/B)\times 10^N$. The |\zz| above produces a somewhat large output: +\begin{everbatim*} +\printnumber{\xintthe\zz }${}\approx{}$\xintFloat{\xintthe\zz} +\end{everbatim*} +\endgroup % pour \z, \zz + + \begin{framed} + By default, computations done by the macros of \xintfracname or within + |\xintexpr..\relax| are exact. Inputs containing decimal points or + scientific parts do not make the package switch to a `floating-point' mode. + The inputs, however long, are converted into exact internal representations. +% + % Floating point evaluations are done via special macros containing + % `Float' in their names, or inside |\xintfloatexpr|-essions. + + Manipulating exactly big fractions quickly leads to \dots bigger fractions. + There is a command \csbxint{Irr} (or the function |reduce| in an expression) + to reduce to smallest terms, but it has to be explicitely requested. Prior + to release |1.1| addition and subtraction blindly multiplied denominators; + they now check if one is a multiple of the other.\IMPORTANT\ But systematic + reduction of the result to its smallest terms would be too + costly.\def\everbatimindent{0pt } +\begin{everbatim*} +\xinttheexpr 27/25+46/50\relax\ is a bit simpler than \xinttheexpr (27*50+25*46)/(25*50)\relax, +but less so than \xinttheexpr reduce(27/25+46/50)\relax. And \xinttheexpr 3/75+4/50+2/100\relax\ +looks weird, but systematically reducing fractions would be too costly. +\end{everbatim*} + \end{framed} + +% +The |A/B[N]| shape is the output format of most \xintfracname macros, it +benefits from accelerated parsing when used on input, compared to the normal +user syntax which has no |[N]| part. An example of valid user input for a +fraction is +% +\leftedline{|-123.45602e78/+765.987e-123|} +% +where both the decimal parts, the scientific exponent parts, and the whole +denominator are optional components. The corresponding semi-private form in this +case would be +% +\leftedline{\xintRaw{-123.45602e78/+765.987e-123}} +% +The forward slash |/| is simply a delimiter to separate numerator and +denominator, in order to allow inputs having such denominators. + +Reduction to the irreducible form of the output must be asked for explicitely +via the \csbxint{Irr} macro or the |reduce| function within +|\xintexpr..\relax|. Elementary operations on fractions do very little of the +simplifications which could be obvious to (some) human beings. + + +\subsection{Floating point macros} + +Floating point macros are provided to work with a given arbitrary precision. The default size for significands is $16$ digits. Working with significands of $24$, $32$, $48$, $64$, or even $80$ digits is well within the reach of the package. But @@ -3947,6 +4021,22 @@ values up to $32767$.\footnote{for a one-shot conversion of a fraction to float overflow may occur if the exponents are a bit too close to the \TeX{} bound \dtt{$\pm$\number"7FFFFFFF}.} +\begin{framed} + Currently, the only transcendental operation is the square root + (\csbxint{FloatSqrt}). The elementary functions are not yet implemented. The + power function (\csbxint{FloatPow}, \csbxint{FloatPower}) accept only + (positive or negative) integer exponents. +\end{framed} + +\begin{framed} + Floating point multiplication of two numbers with |P| digits of precision + evaluates \emph{exactly} the exact product with |2P| or |2P-1| digits, + before rounding to |P| digits: obviously this is very wasteful when |P| is + large. But \xintname is initially an exact algebraic operator, not a + floating point one with a fixed maximal size for operands, and the author + hasn't yet had the opportunity to re-examine that point. +\end{framed} + Here is such a floating point computation: % \leftedline{|\xintFloatPower [48] {1.1547}{\xintiiPow {2}{35}}|} @@ -3993,25 +4083,40 @@ exponents. \section{User interface} -Maintaining complete expandability is not for the faint of heart as it excludes -doing macro definitions in the midst of the computation; in many cases, one does -not need complete expandability, and definitions are allowed. In such contexts, -there is no declaration for the user to be made to the package of a ``typed -variable'' such as a long integer, or a (long) fraction, or possibly an -|\xintexpr|-ession. Rather, the user has at its disposal the general tools of -the \TeX{} language: |\def| and |\edef|. In \LaTeX\ there is |\newcommand| as -wrapper to |\def|, -but \LaTeX\ chose not to provide an analogous wrapper for |\edef|. It can still -be used directly of course.\footnote{I don't know if \LaTeX3 will still allow - direct use of |\def| and |\edef|\dots} - -The \xinttoolsname package provides |\oodef| which expands twice the replacement -text\footnote{only for parameter less undelimited macros.}, hence forces -complete expansion when the top level of this replacement -text is a call to one of the \xintname bundle macros, its arguments being -themselves chains of such macros. There is also |\fdef| which will apply \fexpan -sion to the replacement text. Both are in such uses faster alternatives to -|\edef|. +{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents} + + +% Je ne sais pas ce que faisait ce paragraphe ici: +% +% Maintaining complete expandability is not for the faint of heart as it excludes +% doing macro definitions in the midst of the computation; in many cases, one does +% not need complete expandability, and definitions are allowed. In such contexts, +% there is no declaration for the user to be made to the package of a ``typed +% variable'' such as a long integer, or a (long) fraction, or possibly an +% |\xintexpr|-ession. Rather, the user has at its disposal the general tools of +% the \TeX{} language: |\def| and |\edef|. In \LaTeX\ there is |\newcommand| as +% wrapper to |\def|, +% but \LaTeX\ chose not to provide an analogous wrapper for |\edef|. It can still +% be used directly of course.\footnote{I don't know if \LaTeX3 will still allow +% direct use of |\def| and |\edef|\dots} + +%%%%%%%%%%%%% pas le bon endroit pour cette discussion +% The \xinttoolsname package provides |\oodef|, resp. |\fdef|, which expands +% twice, resp. fully (\fexpan sion), the replacement text\footnote{only for +% parameter less undelimited macros.}, hence forces complete expansion when +% this replacement text is a call to one of the \xintname bundle macros, its +% arguments being either explicit digits or further such macro calls. They are +% useful as sometimes one does not want |\edef| expansion, \emph{e.g.} with a +% macro such as \csbxint{Trim} which acts on lists of braced tokens which one +% might not want to see expanded. Furthermore |\fdef| appears to be faster than +% |\edef| in (non-trivial) situations already with only a few dozens of digits: I +% tested it to be a bit faster than |\edef| for expanding |\xintiiPow {2}{100}| +% (which has $31$ digits). However |\oodef| needs thousands of digits to become +% competitive.% +% % +% \footnote{earlier releases of this manual sort of suggested \csa{oodef} was +% competitive starting with a ``few hundred'' digits but that was perhaps a bit +% optimistic. The better choice is \csa{fdef}.} This section will explain the various inputs which are recognized by the package macros and the format for their outputs. Inputs have mainly five possible @@ -4046,13 +4151,8 @@ Outputs are mostly of the following types: or be used as argument to the package macros. \end{enumerate} -{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents} - \subsection {Input formats}\label{sec:inputs} -% \edef\z {\xintAdd -% {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}} - Some macro arguments are by nature `short' integers,\ntype{\numx} \emph{i.e.} less than (or equal to) in absolute value \np{\number "7FFFFFFF}. This is generally the case for arguments which serve to count or index something. They @@ -4065,13 +4165,13 @@ the allowed input formats for `long numbers' and `fractions' are: \begin{enumerate} \item the strict format\ntype{f} is for some macros of \xintname which only \fexpan d their arguments. After this \fexpan sion the input should be a - string of digits, optionally preceded by a unique minus sign. The first digit - can be zero only if the number is zero. A plus sign is not accepted. |-0| is - not legal in the strict format. A count register can serve as argument of such - a macro only if prefixed by |\the| or |\number|. Most macros of \xintname are - like \csbxint{Add} and accept the extended format described in the next item; - they may have a `strict' variant such as \csbxint{iiAdd} which remains - available even with \xintfracname loaded, for optimization purposes. + string of digits, optionally preceded by a unique minus sign. The first + digit can be zero only if the number is zero. A plus sign is not accepted. + |-0| is not legal in the strict format. A count register can serve as + argument of such a macro only if prefixed by |\the| or |\number|. Macros of + \xintname such as \csbxint{iiAdd} with a double |ii| require this `strict' + format for the inputs. The macros such as \csbxint{iAdd} with a single |i| + will apply the \csbxint{Num} normalizer described in the next item. \item the macro \csbxint{Num} normalizes into strict format an input having arbitrarily many minus and plus signs, followed by a string of zeroes, then @@ -4086,37 +4186,63 @@ the allowed input formats for `long numbers' and `fractions' are: \csbxint{Num}.% % \footnote{A - \LaTeX{} \texttt{\char 92value\{countername\}} is accepted as macro + \LaTeX{} |\value{countername}| is accepted as macro argument.} -\item the fraction format\ntype{\Ff} is what is expected by the macros of - \xintfracname: a fraction is constituted of a numerator |A| and optionally a - denominator |B|, separated by a forward slash |/| and |A| and |B| may be - macros which will be automatically given to \csbxint{Num}. Each of |A| and |B| - may be decimal numbers (the decimal mark must be a |.|). Here is an - example:% - % - \footnote{the square brackets one sees in various outputs are - explained - near the end of this section.} % - % - \leftedline{|\xintAdd - {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}|} - % - Scientific notation is accepted for both numerator and denominator of a - fraction, and is produced on output by \csbxint{Float}: - % - \begin{quote} - |\xintAdd{10.1e1}{101.010e3}|\dtt{=\xintAdd{10.1e1}{101.010e3}}\\ - % - |\xintFloatAdd{10.1e1}{101.010e3}|\dtt{=\xintFloatAdd{10.1e1}{101.010e3}}\\ - % - |\xintPow {2}{100}|\dtt{=\xintPow {2}{100}}\\ - % - |\xintFloat{\xintPow {2}{100}}|\dtt{=\xintFloat{\xintPow {2}{100}}}\\ - % - |\xintFloatPow {2}{100}|\dtt{=\xintFloatPow {2}{100}} - \end{quote} +\item the fraction format\ntype{\Ff} is what is expected on input by the + macros of \xintfracname. It has two variants: + \begin{description} + \item[general:] these are inputs of the shape |A.BeC/D.EeF|. Example: +\begin{everbatim*} +\noindent\xintRaw{+--0367.8920280e17/-++278.289287e-15}\newline +\xintRaw{+--+1253.2782e++--3/---0087.123e---5}\par +\end{everbatim*} + Notice that the input process does not reduce fractions to smallest terms. + Here are the rules of the format:\footnote{Earlier releases were slightly + more strict, the optional decimal parts |B|, |E| were not individually + \fexpan ded.} + \begin{itemize} + \item everything is optional, absent numbers are treated as zero, here are + some extreme cases: +\begin{everbatim*} +\xintRaw{}, \xintRaw{.}, \xintRaw{./1.e}, \xintRaw{-.e}, \xintRaw{e/-1} +\end{everbatim*} + \item |AB| and |DE| may start with pluses and minuses, then leading + zeroes, then digits. + \item |C| and |F| will be given to |\numexpr| and can be anything + recognized as such and not provoking arithmetic overflow (the lengths of + |B| and |E| will also intervene to build the final exponent naturally + which must obeys the \TeX{} bound). + \item the |/|, |.| (numerator and/or denominator) and |e| + (numerator and/or denominator) are all optional components. + \item each of |A|, |B|, |C|, |D|, |E| and |F| may arise from \fexpan sion + of a macro. + \item the whole thing may arise from \fexpan sion, however the |/|, |.|, + and |e| should all come from this initial expansion. The |e| of + scientific notation is mandatorily lowercased. + \end{itemize} + \item[restricted:] these are inputs either of the shape |A[N]| or |A/B[N]| + (representing the fraction |A/B| times |10^N|) where the whole thing or + each of |A|, |B|, |N| (but then not |/| or |[|) may arise from \fexpan + sion, |A| (after expansion) \emph{must} have a unique optional minus sign + and no leading zeroes, |B| (after expansion) if present \emph{must} be a + positive integer with no signs and no leading zeroes, |N| (which may be + empty) will be given to |\numexpr|. This format is parsed with smaller + overhead than the general one, thus allowing more efficient nesting of + macros as it is the one used on output (except for the floating macros). + Any deviation from the rules above will result in errors.\footnote{With + earlier releases the |N| could not be empty and had to be given as + explicit digits, not some macro or expression expanded in |\numexpr|.} + \end{description} + Examples of inputs and outputs: +\begin{everbatim*} + \xintAdd{+--0367.8920280/-++278.289287}{-109.2882/+270.12898}\newline + \xintAdd{10.1e1}{101.010e3}\newline + \xintFloatAdd{10.1e1}{101.010e3}\newline + \xintiiPow {2}{100}\newline + \xintPow {2}{100}\newline + \xintFloatPow {2}{100}\par +\end{everbatim*} % Produced fractions having a denominator equal to one are, as a general rule, nevertheless printed as fractions. In math mode \csbxint{Frac} @@ -4131,45 +4257,55 @@ the allowed input formats for `long numbers' and `fractions' are: \item the \hyperref[xintexpr]{expression format} is for inclusion in an \csbxint{expr}|...\relax|, it uses infix notations, function names, complete - expansion, and is described in \autoref{sec:expr11} and \autoref{sec:expr}. + expansion, recognizes decimal and scientific numbers, and is described in + \autoref{sec:expr11} and \autoref{sec:expr}.% +% +\footnote{The isolated dot |"."| is not legal anymore\MyMarginNote{Changed!} in expressions with + release |1.2|: there must be digits either before or after.} \end{enumerate} +Even with \xintfracname loaded, some macros by their nature cannot accept +fractions on input. Those parsing their inputs through \csbxint{Num} will now +accept fractions, truncating them first to integers. + +% The scientific notation is necessarily (except in |\xintexpr..\relax|) with a +% lowercase |e|. It may appear both at the numerator and at the denominator of a +% fraction. + Generally speaking, there should be no spaces among the digits in the inputs -(in arguments to the package macros). -Although most would be harmless in most macros, there are some cases -where spaces could break havoc. So the best is to avoid them entirely. +(in arguments to the package macros). Although most would be harmless in most +macros, there are some cases where spaces could break havoc.% +\footnote{The \csbxint{Num} macro does not remove spaces between digits beyond + the first non zero ones; however this should not really alter the subsequent + functioning of the arithmetic macros, and besides, since \xintcorename v1.2 + there is an initial parsing of the entire number, during which spaces will + be gobbled. However I have not done a complete review of the legacy code to + be certain of all possibilities after |v1.2| release. One thing to be aware + of is that \csa{numexpr} stops on spaces between digits (although it + provokes an expansion to see if an infix operator follows); the exponent for + \csbxint{iiPow} or the argument of the factorial \csbxint{iFac} are only + subjected to such a \csa{numexpr} (there are a few other macros with such + input types in \xintname). If the input is given as, say |1 2\x| where + \csa{x} is a macro, the macro \csa{x} will not be expanded by the + \csa{numexpr}, and this will surely cause problems afterwards. Perhaps a + later \xintname will force \csa{numexpr} to expand beyond spaces, but I + decided that was not really worth the effort. Another immediate cause of + problems is an input of the type |\xintiiAdd{<space>\x}{\y}|, because the + space will stop the initial expansion; this will most certainly cause an + arithmetic overflow later when the \csa{x} will be expanded in a + \csa{numexpr}. Thus in conclusion, damages due to spaces are unlikely if + only explicit digits are involved in the inputs, or arguments are single + macros with no preceding space.} +% +% j'avais oublié que mon |...| savait gérer les \ dans les footnote pas besoin +% de \char92 ou autre! +% +So the best is to avoid them entirely. This is entirely otherwise inside an |\xintexpr|-ession, where spaces are ignored (except when they occur inside arguments to some macros, thus escaping the |\xintexpr| parser). See the \hyperref[sec:expr]{documentation}. -Even with \xintfracname loaded, some macros by their nature can not accept -fractions on input. Those parsing their inputs through \csbxint{Num} will now -accept fractions, truncating them first to integers. - -With \xintfracname loaded, a number may be empty or start directly with a -decimal point: -\begin{quote} - |\xintRaw{}=\xintRaw{.}|\dtt{=\xintRaw{}}\\ - |\xintPow{-.3/.7}{11}|\dtt{=\xintPow{-.3/+.7}{11}}\\ - |\xinttheexpr (-.3/.7)^11\relax|\dtt{=\xinttheexpr (-.3/.7)^11\relax} -\end{quote} -It is also licit to use |\A/\B| as -input if each of |\A| and |\B| expands (in the sense previously described) to a -``decimal number'' as examplified above by the numerators and denominators -(thus, possibly with a `scientific' exponent part, with a lowercase `e'). Or one -may have just one macro |\C| which expands to such a ``fraction with optional -decimal points'', or mixed things such as |\A 245/7.77|, where the numerator -will be the concatenation of the expansion of |\A| and |245|. But, as explained -already |123\A| is a no-go, \emph{except inside an |\xintexpr|-ession}! - -The scientific notation is necessarily (except in |\xintexpr..\relax|) with a -lowercase |e|. It may appear both at the numerator and at the denominator of a -fraction. -% -\leftedline{|\xintRaw {+--+1253.2782e++--3/---0087.123e---5}|\dtt{=\xintRaw - {+--+1253.2782e++--3/---0087.123e---5}}} - Arithmetic macros of \xintname which parse their arguments automatically through \csbxint{Num} are signaled by a special symbol%\ntype{\Numf{\unskip\kern\dimexpr\FrameSep+\FrameRule\relax}} @@ -4182,79 +4318,39 @@ contain to some extent infix algebra with count registers, see the section full fraction format with no restriction there is the corresponding symbol in the margin\ntype{\Ff}. -The \xintfracname macros generally output -their result in |A/B[n]| format, representing the fraction |A/B| times |10^n|. - -This format with a trailing |[n]| (possibly, |n=0|) is accepted on input -but it presupposes that the numerator and denominator |A| and |B| are in -the strict integer format described above. So |16000/289072[17]| or -|3[-4]| are authorized and it is even possible to use |\A/\B[17]| if -|\A| expands to |16000| and |\B| to |289072|, or |\A| if |\A| expands to -|3[-4]|. However, NEITHER the numerator NOR the denominator may then -have a decimal point\IMPORTANT{}. And, for this format, ONLY the -numerator may carry a UNIQUE minus sign (and no superfluous leading -zeroes; and NO plus sign). - -It is allowed for user input but the parsing is minimal and it is mandatory to -follow the above rules. This reduced flexibility, compared to the format without -the square brackets, allows nesting package macros without too much speed -impact. - \subsection{Output formats} -With package \xintfracname loaded, the routines \csbxint{Add}, \csbxint{Sub}, -\csbxint{Mul}, \csbxint{Pow}, initiallly synonyms in \xintname of -\csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow}, are modified -to become the fraction handling routines.% -% -\footnote{the power function does not accept a fractional - exponent. Or rather, does not expect, and errors will result if one is - provided.} -% -\footnote{as commented upon more later, for that very reason use of - \csbxint{Add} etc\dots when only \xintname is loaded is strongly - discouraged.}\,% -% -% \footnote{macros \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, -% \csbxint{iPow}, are the original ones dealing only with integers. They are -% available as synonyms, also when \xintfracname is not loaded. With -% \xintfracname loaded they accept on input also fractions, which they first -% truncate to integers, and then the output format is the integer one. The macros \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, -% \csbxint{iiPow}, \csbxint{iiSum}, \csbxint{iiPrd} are strictly integer-only: -% they skip the overhead of parsing their arguments via -% \csbxint{Num}.}\,% -% -% \footnote{also \csbxint{Cmp}, \csbxint{Sgn}, \csbxint{Geq}, -% \csbxint{Opp}, \csbxint{Abs}, \csbxint{Max}, \csbxint{Min} are extended to -% fractions; and the last four have the integer-only variants \csbxint{iOpp}, -% \csbxint{iAbs}, \csbxint{iMax}, \csbxint{iMin}.}\,% -% % -% \footnote{and \csbxint{Fac}, -% \csbxint{Quo}, \csbxint{Rem}, \csbxint{Division}, \csbxint{FDg}, -% \csbxint{LDg}, \csbxint{Odd}, \csbxint{MON}, \csbxint{MMON} all accept a -% fractional input as long as it reduces to an integer.} -% -They produce on output a -fractional number |f=A/B[n]| where |A| and |B| are integers, with |B| positive, -and |n| is a ``short'' integer. -% -% (\emph{i.e} less in absolute value than |2^{31}-9|). +Package \xintcorename provides macros \csbxint{iiAdd}, \csbxint{iiSub}, +\csbxint{iiMul}, \csbxint{iiPow}, which only \fexpan d their arguments and +\csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow} which +normalize them first to strict format, thus have a bit of overhead. These +macros always produce integers on output. + +With \xintfracname loaded \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, +... are not modified, and \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, ... +are only extended to the extent of accepting fraction inputs but they will be +truncated to integers.% % -This represents |(A/B)| times |10^n|. +\footnote{the power function does not accept a fractional exponent. Or rather, + does not expect, and errors will result if one is provided.} % -\footnote{at each stage of the - computations, the sum of |n| and the length of |A|, or of the absolute value - of |n| and the length of |B|, must be kept less than - |2^{31}-9|.} +The output will be an integer. \begin{framed} - The fraction output format for most \xintfracname macros is {|A/B[n]|} which - stands for |(A/B)|$\times$|10^n|. The |A| and |B| may end in zeroes - (\emph{i.e}, |n| does not represent all powers of ten), and will generally - have a common factor. The denominator |B| is always strictly positive. - Conversely, this format is accepted on input and is parsed more quickly than - fractions containing decimal points or in scientific notation; the input - denominator is optional. + The fraction handling macros from \xintfracname are called \csbxint{Add}, + \csbxint{Sub}, \csbxint{Mul}, etc... they are \emph{not} defined in the + absence of \xintfracname.\MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{Changed!} + + They produce on output a fractional number |f=A/B[n]| (which stands for + |(A/B)|$\times$|10^n|) where |A| and |B| are integers, with |B| positive, + and |n| is a ``short'' integer (\emph{i.e} less in absolute value than + \dtt{\number"7FFFFFFF}.) + + The output fraction is not reduced to smallest terms. The |A| and |B| may + end in zeroes (\emph{i.e}, |n| does not represent all powers of ten). The + denominator |B| is always strictly positive. There is no |+| sign on output + but only possibly a |-| at the numerator. The output will be expressed as + a fraction even if the inputs are both integers. \end{framed} \begin{itemize} @@ -4291,8 +4387,8 @@ $\xintIrr{273.3734e5/3395.7200e-2}$ \item The \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, \csbxint{iiPow}, and others with `\textcolor{blue}{ii}' in their names accept on input only - integers in the strict format (they skip the overhead of the \csbxint{Num} - parsing) and naturally they output an integer, with no fraction slash nor + integers in the strict format. They skip the overhead of the \csbxint{Num} + parsing and naturally they output integers, with no fraction slash nor trailing |[n]|. \end{itemize} @@ -4658,7 +4754,7 @@ naturally! (but \dtt{F(\xintiiPow2{31}}) would be rather big anyhow...). In practice, whenever one typesets things, one has left the expansion only contexts; hence there is no objection to, on the contrary it is recommended, assign the result of earlier computations to macros via an |\edef| (or an -|\oodef|, see \ref{oodef}), for later use. The above could thus be coded +|\fdef|, see \ref{fdef}), for later use. The above could thus be coded \begin{everbatim} \begingroup \def\A {1859} \def\B {1573} \edef\C {\xintiiGCD\A\B} @@ -4700,28 +4796,33 @@ others are more annoying as they may pass through unsignaled. \item using one pair of braces too many |\xintIrr{{\xintiPow {3}{13}}/243}| (the computation goes through with no error signaled, but the result is completely wrong). +\item things like |\xintiiAdd { \x}{\y}| as the space will cause \csa{x} to be + expanded later, most probably within a |\numexpr| thus provoking possibly an + arithmetic overflow. \item using |[]| and decimal points at the same time |1.5/3.5[2]|, or with a sign in the denominator |3/-5[7]|. The scientific notation has no such restriction, the two inputs |1.5/-3.5e-2| and |-1.5e2/3.5| are equivalent: |\xintRaw{1.5/-3.5e-2}|\dtt{=\xintRaw{1.5/-3.5e-2}}, |\xintRaw{-1.5e2/3.5}|\dtt{=\xintRaw{-1.5e2/3.5}}. -\item specifying numerators and - denominators with macros producing fractions when \xintfracname is loaded: - |\edef\x{|\allowbreak|\xintMul {3}{5}/\xintMul{7}{9}}|. This expands to - \texttt{\x} which is - invalid on input. Using this |\x| in a fraction macro will most certainly - cause a compilation error, with its usual arcane and undecipherable - accompanying message. The fix here would be to use |\xintiMul|. The simpler - alternative with package \xintexprname: - |\xinttheexpr 3*5/(7*9)\relax|. +% \item specifying numerators and +% denominators with macros producing fractions when \xintfracname is loaded: +% |\edef\x{|\allowbreak|\xintMul {3}{5}/\xintMul{7}{9}}|. This expands to +% \texttt{\x} which is +% invalid on input. Using this |\x| in a fraction macro will most certainly +% cause a compilation error, with its usual arcane and undecipherable +% accompanying message. The fix here would be to use |\xintiMul|. The simpler +% alternative with package \xintexprname: +% |\xinttheexpr 3*5/(7*9)\relax|. \item generally speaking, using in a context expecting an integer (possibly restricted to the \TeX{} bound) a macro or expression which returns a fraction: |\xinttheexpr 4/2\relax| outputs \dtt{\xinttheexpr 4/2\relax}, not $2$. Use |\xintNum {\xinttheexpr 4/2\relax}| or |\xinttheiexpr 4/2\relax| (which rounds the result to the nearest integer, here, the result is already - an integer) or |\xinttheiiexpr 4/2\relax|. -\item use of square brackets |[|, |]| in |\xintexpr...\name| has some traps, see - \autoref{sec:expr}. + an integer) or |\xinttheiiexpr 4/2\relax|. Or, divide in your head |4| by + |2| and insert the result directly in the \TeX{} source. +% trop technique +% \item use of square brackets |[|, |]| in |\xintexpr...\name| has some traps, see +% \autoref{sec:expr}. \end{itemize} \subsection{Error messages} @@ -4772,6 +4873,29 @@ sequences: % par ailleurs il y a trop d'espace vertical avant le multicols, mais % bon. +There are now a few more if for example one attempts to use |\xintAdd| without +having loaded \xintfracname (with only \xintname loaded, only |\xintiAdd| and +|\xintiiAdd| are legal).\MyMarginNote{Changed!} +\begin{multicols}{2}\parskip0pt\relax +\begin{everbatim} +\Did_you_mean_iiAbs?or_load_xintfrac +\Did_you_mean_iiOpp?or_load_xintfrac +\Did_you_mean_iiAdd?or_load_xintfrac +\Did_you_mean_iiSub?or_load_xintfrac +\Did_you_mean_iiMul?or_load_xintfrac +\Did_you_mean_iiPow?or_load_xintfrac +\Did_you_mean_iiSqr?or_load_xintfrac +\Did_you_mean_iiMax?or_load_xintfrac +\Did_you_mean_iiMin?or_load_xintfrac +\Did_you_mean_iMaxof?or_load_xintfrac +\Did_you_mean_iMinof?or_load_xintfrac +\Did_you_mean_iiSum?or_load_xintfrac +\Did_you_mean_iiPrd?or_load_xintfrac +\Did_you_mean_iiPrdExpr?or_load_xintfrac +\Did_you_mean_iiSumExpr?or_load_xintfrac +\end{everbatim} +\end{multicols} + Don't forget to set |\errorcontextlines| to at least |2| to get from \LaTeX\ more meaningful error messages. Errors occuring during the parsing of |\xintexpr-essions| try to provide helpful information about the offending @@ -4786,9 +4910,11 @@ daring experienced \TeX/\LaTeX\ user. \subsection{Package namespace, catcodes} -The \xintname bundle packages presuppose that the \csa{space}, \csa{empty} and -|\m@ne| control sequences are pre-defined with meanings as in Plain -\TeX{} or \LaTeX2e. +% note: v1.2 définit \m@ne si ce count n'existe pas. + +The bundle packages needs that the \csa{space} and \csa{empty} control +sequences are pre-defined with the identical meanings as in Plain \TeX{} or +\LaTeX2e. Private macros of \xintkernelname, \xintcorename, \xinttoolsname, \xintname, \xintfracname, \xintexprname, \xintbinhexname, \xintgcdname, @@ -4945,7 +5071,7 @@ $2^{100}$ (=\xintiPow {2}{100}) has \xintLen{\xintiPow {2}{100}} digits and the \end{everbatim*} It would be more efficient to do once and for all -|\oodef\z{\xintiPow {2}{100}}|, and then use |\z| in place of +|\edef\z{\xintiPow {2}{100}}|, and then use |\z| in place of |\xintiPow {2}{100}| everywhere as this would spare the CPU some repetitions. Expandably computing primes is done in \autoref{xintSeq}. @@ -4979,434 +5105,6 @@ expandability. Check it out (\autoref{xintiloop}). % \noindent\kern\parindent\input README.md % \endgroup }\x -\section{New features of the \xintexprname package} -\label{sec:expr11} - -Release |1.1| brought many changes to \xintexprname. This chapter is -for people already familiar with earlier versions. A more systematic -item per item syntax description is provided in \autoref{sec:expr}. But be -sure to come back here as the latter chapter has not been compeletely updated. - -First, there were some breaking changes: -\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, - itemindent=\leftmarginii, leftmargin=0em] - \item in |\xintiiexpr|, |/| does \emph{rounded} division, rather than as - in earlier releases the - Euclidean division (for positive arguments, this is truncated division). - The new |//| operator does truncated division, - \item the |:| operator for three-way branching is gone, replaced with |??|, - \item |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 postfix operators, - \item the |add| and |mul| have a new syntax, old syntax is with |`+`| and - |`*`| (quotes mandatory), |sum| and |prd| are gone, - \item 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 space-stripped in a |\numexpr|, it is not parsed by the - |\xintexpr|-ession scanner). - \item although |&| and \verb+|+ are still available as Boolean operators the - use of |&&| and \verb+||+ is strongly recommended. The single - letter operators might be assigned some other meaning in later releases - (bitwise operations, perhaps). Do not use them. - \item place holders for |\xintNewExpr| - could be denoted |#1|, |#2|, ... or also, for special purposes |$1|, |$2|, - ... Only the first form is now accepted and the special cases previously - treated via the second form are now managed via a |protect(...)| function. -\end{itemize} - -Let's now describle some of the numerous additional functionalities. - -\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, - itemindent=\leftmarginii, leftmargin=0em] - \item |\xintiexpr|, |\xinttheiexpr| admit an optional argument within - brackets |[d]|, it then presents the computation result (or results, if - comma separated) after rounding to |d| digits after decimal mark, (the - whole computation is done exactly, as in |xintexpr|), -\begin{everbatim*} -\xinttheiexpr [32] 1.23^50, 1.231^50\relax -\end{everbatim*} - \item |\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|), -\begin{everbatim*} -\xintDigits:=32;\xintthefloatexpr 1.010101^10-1.0101^10\relax - -\xintDigits:=16;\xintthefloatexpr 1.010101^10-1.0101^10\relax - -\xintthefloatexpr [12] 1.010101^10-1.0101^10\relax -\end{everbatim*} - - \item |\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, - \item the |//| does truncated division and |/:| is the associated modulo, - \item multi-character operators |&&|, \verb+||+, |==|, |<=|, |>=|, |!=|, - |**|, - \item multi-letter infix binary words |'and'|, |'or'|, |'xor'|, |'mod'| - (quotes mandatory), - \item functions |even|, |odd|, |first|, |last|, - \item |\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), - \item generation of comma separated lists |a..b|, |a..[d]..b|, - \item Python syntax-like list extractors |[list][n:]|, |[list][:n]|, |[list][a:b]| - and |[list][n]| (|n=0| for the number of list items), the step is always - |+1|, - \item function |reversed|, to reverse the order of list items, - \item itemwise sequence operations |a*[list]|, etc.., on both sides |a*[list]^b|, - \item dummy variables in |add| and |mul|: |add(x(x+1)(x-1), x=-10..10)|, - \item variable substitutions with |subs|: |subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)|, - \item sequence generation using |seq| with a dummy variable: |seq(x^3, x=-10..10)|, - \item simple recursive sequences with |rseq|, with |@| given the last value, - |rseq(1;2@+1,i=1..10)|, - \item 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)|, - \item iteration with |iter| which is like |rrseq| but outputs only the - last |K| terms, where |K| was the number of initial terms, - \item inside |seq|, |rseq|, |rrseq|, |iter|, possibility to use |omit|, - |abort| and |break| to control termination, - \item |n++| potentially infinite index generation for |seq|, |rseq|, - |rrseq|, and |iter|, it is advised to use |abort| or |break(..)| at - some point, - \item the |add|, |mul|, |seq|, ... are nestable, - \item |\xintthecoords| converts a comma separated list of an even number - of items to the format as expected by the |TikZ| |coordinates| syntax, - \item completely rewritten |\xintNewExpr|, new |protect| function to handle - external macros. However not all constructs are compatible with - |\xintNewExpr|. -\end{itemize} - -% \subsection{Examples with the \texorpdfstring{\unexpanded{\unexpanded{|v1.1|}}}{v1.1} \csh{xintexpr}} - -And now some examples: - -\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, - itemindent=\leftmarginii, leftmargin=0em] -\item One can define variables (the definition itself is a non expandable - step). The allowed names are composed of letters, digits, and underscores. - The variable should not start with a digit and single letters |a..z|, |A..Z| - are predefined for use as dummy variables --- see below. The |@| is - reserved. -\begin{everbatim*} -\begingroup - \xintdefvar a_1 := 3.14159;\xintdefvar a2 := 2.71828;\xinttheiexpr [5] a_1+a2\relax -\endgroup -\end{everbatim*} -\item |add| and |mul| have a new syntax requiring a dummy variable: -\begin{everbatim*} -\xinttheexpr add(x, x=1,3,19), mul(x^2, x=1,3,19), add(x(x+1), x= 1,3,19)\relax -\end{everbatim*} - -Use |`+`| and |`*`| (left ticks mandatory) for syntax without dummy variables: -\begin{everbatim*} -\xinttheexpr `+`(1,3,19), `*`(1^2,3^2,19^2), `+`(1*2,3*4,19*20)\relax -\end{everbatim*} -\item The |seq| function generates sequences according to a given formula: -\begin{everbatim*} -\xinttheexpr seq(x(x+1)(x+2), x=1,3,19), `+`(seq(x(x+1)(x+2), x=1,3,19)), - add(x(x+1)(x+2), x=1,3,19)\relax -\end{everbatim*} -\begin{everbatim*} -And this is nestable! -\xinttheexpr seq(seq(x^y, y=1..5),x=1..5), add(mul(x^y,y=1..5), x=1..5), - add(x^15, x=1..5)\relax % 15 = 1+2+3+4+5 -\end{everbatim*} - -One should use parentheses appropriately. The \csbxint{expr} parser in normal -operation is not bad at identifying missing or extra opening or closing -parentheses, but when it handles |seq|, |add|, |mul| or similar constructs it -switches to another mode of operation (it starts using delimited macros, -something which is almost non-existent in all its other operations) and ill-formed -expressions are much more likely to let the parser fetch tokens from beyond the -mandatory ending |\relax|. Thus, in case of a missing parenthesis in such -circumstances the error message from \TeX{} might be very cryptic, even for -the seasoned \xintname user. - -\item As seen in the last example |a..b| constructs the integers from |a| to - |b|. This is (small) integer only. A more general |a..[d]..b| works with big - integers, or fractions, from |a| to |b| with step |d|. -\begin{everbatim*} -\xinttheexpr seq(2x+1, x=1..10, 100..110, 3/5..[1/5]..7/5)\relax -\end{everbatim*} -\item itemwise operations on lists are possible, as well as item extractions: -\begin{everbatim*} -\xinttheiiexpr 2*[1,10,100]^3, 5+[2*[1,10,100]^3]*100 \relax -\end{everbatim*} -\begin{everbatim*} -\xinttheiiexpr 1+[seq(3^j, j=1..10, 21..30)][17], 1+3^27\relax -\end{everbatim*} - -We used the |[list][n]| construct which gives the nth item from the list. In -this context there are also the functions |last| and |first|. There is no real -concept of a list object, nor list operations, although itemwise manipulation -are made possible as shown above via the |[..]| constructor. The list -manipulation utilities are so far a bit limited. There is no notion of an -``nuple'' object. The variable |nil| is reserved, it represents an empty list. - -\item |subs| is similar to |seq| in syntax but is for variable substitution: -\begin{everbatim*} -\xinttheexpr subs(100*subs(10*subs(3*x+5,x=y+50)+2,y=z^2),z=10)\relax % 100(10(3*150+5)+2) -\end{everbatim*} -\begin{everbatim*} -\xinttheexpr subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)\relax -\end{everbatim*} - -The substituted variable may be a comma separated list (this is impossible -with |seq| which will always pick one item after the other of a list). -\begin{everbatim*} -\xinttheexpr subs([x]^2,x=-123,17,32)\relax -\end{everbatim*} - -\item last but not least, |seq| has variants |rseq| and |rrseq| which allow - recursive definitions. They start with at least one initial value, then a - semi-colon, then the formula, then the list of indices to iterate over. |@| - (or |@1|) evaluates to the last computed item, and |rrseq| keeps the memory - of the |K| last results, where |K| was the number of initial terms. One - accesses them via |@1, @2, @3, @4| and |@@(N)| for |N>4|. It is even - possible to nest them and use |@@@| to access the values of the master - recursion... -\begin{everbatim*} -\xinttheiiexpr rseq(1; 2*@, i=1..10), `+`(rseq(1; 2*@, i=1..10))\relax -\end{everbatim*} -\begin{everbatim*} -\xinttheiiexpr rseq(2; @(@+1)/2, i=1..5)\relax -\end{everbatim*} -\begin{everbatim*} -\xinttheiiexpr rseq(0,1; (@1,add(x,x=@1)), y=2..10)\relax -\end{everbatim*} - -Some Fibonacci fun -\begin{everbatim*} -\xinttheiiexpr rrseq(0,1; @1+@2, x=2..10), last(rrseq(0,1; @1+@2, x=2..100))\relax -\end{everbatim*} -\begin{everbatim*} -Sum of previous last three: \xinttheiiexpr rrseq(0,0,1; @1+@2+@3, i=1..20)\relax -\end{everbatim*} -\begin{everbatim*} -Big numbers: \printnumber{\xinttheexpr rseq(1; @(@+1), j=1..10)\relax } -\end{everbatim*} - -Nested recursion often quickly leads to gigantic outputs. This is an -experimental feature, susceptible to be removed or altered in the future. -\begin{everbatim*} -\xinttheexpr rrseq(1; `+`(rrseq(0,1; @@@(1)+@1+@2, i=1..10)), j=1..5)\relax -\end{everbatim*} - -\item The special keywords |omit|, |abort| and |break(..)| are available - inside |seq|, |rseq|, |rrseq|, as well as the |n++| for a potentially - infinite iteration. The |n++| construct in conjunction with an |abort| or - |break| is often more efficient, because in other cases the list to iterate - over is first completely constructed. -\begin{everbatim*} -First Fibonacci number at least |2^31| and its index -\xinttheiiexpr iter(0,1; (@1>=2^31)?{break(i)}{@2+@1}, i=1++)\relax -\end{everbatim*} - -\begin{everbatim*} -Prime numbers are always cool -\xinttheiiexpr seq((seq((subs((x/:m)?{(m*m>x)?{1}{0}}{-1},m=2n+1)) - ??{break(0)}{omit}{break(1)},n=1++))?{x}{omit}, - x=10001..[2]..10200)\relax -\end{everbatim*} - -The syntax in this last example may look a bit involved. First |x/:m| computes -|x modulo m| (this is the modulo with respect to truncated division, which -here for positive arguments is like Euclidean division; in -|\xintexpr...\relax|, |a/:b| is such that |a = b*(a//b)+a/:b|, with |a//b| the -algebraic quotient |a/b| truncated to an integer.). The |(x)?{yes}{no}| -construct checks if |x| (which \emph{must} be within parentheses) is true or -false, i.e. non zero or zero. It then executes either the |yes| or the |no| -branch, the non chosen branch is \emph{not} evaluated. Thus if |m| divides |x| -we are in the second (``false'') branch. This gives a |-1|. This |-1| is the -argument to a |??| branch which is of the type |(y)??{y<0}{y=0}{y>0}|, thus here -the |y<0|, i.e., |break(0)| is chosen. This |0| is thus given to another |?| -which consequently chooses |omit|, hence the number is not kept in the list. -The numbers which survive are the prime numbers. - -% A006877 In the `3x+1' problem, these values for the starting value set new -% records for number of steps to reach 1. (Formerly M0748) 14 1, 2, 3, 6, 7, -% 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, -% 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, -% 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, -% 626331, 837799 - -\item The |iter| function is like |rrseq| but does not leave a trace of earlier items, - it starts with |K| initial values, then it iterates: either a fixed number of times, - or until aborting or breaking. And ultimately it prints |K| final values. -\begin{everbatim*} -The first Fibonacci number beyond the \TeX{} bound is -\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^31)\relax{} -and the previous number was its index. -\end{everbatim*}But this was a bit too easy, what is the smallest Fibonacci number not representable on 64 bits? -\begin{everbatim*} -The first Fibonacci number beyond |2^64| bound is -\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^64)\relax{} -and the previous number was its index. -\end{everbatim*} - -One more recursion: -\begin{everbatim*} -\def\syr #1{\xinttheiiexpr rseq(#1; (@<=1)?{break(i)}{odd(@)?{3@+1}{@//2}},i=0++)\relax} -The 3x+1 problem: \syr{231}\par -\end{everbatim*} - -Ok, a final one: -\begin{everbatim*} -\def\syrMax #1{\xinttheiiexpr iter(#1,#1;even(i)? - {(@2<=1)?{break(i/2)}{odd(@2)?{3@2+1}{@2//2}}} - {(@1>@2)?{@1}{@2}},i=0++)\relax } -With initial value 1161, the maximal number attained is \syrMax{1161} and that latter -number is the number of steps which was needed to reach 1.\par -\end{everbatim*} - -Well, one more: - -\begin{everbatim*} -\newcommand\GCD [2]{\xinttheiiexpr rrseq(#1,#2; (@1=0)?{abort}{@2/:@1}, i=1++)\relax } -\GCD {13^10*17^5*29^5}{2^5*3^6*17^2} -\end{everbatim*} - -and the ultimate: - -\begin{everbatim*} -\newcommand\Factors [1]{\xinttheiiexpr - subs(seq((i/:3=2)?{omit}{[L][i]},i=1..([L][0])), - L=rseq(#1;(p^2>[@][1])?{([@][1]>1)?{break(1,[@][1],1)}{abort}} - {(([@][1])/:p)?{omit} - {iter(([@][1])//p; (@/:p)?{break(@,p,e)}{@//p},e=1++)}},p=2++))\relax } -\Factors {41^4*59^2*29^3*13^5*17^8*29^2*59^4*37^6} -\end{everbatim*} - -This might look a bit scary, I admit. \xintexprname has minimal tools and -is obstinate about doing everything expandably! We are hampered by absence of a -notion of ``nuple''. The algorithm divides |N| by |2| until no more possible, -then by |3|, then by |4| (which is silly), then by |5|, then by |6| (silly -again), \dots. - -The variable |L=rseq(#1;...)| expands, if one follows the steps, to a comma -separated list starting with the initial (evaluated) |N=#1| and then -pseudo-triplets where the first item is |N| trimmed of small primes, the -second item is the last prime divisor found, and the third item is its -exponent in original |N|. - -The algorithm needs to keep handy the last computed quotient by prime powers, -hence all of them, but at the very end it will be cleaner to get rid of them -(this corresponds to the first line in the code above). This is achieved in a -cumbersome inefficient way; indeed each item extraction |[L][i]| is costly: it -is not like accessing an array stored in memory, due to expandability, nothing -can be stored in memory! Nevertheless, this step could be done here in a far -less inefficient manner if there was a variant of |seq| which, in the spirit -of \csbxint{iloopindex}, would know how many steps it had been through so far. -This is a feature to be added to |\xintexpr|! (as well as a |++| construct -allowing a non unit step). - -Notice that in |iter(([@][1])//p;| the |@| refers to the previous triplet (or -in the first step to |N|), but the latter |@| showing up in |(@/:p)?| refers -to the previous value computed by |iter|. - -\begin{snugframed} - Parentheses are essential in |..([y][0])| else the parser will see |..[| and - end up in ultimate confusion, and also in |([@][1])/:p| else the parser will - see the itemwise operator |]/| on lists and again be very confused (I could - implement a |]/:| on lists, but in this situation this would also be very - confusing to the parser.) -\end{snugframed} - -For comparison, here is an \fexpan dable macro expanding to the same result, -but coded directly with the \xintname macros. Here |#1| can not be itself an -expression, naturally. But at least we let |\Factorize| \fexpan d its -argument. -\begin{everbatim} -\makeatletter -\newcommand\Factorize [1] - {\romannumeral0\expandafter\factorize\expandafter{\romannumeral-`0#1}}% -\newcommand\factorize [1]{\xintiiifOne{#1}{ 1}{\factors@a #1.{#1};}}% -\def\factors@a #1.{\xintiiifOdd{#1} - {\factors@c 3.#1.}% - {\expandafter\factors@b \expandafter1\expandafter.\romannumeral0\xinthalf{#1}.}}% -\def\factors@b #1.#2.{\xintiiifOne{#2} - {\factors@end {2, #1}}% - {\xintiiifOdd{#2}{\factors@c 3.#2.{2, #1}}% - {\expandafter\factors@b \the\numexpr #1+\@ne\expandafter.% - \romannumeral0\xinthalf{#2}.}}% -}% -\def\factors@c #1.#2.{% - \expandafter\factors@d\romannumeral0\xintiidivision {#2}{#1}{#1}{#2}% -}% -\def\factors@d #1#2#3#4{\xintiiifNotZero{#2} - {\xintiiifGt{#3}{#1} - {\factors@end {#4, 1}}% ultimate quotient is a prime with power 1 - {\expandafter\factors@c\the\numexpr #3+\tw@.#4.}}% - {\factors@e 1.#3.#1.}% -}% -\def\factors@e #1.#2.#3.{\xintiiifOne{#3} - {\factors@end {#2, #1}}% - {\expandafter\factors@f\romannumeral0\xintiidivision {#3}{#2}{#1}{#2}{#3}}% -}% -\def\factors@f #1#2#3#4#5{\xintiiifNotZero{#2} - {\expandafter\factors@c\the\numexpr #4+\tw@.#5.{#4, #3}}% - {\expandafter\factors@e\the\numexpr #3+\@ne.#4.#1.}% -}% -\def\factors@end #1;{\xintlistwithsep{, }{\xintRevWithBraces {#1}}}% -\makeatother -\end{everbatim} - -The macro |\Factorize| puts a little stress on the input save stack in order -not be bothered with previously gathered things. I timed it to be about eight -times faster than |\Factors| in test cases such as -|16246355912554185673266068721806243461403654781833| and others. Among the -various things explaining the speed-up, there is fact that we step by -increments of two, not one, and also that we divide only once, obtaining -quotient and remainder in one go. These two things already make for a speed-up -factor of about four. Thus, our earlier |\Factors| was not completely -inefficient, and was quite easier to come up with than |\Factorize|. - -If we only considered small integers, we could write pure |\numexpr| methods -which would be very much faster (especially if we had a table of small primes -prepared first) but still ridiculously slow compared to any non expandable -implementation, not to mention use of programming languages directly accessing -the CPU registers\dots -\end{itemize} - -%\phantomsection -\phantomsection\label{sec:expr11coords} - -To conclude with this overview of the new features in \xintexprname |1.1|, I -will mention {\bfseries |\xintthecoords|} which converts a comma separated -list as produced by |\xintfloatexpr| or |\xintiexpr [d]| to the format -expected by the |TikZ| |coordinates| syntax. -\begin{everbatim*} -{\centering\begin{tikzpicture}[scale=10]\xintDigits:=8; - \clip (-1.1,-.25) rectangle (.3,.25); - \draw [blue] (-1.1,0)--(1,0); - \draw [blue] (0,-1)--(0,+1); - \draw [red] plot[smooth] coordinates {\xintthecoords - % converts into (x1, y1) (x2, y2)... format - \xintfloatexpr seq((x^2-1,mul(x-t,t=-1+[0..4]/2)),x=-1.2..[0.1]..+1.2) \relax }; -\end{tikzpicture}\par } -\end{everbatim*} - -% Notice: if x goes no take exactly value 1 or -1, the origin appears slightly -% off the curve, not MY fault!!! - -\csbxint{thecoords} should be followed immediately by \csbxint{floatexpr} or -\csbxint{iexpr} or \csbxint{iiexpr}, but not |\xintthefloatexpr|, etc\dots - -Besides, as |TikZ| will not understand the |A/B[N]| format which is used on -output by |\xintexpr|, |\xintexpr| is not really usable with |\xintthecoords| -for a |TikZ| picture, but one may use it on its own, and the reason for the -spaces in and between coordinate pairs is to allow if necessary to print on -the page for examination with about correct line-breaks. - -\begin{everbatim*} -\oodef\x{\xintthecoords \xintexpr rrseq(1/2,1/3; @1+@2, x=1..20)\relax } -\meaning\x +++ -\end{everbatim*} - \etocdepthtag.toc {commands} \indescriptionfalse \addtocontents{toc}{\gdef\string\sectioncouleur{{joli}}} @@ -5459,12 +5157,16 @@ them). There is a similar macro |\odef| with only one expansion of the replacement text |<stuff>|, and |\fdef| which expands fully |<stuff>| using |\romannumeral-`0|. -These tools are provided as it is sometimes wasteful (from the point of view -of running time) to do an |\edef| when one knows that the contents expand in -only two steps for example, as is the case with all (except \csbxint{loop} and -\csbxint{iloop}) the expandable macros of the \xintname packages. Each will be -defined only if \xintkernelname finds them currently undefined. They can be -prefixed with |\global|. +% These tools are provided as it is sometimes wasteful (from the point of view +% of running time) to do an |\edef| when one knows that the contents expand in +% only two steps for example, as is the case with all (except \csbxint{loop} +% and \csbxint{iloop}) the expandable macros of the \xintname packages. Each +% will be defined only if \xintkernelname finds them currently undefined. + +They can be prefixed with |\global|. It appears than |\fdef| is generally a bit +faster than |\edef| when expanding macros from the \xintname bundle, when the +result has a few dozens of digits. |\oodef| needs thousands of digits it seems +to become competitive. \subsection{\csbh{xintReverseOrder}}\label{xintReverseOrder} @@ -5869,7 +5571,7 @@ then |\xintNthElt| returns nothing. \subsection{\csbh{xintKeep}}\label{xintKeep} \csa{xintKeep\x}\marg{list}\etype{\numx f} expands the list argument and returns -a new list containing only the first |x| elements.\NewWith {1.09m} If |x<0| the +a new list containing only the first |x| elements. If |x<0| the macro returns the last \verb+|x|+ elements (in the same order as in the initial list). If \verb+|x|+ equals or exceeds the length of the list, the list (as arising from expansion of the second argument) is returned. For |x=0| the empty @@ -5884,7 +5586,7 @@ brace pairs (either added to a naked token, or initially present), one may use argument. % \begin{everbatim*} -\oodef\test {\xintKeep {17}{\xintKeep {-69}{\xintSeq {1}{100}}}}\meaning\test +\fdef\test {\xintKeep {17}{\xintKeep {-69}{\xintSeq {1}{100}}}}\meaning\test \end{everbatim*} % @@ -5892,7 +5594,7 @@ argument. \csa{xintTrim\x}\marg{list}\etype{\numx f} expands the list argument and gobbles its first |x| elements. If |x<0| the macro gobbles the last -\verb+|x|+ elements.\NewWith {1.09m} If \verb+|x|+ equals or exceeds +\verb+|x|+ elements. If \verb+|x|+ equals or exceeds the length of the list, the empty list is returned. For |x=0| the full list is returned. @@ -5902,7 +5604,7 @@ up braced in the output (if present there). \csa{xintTrimNoExpand} does the same without first \fexpan ding its list argument. \begin{everbatim*} -\oodef\test {\xintTrim {17}{\xintTrim {-69}{\xintSeq {1}{100}}}}\meaning\test +\fdef\test {\xintTrim {17}{\xintTrim {-69}{\xintSeq {1}{100}}}}\meaning\test \end{everbatim*} \subsection{\csbh{xintListWithSep}}\label{xintListWithSep} @@ -6569,7 +6271,6 @@ to illustrate use of the general purpose \csbxint{iloop}. A little table giving the first values of |\SmallestFactor| follows, its coding uses \csbxint{For}, which is described later; none of this uses count registers. % -%\tracingmacros1 \begin{everbatim*} \let\IsPrime\undefined \let\SmallestFactor\undefined % clean up possible previous mess @@ -7673,9 +7374,9 @@ an optional parameter to \csa{xintAssign}, see \emph{infra}. \noindent\csa{xintAssign} admits since |1.09i| an optional parameter, for example |\xintAssign [e]...| or |\xintAssign [oo] -...|. The latter means that the definitions of the macros initially on the -right of |\to| will be made with \hyperref[oodef]{\ttfamily\char92oodef} which -expands twice the replacement text. The default is simply to make the +...|. With |[f]| for example the definitions of the macros initially on the +right of |\to| will be made with \hyperref[fdef]{\ttfamily\char92fdef} which +\fexpan ds the replacement text. The default is simply to make the definitions with |\def|, corresponding to an empty optional paramter |[]|. Possibilities: |[], [g], [e], [x], [o], [go], [oo], [goo], [f], [gf]|. @@ -7739,6 +7440,19 @@ here to make the definitions global. For this, one should rather do Note that prior to release |1.09j| each item (token or braced material) was submitted to an |\edef|, but the default is now to use |\def|. +\subsection{\csbh{xintDigitsOf}}\label{xintDigitsOf} + +This is a synonym for \csbxint{AssignArray},\ntype{fN} to be used to define +an array giving all the digits of a given (positive, else the minus sign will +be treated as first item) number. +\begingroup\xintDigitsOf\xintiPow {7}{500}\to\digits +% +\leftedline{|\xintDigitsOf\xintiPow {7}{500}\to\digits|} +\noindent $7^{500}$ has |\digits{0}=|\digits{0} digits, and the 123rd among them +(starting from the most significant) is +|\digits{123}=|\digits{123}. +\endgroup + \subsection{\csbh{xintRelaxArray}}\label{xintRelaxArray} \csa{xintRelaxArray}\csa{myArray} %\ntype{N} @@ -7986,33 +7700,30 @@ expression, in the arguments to the package macros, see the \hyperref[sec:useofcount]{Use of count} section. \begin{framed} - \xintcorename also provides macros |\xintAdd|, |\xintMul|,\dots as synonyms - to |\xintiAdd|, |\xintiMul|,\dots. Their usage is - \fbox{deprecated}\IMPORTANT{} for the following reason: with \xintfracname - loaded they become the routines dealing fully with fractions on input. But - this means that they now use fraction format on output, even with integer - arguments. - - Due to this variability of the output format on whether the document uses - only \xintname or loads additionally \xintfracname, code using these macros - is fragile, because loading at some later date a package which itself loads - \xintfracname or \xintexprname will modify their output format, and this is - catastrophic for example in locations expanded by |\ifnum|, or even in - arguments to those other macros of \xintname with |ii| in their names. - - Prefer thus, when writing code loading only \xintcorename or \xintname, to - use the macros \csbxint{iAdd}, \csbxint{iMul}, \dots, or \csbxint{iiAdd}, - \csbxint{iiMul}, \dots. They are guaranteed to always output an integer - without a trailing |/B[n]|. The latter have the less overhead, and the - former do not complain, if \xintfracname is loaded, even if used with true - fractions, as they will then truncate their arguments to - integers. - -%\MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{Changed} - - It was an error for the \xintname package (now \xintcorename) to provide - macros |\xintAdd|, |\xintMul|, |\xintSub| \dots. They should be used only - with \xintfracname loaded. + Earlier releases of \xintcorename also provided macros |\xintAdd|, + |\xintMul|,\dots as synonyms to |\xintiAdd|, |\xintiMul|,\dots, destined to + be re-defined by \xintfracname.\IMPORTANT{} It was announced some time ago + that their usage was deprecated, because the output formats depended on + whether \xintfracname was loaded or not. They now have been \fbox{removed.} + \MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{Changed} + + % Due to this variability of the output format on whether the document uses + % only \xintname or loads additionally \xintfracname, code using these macros + % is fragile, because loading at some later date a package which itself loads + % \xintfracname or \xintexprname will modify their output format, and this is + % catastrophic for example in locations expanded by |\ifnum|, or even in + % arguments to those other macros of \xintname with |ii| in their names. + + The macros \csbxint{iAdd}, \csbxint{iMul}, \dots, or \csbxint{iiAdd}, + \csbxint{iiMul}, \dots which come with \xintcorename are guaranteed to + always output an integer without a trailing |/B[n]|. The latter have the + lesser overhead, and the former do not complain, if \xintfracname is loaded, + even if used with true fractions, as they will then truncate their arguments + to integers. But their output format remains unmodified: integers with no + fraction slash nor |[N]| thingy. + % It was an error for the \xintname package (now \xintcorename) to provide + % macros |\xintAdd|, |\xintMul|, |\xintSub| \dots. They should be used only + % with \xintfracname loaded. \end{framed} The {\color[named]{PineGreen}$\star$}'s in the margin are there to remind of @@ -8045,15 +7756,13 @@ extended by \xintfracname to fractions. \subsection{\csbh{xintiOpp}, \csbh{xintiiOpp}}\label{xintiOpp}\label{xintiiOpp} |\xintiOpp|\n\etype{\Numf} return the opposite |-N| of the number |N|. -\csbxint{Opp} is initially a synonym but gets extended by \xintfracname to -fractions (its output format will be a fraction even if the argument is an -integer) and \csa{xintiiOpp} is the strict integer-only variant which skips +\csa{xintiiOpp} is the strict integer-only variant which skips the \csbxint{Num} overhead.\etype{f} \subsection{\csbh{xintiAbs}, \csbh{xintiiAbs}}\label{xintiAbs}\label{xintiiAbs} |\xintiAbs|\n\etype{\Numf} returns the absolute value of the number. -\csbxint{Abs} is a synonym but gets modified by \xintfracname. \csa{xintiiAbs} +\csa{xintiiAbs} skips the \csbxint{Num} overhead.\etype{f} \subsection{\csbh{xintiiFDg}}\label{xintFDg}\label{xintiiFDg} @@ -8090,67 +7799,63 @@ of parsing via \csbxint{Num}. \subsection{\csbh{xintiAdd}, \csbh{xintiiAdd}}\label{xintiAdd}\label{xintiiAdd} |\xintiAdd|\n\m\etype{\Numf\Numf} returns the sum of the two numbers. -\csbxint{Add} is initially a synonym but gets extended by \xintfracname. \csa{xintiiAdd} skips the \csbxint{Num} overhead.\etype{ff} \subsection{\csbh{xintiSub}, \csbh{xintiiSub}}\label{xintiSub}\label{xintiiSub} -|\xintiSub|\n\m\etype{\Numf\Numf} returns the difference |N-M|. \csbxint{Sub} -is initially a synonym but gets extended by \xintfracname. +|\xintiSub|\n\m\etype{\Numf\Numf} returns the difference |N-M|. \csa{xintiiSub} skips the \csbxint{Num} overhead.\etype{ff} \subsection{\csbh{xintiMul}, \csbh{xintiiMul}}\label{xintiMul}\label{xintiiMul} %{\small Modified in release |1.03|.\par} |\xintiMul|\n\m\etype{\Numf\Numf} returns the product of the two numbers. -\csbxint{Mul} is the initial synonym modified by \xintfracname, and \csa{xintiiMul} skips the \csbxint{Num} overhead.\etype{ff} \subsection{\csbh{xintiSqr}, \csbh{xintiiSqr}}\label{xintiSqr}\label{xintiiSqr} -|\xintiSqr|\n\etype{\Numf} returns the square. \csbxint{Sqr} is the initial -synonym extended by \xintfracname to fractions. \csa{xintiiSqr} skips the +|\xintiSqr|\n\etype{\Numf} returns the square. \csa{xintiiSqr} skips the \csbxint{Num} overhead.\etype{f} \subsection{\csbh{xintiPow}, \csbh{xintiiPow}}\label{xintiPow}\label{xintiiPow} |\xintiPow|\n\x\etype{\Numf\numx} returns |N^x|. When |x| is zero, this is 1. -If |N=0| and |x<0|, if \verb+|N|>1+ and |x<0|, or if \verb+|N|>1+ -and |x>100000|, then an error is raised. +If |N=0| and |x<0|, if \verb+|N|>1+ and |x<0|, an error is raised. There will +also be an error naturally if |x| exceeds the maximal \eTeX{} number +\dtt{\number"7FFFFFFF}, but the real limit for huge exponents comes from +either the computation time or the settings of some tex memory parameters. + +\begin{framed} + Indeed, the maximal power of $2$ which \xintname is able to compute + explicitely is |2^(2^17)=2^131072| which has \dtt{39457} digits. This + exceeds the maximal size on input for the \xintcorename multiplication, hence + any |2^N| with a higher |N| will fail. On the other hand |2^(2^16)| has + \dtt{19729} digits, thus it can be squared once to obtain |2^(2^17)| or + multiplied by anything smaller, thus all exponents up and including |2^17| + are allowed (because the power operation works by squaring things and making + products). +\end{framed} -The |x>100000| condition should perhaps be made more strict: as it stands it -allows launching operations taking hours to complete. Indeed, observe that -|2^50000| already has \dtt{\xintLen{\xintFloatPow [1]{2}{50000}}} digits; as -it turns out each exact multiplication done via \csbxint{iiMul} of two -numbers with one thousand digits each already takes of the order of seconds, -and it would take hours for arguments each with circa $15000$ digits. Perhaps -some completely expandable but not \fexpan dable variants could fare better? +Side remark: after all it does pay to think! I almost melted my CPU trying by +dichotomy to pin-point the exact maximal allowable |N| for |\xintiiPow 2{N}| +before finally making the reasoning above. Indeed, each such computation with +|N>130000| activates the fan of my laptop and results in so warm a keyboard +that I can hardly go on working on it! And it takes about 12 minutes for each +|\xintiiPow2{N}| with such |N|'s of the order of $130000$ (a.t.t.o.w.). \csa{xintiiPow} is an integer only variant skipping the \csbxint{Num} overhead\etype{f\numx}, it produces the same result as \csa{xintiPow} with stricter assumptions on the inputs, and is thus a tiny bit faster. -\csbxint{Pow} is the initial synonym of \csa{xintiPow} which gets extended by -\xintfracname to fractions (see -also \csbxint{FloatPow} for which the exponent must still obey the \TeX{} bound -and \csbxint{FloatPower} which has no restriction at all on the size of the -exponent). Negative exponents do not then raise errors anymore. The float -version is able to deal with things such as |2^999999999| without any problem. -For example |\xintFloatPow[4]{2}{50000}|\dtt{=\xintFloatPow[4]{2}{50000}} -and |\xintFloatPow[4]{2}{999999999}| -\dtt{=\xintFloatPow[4]{2}{999999999}}.% -% -\footnote{On my laptop \texttt{\detokenize{\xintiiPow {2}{9999}}} - obtains all |3010| digits in about ten or eleven seconds. In contrast, - the float versions for |8|, |16|, |24|, or even more significant - figures, do their jobs in less than one hundredth of a second - (|1.09j|; we used in the text only four significant digits only for - reasons of space, not time.) This is done without |log|/|exp| which - are not (yet?) implemented in \xintfracname. The \LaTeX3 - \href{http://www.ctan.org/pkg/l3kernel}{l3fp} package does this with - |log|/|exp| and is ten times faster, but allows only |16| significant - figures and the (exactly represented) floating point numbers must have - their exponents limited to $\pm$\dtt{9999}.} +\xintfracname also provides the floating variants \csbxint{FloatPow} (for +which the exponent must still obey the \TeX{} bound) and \csbxint{FloatPower} +(which has no restriction at all on the size of the exponent). Negative +exponents do not then raise errors anymore. The float version is able to deal +with things such as |2^999999999| without any problem. +\begin{everbatim*} +$\xintFloatPow[32]{2}{50000}<\xintFloatPow[32]{2}{999999999}$ +\end{everbatim*}% +and both are computed swiftly!\footnote{see however \autoref{fn:floatpow}.} Within an \csbxint{iiexpr}|..\relax| the infix operator |^| is mapped to \csa{xintiiPow}; within an \csbxint{expr}-ession it is mapped to \csbxint{Pow} @@ -8254,18 +7959,31 @@ significance of the \textcolor[named]{PineGreen}{\Numf}, and \textcolor[named]{PineGreen}{$\star$} margin annotations and some important background information. -\subsection{\csbh{xintRev}} \label{xintRev} +\subsection{\csbh{xintReverseDigits}} \label{xintReverseDigits} -|\xintRev|\n\etype{f} will reverse the order of the digits of the number, -keeping the optional sign. Leading zeroes -resulting from the operation are not removed (see the -\csa{xintNum} macro for this). This macro and all other -macros dealing with numbers first expand `fully' their arguments. -% -\leftedline{|\xintRev{-123000}|\dtt{=\xintRev{-123000}}} -% -\leftedline{|\xintNum{\xintRev{-123000}}|% - \dtt{=\xintNum{\xintRev{-123000}}}} +|\xintReverseDigits|\n\etype{f} will reverse the order of the digits of the +number, preserving an optional upfront minus sign. \csa{xintRev} is the former +denomination and is kept as an alias to it. Leading zeroes resulting from the +operation are not removed. Contrarily to \csbxint{ReverseOrder} this macro can +only be used with digits and it first expands its argument (but beware that +|-\x| will give an unexpected result as the minus sign immediately stops this +expansion; one can use |\xintiiOpp{\x}| as argument.) + +This command has been rewritten for |1.2| and is faster for very long inputs. +It is (almost) not used internally by the \xintcorename code, but the use +of related routines explains to some extent the higher speed of release |1.2|. + +\begingroup +\begin{everbatim*} +\fdef\x{\xintReverseDigits + {-98765432109876543210987654321098765432109876543210}}\meaning\x\par +\noindent\fdef\x{\xintReverseDigits {\xintReverseDigits + {-98765432109876543210987654321098765432109876543210}}}\meaning\x\par +\end{everbatim*} +\endgroup + +Notice that the output in this case with its leading zero is not in the strict +integer format expected by the `|ii|' arithmetic macros. \subsection{\csbh{xintLen}}\label{xintiLen} @@ -8288,22 +8006,10 @@ have been returned by \csbxint{Raw}: |\xintRaw {-1e3/5.425}|\dtt{=\xintRaw Let's point out that the whole thing should sum up to less than circa $2^{31}$, but this is a bit theoretical. -|\xintLen| is only for numbers or fractions. See \csbxint{Length} for counting +|\xintLen| is only for numbers or fractions. See also \csbxint{NthElt} from +\xinttoolsname. See also \csbxint{Length} from \xintkernelname for counting tokens (or rather braced groups), more generally. -\subsection{\csbh{xintDigitsOf}}\label{xintDigitsOf} - -This is a synonym for \csbxint{AssignArray},\ntype{fN} to be used to define -an array giving all the digits of a given (positive, else the minus sign will -be treated as first item) number. -\begingroup\xintDigitsOf\xintiPow {7}{500}\to\digits -% -\leftedline{|\xintDigitsOf\xintiPow {7}{500}\to\digits|} -\noindent $7^{500}$ has |\digits{0}=|\digits{0} digits, and the 123rd among them -(starting from the most significant) is -|\digits{123}=|\digits{123}. -\endgroup - \subsection{\csbh{xintCmp}, \csbh{xintiiCmp}} |\xintCmp|\n\m\etype{\Numf\Numf} returns \dtt{1} if |N>M|, \dtt{0} if |N=M|, @@ -8441,9 +8147,7 @@ Important: the macro compares \emph{absolute values}. |\xintiMax|\n\m\etype{\Numf\Numf} returns the largest of the two in the sense of the order structure on the relative integers (\emph{i.e.} the right-most number if they are put on a line with positive numbers on the right): -|\xintiMax {-5}{-6}|\dtt{=\xintiMax{-5}{-6}}. The initial synonym \csbxint{Max} -gets modified by \xintfracname which extends it to fractions. Its usage when -only \xintname is loaded is discouraged. +|\xintiMax {-5}{-6}|\dtt{=\xintiMax{-5}{-6}}. The |\xintiiMax| macro skips the overhead of parsing the operands with \csbxint{Num}.\etype{ff} @@ -8453,9 +8157,7 @@ The |\xintiiMax| macro skips the overhead of parsing the operands with |\xintiMin|\n\m\etype{\Numf\Numf} returns the smallest of the two in the sense of the order structure on the relative integers (\emph{i.e.} the left-most number if they are put on a line with positive numbers on the right): |\xintiMin -{-5}{-6}|\dtt{=\xintiMin{-5}{-6}}. The initial synonym \csbxint{Min} -gets modified by \xintfracname which extends it to fractions. Its usage when -only \xintname is loaded is discouraged. +{-5}{-6}|\dtt{=\xintiMin{-5}{-6}}. The |\xintiiMin| macro skips the overhead of parsing the operands with \csbxint{Num}.\etype{ff} @@ -8465,29 +8167,22 @@ The |\xintiiMin| macro skips the overhead of parsing the operands with \csa{xintiMaxof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the maximum. The list argument may be a macro, it is \fexpan ded first. Each item -is submitted to |\xintNum| normalization. \csbxint{Maxof} is the initial -synonym which gets extended by \xintfracname to fractions. Do not use it when -only \xintname is loaded. +is submitted to |\xintNum| normalization. \subsection{\csbh{xintiMinof}}\label{xintiMinof} %{\small New with release |1.09a|.\par} \csa{xintiMinof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the minimum. The list argument may be a macro, it is \fexpan ded first. Each item -is submitted to |\xintNum| normalization. \csbxint{Minof} is the initial -synonym which gets extended by \xintfracname to fractions. Do not use it when -only \xintname is loaded. +is submitted to |\xintNum| normalization. \subsection{\csbh{xintiiSum}}\label{xintiiSum} \csa{xintiiSum}\marg{braced things}\etype{{\lowast f}} after expanding its argument expects to find a sequence of tokens (or braced material). Each is -expanded (with the usual meaning), and the sum of all these numbers is returned. +\fexpan ded, and the sum of all these numbers is returned. Note: the summands are \emph{not} parsed by \csbxint{Num}. -\csbxint{Sum} is initially a synonym, it gets extended by \xintfracname to -fractions. - % \leftedline{% \csa{xintiiSum}|{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}|% @@ -8533,7 +8228,6 @@ With \xintexprname, this would be easier: % \leftedline {|\xinttheiiexpr 2^200*3^100*7^100\relax |} -The initial synonym \csbxint{Prd} is extended by \xintfracname to fractions. % \subsection{\csbh{xintPrdExpr}}\label{xintiiPrdExpr} @@ -8668,13 +8362,44 @@ odd integer and in that case executes the |YES| branch. \subsection{\csbh{xintiFac}}\label{xintiFac} -|\xintiFac|\x\etype{\numx} returns the factorial. It is an error if the -argument is negative or at least $10^5$.% avant 1.09j c'était 1000000. - -|\xintFac| is a variant using |\xintNum| and thus, when \xintfracname is -loaded, accepting a fraction on input (but it truncates it first). +|\xintiFac|\x\etype{\numx} returns the factorial. It is an error on input if +the argument is negative. -% temps obsolètes, mettre à jour +\begin{framed} + The macro will limits the acceptable inputs to a maximum of $9999$. However + the maximal computation depends on the values of some memory parameters of + the |tex| executable: with the the current default settings of TeXLive 2015, + the maximal computable factorial (a.t.t.o.w. 2015/10/06) turns out to be + $5971!$ which has $19956$ digits.%\footnotemark +\end{framed} +% \footnotetext{The computation with \xintname 1.2 of $5971!$ takes of the order +% of 27 seconds on my laptop. And about half a second for the $2568$ digits of +% $1000!$.} + +Package \xintfracname provides \csbxint{FloatFac} which allows to evaluate +faster significant digits of big factorials and accepts (theoretically) inputs +up to $99999999$. See \autoref{sec:examples} for the example of $2000!$ with +$50$ significant digits. + +% avant 1.09j c'était 1000000. +% avant 1.2 c'était 100000. (n'importe quoi!) + +|\xintFac| is the variant applying |\xintNum| on his input and thus, when +\xintfracname is loaded, accepting a fraction on input (but it truncates it +first). + +% avec xint1.2: 1000!, 2000!, 3000! +% Mercredi 07 octobre 2015 à 14:34:20 +% (0.534s) +% 402387260077093773543702, 2568, 4.023872600770938e2567. +% (2.521s) +% 331627509245063324117539, 5736, 3.316275092450633e5735. +% (6.097s) +% 414935960343785408555686, 9131, 4.149359603437854e9130. + +% ATTENTION TOTALEMENT MAIS TOTALEMENT OBSOLETE +% JE CONSERVE UNIQUEMENT POUR ME SOUVENIR DU PASSÉ +% ---- obsolète, remonte au premier xint % On my laptop $1000!$ (2568 digits) % is computed in a little less than ten seconds, $2000!$ (5736 % digits) is computed in a little less than one hundred seconds, and @@ -8684,6 +8409,12 @@ loaded, accepting a fraction on input (but it truncates it first). % transition from $N=9999$ to $10000$ and higher; $10000!$ has 35660 % digits). Not to mention $100000!$ which, from the Stirling formula, % should have 456574 digits. +% ---- (je rêvais à l'époque avec 100000! ... +% +% Je me souviens qu'au tout début je ne m'attendais pas du tout à rencontrer +% de tels problèmes dès des nombres de quelques milliers de chiffres, car je +% n'étais pas imprégné de la pénalité liée à parcourir par des macros +% délimités de longues séquences de tokens \subsection{\csbh{xintiiMON}, \csbh{xintiiMMON}} \label{xintMON}\label{xintMMON}\label{xintiiMON}\label{xintiiMMON} @@ -8884,6 +8615,7 @@ of \csa{xintDecSplit}. \xintiiE {123}{89} \end{everbatim*} +\pagebreak \section{Commands of the \xintfracname package} \label{sec:frac} @@ -9309,10 +9041,21 @@ xxx:_xint $ | \endgroup -Using |\xintTrunc| rather than |\xintXTrunc| would be hopeless on such long -outputs (and even |\xintXTrunc| needed of the order of seconds to complete -here). But it is not worth it to use |\xintXTrunc| for less than hundreds of -digits. +% \emph{Outdated note: Using |\xintTrunc| rather than |\xintXTrunc| would be +% hopeless on such long outputs (and even |\xintXTrunc| needed of the order of +% seconds to complete here). But it is not worth it to use |\xintXTrunc| for +% less than hundreds of digits.} + +\begin{framed} + The |\xintiiMul {\ZA}{66049}| above can sadly \emph{not} be executed with + \xintname 1.2, due to the new limitation to at most about $19950$ digits for + multiplication. On the other hand |\edef\W {\xintXTrunc {131584}{1/66049}}| + produces the $131584$ digits four times faster. The macro \csbxint{XTrunc} + has not yet been adapted to the new integer model underlying the 1.2 + \xintcorename macros, and perhaps some future improvements are possible. So + far it only benefits from a faster division routine, in that specific case + for a divisor having more than four but less than nine digits. +\end{framed} Fraction arguments to |\xintXTrunc| corresponding to a |A/B[N]| with a negative |N| are treated somewhat less efficiently (additional memory impact) than for positive or zero |N|. This is because the algorithm tries to work with the @@ -9600,6 +9343,17 @@ an integer format on output use \csbxint{iSqr}. exactly and outputs in float format with precision |P| (which is optional), or |\xintDigits| if |P| was absent, the result of this computation. +\begin{framed} + It is obviously much needed that the author improves its algorithms to avoid + going through the exact |2P| or |2P-1| digits (plus safety digits) before + throwing to the waste-bin half of those digits ! + + \xintname initially was purely an \emph{exact} arbitrary precision + arithmetic machine, and the introduction of floating point numbers was an + after-thought. I got it working in release |1.07 (2013/05/25)| and never had + time to come back to it. +\end{framed} + \subsection{\csbh{xintDiv}}\label{xintDiv} Computes the algebraic quotient \etype{\Ff\Ff} of two fractions. @@ -9621,13 +9375,34 @@ exactly and outputs in float format with precision |P| (which is optional), or %{\small Modified in |1.08b| (to allow fractions on input).\par} The original\etype{\Numf} is extended to allow a fraction |f| which will be -truncated first to an integer |n| (non negative and at most |999999|, but -already |100000!| is prohibitively time-costly). On output |n!| (with no -trailing |/1[0]|). +truncated first to an integer |n|. See \csbxint{iFac} for a discussion of the +maximal allowed input. + +Output format is an integer without trailing |/1[0]|. The original macro\etype{\numx} (which parses its input via |\numexpr|) is still available as \csbxint{iFac}. +\subsection{\csbh{xintFloatFac}}\label{xintFloatFac} + +\csa{xintFloatFac}|[P]{f}|\etype{{\upshape[\numx]}\Ff} returns the +factorial. +\begin{everbatim*} +$1000!\approx{}$\xintFloatFac [30]{1000} +\end{everbatim*} +The computation\NewWith{1.2 !} proceeds via doing explicitely the product, as +the Stirling formula cannot be used for lack so far of |exp/\log|. +% \footnote{The computation of $100000!$ with $16$ digits of precision takes +% about three or four seconds and for $1000000!$ it is about fifty seconds on +% my laptop (2015/10/06).} +% +There is no a priori limit set on the |P| optional argument, thus the Stirling +approach would become complicated if that freedom was to be obeyed. + +The macro |\xintFloatFac| chooses dynamically an appropriate number of +digits for the intermediate computations, large enough to achieve the desired +accuracy (hopefully). + \subsection{\csbh{xintPow}}\label{xintPow} \csa{xintPow}{|{f}{g}|}:\etype{\Ff\Numf} computes |f^g| with |f| a fraction @@ -9639,10 +9414,11 @@ vanishes: |\xintPow {2/3}{0}|\dtt{=\xintPow{2/3}{0}}). The original is available as \csbxint{iPow}. -The exponent (after truncation to an integer) will be checked to not exceed -|100000|. Indeed |2^50000| already has \dtt{\xintLen {\xintFloatPow - [1]{2}{50000}}} digits, and squaring such a number would take hours (I -think) with the expandable routine of \xintname. +%%%%% OBSOLETE +% The exponent (after truncation to an integer) will be checked to not exceed +% |100000|. Indeed |2^50000| already has \dtt{\xintLen {\xintFloatPow +% [1]{2}{50000}}} digits, and squaring such a number would take hours (I +% think) with the expandable routine of \xintname. \subsection{\csbh{xintFloatPow}}\label{xintFloatPow} %{\small New with |1.07|.\par} @@ -9688,7 +9464,16 @@ which is, in disguise, an integer. The intermediate multiplications are done with a higher precision than |\xintDigits| or the optional |P| argument, in order for the -final result to hopefully have the desired accuracy. +final result to hopefully have the desired accuracy.% +% +\footnote{\label{fn:floatpow}% + Release |1.2| did not change a single line of code to these macros because + they don't access low-level entry points. There is some sure important + efficiency gains to be obtained in maintaining internally the best inner + format for the successive squarings and multiplications, but I decided to + postpone that, as the more urgent issue is to improve \csbxint{FloatMul} to + not compute exactly with all digits the product before keeping only the + required digits.} \subsection{\csbh{xintFloatSqrt}}\label{xintFloatSqrt} %{\small New with |1.08|.\par} @@ -9870,9 +9655,17 @@ be used when one is dealing exclusively with (big) integers. \localtableofcontents The \xintexprname package was first released with version |1.07| -(|2013/05/25|) of the \xintname bundle. The package loads automatically -\xintfracname and \xinttoolsname (it is now the only arithmetic package from the \xintname -bundle which loads \xinttoolsname). +(|2013/05/25|) of the \xintname bundle. It was substantially enhanced with +release |1.1| from |2014/10/28|. + +Release |1.2| removed a limitation to numbers of at most $5000$ digits, and +there is now a float variant of the factorial. Also the ``pseudo-functions'' +|qint|, |qfrac|, |qfloat| (|'q'| for quick), were added to handle very big +inputs and avoid scanning it digit per digit. + +The package loads automatically \xintfracname and \xinttoolsname (it is now +the only arithmetic package from the \xintname bundle which loads +\xinttoolsname). \begin{itemize} \item for using the |gcd| and |lcm| functions, it is necessary to load package \xintgcdname. @@ -9886,30 +9679,32 @@ bundle which loads \xinttoolsname). \end{everbatim*} \end{itemize} -Release |1.1| has brought many changes to \xintexprname. -See \autoref{sec:expr11} if you are already familiar with the earlier versions. +\begin{framed} + This documentation has repetitions, is a.t.t.of.w generally speaking not + well structured, and mixes old explanations dating back to the first release + and some more recent ones. +\end{framed} + \subsection{The \csbh{xintexpr} expressions}\label{xintexpr}% \label{xinttheexpr}\label{xintthe} An \xintexprname{}ession is a construct -\csbxint{expr}\meta{expandable\_expression}|\relax|\etype{x} where the expandable -expression is read and completely expanded from left to right. - -During this parsing, braced sub-content may be serving as usual as a macro -parameter, or as a branch to the |?| two-way and |??| three-way operators. -Prior to release |1.1|, there were also some other usage, but this has been -removed. It was mainly needed because there was no other way to feed the -number parser directtly with fractions in the |A/B[N]| notation which is the -output format of the \xintfracname macros. There was no real need to use such -macros anyhow. If one really wants to, one can now directly: -\begin{everbatim*} -\xinttheexpr \xintAdd{3/5[2]}{7/13[2]}+199/13[1]\relax -\end{everbatim*} - -Notice in passing that the expressions benefit from the improved handling of -denominators by \csbxint{Add} and \csbxint{Sub} from \xintfracname, which are -the macros to which naturally |+| and |-| are mapped. +\csbxint{expr}\meta{expandable\_expression}|\relax|\etype{x} where the +expandable expression is read and completely expanded from left to right. + +% NOT THE GOOD LOCATION FOR THIS + +% During this parsing, braced sub-content may appear as usual as a macro +% parameter, or as a branch to the |?| two-way and |??| three-way operators. +% Prior to release |1.1|, there were also some other usage, but this has been +% removed. It was mainly needed because there was no other way to feed the +% number parser directtly with fractions in the |A/B[N]| notation which is the +% output format of the \xintfracname macros. There was no real need to use such +% macros anyhow. If one really wants to, one can now directly: +% \begin{everbatim*} +% \xinttheexpr \xintAdd{3/5[2]}{7/13[2]}+199/13[1]\relax +% \end{everbatim*} An |\xintexpr..\relax| \emph{must} end in a |\relax| (which will be absorbed). Like a |\numexpr| expression, it is not printable as is, nor can it be directly @@ -9928,7 +9723,7 @@ or |reduce|.% number of digits of the fractional part; in |float| it is the total number of digits of the mantissa.} % -Here are some examples\par +Here are some examples\par % DO BETTER EXAMPLES !!!!!!!!!!!!!!! \leftedline{|\xinttheexpr 1/5!-1/7!-1/9!\relax|% \dtt{=\xinttheexpr 1/5!-1/7!-1/9!\relax}} \leftedline{|\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax|% @@ -9974,7 +9769,438 @@ choice between parentheses or |\xintexpr...\relax|: |\def\x {(\a+\b)}| or |\def\x {\xintexpr \a+\b\relax}|. The latter is the better choice as it allows also to be prefixed with |\xintthe|. Furthemore, if |\a| and |\b| are already defined |\edef\x {\xintexpr \a+\b\relax}| will do the -computation on the spot. Rather than |\edef| one can use |\oodef|. +computation on the spot.% Rather than |\edef| one can use |\oodef|. + + +\subsection{Some features of the 1.1 release of \xintexprname} +\label{sec:expr11} + +Release |1.1| brought many changes to \xintexprname. This chapter is for +people already familiar with earlier versions. A more systematic item per item +syntax description is provided in the next \autoref{sec:expr}. Both this +section and the next are in need of being improved. + +First, there were some breaking changes: +\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, + itemindent=0pt, leftmargin=\leftmarginii] + \item in |\xintiiexpr|, |/| does \emph{rounded} division, rather than as + in earlier releases the + Euclidean division (for positive arguments, this is truncated division). + The new |//| operator does truncated division, + \item the |:| operator for three-way branching is gone, replaced with |??|, + \item |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 postfix operators, + \item the |add| and |mul| have a new syntax, old syntax is with |`+`| and + |`*`| (quotes mandatory), |sum| and |prd| are gone, + \item 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 space-stripped in a |\numexpr|, it is not parsed by the + |\xintexpr|-ession scanner). + \item although |&| and \verb+|+ are still available as Boolean operators the + use of |&&| and \verb+||+ is strongly recommended. The single + letter operators might be assigned some other meaning in later releases + (bitwise operations, perhaps). Do not use them. + \item place holders for |\xintNewExpr| + could be denoted |#1|, |#2|, ... or also, for special purposes |$1|, |$2|, + ... Only the first form is now accepted and the special cases previously + treated via the second form are now managed via a |protect(...)| function. +\end{itemize} + +Let's now describle some of the numerous additional functionalities. + +\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, + itemindent=\leftmarginii, leftmargin=0em] + \item |\xintiexpr|, |\xinttheiexpr| admit an optional argument within + brackets |[d]|, it then presents the computation result (or results, if + comma separated) after rounding to |d| digits after decimal mark, (the + whole computation is done exactly, as in |xintexpr|), +\begin{everbatim*} +\xinttheiexpr [32] 1.23^50, 1.231^50\relax +\end{everbatim*} + \item |\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|), +\begin{everbatim*} +\xintDigits:=32;\xintthefloatexpr 1.010101^10-1.0101^10\relax + +\xintDigits:=16;\xintthefloatexpr 1.010101^10-1.0101^10\relax + +\xintthefloatexpr [12] 1.010101^10-1.0101^10\relax +\end{everbatim*} + + \item |\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, + \item the |//| does truncated division and |/:| is the associated modulo, + \item multi-character operators |&&|, \verb+||+, |==|, |<=|, |>=|, |!=|, + |**|, + \item multi-letter infix binary words |'and'|, |'or'|, |'xor'|, |'mod'| + (quotes mandatory), + \item functions |even|, |odd|, |first|, |last|, + \item |\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), + \item generation of comma separated lists |a..b|, |a..[d]..b|, + \item Python syntax-like list extractors |[list][n:]|, |[list][:n]|, |[list][a:b]| + and |[list][n]| (|n=0| for the number of list items), the step is always + |+1|, + \item function |reversed|, to reverse the order of list items, + \item itemwise sequence operations |a*[list]|, etc.., on both sides |a*[list]^b|, + \item dummy variables in |add| and |mul|: |add(x(x+1)(x-1), x=-10..10)|, + \item variable substitutions with |subs|: |subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)|, + \item sequence generation using |seq| with a dummy variable: |seq(x^3, x=-10..10)|, + \item simple recursive sequences with |rseq|, with |@| given the last value, + |rseq(1;2@+1,i=1..10)|, + \item 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)|, + \item iteration with |iter| which is like |rrseq| but outputs only the + last |K| terms, where |K| was the number of initial terms, + \item inside |seq|, |rseq|, |rrseq|, |iter|, possibility to use |omit|, + |abort| and |break| to control termination, + \item |n++| potentially infinite index generation for |seq|, |rseq|, + |rrseq|, and |iter|, it is advised to use |abort| or |break(..)| at + some point, + \item the |add|, |mul|, |seq|, ... are nestable,% +\footnote{but |add(seq(x,x=1..t),t=1..2)| fails for the reason that |add| will + receive not a list of numbers but a list of lists.} + \item |\xintthecoords| converts a comma separated list of an even number + of items to the format as expected by the |TikZ| |coordinates| syntax, + \item completely rewritten |\xintNewExpr|, new |protect| function to handle + external macros. However not all constructs are compatible with + |\xintNewExpr|. +\end{itemize} + +% \subsection{Examples with the \texorpdfstring{\unexpanded{\unexpanded{|v1.1|}}}{v1.1} \csh{xintexpr}} + +And now some examples: + +\begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, + itemindent=\leftmarginii, leftmargin=0em] +\item One can define variables (the definition itself is a non expandable + step). The allowed names are composed of letters, digits, and underscores. + The variable should not start with a digit and single letters |a..z|, |A..Z| + are predefined for use as dummy variables --- see below. The |@| is + reserved. +\begin{everbatim*} +\begingroup + \xintdefvar a_1 := 3.14159;\xintdefvar a2 := 2.71828;\xinttheiexpr [5] a_1+a2\relax +\endgroup +\end{everbatim*} +\item |add| and |mul| have a new syntax requiring a dummy variable: +\begin{everbatim*} +\xinttheexpr add(x, x=1,3,19), mul(x^2, x=1,3,19), add(x(x+1), x= 1,3,19)\relax +\end{everbatim*} + +Use |`+`| and |`*`| (left ticks mandatory) for syntax without dummy variables: +\begin{everbatim*} +\xinttheexpr `+`(1,3,19), `*`(1^2,3^2,19^2), `+`(1*2,3*4,19*20)\relax +\end{everbatim*} +\item The |seq| function generates sequences according to a given formula: +\begin{everbatim*} +\xinttheexpr seq(x(x+1)(x+2), x=1,3,19), `+`(seq(x(x+1)(x+2), x=1,3,19)), + add(x(x+1)(x+2), x=1,3,19)\relax +\end{everbatim*} +\begin{everbatim*} +And this is nestable! +\xinttheexpr seq(seq(x^y, y=1..5),x=1..5), add(mul(x^y,y=1..5), x=1..5), + add(x^15, x=1..5)\relax % 15 = 1+2+3+4+5 +\end{everbatim*} + +One should use parentheses appropriately. The \csbxint{expr} parser in normal +operation is not bad at identifying missing or extra opening or closing +parentheses, but when it handles |seq|, |add|, |mul| or similar constructs it +switches to another mode of operation (it starts using delimited macros, +something which is almost non-existent in all its other operations) and ill-formed +expressions are much more likely to let the parser fetch tokens from beyond the +mandatory ending |\relax|. Thus, in case of a missing parenthesis in such +circumstances the error message from \TeX{} might be very cryptic, even for +the seasoned \xintname user. + +\item As seen in the last example |a..b| constructs the integers from |a| to + |b|. This is (small) integer only. A more general |a..[d]..b| works with big + integers, or fractions, from |a| to |b| with step |d|. +\begin{everbatim*} +\xinttheexpr seq(2x+1, x=1..10, 100..110, 3/5..[1/5]..7/5)\relax +\end{everbatim*} +\item itemwise operations on lists are possible, as well as item extractions: +\begin{everbatim*} +\xinttheiiexpr 2*[1,10,100]^3, 5+[2*[1,10,100]^3]*100 \relax +\end{everbatim*} +\begin{everbatim*} +\xinttheiiexpr 1+[seq(3^j, j=1..10, 21..30)][17], 1+3^27\relax +\end{everbatim*} + +We used the |[list][n]| construct which gives the nth item from the list. In +this context there are also the functions |last| and |first|. There is no real +concept of a list object, nor list operations, although itemwise manipulation +are made possible as shown above via the |[..]| constructor. The list +manipulation utilities are so far a bit limited. There is no notion of an +``nuple'' object. The variable |nil| is reserved, it represents an empty list. + +\item |subs| is similar to |seq| in syntax but is for variable substitution: +\begin{everbatim*} +\xinttheexpr subs(100*subs(10*subs(3*x+5,x=y+50)+2,y=z^2),z=10)\relax % 100(10(3*150+5)+2) +\end{everbatim*} +\begin{everbatim*} +\xinttheexpr subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)\relax +\end{everbatim*} + +The substituted variable may be a comma separated list (this is impossible +with |seq| which will always pick one item after the other of a list). +\begin{everbatim*} +\xinttheexpr subs([x]^2,x=-123,17,32)\relax +\end{everbatim*} + +\item last but not least, |seq| has variants |rseq| and |rrseq| which allow + recursive definitions. They start with at least one initial value, then a + semi-colon, then the formula, then the list of indices to iterate over. |@| + (or |@1|) evaluates to the last computed item, and |rrseq| keeps the memory + of the |K| last results, where |K| was the number of initial terms. One + accesses them via |@1, @2, @3, @4| and |@@(N)| for |N>4|. It is even + possible to nest them and use |@@@| to access the values of the master + recursion... +\begin{everbatim*} +\xinttheiiexpr rseq(1; 2*@, i=1..10), `+`(rseq(1; 2*@, i=1..10))\relax +\end{everbatim*} +\begin{everbatim*} +\xinttheiiexpr rseq(2; @(@+1)/2, i=1..5)\relax +\end{everbatim*} +\begin{everbatim*} +\xinttheiiexpr rseq(0,1; (@1,add(x,x=@1)), y=2..10)\relax +\end{everbatim*} + +Some Fibonacci fun +\begin{everbatim*} +\xinttheiiexpr rrseq(0,1; @1+@2, x=2..10), last(rrseq(0,1; @1+@2, x=2..100))\relax +\end{everbatim*} +\begin{everbatim*} +Sum of previous last three: \xinttheiiexpr rrseq(0,0,1; @1+@2+@3, i=1..20)\relax +\end{everbatim*} +\begin{everbatim*} +Big numbers: \printnumber{\xinttheexpr rseq(1; @(@+1), j=1..10)\relax } +\end{everbatim*} + +Nested recursion often quickly leads to gigantic outputs. This is an +experimental feature, susceptible to be removed or altered in the future. +\begin{everbatim*} +\xinttheexpr rrseq(1; `+`(rrseq(0,1; @@@(1)+@1+@2, i=1..10)), j=1..5)\relax +\end{everbatim*} + +\item The special keywords |omit|, |abort| and |break(..)| are available + inside |seq|, |rseq|, |rrseq|, as well as the |n++| for a potentially + infinite iteration. The |n++| construct in conjunction with an |abort| or + |break| is often more efficient, because in other cases the list to iterate + over is first completely constructed. +\begin{everbatim*} +First Fibonacci number at least |2^31| and its index +\xinttheiiexpr iter(0,1; (@1>=2^31)?{break(i)}{@2+@1}, i=1++)\relax +\end{everbatim*} + +\begin{everbatim*} +Prime numbers are always cool +\xinttheiiexpr seq((seq((subs((x/:m)?{(m*m>x)?{1}{0}}{-1},m=2n+1)) + ??{break(0)}{omit}{break(1)},n=1++))?{x}{omit}, + x=10001..[2]..10200)\relax +\end{everbatim*} + +The syntax in this last example may look a bit involved. First |x/:m| computes +|x modulo m| (this is the modulo with respect to truncated division, which +here for positive arguments is like Euclidean division; in +|\xintexpr...\relax|, |a/:b| is such that |a = b*(a//b)+a/:b|, with |a//b| the +algebraic quotient |a/b| truncated to an integer.). The |(x)?{yes}{no}| +construct checks if |x| (which \emph{must} be within parentheses) is true or +false, i.e. non zero or zero. It then executes either the |yes| or the |no| +branch, the non chosen branch is \emph{not} evaluated. Thus if |m| divides |x| +we are in the second (``false'') branch. This gives a |-1|. This |-1| is the +argument to a |??| branch which is of the type |(y)??{y<0}{y=0}{y>0}|, thus here +the |y<0|, i.e., |break(0)| is chosen. This |0| is thus given to another |?| +which consequently chooses |omit|, hence the number is not kept in the list. +The numbers which survive are the prime numbers. + +% A006877 In the `3x+1' problem, these values for the starting value set new +% records for number of steps to reach 1. (Formerly M0748) 14 1, 2, 3, 6, 7, +% 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, +% 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, +% 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, +% 626331, 837799 + +\item The |iter| function is like |rrseq| but does not leave a trace of earlier items, + it starts with |K| initial values, then it iterates: either a fixed number of times, + or until aborting or breaking. And ultimately it prints |K| final values. +\begin{everbatim*} +The first Fibonacci number beyond the \TeX{} bound is +\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^31)\relax{} +and the previous number was its index. +\end{everbatim*}But this was a bit too easy, what is the smallest Fibonacci number not representable on 64 bits? +\begin{everbatim*} +The first Fibonacci number beyond |2^64| bound is +\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^64)\relax{} +and the previous number was its index. +\end{everbatim*} + +One more recursion: +\begin{everbatim*} +\def\syr #1{\xinttheiiexpr rseq(#1; (@<=1)?{break(i)}{odd(@)?{3@+1}{@//2}},i=0++)\relax} +The 3x+1 problem: \syr{231}\par +\end{everbatim*} + +Ok, a final one: +\begin{everbatim*} +\def\syrMax #1{\xinttheiiexpr iter(#1,#1;even(i)? + {(@2<=1)?{break(i/2)}{odd(@2)?{3@2+1}{@2//2}}} + {(@1>@2)?{@1}{@2}},i=0++)\relax } +With initial value 1161, the maximal number attained is \syrMax{1161} and that latter +number is the number of steps which was needed to reach 1.\par +\end{everbatim*} + +Well, one more: + +\begin{everbatim*} +\newcommand\GCD [2]{\xinttheiiexpr rrseq(#1,#2; (@1=0)?{abort}{@2/:@1}, i=1++)\relax } +\GCD {13^10*17^5*29^5}{2^5*3^6*17^2} +\end{everbatim*} + +and the ultimate: + +\begin{everbatim*} +\newcommand\Factors [1]{\xinttheiiexpr + subs(seq((i/:3=2)?{omit}{[L][i]},i=1..([L][0])), + L=rseq(#1;(p^2>[@][1])?{([@][1]>1)?{break(1,[@][1],1)}{abort}} + {(([@][1])/:p)?{omit} + {iter(([@][1])//p; (@/:p)?{break(@,p,e)}{@//p},e=1++)}},p=2++))\relax } +\Factors {41^4*59^2*29^3*13^5*17^8*29^2*59^4*37^6} +\end{everbatim*} + +This might look a bit scary, I admit. \xintexprname has minimal tools and +is obstinate about doing everything expandably! We are hampered by absence of a +notion of ``nuple''. The algorithm divides |N| by |2| until no more possible, +then by |3|, then by |4| (which is silly), then by |5|, then by |6| (silly +again), \dots. + +The variable |L=rseq(#1;...)| expands, if one follows the steps, to a comma +separated list starting with the initial (evaluated) |N=#1| and then +pseudo-triplets where the first item is |N| trimmed of small primes, the +second item is the last prime divisor found, and the third item is its +exponent in original |N|. + +The algorithm needs to keep handy the last computed quotient by prime powers, +hence all of them, but at the very end it will be cleaner to get rid of them +(this corresponds to the first line in the code above). This is achieved in a +cumbersome inefficient way; indeed each item extraction |[L][i]| is costly: it +is not like accessing an array stored in memory, due to expandability, nothing +can be stored in memory! Nevertheless, this step could be done here in a far +less inefficient manner if there was a variant of |seq| which, in the spirit +of \csbxint{iloopindex}, would know how many steps it had been through so far. +This is a feature to be added to |\xintexpr|! (as well as a |++| construct +allowing a non unit step). + +Notice that in |iter(([@][1])//p;| the |@| refers to the previous triplet (or +in the first step to |N|), but the latter |@| showing up in |(@/:p)?| refers +to the previous value computed by |iter|. + +\begin{snugframed} + Parentheses are essential in |..([y][0])| else the parser will see |..[| and + end up in ultimate confusion, and also in |([@][1])/:p| else the parser will + see the itemwise operator |]/| on lists and again be very confused (I could + implement a |]/:| on lists, but in this situation this would also be very + confusing to the parser.) +\end{snugframed} + +For comparison, here is an \fexpan dable macro expanding to the same result, +but coded directly with the \xintname macros. Here |#1| can not be itself an +expression, naturally. But at least we let |\Factorize| \fexpan d its +argument. +\begin{everbatim} +\makeatletter +\newcommand\Factorize [1] + {\romannumeral0\expandafter\factorize\expandafter{\romannumeral-`0#1}}% +\newcommand\factorize [1]{\xintiiifOne{#1}{ 1}{\factors@a #1.{#1};}}% +\def\factors@a #1.{\xintiiifOdd{#1} + {\factors@c 3.#1.}% + {\expandafter\factors@b \expandafter1\expandafter.\romannumeral0\xinthalf{#1}.}}% +\def\factors@b #1.#2.{\xintiiifOne{#2} + {\factors@end {2, #1}}% + {\xintiiifOdd{#2}{\factors@c 3.#2.{2, #1}}% + {\expandafter\factors@b \the\numexpr #1+\@ne\expandafter.% + \romannumeral0\xinthalf{#2}.}}% +}% +\def\factors@c #1.#2.{% + \expandafter\factors@d\romannumeral0\xintiidivision {#2}{#1}{#1}{#2}% +}% +\def\factors@d #1#2#3#4{\xintiiifNotZero{#2} + {\xintiiifGt{#3}{#1} + {\factors@end {#4, 1}}% ultimate quotient is a prime with power 1 + {\expandafter\factors@c\the\numexpr #3+\tw@.#4.}}% + {\factors@e 1.#3.#1.}% +}% +\def\factors@e #1.#2.#3.{\xintiiifOne{#3} + {\factors@end {#2, #1}}% + {\expandafter\factors@f\romannumeral0\xintiidivision {#3}{#2}{#1}{#2}{#3}}% +}% +\def\factors@f #1#2#3#4#5{\xintiiifNotZero{#2} + {\expandafter\factors@c\the\numexpr #4+\tw@.#5.{#4, #3}}% + {\expandafter\factors@e\the\numexpr #3+\@ne.#4.#1.}% +}% +\def\factors@end #1;{\xintlistwithsep{, }{\xintRevWithBraces {#1}}}% +\makeatother +\end{everbatim} + +The macro |\Factorize| puts a little stress on the input save stack in order +not be bothered with previously gathered things. I timed it to be about eight +times faster than |\Factors| in test cases such as +|16246355912554185673266068721806243461403654781833| and others. Among the +various things explaining the speed-up, there is fact that we step by +increments of two, not one, and also that we divide only once, obtaining +quotient and remainder in one go. These two things already make for a speed-up +factor of about four. Thus, our earlier |\Factors| was not completely +inefficient, and was quite easier to come up with than |\Factorize|. + +If we only considered small integers, we could write pure |\numexpr| methods +which would be very much faster (especially if we had a table of small primes +prepared first) but still ridiculously slow compared to any non expandable +implementation, not to mention use of programming languages directly accessing +the CPU registers\dots +\end{itemize} + +%\phantomsection +\phantomsection\label{sec:expr11coords} + +To conclude with this overview of the new features in \xintexprname |1.1|, I +will mention {\bfseries |\xintthecoords|} which converts a comma separated +list as produced by |\xintfloatexpr| or |\xintiexpr [d]| to the format +expected by the |TikZ| |coordinates| syntax. +\begin{everbatim*} +{\centering\begin{tikzpicture}[scale=10]\xintDigits:=8; + \clip (-1.1,-.25) rectangle (.3,.25); + \draw [blue] (-1.1,0)--(1,0); + \draw [blue] (0,-1)--(0,+1); + \draw [red] plot[smooth] coordinates {\xintthecoords + % converts into (x1, y1) (x2, y2)... format + \xintfloatexpr seq((x^2-1,mul(x-t,t=-1+[0..4]/2)),x=-1.2..[0.1]..+1.2) \relax }; +\end{tikzpicture}\par } +\end{everbatim*} + +% Notice: if x goes no take exactly value 1 or -1, the origin appears slightly +% off the curve, not MY fault!!! + +\csbxint{thecoords} should be followed immediately by \csbxint{floatexpr} or +\csbxint{iexpr} or \csbxint{iiexpr}, but not |\xintthefloatexpr|, etc\dots + +Besides, as |TikZ| will not understand the |A/B[N]| format which is used on +output by |\xintexpr|, |\xintexpr| is not really usable with |\xintthecoords| +for a |TikZ| picture, but one may use it on its own, and the reason for the +spaces in and between coordinate pairs is to allow if necessary to print on +the page for examination with about correct line-breaks. + +\begin{everbatim*} +\edef\x{\xintthecoords \xintexpr rrseq(1/2,1/3; @1+@2, x=1..20)\relax } +\meaning\x +++ +\end{everbatim*} \subsection{The syntax}\label{ssec:syntax} @@ -10001,12 +10227,13 @@ two |t|'s. |\xintexpr|-essions and |\xinttheexpr|-essions are completely expandable, in two steps. \begin{itemize}[parsep=0pt, labelwidth=\leftmarginii, - itemindent=\leftmarginii, leftmargin=0em] + itemindent=0pt, leftmargin=\leftmarginii] \item An expression is built the standard way with opening and closing parentheses, infix operators, and (big) numbers, with possibly a fractional part, and/or scientific notation (except for \csbxint{iiexpr} which only admits big integers). All variants work with comma separated expressions. On - output each comma will be followed by a space. + output each comma will be followed by a space. A decimal number must have + digits either before or after the decimal mark.\MyMarginNote{Changed!} \item as everything gets expanded, the characters |.|, |+|, |-|, |*|, |/|, |^|, |!|, |&|, \verb+|+, |?|, |:|, |<|, |>|, |=|, |(|, |)|, |"|, |]|, |[|, |@| @@ -10031,7 +10258,8 @@ two |t|'s. \item The |!| is either a function (the logical not) requiring an argument within parentheses, or a - post-fix operator which does the factorial (so far, no float version). + post-fix operator which does the factorial. In \csbxint{floatexpr} it is + mapped to \csbxint{FloatFac}, else it computes the exact factorial. \item The |?| may serve either as a function (the truth value) requiring an argument within parentheses), or as two-way post-fix branching operator @@ -10264,10 +10492,10 @@ operators and functions. |?| and |!| (as prefix) require parentheses around their arguments. \begin{snugframed} - \ctexttt{num, reduce, abs, sgn, frac, floor, ceil, sqr, sqrt, sqrtr, float, - round, trunc, mod, quo, rem, gcd, lcm, max, min, `+`, `*`, ?, !, not, - all, any, xor, if, ifsgn, even, odd, first, last, reversed, bool, - togl, add, mul, seq, subs, rseq, rrseq, iter} + \ctexttt{num, qint, qfrac, qfloat, reduce, abs, sgn, frac, floor, ceil, + sqr, sqrt, sqrtr, float, round, trunc, mod, quo, rem, gcd, lcm, max, + min, `+`, `*`, ?, !, not, all, any, xor, if, ifsgn, even, odd, first, + last, reversed, bool, togl, add, mul, seq, subs, rseq, rrseq, iter} |quo|, |rem|, |even|, |odd|, |gcd| and |lcm| will first truncate their arguments to integers; the latter two require package \xintgcdname; @@ -10275,8 +10503,11 @@ operators and functions. |`*`|, |max| and |min| are functions with arbitrarily many comma separated arguments. - |bool| and |togl| use delimited macros to fetch their argument whose - closing parenthesis thus must be explicit, not arising from expansion. + |bool|, |togl| use delimited macros to fetch their argument and the + closing parenthesis which thus must be explicit, not arising from + expansion. + + The same holds for |qint|, |qfrac|, |qfloat|.\NewWith{1.2} Similarly |add|, |mul|, |subs|, |seq|, |rseq|, |rrseq|, |iter| use at some stages delimited macros. They work with \emph{dummy variables}, @@ -10298,11 +10529,34 @@ operators and functions. \item[functions with a single (numeric) argument] \noindent\par \begin{description} - \item[num] truncates to the nearest integer (truncation towards zero) + \item[num] truncates to the nearest integer (truncation towards zero). \begin{everbatim*} \xinttheexpr num(3.1415^20)\relax \end{everbatim*} + \item[qint] skips the token by token parsing of the input. The ending + parenthesis must be physically present rather than arising from + expansion.\NewWith{1.2} The |q| stands for ``quick''. This ``function'' + handles the input exactly like do the |i| macros of \xintcorename, via + \csbxint{iNum}. Hence leading signs and the leading zeroes (coming next) + will be handled appropriately but spaces will not be systematically + stripped. They should cause no harm and will be removed as soon as the + number is used with one of the basic operators. This input form \emph{does + not accept decimal part or scientific part}. +\begin{everbatim} +\def\x{....many many many ... digits}\def\y{....also many many many digits...} +\xinttheiiexpr qint(\x)*qint(\y)+qint(\y)^2\relax +\end{everbatim} + + \item[qfrac] does the same as \dtt{qint} excepts that it accepts fractions, + decimal numbers, scientific numbers as they are understood by the macros of + package\NewWith{1.2} \xintfracname. Not to be used within an + |\xintiiexpr|-ession, except if hidden inside functions such as + \dtt{round} or \dtt{trunc} which produce integers from fractions. + + \item[qfloat] does the same as \dtt{qfrac} and converts to a float with the + precision given by the setting of |\xintDigits|. + \item[reduce] reduces a fraction to smallest terms \begin{everbatim*} \xinttheexpr reduce(50!/20!/20!/10!)\relax @@ -10567,7 +10821,6 @@ Refer to \autoref{sec:expr11} for more examples. \item The postfix operators \ctexttt{!} and the branching conditionals \ctexttt{?, ??}. \begin{description} \item[{\color[named]{DarkOrchid}!}] computes the factorial of an integer. - This is the exact factorial even when used inside |\xintfloatexpr|. \item[{\color[named]{DarkOrchid}?}] is used as |(cond)?{yes}{no}|. It evaluates the (numerical) condition (any non-zero value counts as @@ -10610,13 +10863,16 @@ Refer to \autoref{sec:expr11} for more examples. optional and unique component of a being formed number. One can do things such as % - \leftedline{\restoreMicroFont|\xinttheexpr .^2+2^.\relax|} + \leftedline{\restoreMicroFont|\xinttheexpr 0.^2+2^.0\relax|} % - which is |0^2+2^0| and produces \dtt{\xinttheexpr .^2+2^.\relax}. + which is |0^2+2^0| and produces \dtt{\xinttheexpr 0.^2+2^.0\relax}. + + However a single dot |"."| as in |\xinttheexpr .^2\relax| is now illegal + input.\MyMarginNote{Changed!} \item The |e| and |E| for scientific notation. They are parsed - like the decimal mark is. Thus |1e(3+2)| is no legal syntax anymore, one - must use |10^(3+2)| in such cases. + like the decimal mark is.% Thus |1e(3+2)| is no legal syntax anymore, one + % must use |10^(3+2)| in such cases. \begingroup \restoreMicroFont |1e3^2| is \dtt{\xinttheexpr 1e3^2\relax} \endgroup @@ -10726,22 +10982,39 @@ one obtains as output \xinttheexpr 2^3,3^4,5^6\relax{}. dimexpr} expressions, count and dimension registers and variables} \label{ssec:countinexpr} -Count registers, count control sequences, dimen registers, -dimen control sequences, skips and skip control sequences, |\numexpr|, -|\dimexpr|, |\glueexpr| can be inserted directly, they will be unpacked using -|\number| (which gives the internal value in terms of scaled points for the -dimensional variables: $1$\,|pt|${}=65536$\,|sp|; stretch and shrink -components are thus discarded). Tacit multiplication is implied, when a -number or decimal number prefixes such a register or control sequence. - -\LaTeX{} lengths are skip control sequences and \LaTeX{} counters should be -inserted using |\value|. +Count registers, count control sequences, dimen registers, dimen control +sequences (like |\parindent|), skips and skip control sequences, |\numexpr|, +|\dimexpr|, |\glueexpr|, |\fontdimen| can be inserted directly, they will be +unpacked using |\number| which gives the internal value in terms of scaled +points for the dimensional variables: $1$\,|pt|${}=65536$\,|sp| (stretch and +shrink components are thus discarded). + +Tacit multiplication is implied, when a number or decimal number prefixes such +a register or control sequence. \LaTeX{} lengths are skip control sequences +and \LaTeX{} counters should be inserted using |\value|. + +Release |1.2| of the |\xintexpr| parser also recognizes and prefixes with +|\number| the |\ht|, |\dp|, and |\wd| \TeX{} primitives as well as the +|\fontcharht|, |\fontcharwd|, |\fontchardp| and |\fontcharit| \eTeX{} +primitives. + +In the case of numbered registers like |\count255| or |\dimen0| (or |\ht0|), +the resulting digits will be re-parsed, so for example |\count255 0| is like +|100| if |\the\count255| would give |10|. The same happens with inputs such +as |\fontdimen6\font|. And |\numexpr 35+52\relax| will be exactly as if |87| +as been encountered by the parser, thus more digits may follow: |\numexpr +35+52\relax 000| is like |87000|. If a new |\numexpr| follows, it is treated +as what would happen when |\xintexpr| scans a number and finds a non-digit: it +does a tacit multiplication. +\begin{everbatim*} +\xinttheexpr \numexpr 351+877\relax\numexpr 1000-125\relax\relax{} is the same +as \xinttheexpr 1228*875\relax. +\end{everbatim*} -In the case of numbered registers like |\count255| or |\dimen0|, the resulting -digits will be re-parsed, so for example |\count255 0| is like |100| if -|\the\count255| would give |10|. Control sequences define complete numbers, thus -cannot be extended that way with more digits, on the other hand they are more -efficient as they avoid the re-parsing of their unpacked contents. +Control sequences however (such as |\parindent|) are picked up as a whole by +|\xintexpr|, and the numbers they define cannot be extended extra digits, a +syntax error is raised if the parser finds digits rather than a legal +operation after such a control sequence. A token list variable must be prefixed by |\the|, it will not be unpacked automatically (the parser will actually try |\number|, and thus fail). Do not @@ -10750,17 +11023,8 @@ doesn't understand |pt| and its presence is a syntax error. To use a dimension expressed in terms of points or other \TeX{} recognized units, incorporate it in |\dimexpr...\relax|. -% REVOIR -% If one needs to optimize, |1.72\dimexpr 3.2pt\relax| is less efficient -% than |1.72*{\number\dimexpr 3.2pt\relax}| as the latter avoids re-parsing the -% digits of the representation of the dimension as scaled points. -% \centeredline{|\xinttheexpr 1.72\dimexpr 3.2pt\relax/2.71828\relax=|} -% \centeredline{|\xinttheexpr 1.72*{\number\dimexpr 3.2pt\relax}/2.71828\relax|} -% \centeredline{\dtt{\xinttheexpr 1.72\dimexpr -% 3.2pt\relax/2.71828\relax=\xinttheexpr 1.72*{\number\dimexpr -% 3.2pt\relax}/2.71828\relax}} Regarding how dimensional expressions are converted by \TeX{} into scaled points -see \autoref{sec:Dimensions}. +see also \autoref{sec:Dimensions}. \subsection{Catcodes and spaces} @@ -10801,10 +11065,15 @@ The characters |+|, |-|, |*|, |/|, |^|, |!|, |&|, \verb+|+, |?|, |:|, |<|, |>|, in the expression, as everything is expanded along the way. If one of them is active, it should be prefixed with |\string|. -The |!| as either logical negation or postfix factorial operator must be a -standard (\emph{i.e.} catcode $12$) |!|, more precisely a catcode $11$ -exclamation point |!| must be avoided as it is used internally by |\xintexpr| -for various special purposes. +The exclamation mark should have its standard catcode, because it is used for +internal purposes with a different one. + +% TOO TECHNICAL +% +% The |!| as either logical negation or postfix factorial operator must be a +% standard (\emph{i.e.} catcode $12$) |!|, more precisely a catcode $11$ +% exclamation point |!| must be avoided as it is used internally by |\xintexpr| +% for various special purposes. Digits, slash, square brackets, minus sign, in the output from an |\xinttheexpr| are all of catcode 12. For |\xintthefloatexpr| the `e' in the output has its standard catcode ``letter''. @@ -11010,7 +11279,7 @@ And this works: \item \FA{1}{3}{90}{-40}{-15} \item \FA{1.234}{-0.123}{-10}{3}{7} \end{itemize} -\oodef\test {\FA {0}{10}{100}{3}{6}}\meaning\test +++ +\fdef\test {\FA {0}{10}{100}{3}{6}}\meaning\test +++ \end{everbatim*} In the last example though, do not hope to use empty |#4| or |#5|: this is @@ -11157,30 +11426,34 @@ multiplication, power, square, sums, products, euclidean quotient and remainder. The |round|, |trunc|, |floor|, |ceil| functions are still available, and are -about the only places where fractions can be used, but |/| can not be used! -This dilemma is solved using |protect|. For understanding the -next example, recall that |round| and |trunc| have a second (non negative) -optional argument. In a normal \csbxint{expr}-essions, |round| and |trunc| are -mapped to \csbxint{Round} and \csbxint{Trunc}, in \csbxint{iiexpr}-essions, -they are mapped to \csbxint{iRound} and \csbxint{iTrunc}. +about the only places where fractions can be used, but |/| within, if not +somehow hidden will be executed as integer rounded division. To avoid this one +can wrap the input in \dtt{qfrac}: this means however that none of the normal +expression parsing will be executed on the argument. +To understand the illustrative examples, recall that |round| and |trunc| have +a second (non negative) optional argument. In a normal \csbxint{expr}-essions, +|round| and |trunc| are mapped to \csbxint{Round} and \csbxint{Trunc}, in +\csbxint{iiexpr}-essions, they are mapped to \csbxint{iRound} and +\csbxint{iTrunc}. -\begin{everbatim*} -\xinttheiiexpr 5/3, round(5/3,3), trunc(5/3,3), trunc(\xintDiv {5}{3},3), -trunc(\xintRaw {5/3},3)\relax{}, -but \xinttheiiexpr 5/3, round(protect(5/3),3), trunc(protect(5/3),3), floor(protect(5/3)), -ceil(protect(5/3))\relax{} works! -\noindent And with negative numbers: \xinttheiiexpr -5/3, round(protect(-5/3),3), -trunc(protect(-5/3),3), floor(protect(-5/3)), ceil(protect(-5/3))\relax. +\begin{everbatim*} +\xinttheiiexpr 5/3, round(5/3,3), trunc(5/3,3), trunc(\xintDiv {5}{3},3), +trunc(\xintRaw {5/3},3)\relax{} are problematic, but +% +\xinttheiiexpr 5/3, round(qfrac(5/3),3), trunc(qfrac(5/3),3), floor(qfrac(5/3)), +ceil(qfrac(5/3))\relax{} work! \end{everbatim*} +On the other hand decimal numbers and scientific numbers can be used directly +as arguments to the |num|, |round|, or any function producing an integer. -Decimal numbers and numbers using scientific notations must be given as -arguments to one of the |num|, or |round|, or etc\dots functions, which will truncate -them to an integer.% - Internally the number will be represented with as many zeros -as is necessary, thus one does not want to do |num(1e100000)| for example! +\begin{framed} + Scientific numbers are either rounded (in case of negative exponent) or + represented with as many zeroes as necessary, thus one does not want to + insert \dtt{num(1e100000)} for example in an \csa{xintiiexpr}ession ! +\end{framed} % \begin{everbatim*} @@ -11193,8 +11466,8 @@ function also. The |sqrt| function is mapped to \csbxint{iiSqrt} which gives a truncated square root. The |sqrtr| function is mapped to \csbxint{iiSqrtR} which gives a rounded square root. -One can use the Float macros if one is careful to use |num|, or |round| on -their output, +One can use the Float macros if one is careful to use |num|, or |round| +etc\dots on their output. \begin{everbatim*} \xinttheiiexpr \xintFloatSqrt [20]{2}, \xintFloatSqrt [20]{3}\relax % no operations @@ -11207,14 +11480,17 @@ their output, decimal mark one should keep.) \end{everbatim*} -The whole point of \csbxint{iiexpr} is to gain some speed in \emph{integer-only} -algorithms, and the above explanations related to how to nevertheless use -fractions therein are a bit peripheral. We observed of the order of -$30$\% speed gain when dealing with numbers with circa one hundred digits, but this -gain decreases the longer the manipulated numbers become and becomes negligible -for numbers with thousand digits: the overhead from parsing fraction format is -little compared to other expensive aspects of the expandable shuffling of -tokens. +The whole point of \csbxint{iiexpr} is to gain some speed in +\emph{integer-only} algorithms, and the above explanations related to how to +nevertheless use fractions therein are a bit peripheral. We observed +(2013/12/18) of the order of $30$\% speed gain when dealing with numbers with +circa one hundred digits (v1.2: this info may be obsolete). + +% but this gain decreases the longer the manipulated +% numbers become and becomes negligible for numbers with thousand digits: the +% overhead from parsing fraction format is little compared to other expensive +% aspects of the expandable shuffling of tokens + \subsection{\csbh{xintboolexpr}, \csbh{xinttheboolexpr}}\label{xintboolexpr}\label{xinttheboolexpr} @@ -11239,10 +11515,6 @@ An optional parameter within brackets is allowed: the final float will have that many digits of precision. This is provided to get rid of non-relevant last digits. -Currently, the factorial function hasn't yet a float version; so inside -|\xintthefloatexpr . . . \relax|, |n!| will be computed exactly. Perhaps this -will be improved in a future release. - \xintDigits:= 9; Note that |1.000000001| and |(1+1e-9)| will not be equivalent for @@ -11414,7 +11686,7 @@ The |\escapechar| setting may be arbitrary when using |\xintexpr|. The format of the output of |\xintexpr|\meta{stuff}|\relax| is a |!| (with catcode 11) followed by various things: \begin{everbatim*} -\oodef\f {\xintexpr 1.23^10\relax }\meaning\f +\fdef\f {\xintexpr 1.23^10\relax }\meaning\f \end{everbatim*} \begin{framed} @@ -11443,11 +11715,6 @@ his/her expansion control. possibility. \end{framed} -% \begin{framed} -% This implementation and user interface are still to be considered -% \emph{experimental}. -% \end{framed} - Syntax errors in the input such as using a one-argument function with two arguments will generate low-level \TeX{} processing unrecoverable errors, with cryptic accompanying message. @@ -11748,8 +12015,8 @@ needed.% \begin{everbatim*} \def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2) -\oodef\w {\xintSeries {0}{50}{\coeff}} % we want to re-use it -\oodef\z {\xintJrr {\w}[0]} % the [0] for a microsecond gain. +\fdef\w {\xintSeries {0}{50}{\coeff}} % we want to re-use it +\fdef\z {\xintJrr {\w}[0]} % the [0] for a microsecond gain. % \xintJrr preferred to \xintIrr: a big common factor is suspected. % But numbers much bigger would be needed to show the greater efficiency. \[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \] @@ -11773,7 +12040,7 @@ A more efficient way to code |\coeff| is illustrated next. \def\coeff #1{\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}% % The [0] in \coeff is a tiny optimization: in its presence the \xintfracname parser % sees something which is already in internal format. -\oodef\w {\xintSeries {0}{50}{\coeff}} +\fdef\w {\xintSeries {0}{50}{\coeff}} \[\sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12}=\xintFrac\w\] \end{everbatim*} The reduced form |\z| as displayed above only differs from this one by a @@ -11882,7 +12149,7 @@ expressions built with such; they must obey the \TeX{} bound. The initial term \def\ratio #1{2/#1[0]}% 2/n, to compute exp(2) \cnta 0 % previously declared count \begin{quote} -\loop \oodef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% +\loop \fdef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% \noindent$\sum_{n=0}^{\the\cnta} \frac{2^n}{n!}= \xintTrunc{12}\z\dots= \xintFrac\z=\xintFrac{\xintIrr\z}$\vtop to 5pt{}\par @@ -11895,7 +12162,7 @@ expressions built with such; they must obey the \TeX{} bound. The initial term \cnta 0 % previously declared count \begin{quote} \loop -\oodef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% +\fdef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% \noindent$\sum_{n=0}^{\the\cnta} \frac{(-1)^n}{n!}= \xintTrunc{20}\z\dots=\xintFrac{\z}=\xintFrac{\xintIrr\z}$% \vtop to 5pt{}\par @@ -11942,12 +12209,12 @@ Here is a slightly more complicated evaluation: \begin{everbatim*} \cnta 1 \begin{multicols}{2} -\loop \oodef\z {\xintRationalSeries +\loop \fdef\z {\xintRationalSeries {\cnta} {2*\cnta-1} {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}} {\ratioexp{\the\cnta}}}% -\oodef\w {\xintRationalSeries {0}{2*\cnta-1}{1}{\ratioexp{\the\cnta}}}% +\fdef\w {\xintRationalSeries {0}{2*\cnta-1}{1}{\ratioexp{\the\cnta}}}% \noindent $\sum_{n=\the\cnta}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!}/% \sum_{n=0}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!} = @@ -12012,19 +12279,19 @@ next section. These completely exact operations rapidly create numbers with many digits. Let us print in full the raw fractions created by the operation illustrated above: -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1[-1]}}} |E(L(1[-1]))=|\dtt{\printnumber{\z}} (length of numerator: \xintLen {\xintNumerator \z}) -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{12[-2]}}} |E(L(12[-2]))=|\dtt{\printnumber{\z}} (length of numerator: \xintLen {\xintNumerator \z}) -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{123[-3]}}} |E(L(123[-3]))=|\dtt{\printnumber{\z}} (length of numerator: @@ -12036,21 +12303,21 @@ only, as we can see) powers of ten. Notice that 1 more digit in an input denominator seems to mean 90 more in the raw output. We can check that with some other test cases: -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/7}}} |E(L(1/7))=|\dtt{\printnumber{\z}} (length of numerator: \xintLen {\xintNumerator \z}; length of denominator: \xintLen {\xintDenominator \z}) -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/71}}} |E(L(1/71))=|\dtt{\printnumber{\z}} (length of numerator: \xintLen {\xintNumerator \z}; length of denominator: \xintLen {\xintDenominator \z}) -\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm} +\fdef\z{\xintRationalSeriesX {0}{9}{\firstterm} {\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/712}}} |E(L(1/712))=|\dtt{\printnumber{\z}} (length of numerator: @@ -12245,7 +12512,7 @@ $\xintFxPtPowerSeries {0}{\cnta}{\coeffexp}{\f}{20}$\\ % \leftedline{|\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}=| \dtt{\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}}} -\oodef\z{\xintIrr {\xintPowerSeries {0}{19}{\coeffexp}{\f}}} +\fdef\z{\xintIrr {\xintPowerSeries {0}{19}{\coeffexp}{\f}}} % \texttt{\hyphenchar\font45 }% @@ -12593,7 +12860,7 @@ You want more digits and have some time? compile this copy of the {\xintMul {4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}% }}% \pdfresettimer -\oodef\Z {\Machin {1000}} +\fdef\Z {\Machin {1000}} \odef\W {\the\pdfelapsedtime} \message{\Z} \message{computed in \xintRound {2}{\W/65536} seconds.} @@ -12601,12 +12868,13 @@ You want more digits and have some time? compile this copy of the | This will log the first 1000 digits of $\pi$ after the decimal point. On my -laptop (a 2012 model) this took about $16$ seconds last time I tried.% +laptop (a 2012 model) this took about $5.6$ seconds last time I tried.% % -\footnote{With \texttt{1.09i} and earlier \xintname releases, this used - to be \dtt{42} seconds; the \texttt{1.09j} division is much faster - with small denominators as occurs here with \dtt{\char92xa=1/25}, and - I believe this to be the main explanation for the speed gain.} +\footnote{With \texttt{v1.09i} and earlier \xintname, this used to be \dtt{42} + seconds; starting with \texttt{v1.09j}, and prior to \texttt{v1.2}, it was + \dtt{16} seconds (this was probably due to a more efficient division with + denominators at most $9999$). The |v1.2| \xintcorename achieves a further + gain.} % As mentioned in the introduction, the file \href{http://www.ctan.org/pkg/pi}{pi.tex} by \textsc{D. @@ -12951,7 +13219,7 @@ $+$ or $-$. \csa{xintGGCFrac}|{a+b/c+d/e+f/g+h/...+x/y}|\ntype{f} is a clone of \csbxint{GCFrac}, hence again \LaTeX{} specific with package -|amsmath|.\NewWith {1.09m} +|amsmath|. It does not assume the coefficients to be numbers as understood by \xintfracname. The macro can be used for displaying arbitrary content as a continued fraction with |\cfrac|, using only plus signs though. Note @@ -12986,10 +13254,10 @@ $$\xintGCtoGCx {+\cfrac\xintFwOver}{\\\xintFwOver}{a+b/...}\endcfrac$$ \csa{xintFtoC}|{f}|\etype{\Ff} computes the coefficients of the simple continued fraction of |f| and returns them as a list -(sequence) of braced items.\NewWith {1.09m} +(sequence) of braced items. \begin{everbatim*} -\oodef\test{\xintFtoC{-5262046/89233}}\texttt{\meaning\test} +\fdef\test{\xintFtoC{-5262046/89233}}\texttt{\meaning\test} \end{everbatim*} \subsection{\csbh{xintFtoCs}}\label{xintFtoCs} @@ -13049,20 +13317,20 @@ format'. \csa{xintFGtoC}|{f}{g}|\etype{\Ff\Ff} computes the common initial coefficients to -two given fractions |f| and |g|. Notice\NewWith {1.09m} that any real number |f<x<g| or |f>x>g| +two given fractions |f| and |g|. Notice that any real number |f<x<g| or |f>x>g| will then necessarily share with |f| and |g| these common initial coefficients for its regular continued fraction. The coefficients are output as a sequence of braced numbers. This list can then be manipulated via macros from \xinttoolsname, or other macros of \xintcfracname. \begin{everbatim*} -\oodef\test{\xintFGtoC{-5262046/89233}{-5314647/90125}}\texttt{\meaning\test} +\fdef\test{\xintFGtoC{-5262046/89233}{-5314647/90125}}\texttt{\meaning\test} \end{everbatim*} \begin{everbatim*} -\oodef\test{\xintFGtoC{3.141592653}{3.141592654}}\texttt{\meaning\test} +\fdef\test{\xintFGtoC{3.141592653}{3.141592654}}\texttt{\meaning\test} \end{everbatim*} \begin{everbatim*} -\oodef\test{\xintFGtoC{3.1415926535897932384}{3.1415926535897932385}}\meaning\test +\fdef\test{\xintFGtoC{3.1415926535897932384}{3.1415926535897932385}}\meaning\test \end{everbatim*} \begin{everbatim*} \xintRound {30}{\xintCstoF{\xintListWithSep{,}{\test}}} @@ -13071,7 +13339,7 @@ braced numbers. This list can then be manipulated via macros from \xintRound {30}{\xintCtoF{\test}} \end{everbatim*} \begin{everbatim*} -\oodef\test{\xintFGtoC{1.41421356237309}{1.4142135623731}}\meaning\test +\fdef\test{\xintFGtoC{1.41421356237309}{1.4142135623731}}\meaning\test \end{everbatim*} \subsection{\csbh{xintFtoCC}}\label{xintFtoCC} @@ -13109,7 +13377,7 @@ simplification by 3 in the result above). \subsection{\csbh{xintCtoF}}\label{xintCtoF} \csa{xintCtoF}|{{a}{b}{c}...{z}}|\etype{f} computes the fraction corresponding -to the coefficients, which may be fractions or even macros.\NewWith {1.09m} +to the coefficients, which may be fractions or even macros. \begin{everbatim*} \xintCtoF {\xintApply {\xintiiPow 3}{\xintSeq {1}{5}}} \end{everbatim*} @@ -13166,9 +13434,9 @@ fractions, but otherwise it is not necessarily the case. \subsection{\csbh{xintCtoCv}}\label{xintCtoCv} \csa{xintCtoCv}|{{a}{b}{c}...{z}}|\etype{f} returns the sequence of the -corresponding convergents, each one within braces.\NewWith {1.09m} +corresponding convergents, each one within braces. \begin{everbatim*} -\oodef\test{\xintCtoCv {11111111111}}\texttt{\meaning\test} +\fdef\test{\xintCtoCv {11111111111}}\texttt{\meaning\test} \end{everbatim*} \subsection{\csbh{xintGCtoCv}}\label{xintGCtoCv} @@ -13263,7 +13531,7 @@ The coefficients, after expansion, are, as shown, being enclosed in an added pair of braces, they may thus be fractions. \begin{everbatim*} \def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/\the\numexpr 1+#1*#1\relax} -\oodef\x{\xintCntoGC {5}{\macro}}\meaning\x +\fdef\x{\xintCntoGC {5}{\macro}}\meaning\x \[\xintGCFrac{\xintCntoGC {5}{\macro}}\] \end{everbatim*} @@ -13309,7 +13577,7 @@ fraction of the same type, each expanded coefficient being enclosed within braces. % \begin{everbatim*} -\oodef\x {\xintGCtoGC {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/% +\fdef\x {\xintGCtoGC {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/% \xintFac {6}+\xintCstoF {2,-7,-5}/16}} \meaning\x \end{everbatim*} @@ -13374,7 +13642,7 @@ convergent. % (getting the 500th took about 1.2s on my laptop last time I tried, % and the 200th convergent is obtained ten times faster). \begin{everbatim*} -\oodef\z {\xintCntoF {199}{\cn}}% +\fdef\z {\xintCntoF {199}{\cn}}% \begingroup\parindent 0pt \leftskip 2.5cm \indent\llap {Numerator = }\printnumber{\xintNumerator\z}\par \indent\llap {Denominator = }\printnumber{\xintDenominator\z}\par @@ -13463,15 +13731,29 @@ $1$ or $-1$. % \bigskip % This is \expandafter|\xintbndlversion| of \expandafter|\xintbndldate|. % -% Extensive changes in release |1.1| of |2014/10/28| were located in -% \xintexprnameimp. Also with that release, -% packages \xintkernelnameimp and \xintcorenameimp were extracted from -% \xinttoolsnameimp and \xintnameimp, and |\xintAdd| was modified to not -% multiply denominators blindly. -% -% \smallskip -% \noindent\fbox{\xinttoolsnameimp is not loaded anymore by -% \xintnameimp, nor by \xintfracnameimp. It is loaded by \xintexprnameimp.} +% \begin{itemize} +% \item Release |1.2| of |2015/10/10| has entirely rewritten the core +% arithmetic routines in \xintcorenameimp. Many macros benefit indirectly +% from the faster core routines. The new model is yet to be extended to +% other portions of the code: for example the routines of \xintbinhexnameimp +% could be made faster for very big inputs if they adopted some of the style +% used now for the basic arithmetic routines. +% +% The parser of \xintexprnameimp is also faster at gathering digits and does +% not have a limit at |5000| digits per number anymore. +% +% \item Extensive changes in release |1.1| of |2014/10/28| were located in +% \xintexprnameimp. Also with that release, packages \xintkernelnameimp and +% \xintcorenameimp were extracted from \xinttoolsnameimp and \xintnameimp, +% and |\xintAdd| was modified to not multiply denominators blindly. +% +% \xinttoolsnameimp is not loaded anymore by \xintnameimp, nor by +% \xintfracnameimp. It is loaded by \xintexprnameimp. +% \end{itemize} +% +% Large portions of the code date back to the initial release, and at that +% time I was learning my trade in expandable TeX macro programming. At some +% point in the future, I will have to re-examine the older parts of the code. % % \section {Package \xintkernelnameimp implementation} % \label{sec:kernelimp} @@ -13479,8 +13761,10 @@ $1$ or $-1$. % \localtableofcontents % % This package provides the common minimal code base for loading management -% and catcode control and also a few programming utilities. It is loaded by both -% |xintcore.sty| and |xinttools.sty| hence by all other packages. +% and catcode control and also a few programming utilities. With |1.2| a few +% more helper macros and all |\chardef|'s have been moved here. The package is +% loaded by both |xintcore.sty| and |xinttools.sty| hence by all other +% packages. % % First appeared as a separate package with release |1.1|. % @@ -13524,19 +13808,36 @@ $1$ or $-1$. \else \def\y#1#2{\PackageInfo{#1}{#2}}% \fi + \let\z\relax \expandafter \ifx\csname numexpr\endcsname\relax \y{xintkernel}{\numexpr not available, aborting input}% - \aftergroup\endinput + \def\z{\endgroup\endinput}% \else \expandafter \ifx\csname XINTsetupcatcodes\endcsname\relax \else \y{xintkernel}{I was already loaded, aborting input}% - \aftergroup\endinput + \def\z{\endgroup\endinput}% \fi \fi - \def\SetCatcodesIfInputNotAborted + \ifx\z\relax\else\expandafter\z\fi% +% \end{macrocode} +% |1.2| corrects a long-standing somewhat subtle bug, of which the author +% became aware only on |15/09/13|: earlier releases had |\aftergroup\endinput| +% above, rather than |\def\z{\endgroup\endinput}| and the |\ifx| test. The +% |\endinput| token was indeed inserted after the |\endgroup| from +% |\PrepareCatcodes|, but all material and in particular |\XINT_setupcatcodes| +% from the macro now called |\PrepareCatcodes| was expanded before the +% |\endinput| had come into effect ! as a result the catcodes would be +% modified in unwanted ways, in Plain \TeX, if the source had for example +% |\input xint.sty| followed by |\input xintkernel.sty|: the catcode changes +% would be done before the second input of |xintkernel.sty| had been aborted. +% One didn't see the situation under \LaTeX{} (in normal circumstances), +% because a second |\usepackage{xintkernel}| would not do any input of +% |xintkernel.sty| to start with. +% \begin{macrocode} + \def\PrepareCatcodes {% \endgroup \def\XINT_restorecatcodes @@ -13605,19 +13906,19 @@ $1$ or $-1$. \catcode36=3 % $ \catcode91=12 % [ \catcode93=12 % ] - \catcode33=11 % ! LETTER + \catcode33=12 % ! \catcode64=11 % @ LETTER \catcode38=12 % & \catcode124=12 % | \catcode63=11 % ? LETTER \catcode34=12 % " \catcode39=12 % ' - \catcode126=3 % ~ + \catcode126=3 % ~ MATH \catcode59=12 % ; }% \XINT_setcatcodes }% -\SetCatcodesIfInputNotAborted +\PrepareCatcodes % \end{macrocode} % Other modules could possibly be loaded under a different catcode regime. % \begin{macrocode} @@ -13648,10 +13949,55 @@ $1$ or $-1$. \fi \XINT_providespackage \ProvidesPackage {xintkernel}% - [2015/09/12 v1.1c Paraphernalia for the xint packages (jfB)]% + [2015/10/10 v1.2 Paraphernalia for the xint packages (jfB)]% +% \end{macrocode} +% \subsection{Constants} +% |v1.2| decides to move them to \xintkernelnameimp from \xintcorenameimp and +% \xintnameimp. The |\count|'s are left in their respective packages. +% \begin{macrocode} +\chardef\xint_c_ 0 +\chardef\xint_c_i 1 +\chardef\xint_c_ii 2 +\chardef\xint_c_iii 3 +\chardef\xint_c_iv 4 +\chardef\xint_c_v 5 +\chardef\xint_c_vi 6 +\chardef\xint_c_vii 7 +\chardef\xint_c_viii 8 +\chardef\xint_c_ix 9 +\chardef\xint_c_x 10 +\chardef\xint_c_xiv 14 +\chardef\xint_c_xvi 16 +\chardef\xint_c_xviii 18 +\chardef\xint_c_xxii 22 +\chardef\xint_c_ii^v 32 +\chardef\xint_c_ii^vi 64 +\chardef\xint_c_ii^vii 128 +\mathchardef\xint_c_ii^viii 256 +\mathchardef\xint_c_ii^xii 4096 +\mathchardef\xint_c_x^iv 10000 % \end{macrocode} % \subsection{Token management utilities} % \begin{macrocode} +\def\XINT_tmpa { }% +\ifx\XINT_tmpa\space\else + \immediate\write-1{Package xintkernel Warning: ATTENTION!}% + \immediate\write-1{\string\space\XINT_tmpa macro does not have its normal + meaning.}% + \immediate\write-1{\XINT_tmpa\XINT_tmpa\XINT_tmpa\XINT_tmpa + All kinds of catastrophes will ensue!!!!}% +\fi +\def\XINT_tmpb {}% +\ifx\XINT_tmpb\empty\else + \immediate\write-1{Package xintkernel Warning: ATTENTION!}% + \immediate\write-1{\string\empty\XINT_tmpa macro does not have its normal + meaning.}% + \immediate\write-1{\XINT_tmpa\XINT_tmpa\XINT_tmpa\XINT_tmpa + All kinds of catastrophes will ensue!!!!}% +\fi +\let\XINT_tmpa\relax \let\XINT_tmpb\relax +\ifdefined\space\else\def\space { }\fi +\ifdefined\empty\else\def\empty {}\fi \long\def\xint_gobble_ {}% \long\def\xint_gobble_i #1{}% \long\def\xint_gobble_ii #1#2{}% @@ -13667,14 +14013,33 @@ $1$ or $-1$. \long\def\xint_firstofone_thenstop #1{ #1}% \long\def\xint_firstoftwo_thenstop #1#2{ #1}% \long\def\xint_secondoftwo_thenstop #1#2{ #2}% +\def\xint_minus_thenstop { -}% +\def\xint_exchangetwo_keepbraces #1#2{{#2}{#1}}% % \end{macrocode} -% \subsection{gob til macros and UD style fork} +% \subsection{``gob til'' macros and UD style fork} +% Some moved here from \xintcorenameimp by release |1.2|. % \begin{macrocode} -\def\xint_gob_til_zero #10{}% -\def\xint_UDzerominusfork #10-#2#3\krof {#2}% \long\def\xint_gob_til_R #1\R {}% \long\def\xint_gob_til_W #1\W {}% \long\def\xint_gob_til_Z #1\Z {}% +\def\xint_gob_til_zero #10{}% +\def\xint_gob_til_one #11{}% +\def\xint_gob_til_zeros_iii #1000{}% +\def\xint_gob_til_zeros_iv #10000{}% +\def\xint_gob_til_eightzeroes #100000000{}% +\def\xint_gob_til_exclam #1!{}% catcode 12 exclam +\def\xint_gob_til_dot #1.{}% +\def\xint_gob_til_G #1G{}% +\def\xint_gob_til_minus #1-{}% +\def\xint_gob_til_relax #1\relax {}% +\def\xint_UDzerominusfork #10-#2#3\krof {#2}% +\def\xint_UDzerofork #10#2#3\krof {#2}% +\def\xint_UDsignfork #1-#2#3\krof {#2}% +\def\xint_UDwfork #1\W#2#3\krof {#2}% +\def\xint_UDXINTWfork #1\XINT_W#2#3\krof {#2}% +\def\xint_UDzerosfork #100#2#3\krof {#2}% +\def\xint_UDonezerofork #110#2#3\krof {#2}% +\def\xint_UDsignsfork #1--#2#3\krof {#2}% \let\xint_relax\relax \def\xint_brelax {\xint_relax }% \long\def\xint_gob_til_xint_relax #1\xint_relax {}% @@ -13687,19 +14052,20 @@ $1$ or $-1$. % \begin{macrocode} \long\def\xint_bye #1\xint_bye {}% % \end{macrocode} -% \subsection{\csh{xint_dothis}, \csh{xint_orthat}} -% \lverb|New with 1.1. Used as \if..\xint_dothis{..}\fi <multiple times> -% followed by \xint_orthat{...}. To be used with less probable things first.| +% \subsection{\csh{xintdothis}, \csh{xintorthat}} +% \lverb|New with 1.1. Public names without underscores with 1.2. Used as +% \if..\xint_dothis{..}\fi <multiple times> followed by \xint_orthat{...}. To +% be used with less probable things first.| % \begin{macrocode} \long\def\xint_dothis #1#2\xint_orthat #3{\fi #1}% v1.1 \let\xint_orthat \xint_firstofone +\long\def\xintdothis #1#2\xintorthat #3{\fi #1}% +\let\xintorthat \xint_firstofone % \end{macrocode} % \subsection{\csh{xint_zapspaces}} -% \lverb|& -% 1.1. Zaps leading, intermediate, trailing, spaces in completely -% expanding context -% (\edef, \csname . . . \endcsname). To be used as:$\ -% $null$ $ $ $ \xint_zapspaces foo \xint_gobble_i $% notice the mandatory space after foo +% \lverb|1.1. This little utility zaps leading, intermediate, trailing, +% spaces in completely expanding context (\edef, \csname . . . \endcsname). +% $centeredline$bgroup\xint_zapspaces foo<space>\xint_gobble_i$egroup % % Will remove some brace pairs (but not spaces inside them). By the way the % \zap@spaces of LaTeX2e handles unexpectedly things such as \zap@spaces 1 @@ -13712,28 +14078,25 @@ $1$ or $-1$. % brace-stripping may occur. And this iterates: each time a #2 is removed, % either we then have spaces and next #1 will be empty, or we have no spaces % and #1 will end at the first space. Ultimately #2 will be \xint_gobble_i. -% -% Code comment from 1.1 release said to do: % -% \xint_zapspaces foo \xint_bye\xint_bye -% -% perhaps because it was pretty. It works also, but \xint_gobble_i is one -% token less. Compatible with an empty foo.| +% This is not really robust as it may switch the expansion order of macros, +% and the \xint_zapspaces token might end up being fetched up by a macro. But +% it is enough for our purposes, for example: +% $centeredline +% $bgroup\the\numexpr \xint_zapspaces 1 2 \xint_gobble_i\relax$egroup +% expands to 12, and not 12\relax. Imagine also: +% $centeredline +% $bgroup\the\numexpr 1 2\expandafter.\the\numexpr ...$egroup +% +% The space will delay the \expandafter. Thus we have to get rid of spaces in +% contexts where arguments are fetched by delimited macros and fed to +% \numexpr (or for any reason can contain spaces). I apply this corrective +% treatment so far only in $xintexprnameimp but perhaps I should in +% $xintfracnameimp too. As said above, perhaps the zapspaces should force +% expansion too, but I leave it standing.| % \begin{macrocode} \def\xint_zapspaces #1 #2{#1#2\xint_zapspaces }% v1.1 % \end{macrocode} -% \subsection{Constants} -% \begin{macrocode} -\chardef\xint_c_ 0 -\chardef\xint_c_i 1 -\chardef\xint_c_ii 2 -\chardef\xint_c_iii 3 -\chardef\xint_c_iv 4 -\chardef\xint_c_v 5 -\chardef\xint_c_vi 6 -\chardef\xint_c_vii 7 -\chardef\xint_c_viii 8 -% \end{macrocode} % \subsection{\csh{odef}, \csh{oodef}, \csh{fdef}} % \lverb|May be prefixed with \global. No parameter text.| % \begin{macrocode} @@ -13748,7 +14111,10 @@ $1$ or $-1$. \ifdefined\fdef\else\let\fdef\xintfdef\fi % \end{macrocode} % \subsection{\csh{xintReverseOrder}} -% \lverb|\xintReverseOrder: does NOT expand its argument.| +% \lverb|\xintReverseOrder: does NOT expand its argument. Thus one must use +% some \expandafter if argument is a macro. A faster reverse, but only +% applicable to (many) digit tokens has been provided with +% \csh{xintReverseDigits} from 1.2 xintcore.| % \begin{macrocode} \def\xintReverseOrder {\romannumeral0\xintreverseorder }% \long\def\xintreverseorder #1% @@ -13879,7 +14245,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xinttools}% - [2015/09/12 v1.1c Expandable and non-expandable utilities (jfB)]% + [2015/10/10 v1.2 Expandable and non-expandable utilities (jfB)]% % \end{macrocode} % \lverb|\XINT_toks is used in macros such as \xintFor. It is not used % elsewhere in the xint bundle.| @@ -15160,17 +15526,24 @@ $1$ or $-1$. % % \localtableofcontents % -% Got split off from \xintnameimp with release |1.1| (macros |\XINT_SQ|, -% |\xintLDg|, |\xintHalf| which are dependencies of |\XINT_div_prepare| were -% forgotten and they were added to the package only later with |1.1b|). -% Release |1.1| also added the new macro |\xintiiDivRound|. The package does -% not load \xinttoolsnameimp. +% Got split off from \xintnameimp with release |1.1|. Release |1.1| also added +% the new macro |\xintiiDivRound|. The package does not load +% \xinttoolsnameimp. +% +% \begin{framed} +% The core arithmetic routines have been entirely rewritten for release +% |1.2|. +% +% The commenting continues (\xintdocdate) to be very sparse: actually it got +% worse than ever with release |1.2|. I will possibly add comments at a +% later date, but for the time being the new routines are not commented at +% all. +% \end{framed} % -% Since release |xint 1.09a| the macros doing arithmetic operations -% apply systematically |\xintnum| to their arguments; this adds a little -% overhead but this is more convenient for using count registers even -% with infix notation; also this is what |xintfrac.sty| did all along. -% It simplifies the discussion in the documentation too. +% Also, with |1.2|, |\xintAdd| etc... have been left undefined control +% sequences: only |\xintiAdd| and |\xintiiAdd| (etc...) are provided via +% \xintcorenameimp. It was announced a long time ago that |\xintAdd| etc... +% were to be removed from \xintnameimp and only defined by \xintfracnameimp. % % \subsection{Catcodes, \protect\eTeX{} and reload detection} % @@ -15229,246 +15602,278 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintcore}% - [2015/09/12 v1.1c Expandable arithmetic on big integers (jfB)]% + [2015/10/10 v1.2 Expandable arithmetic on big integers (jfB)]% % \end{macrocode} -% \subsection{More token management, constants} +% \subsection{Counts for holding needed constants} % \begin{macrocode} -\def\xint_minus_thenstop { -}% -\def\xint_gob_til_zeros_iii #1000{}% -\def\xint_gob_til_zeros_iv #10000{}% -\def\xint_gob_til_one #11{}% -\def\xint_gob_til_G #1G{}% -\def\xint_gob_til_minus #1-{}% -\def\xint_gob_til_relax #1\relax {}% -\def\xint_exchangetwo_keepbraces #1#2{{#2}{#1}}% -\def\xint_exchangetwo_keepbraces_thenstop #1#2{ {#2}{#1}}% -\def\xint_UDzerofork #10#2#3\krof {#2}% -\def\xint_UDsignfork #1-#2#3\krof {#2}% -\def\xint_UDwfork #1\W#2#3\krof {#2}% -\def\xint_UDzerosfork #100#2#3\krof {#2}% -\def\xint_UDonezerofork #110#2#3\krof {#2}% -\def\xint_UDsignsfork #1--#2#3\krof {#2}% -\chardef\xint_c_ix 9 -\chardef\xint_c_x 10 -\chardef\xint_c_ii^v 32 % not used in xint, common to xintfrac and xintbinhex -\chardef\xint_c_ii^vi 64 -\mathchardef\xint_c_ixixixix 9999 -\mathchardef\xint_c_x^iv 10000 -\newcount\xint_c_x^viii \xint_c_x^viii 100000000 +\ifdefined\m@ne\let\xint_c_mone\m@ne + \else\csname newcount\endcsname\xint_c_mone \xint_c_mone -1 \fi +\newcount\xint_c_x^viii \xint_c_x^viii 100000000 +\newcount\xint_c_x^ix \xint_c_x^ix 1000000000 +\newcount\xint_c_x^viii_mone \xint_c_x^viii_mone 99999999 +\newcount\xint_c_xii_e_viii \xint_c_xii_e_viii 1200000000 +\newcount\xint_c_xi_e_viii_mone \xint_c_xi_e_viii_mone 1099999999 +\newcount\xint_c_xii_e_viii_mone\xint_c_xii_e_viii_mone 1199999999 % \end{macrocode} -% \subsection{\csh{XINT_RQ}} -% \lverb|Cette macro renverse et ajoute le nombre minimal de zéros à -% la fin pour que la longueur soit alors multiple de 4$\ -% \romannumeral0\XINT_RQ {}<le truc à renverser>\R\R\R\R\R\R\R\R\Z$\ -% Attention, ceci n'est utilisé que pour des chaînes de chiffres, et donc le -% comportement avec des {..} ou autres espaces n'a fait l'objet d'aucune -% attention.| +% \subsection{\csh{xintNum}} +% \lverb|& +% For example \xintNum {----+-+++---+----000000000000003}$\ +% 1.05 defines \xintiNum, which allows redefinition of \xintNum by xintfrac.sty +% Slightly modified in 1.06b (\R->\xint_relax) to avoid initial re-scan of +% input stack (while still allowing empty #1). In versions earlier than 1.09a +% it was entirely up to the user to apply \xintnum; starting with 1.09a +% arithmetic +% macros of xint.sty (like earlier already xintfrac.sty with its own \xintnum) +% make use of \xintnum. This allows arguments to +% be count registers, or even \numexpr arbitrary long expressions (with the +% trick of braces, see the user documentation). +% +% Note (10/2015): I should take time to revisit this.| % \begin{macrocode} -\def\XINT_RQ #1#2#3#4#5#6#7#8#9% +\def\xintiNum {\romannumeral0\xintinum }% +\def\xintinum #1% +{% + \expandafter\XINT_num_loop + \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z +}% +\let\xintNum\xintiNum \let\xintnum\xintinum +\def\XINT_num #1% {% - \xint_gob_til_R #9\XINT_RQ_end_a\R\XINT_RQ {#9#8#7#6#5#4#3#2#1}% + \XINT_num_loop #1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z }% -\def\XINT_RQ_end_a\R\XINT_RQ #1#2\Z +\def\XINT_num_loop #1#2#3#4#5#6#7#8% {% - \XINT_RQ_end_b #1\Z + \xint_gob_til_xint_relax #8\XINT_num_end\xint_relax + \XINT_num_NumEight #1#2#3#4#5#6#7#8% }% -\def\XINT_RQ_end_b #1#2#3#4#5#6#7#8% +\edef\XINT_num_end\xint_relax\XINT_num_NumEight #1\xint_relax #2\Z {% - \xint_gob_til_R - #8\XINT_RQ_end_viii - #7\XINT_RQ_end_vii - #6\XINT_RQ_end_vi - #5\XINT_RQ_end_v - #4\XINT_RQ_end_iv - #3\XINT_RQ_end_iii - #2\XINT_RQ_end_ii - \R\XINT_RQ_end_i - \Z #2#3#4#5#6#7#8% + \noexpand\expandafter\space\noexpand\the\numexpr #1+\xint_c_\relax }% -\def\XINT_RQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_RQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#9000}% -\def\XINT_RQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#900}% -\def\XINT_RQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#90}% -\def\XINT_RQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#9}% -\def\XINT_RQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_RQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_RQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% +\def\XINT_num_NumEight #1#2#3#4#5#6#7#8% +{% + \ifnum \numexpr #1#2#3#4#5#6#7#8+\xint_c_= \xint_c_ + \xint_afterfi {\expandafter\XINT_num_keepsign_a + \the\numexpr #1#2#3#4#5#6#7#81\relax}% + \else + \xint_afterfi {\expandafter\XINT_num_finish + \the\numexpr #1#2#3#4#5#6#7#8\relax}% + \fi +}% +\def\XINT_num_keepsign_a #1% +{% + \xint_gob_til_one#1\XINT_num_gobacktoloop 1\XINT_num_keepsign_b +}% +\def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }% +\def\XINT_num_keepsign_b #1{\XINT_num_loop -}% +\def\XINT_num_finish #1\xint_relax #2\Z { #1}% % \end{macrocode} -% \subsection{\csh{XINT_OQ}} +% \subsection{Zeroes} +% \lverb|Changed for 1.2 which has a base model of eight digits rather than +% four for the basic operations.| % \begin{macrocode} -\def\XINT_OQ #1#2#3#4#5#6#7#8#9% +\edef\XINT_cuz_small #1#2#3#4#5#6#7#8% {% - \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}% + \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax }% -\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z +%%%%%%%%%%%% +\def\XINT_cuz #1#2#3#4#5#6#7#8#9% {% - \XINT_OQ_end_b #1\Z + \xint_gob_til_R #9\XINT_cuz_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_cuz_z 00000000% + \XINT_cuz_clean #1#2#3#4#5#6#7#8#9% }% -\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8% +\edef\XINT_cuz_clean #1#2#3#4#5#6#7#8#9\R + {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax #9}% +\edef\XINT_cuz_e\R #1\XINT_cuz_clean #2\R + {\noexpand\expandafter\space\noexpand\the\numexpr #2\relax }% +\def\XINT_cuz_z 00000000\XINT_cuz_clean 00000000{\XINT_cuz }% +%%%%%%%%%%%% +\def\XINT_cuz_byviii #1#2#3#4#5#6#7#8#9% {% - \xint_gob_til_R - #8\XINT_OQ_end_viii - #7\XINT_OQ_end_vii - #6\XINT_OQ_end_vi - #5\XINT_OQ_end_v - #4\XINT_OQ_end_iv - #3\XINT_OQ_end_iii - #2\XINT_OQ_end_ii - \R\XINT_OQ_end_i - \Z #2#3#4#5#6#7#8% + \xint_gob_til_R #9\XINT_cuz_byviii_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_cuz_byviii_z 00000000% + \XINT_cuz_byviii_clean #1#2#3#4#5#6#7#8#9% }% -\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_OQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}% -\def\XINT_OQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}% -\def\XINT_OQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}% -\def\XINT_OQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}% -\def\XINT_OQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_OQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_OQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% +\def\XINT_cuz_byviii_clean #1\R { #1}% +\def\XINT_cuz_byviii_e\R #1\XINT_cuz_byviii_clean #2\R{ #2}% +\def\XINT_cuz_byviii_z 00000000\XINT_cuz_byviii_clean 00000000{\XINT_cuz_byviii}% % \end{macrocode} -% \subsection{\csh{XINT_SQ}} +% \subsection{Blocks of eight digits} +% \lverb|Lingua of release 1.2.| % \begin{macrocode} -\def\XINT_SQ #1#2#3#4#5#6#7#8% +\def\XINT_zeroes_forviii #1#2#3#4#5#6#7#8% {% - \xint_gob_til_R #8\XINT_SQ_end_a\R\XINT_SQ {#8#7#6#5#4#3#2#1}% + \xint_gob_til_R #8\XINT_zeroes_forviii_end\R\XINT_zeroes_forviii }% -\def\XINT_SQ_end_a\R\XINT_SQ #1#2\Z +\edef\XINT_zeroes_forviii_end\R\XINT_zeroes_forviii #1#2#3#4#5#6#7#8#9\W {% - \XINT_SQ_end_b #1\Z + \noexpand\expandafter\space\noexpand\xint_gob_til_one #2#3#4#5#6#7#8% }% -\def\XINT_SQ_end_b #1#2#3#4#5#6#7% +%%%%%%%%%%%% +\def\XINT_rsepbyviii #1#2#3#4#5#6#7#8% {% - \xint_gob_til_R - #7\XINT_SQ_end_vii - #6\XINT_SQ_end_vi - #5\XINT_SQ_end_v - #4\XINT_SQ_end_iv - #3\XINT_SQ_end_iii - #2\XINT_SQ_end_ii - \R\XINT_SQ_end_i - \Z #2#3#4#5#6#7% -}% -\def\XINT_SQ_end_vii #1\Z #2#3#4#5#6#7#8\Z { #8}% -\def\XINT_SQ_end_vi #1\Z #2#3#4#5#6#7#8\Z { #7#8000000}% -\def\XINT_SQ_end_v #1\Z #2#3#4#5#6#7#8\Z { #6#7#800000}% -\def\XINT_SQ_end_iv #1\Z #2#3#4#5#6#7#8\Z { #5#6#7#80000}% -\def\XINT_SQ_end_iii #1\Z #2#3#4#5#6#7#8\Z { #4#5#6#7#8000}% -\def\XINT_SQ_end_ii #1\Z #2#3#4#5#6#7#8\Z { #3#4#5#6#7#800}% -\def\XINT_SQ_end_i \Z #1#2#3#4#5#6#7\Z { #1#2#3#4#5#6#70}% -% \end{macrocode} -% \subsection{\csh{XINT_cuz}} -% \begin{macrocode} -\edef\xint_cleanupzeros_andstop #1#2#3#4% + \XINT_rsepbyviii_b {#1#2#3#4#5#6#7#8}% +}% +\def\XINT_rsepbyviii_b #1#2#3#4#5#6#7#8#9% {% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax + #2#3#4#5#6#7#8#9\expandafter!\the\numexpr + 1#1\expandafter.\the\numexpr 1\XINT_rsepbyviii }% -\def\xint_cleanupzeros_nostop #1#2#3#4% +\def\XINT_rsepbyviii_end_B #1\relax #2#3{#2.}% +\def\XINT_rsepbyviii_end_A #11#2\expandafter #3\relax #4#5{#2.1#5.}% +%%%%%%%%%%%% +\def\XINT_sepandrev {% - \the\numexpr #1#2#3#4\relax + \expandafter\XINT_sepandrev_a\the\numexpr 1\XINT_rsepbyviii }% -\def\XINT_rev_andcuz #1% +\def\XINT_sepandrev_a {\XINT_sepandrev_b {}}% +\def\XINT_sepandrev_b #1#2.#3.#4.#5.#6.#7.#8.#9.% {% - \expandafter\xint_cleanupzeros_andstop - \romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax + \xint_gob_til_R #9\XINT_sepandrev_end\R + \XINT_sepandrev_b {#9!#8!#7!#6!#5!#4!#3!#2!#1}% +}% +\def\XINT_sepandrev_end\R\XINT_sepandrev_b #1#2\W {\XINT_sepandrev_done #1}% +\def\XINT_sepandrev_done #11#2!{ }% +%%%%%%%%%%%% +\def\XINT_sepandrev_andcount +{% + \expandafter\XINT_sepandrev_andcount_a\the\numexpr 1\XINT_rsepbyviii +}% +\def\XINT_sepandrev_andcount_a {\XINT_sepandrev_andcount_b 0.{}}% +\def\XINT_sepandrev_andcount_b #1.#2#3.#4.#5.#6.#7.#8.#9.% +{% + \xint_gob_til_R #9\XINT_sepandrev_andcount_end\R + \expandafter\XINT_sepandrev_andcount_b \the\numexpr #1+\xint_c_xiv.% + {#9!#8!#7!#6!#5!#4!#3!#2}% }% +\def\XINT_sepandrev_andcount_end\R + \expandafter\XINT_sepandrev_andcount_b\the\numexpr #1+\xint_c_xiv.#2#3#4\W +{\expandafter\XINT_sepandrev_andcount_done\the\numexpr \xint_c_ii*#3+#1.#2}% +\edef\XINT_sepandrev_andcount_done #1.#21#3!% + {\noexpand\expandafter\space\noexpand\the\numexpr #1-#3.}% % \end{macrocode} -% \lverb|& -% routine CleanUpZeros. Utilisée en particulier par la -% soustraction.$\ -% INPUT: longueur **multiple de 4** (<-- ATTENTION)$\ -% OUTPUT: on a retiré tous les leading zéros, on n'est **plus* -% nécessairement de longueur 4n$\ -% Délimiteur pour _main: \W\W\W\W\W\W\W\Z avec SEPT \W| +% \subsection{Blocks of eight, for needs of v1.2 \csh{xintiiDivision}.} % \begin{macrocode} -\def\XINT_cuz #1% +\def\XINT_sepbyviii_andcount {% - \XINT_cuz_loop #1\W\W\W\W\W\W\W\Z% + \expandafter\XINT_sepbyviii_andcount_a\the\numexpr\XINT_sepbyviii }% -\def\XINT_cuz_loop #1#2#3#4#5#6#7#8% +\def\XINT_sepbyviii #1#2#3#4#5#6#7#8% {% - \xint_gob_til_W #8\xint_cuz_end_a\W - \xint_gob_til_Z #8\xint_cuz_end_A\Z - \XINT_cuz_check_a {#1#2#3#4#5#6#7#8}% + 1#1#2#3#4#5#6#7#8\expandafter!\the\numexpr\XINT_sepbyviii }% -\def\xint_cuz_end_a #1\XINT_cuz_check_a #2% +\def\XINT_sepbyviii_end #1\relax {\relax\XINT_sepbyviii_andcount_end!}% +\def\XINT_sepbyviii_andcount_a {\XINT_sepbyviii_andcount_b \xint_c_.}% +\def\XINT_sepbyviii_andcount_b #1.#2!#3!#4!#5!#6!#7!#8!#9!% {% - \xint_cuz_end_b #2% + #2\expandafter!\the\numexpr#3\expandafter!\the\numexpr#4\expandafter + !\the\numexpr#5\expandafter!\the\numexpr#6\expandafter!\the\numexpr + #7\expandafter + !\the\numexpr#8\expandafter!\the\numexpr#9\expandafter!\the\numexpr + \expandafter\XINT_sepbyviii_andcount_b\the\numexpr #1+\xint_c_viii.% }% -\edef\xint_cuz_end_b #1#2#3#4#5\Z +\def\XINT_sepbyviii_andcount_end #1\XINT_sepbyviii_andcount_b\the\numexpr + #2+\xint_c_viii.#3#4\W {\expandafter.\the\numexpr #2+#3.}% +%%%%%%%%%%%% +\def\XINT_rev_nounsep #1#2!#3!#4!#5!#6!#7!#8!#9!% {% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax + \xint_gob_til_R #9\XINT_rev_nounsep_end\R + \XINT_rev_nounsep {#9!#8!#7!#6!#5!#4!#3!#2!#1}% }% -\def\xint_cuz_end_A \Z\XINT_cuz_check_a #1{ 0}% -\def\XINT_cuz_check_a #1% +\def\XINT_rev_nounsep_end\R\XINT_rev_nounsep #1#2\W {\XINT_rev_nounsep_done #1}% +\def\XINT_rev_nounsep_done #11{ 1}% +%%%%%%%%%%%% +\def\XINT_sepbyviii_Z #1#2#3#4#5#6#7#8% {% - \expandafter\XINT_cuz_check_b\the\numexpr #1\relax + 1#1#2#3#4#5#6#7#8\expandafter!\the\numexpr\XINT_sepbyviii_Z }% -\def\XINT_cuz_check_b #1% +\def\XINT_sepbyviii_Z_end #1\relax {\relax\Z!}% +%%%%%%%%%%%% +\def\XINT_unsep_cuzsmall #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% {% - \xint_gob_til_zero #1\xint_cuz_backtoloop 0\XINT_cuz_stop #1% + \xint_gob_til_R #9\XINT_unsep_cuzsmall_end\R + \XINT_unsep_cuzsmall {#1#2#3#4#5#6#7#8#9}% }% -\def\XINT_cuz_stop #1\W #2\Z{ #1}% -\def\xint_cuz_backtoloop 0\XINT_cuz_stop 0{\XINT_cuz_loop }% -% \end{macrocode} -% \subsection{\csh{xintNum}} -% \lverb|& -% For example \xintNum {----+-+++---+----000000000000003}$\ -% 1.05 defines \xintiNum, which allows redefinition of \xintNum by xintfrac.sty -% Slightly modified in 1.06b (\R->\xint_relax) to avoid initial re-scan of -% input stack (while still allowing empty #1). In versions earlier than 1.09a -% it was entirely up to the user to apply \xintnum; starting with 1.09a -% arithmetic -% macros of xint.sty (like earlier already xintfrac.sty with its own \xintnum) -% make use of \xintnum. This allows arguments to -% be count registers, or even \numexpr arbitrary long expressions (with the -% trick of braces, see the user documentation). -% -% Note (22/06/14): \xintiNum jamais utilisé sous ce nom, le supprimer? -% \XINT_num maintenant utilisé par le parseur de xintexpr.| -% \begin{macrocode} -\def\xintiNum {\romannumeral0\xintinum }% -\def\xintinum #1% +\def\XINT_unsep_cuzsmall_end\R + \XINT_unsep_cuzsmall #1{\XINT_unsep_cuzsmall_done #1}% +\def\XINT_unsep_cuzsmall_done #1\R #2\W{\XINT_cuz_small #1}% +\def\XINT_unsep_delim {1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W}% +%%%%%%%%%%%% +\def\XINT_div_unsepQ #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% {% - \expandafter\XINT_num_loop - \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z + \xint_gob_til_R #9\XINT_div_unsepQ_end\R + \XINT_div_unsepQ {#1#2#3#4#5#6#7#8#9}% }% -\let\xintNum\xintiNum \let\xintnum\xintinum -\def\XINT_num #1% +\def\XINT_div_unsepQ_end\R\XINT_div_unsepQ #1{\XINT_div_unsepQ_x #1}% +\def\XINT_div_unsepQ_x #1#2#3#4#5#6#7#8#9% {% - \XINT_num_loop #1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z + \xint_gob_til_R #9\XINT_div_unsepQ_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_div_unsepQ_y 00000000% + \expandafter\XINT_div_unsepQ_done \the\numexpr #1#2#3#4#5#6#7#8.#9% }% -\def\XINT_num_loop #1#2#3#4#5#6#7#8% +\def\XINT_div_unsepQ_e\R\xint_gob_til_eightzeroes #1\XINT_div_unsepQ_y #2\W + {\the\numexpr #1\relax \Z}% +\def\XINT_div_unsepQ_y #1.#2\R #3\W{\XINT_cuz_small #2\Z}% +\def\XINT_div_unsepQ_done #1.#2\R #3\W { #1#2\Z}% +%%%%%%%%%%%% +\def\XINT_div_unsepR #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% {% - \xint_gob_til_xint_relax #8\XINT_num_end\xint_relax - \XINT_num_NumEight #1#2#3#4#5#6#7#8% + \xint_gob_til_R #9\XINT_div_unsepR_end\R + \XINT_div_unsepR {#1#2#3#4#5#6#7#8#9}% }% -\edef\XINT_num_end\xint_relax\XINT_num_NumEight #1\xint_relax #2\Z +\def\XINT_div_unsepR_end\R\XINT_div_unsepR #1{\XINT_div_unsepR_done #1}% +\def\XINT_div_unsepR_done #1\R #2\W {\XINT_cuz #1\R}% +%%%%%%%%%%%% +\def\XINT_unrevbyviii #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% {% - \noexpand\expandafter\space\noexpand\the\numexpr #1+\xint_c_\relax + \xint_gob_til_R #9\XINT_unrevbyviii_a\R + \XINT_unrevbyviii {#9#8#7#6#5#4#3#2#1}% }% -\def\XINT_num_NumEight #1#2#3#4#5#6#7#8% +\edef\XINT_unrevbyviii_a\R\XINT_unrevbyviii #1#2\W + {\noexpand\expandafter\space + \noexpand\romannumeral-`0\noexpand\xint_gob_til_Z #1}% +\def\XINT_smallunrevbyviii 1#1!1#2!1#3!1#4!1#5!1#6!1#7!1#8!#9\W% {% - \ifnum \numexpr #1#2#3#4#5#6#7#8+\xint_c_= \xint_c_ - \xint_afterfi {\expandafter\XINT_num_keepsign_a - \the\numexpr #1#2#3#4#5#6#7#81\relax}% - \else - \xint_afterfi {\expandafter\XINT_num_finish - \the\numexpr #1#2#3#4#5#6#7#8\relax}% - \fi + \expandafter\XINT_cuz_small\xint_gob_til_Z #8#7#6#5#4#3#2#1% }% -\def\XINT_num_keepsign_a #1% +% \end{macrocode} +% \subsection{\csh{xintReverseDigits}} +% \lverb|v1.2. Needed now by \xintLDg.| +% \begin{macrocode} +\def\XINT_microrevsep #1#2#3#4#5#6#7#8% {% - \xint_gob_til_one#1\XINT_num_gobacktoloop 1\XINT_num_keepsign_b + 1#8#7#6#5#4#3#2#1\expandafter!\the\numexpr\XINT_microrevsep }% -\def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }% -\def\XINT_num_keepsign_b #1{\XINT_num_loop -}% -\def\XINT_num_finish #1\xint_relax #2\Z { #1}% +\def\XINT_microrevsep_end #1\W #2\expandafter #3\Z{#2!}% +\def\xintReverseDigits {\romannumeral0\xintreversedigits }% +\def\xintreversedigits #1{\expandafter\XINT_reversedigits\romannumeral-`0#1\Z}% +\def\XINT_reversedigits #1% +{% + \xint_UDsignfork + #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_reversedigits_a}% + -{\XINT_reversedigits_a #1}% + \krof +}% +\def\XINT_reversedigits_a #1\Z +{% + \expandafter\XINT_revdigits_a\the\numexpr\expandafter\XINT_microrevsep + \romannumeral-`0#1{\XINT_microrevsep_end\W}\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end\Z + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_revdigits_a {\XINT_revdigits_b {}}% +\def\XINT_revdigits_b #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_revdigits_end\R + \XINT_revdigits_b {#9#8#7#6#5#4#3#2#1}% +}% +\edef\XINT_revdigits_end\R\XINT_revdigits_b #1#2\W + {\noexpand\expandafter\space\noexpand\xint_gob_til_Z #1}% % \end{macrocode} % \subsection{\csh{xintSgn}, \csh{xintiiSgn}, \csh{XINT_Sgn}, \csh{XINT_cntSgn}} % \lverb|& @@ -15510,13 +15915,12 @@ $1$ or $-1$. {% \xint_UDzerominusfork #1-\xint_c_ - 0#1\m@ne % I will not allocate a count only for -1? + 0#1\xint_c_mone 0-\xint_c_i \krof }% % \end{macrocode} -% \subsection{\csh{xintiOpp}} -% \lverb|\xintnum added in 1.09a| +% \subsection{\csh{xintiOpp}, \csh{xintiiOpp}} % \begin{macrocode} \def\xintiiOpp {\romannumeral0\xintiiopp }% \def\xintiiopp #1% @@ -15528,7 +15932,6 @@ $1$ or $-1$. {% \expandafter\XINT_opp \romannumeral0\xintnum{#1}% }% -\let\xintOpp\xintiOpp \let\xintopp\xintiopp \def\XINT_Opp #1{\romannumeral0\XINT_opp #1}% \def\XINT_opp #1% {% @@ -15556,7 +15959,6 @@ $1$ or $-1$. {% \expandafter\XINT_abs \romannumeral0\xintnum{#1}% }% -\let\xintAbs\xintiAbs \let\xintabs\xintiabs \def\XINT_Abs #1{\romannumeral0\XINT_abs #1}% \def\XINT_abs #1% {% @@ -15600,33 +16002,34 @@ $1$ or $-1$. % defining \xintiiOdd which is used once (currently) elsewhere . % % bug fix (1.1b): \xintiiLDg is needed by the division macros next, thus -% it needs to be in the xintcore.sty| +% it needs to be in the xintcore.sty. +% +% Rewritten for 1.2.| % \begin{macrocode} +\def\xintLDg {\romannumeral0\xintldg }% +\def\xintldg #1{\xintiildg {\xintNum{#1}}}% \def\xintiiLDg {\romannumeral0\xintiildg }% \def\xintiildg #1% {% - \expandafter\XINT_ldg\expandafter {\romannumeral-`0#1}% -}% -\def\xintLDg {\romannumeral0\xintldg }% -\def\xintldg #1% -{% - \expandafter\XINT_ldg\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\XINT_LDg #1{\romannumeral0\XINT_ldg {#1}}% -\def\XINT_ldg #1% -{% - \expandafter\XINT_ldg_\romannumeral0\xintreverseorder {#1}\Z + \expandafter\XINT_ldg_done\romannumeral0% + \expandafter\XINT_revdigits_a\the\numexpr\expandafter\XINT_microrevsep + \romannumeral0\expandafter\XINT_abs + \romannumeral-`0#1{\XINT_microrevsep_end\W}\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end\Z + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W + \Z }% -\def\XINT_ldg_ #1#2\Z{ #1}% +\def\XINT_ldg_done #1#2\Z { #1}% % \end{macrocode} % \subsection{\csh{xintDouble}} -% \lverb|v1.08| +% \lverb|v1.08. Rewritten for v1.2.| % \begin{macrocode} \def\xintDouble {\romannumeral0\xintdouble }% \def\xintdouble #1% {% - \expandafter\XINT_dbl\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W + \expandafter\XINT_dbl\romannumeral-`0#1\Z }% \def\XINT_dbl #1% {% @@ -15636,44 +16039,31 @@ $1$ or $-1$. 0-{\XINT_dbl_pos #1}% \krof }% -\def\XINT_dbl_zero #1\Z \W\W\W\W\W\W\W { 0}% +\def\XINT_dbl_zero #1\Z { 0}% \def\XINT_dbl_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dbl_pos }% -\def\XINT_dbl_pos -{% - \expandafter\XINT_dbl_a \expandafter{\expandafter}\expandafter 0% - \romannumeral0\XINT_SQ {}% -}% -\def\XINT_dbl_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_dbl_end_a\W - \expandafter\XINT_dbl_b - \the\numexpr \xint_c_x^viii+#2+\xint_c_ii*#9#8#7#6#5#4#3\relax {#1}% -}% -\def\XINT_dbl_b 1#1#2#3#4#5#6#7#8#9% +\def\XINT_dbl_pos #1\Z {% - \XINT_dbl_a {#2#3#4#5#6#7#8#9}{#1}% + \expandafter\XINT_dbl_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W 1\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_dbl_end_a #1+#2+#3\relax #4% +\def\XINT_dbl_pos_aa {% - \expandafter\XINT_dbl_end_b #2#4% -}% -\edef\XINT_dbl_end_b #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax + \expandafter\XINT_mul_out\the\numexpr\XINT_verysmallmul 0.2!% }% % \end{macrocode} % \subsection{\csh{xintHalf}} -% \lverb|v1.08. Relase 1.1 left it in xint.sty, but it is needed by the -% division routines included in xintcore.sty. Thus moved here for bugfix -% release 1.1c. -% Also \XINT_SQ which it uses. Moved here \xintDouble as well by sympathy.| +% \lverb|v1.08. Rewritten for v1.2.| % \begin{macrocode} \def\xintHalf {\romannumeral0\xinthalf }% \def\xinthalf #1% {% - \expandafter\XINT_half\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W + \expandafter\XINT_half\romannumeral-`0#1\Z }% \def\XINT_half #1% {% @@ -15683,52 +16073,38 @@ $1$ or $-1$. 0-{\XINT_half_pos #1}% \krof }% -\def\XINT_half_zero #1\Z \W\W\W\W\W\W\W { 0}% +\def\XINT_half_zero #1\Z { 0}% \def\XINT_half_neg {\expandafter\XINT_opp\romannumeral0\XINT_half_pos }% -\def\XINT_half_pos {\expandafter\XINT_half_a\romannumeral0\XINT_SQ {}}% -\def\XINT_half_a #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W #8\XINT_half_dont\W - \expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#7#6#5#4#3#2#1\relax #8% -}% -\edef\XINT_half_dont\W\expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#1#2#3#4#5#6#7\relax \W\W\W\W\W\W\W -{% - \noexpand\expandafter\space - \noexpand\the\numexpr (#1#2#3#4#5#6#7+\xint_c_i)/\xint_c_ii-\xint_c_i \relax -}% -\def\XINT_half_b 1#1#2#3#4#5#6#7#8% -{% - \XINT_half_c {#2#3#4#5#6#7}{#1}% -}% -\def\XINT_half_c #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #3\XINT_half_end_a #2\W - \expandafter\XINT_half_d - \the\numexpr \xint_c_x^viii+\xint_c_v*#9#8#7#6#5#4#3+#2\relax {#1}% -}% -\def\XINT_half_d 1#1#2#3#4#5#6#7#8#9% +\def\XINT_half_pos #1\Z {% - \XINT_half_c {#2#3#4#5#6#7#8#9}{#1}% + \expandafter\XINT_half_pos_a + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + 1\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_half_end_a #1\W #2\relax #3% +\def\XINT_half_pos_a + {\expandafter\XINT_half_pos_b\the\numexpr\XINT_verysmallmul 0.5!}% +\def\XINT_half_pos_b 1#1#2#3#4#5#6#7#8!1#9% {% - \xint_gob_til_zero #1\XINT_half_end_b 0\space #1#3% + \xint_gob_til_Z #9\XINT_half_small \Z + \XINT_mul_out 1#1#2#3#4#5#6#7!1#9% }% -\edef\XINT_half_end_b 0\space 0#1#2#3#4#5#6#7% +\edef\XINT_half_small \Z\XINT_mul_out 1#1!#2\W {% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7\relax + \noexpand\expandafter\space\noexpand\the\numexpr #1\relax }% % \end{macrocode} % \subsection{\csh{xintDec}} -% \lverb!v1.08! +% \lverb|v1.08. Rewritten for v1.2.| % \begin{macrocode} \def\xintDec {\romannumeral0\xintdec }% \def\xintdec #1% {% - \expandafter\XINT_dec\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W + \expandafter\XINT_dec\romannumeral-`0#1\Z }% \def\XINT_dec #1% {% @@ -15738,47 +16114,28 @@ $1$ or $-1$. 0-{\XINT_dec_pos #1}% \krof }% -\def\XINT_dec_zero #1\W\W\W\W\W\W\W\W { -1}% +\def\XINT_dec_zero #1\Z { -1}% \def\XINT_dec_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_inc_pos }% -\def\XINT_dec_pos -{% - \expandafter\XINT_dec_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_dec_a #1#2#3#4#5#6#7#8#9% +\def\XINT_dec_pos #1\Z {% - \expandafter\XINT_dec_b - \the\numexpr 11#9#8#7#6#5#4#3#2-\xint_c_i\relax {#1}% -}% -\def\XINT_dec_b 1#1% -{% - \xint_gob_til_one #1\XINT_dec_A 1\XINT_dec_c -}% -\def\XINT_dec_c #1#2#3#4#5#6#7#8#9{\XINT_dec_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_A 1\XINT_dec_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_B #1#2\W\W\W\W\W\W\W\W -{% - \expandafter\XINT_dec_cleanup - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% + \expandafter\XINT_dec_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W }% -\edef\XINT_dec_cleanup #1#2#3#4#5#6#7#8% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax }% +\def\XINT_dec_pos_aa {\XINT_sub_aa 100000001!\Z!\Z!\Z!\Z!\Z!\W }% % \end{macrocode} % \subsection{\csh{xintInc}} -% \lverb!v1.08! +% \lverb!v1.08. Rewritten for v1.2.! % \begin{macrocode} \def\xintInc {\romannumeral0\xintinc }% \def\xintinc #1% {% - \expandafter\XINT_inc\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W + \expandafter\XINT_inc\romannumeral-`0#1\Z }% \def\XINT_inc #1% {% @@ -15788,520 +16145,231 @@ $1$ or $-1$. 0-{\XINT_inc_pos #1}% \krof }% -\def\XINT_inc_zero #1\W\W\W\W\W\W\W\W { 1}% +\def\XINT_inc_zero #1\Z { 1}% \def\XINT_inc_neg {\expandafter\XINT_opp\romannumeral0\XINT_dec_pos }% -\def\XINT_inc_pos -{% - \expandafter\XINT_inc_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_inc_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_inc_end\W - \expandafter\XINT_inc_b - \the\numexpr 10#9#8#7#6#5#4#3#2+\xint_c_i\relax {#1}% -}% -\def\XINT_inc_b 1#1% -{% - \xint_gob_til_zero #1\XINT_inc_A 0\XINT_inc_c -}% -\def\XINT_inc_c #1#2#3#4#5#6#7#8#9{\XINT_inc_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_A 0\XINT_inc_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_end\W #1\relax #2{ 1#2}% -% \end{macrocode} -% \subsection{Variants for addition sub-routines} -% \lverb|& -% Release 1.03 re-organizes sub-routines to facilitate future developments: the -% diverse variants of addition, with diverse conditions on inputs and output are -% first listed; they will be used in multiplication, or in the summation, or in -% the power routines. I am aware that the commenting is close to non-existent, -% sorry about that. -% -% Addition and multiplication each have multiple implementations corresponding -% to slightly differing formats on input and on output.| -% -% \subsubsection{Addition vI: \csh{XINT_add_A}} -% \lverb|& -% INPUT:$\ -% \romannumeral0\XINT_add_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés $\ -% 2. de longueur 4n (avec des leading zéros éventuels)$\ -% 3. l'un des deux ne doit pas se terminer par 0000$\$relax -% [Donc on peut avoir 0000 comme input si l'autre est >0 et ne se termine pas en -% 0000 bien sûr]. On peut avoir l'un des deux vides. Mais alors l'autre ne doit -% être ni vide ni 0000. -% -% OUTPUT: la somme <N1>+<N2>, ordre normal, plus sur 4n, pas de leading zeros -% La procédure est plus rapide lorsque <N1> est le plus court des deux.$\ -% Nota bene: (30 avril 2013). J'ai une version qui est deux fois plus rapide sur -% des nombres d'environ 1000 chiffres chacun, et qui commence à être avantageuse -% pour des nombres d'au moins 200 chiffres. Cependant il serait vraiment -% compliqué d'en étendre l'utilisation aux emplois de l'addition dans les -% autres routines, comme celle de multiplication ou celle de division; et son -% implémentation ajouterait au minimum la mesure de la longueur des summands.| -% \begin{macrocode} -\def\XINT_add_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_add_az\W - \XINT_add_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_add_az\W\XINT_add_AB #1#2% -{% - \XINT_add_AC_checkcarry #1% -}% -% \end{macrocode} -% \lverb|& -% ici #2 est prévu pour l'addition, mais attention il devra être renversé -% pour \numexpr. #3 = résultat partiel. #4 = chiffres qui restent. On vérifie si -% le deuxième nombre s'arrête.| -% \begin{macrocode} -\def\XINT_add_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_add_bz\W - \XINT_add_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_add_ABEA #1#2#3.#4% -{% - \XINT_add_A #2{#3#4}% -}% -% \end{macrocode} -% \lverb|& -% ici le deuxième nombre est fini -% #6 part à la poubelle, #2#3#4#5 est le #2 dans \XINT_add_AB -% on ne vérifie pas la retenue cette fois, mais les fois suivantes| -% \begin{macrocode} -\def\xint_add_bz\W\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_CC\the\numexpr #1+10#5#4#3#2.% -}% -\def\XINT_add_CC #1#2#3.#4% -{% - \XINT_add_AC_checkcarry #2{#3#4}% on va examiner et \'eliminer #2 -}% -% \end{macrocode} -% \lverb|& -% retenue plus chiffres qui restent de l'un des deux nombres. -% #2 = résultat partiel -% #3#4#5#6 = summand, avec plus significatif à droite| -% \begin{macrocode} -\def\XINT_add_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_add_AC_nocarry 0\XINT_add_C -}% -\def\xint_add_AC_nocarry 0\XINT_add_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_add_C #1#2#3#4#5% -{% - \xint_gob_til_W #2\xint_add_cz\W - \XINT_add_CD {#5#4#3#2}{#1}% -}% -\def\XINT_add_CD #1% -{% - \expandafter\XINT_add_CC\the\numexpr 1+10#1.% -}% -\def\xint_add_cz\W\XINT_add_CD #1#2{ 1#2}% -% \end{macrocode} -% \subsubsection{Addition vII: \csh{XINT_addr_A}} -% \lverb|& -% INPUT: \romannumeral0\XINT_addr_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z -% -% Comme \XINT_add_A, la différence principale c'est qu'elle donne son résultat -% aussi *sur 4n*, renversé. De plus cette variante accepte que l'un ou même les -% deux inputs soient vides. Utilisé par la sommation et par la division (pour -% les quotients). Et aussi par la multiplication d'ailleurs.$\ -% INPUT: comme pour \XINT_add_A$\ -% 1. <N1> et <N2> renversés $\ -% 2. de longueur 4n (avec des leading zéros éventuels)$\ -% 3. l'un des deux ne doit pas se terminer par 0000$\ -% OUTPUT: la somme <N1>+<N2>, *aussi renversée* et *sur 4n*| -% \begin{macrocode} -\def\XINT_addr_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addr_az\W - \XINT_addr_B #1{#3#4#5#6}{#2}% -}% -\def\xint_addr_az\W\XINT_addr_B #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_addr_B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_addr_bz\W - \XINT_addr_E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT_addr_ABEA #1#2#3#4#5#6#7% -{% - \XINT_addr_A #2{#7#6#5#4#3}% -}% -\def\xint_addr_bz\W\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_CC\the\numexpr #1+10#5#4#3#2\relax -}% -\def\XINT_addr_CC #1#2#3#4#5#6#7% -{% - \XINT_addr_AC_checkcarry #2{#7#6#5#4#3}% -}% -\def\XINT_addr_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addr_AC_nocarry 0\XINT_addr_C -}% -\def\xint_addr_AC_nocarry 0\XINT_addr_C #1#2\W\X\Y\Z { #1#2}% -\def\XINT_addr_C #1#2#3#4#5% -{% - \xint_gob_til_W #2\xint_addr_cz\W - \XINT_addr_D {#5#4#3#2}{#1}% -}% -\def\XINT_addr_D #1% -{% - \expandafter\XINT_addr_CC\the\numexpr 1+10#1\relax -}% -\def\xint_addr_cz\W\XINT_addr_D #1#2{ #21000}% -% \end{macrocode} -% \subsubsection{Addition vIII: \csh{XINT_addm_A}} -% \lverb|& -% INPUT:\romannumeral0\XINT_addm_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés$\ -% 2. <N1> de longueur 4n ; <N2> non$\ -% 3. <N2> est *garanti au moins aussi long* que <N1>$\ -% OUTPUT: la somme <N1>+<N2>, ordre normal, pas sur 4n, leading zeros retirés. -% Utilisé par la multiplication.| -% \begin{macrocode} -\def\XINT_addm_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addm_az\W - \XINT_addm_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addm_az\W\XINT_addm_AB #1#2% -{% - \XINT_addm_AC_checkcarry #1% -}% -\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addm_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_addm_ABEA #1#2#3.#4% -{% - \XINT_addm_A #2{#3#4}% -}% -\def\XINT_addm_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C -}% -\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_addm_C #1#2#3#4#5% -{% - \xint_gob_til_W - #5\xint_addm_cw - #4\xint_addm_cx - #3\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD {#5#4#3#2}{#1}% -}% -\def\XINT_addm_CD #1% -{% - \expandafter\XINT_addm_CC\the\numexpr 1+10#1.% -}% -\def\XINT_addm_CC #1#2#3.#4% -{% - \XINT_addm_AC_checkcarry #2{#3#4}% -}% -\def\xint_addm_cw - #1\xint_addm_cx - #2\xint_addm_cy - #3\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.% -}% -\def\XINT_addm_CDw #1.#2#3\X\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cx - #1\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.% -}% -\def\XINT_addm_CDx #1.#2#3\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cy - #1\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDy\the\numexpr 1+#1.% -}% -\def\XINT_addm_CDy #1.#2#3\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}% -\edef\XINT_addm_end #1#2#3#4#5% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}% -% \end{macrocode} -% \subsubsection{Addition vIV: \csh{XINT_addp_A}} -% \lverb|& -% INPUT: -% \romannumeral0\XINT_addp_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés$\ -% 2. <N1> de longueur 4n ; <N2> non$\ -% 3. <N2> est *garanti au moins aussi long* que <N1>$\ -% OUTPUT: la somme <N1>+<N2>, dans l'ordre renversé, sur 4n, et en faisant -% attention de ne pas terminer en 0000. -% Utilisé par la multiplication servant pour le calcul des puissances.| +\def\XINT_inc_pos #1\Z +{% + \expandafter\XINT_inc_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_inc_pos_aa {\XINT_add_aa 100000001!\Z!\Z!\Z!\Z!\Z!\W }% +% \end{macrocode} +% \subsection{Core arithmetic} +% \lverb|The four operations have been rewritten entirely for release v1.2. +% The new routines works with separated blocks of eight digits. They all measure +% first the lengths of the arguments, even addition and subtraction (this was +% not the case with xintcore.sty 1.1 or earlier.) +% +% The technique of chaining \the\numexpr induces a limitation on the +% maximal size depending on the size of the input save stack and the maximum +% expansion depth. For the current (TL2015) settings (5000, resp. 10000), the +% induced limit for addition of numbers is at 19968 and for multiplication +% it is observed to be 19959 (valid as of 2015/10/07). +% +% Side remark: I tested that \the\numexpr was more efficient than \number. But +% it reduced the allowable numbers for addition from 19976 digits to 19968 +% digits.| +% +% \subsection{\csbh{xintiAdd}, \csbh{xintiiAdd}} % \begin{macrocode} -\def\XINT_addp_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addp_az\W - \XINT_addp_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addp_az\W\XINT_addp_AB #1#2% +\def\xintiAdd {\romannumeral0\xintiadd }% +\def\xintiadd #1{\expandafter\XINT_iadd\romannumeral0\xintnum{#1}\Z }% +\def\xintiiAdd {\romannumeral0\xintiiadd }% +\def\xintiiadd #1{\expandafter\XINT_iiadd\romannumeral-`0#1\Z }% +\def\XINT_iiadd #1#2\Z #3% {% - \XINT_addp_AC_checkcarry #1% + \expandafter\XINT_add_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\XINT_addp_AC_checkcarry #1% +\def\XINT_iadd #1#2\Z #3% {% - \xint_gob_til_zero #1\xint_addp_AC_nocarry 0\XINT_addp_C + \expandafter\XINT_add_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xint_addp_AC_nocarry 0\XINT_addp_C +\def\XINT_add_fork #1#2\Z #3\Z {\XINT_add_nfork #1#3\Z #2\Z}% +\def\XINT_add_nfork #1#2% {% - \XINT_addp_F -}% -\def\XINT_addp_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addp_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z + \xint_UDzerofork + #1\XINT_add_firstiszero + #2\XINT_add_secondiszero + 0{}% + \krof + \xint_UDsignsfork + #1#2\XINT_add_minusminus + #1-\XINT_add_minusplus + #2-\XINT_add_plusminus + --\XINT_add_plusplus + \krof #1#2% }% -\def\XINT_addp_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addp_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax +\def\XINT_add_firstiszero #1\krof 0#2#3\Z #4\Z { #2#3}% +\def\XINT_add_secondiszero #1\krof #20#3\Z #4\Z { #2#4}% +\def\XINT_add_minusminus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pp_a {}{}}% +\def\XINT_add_minusplus #1#2{\XINT_sub_mm_a {}#2}% +\def\XINT_add_plusminus #1#2% + {\expandafter\XINT_opp\romannumeral0\XINT_sub_mm_a #1{}}% +\def\XINT_add_pp_a #1#2#3\Z +{% + \expandafter\XINT_add_pp_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\let\XINT_add_plusplus \XINT_add_pp_a +\def\XINT_add_pp_b #1.#2\X #3\Z +{% + \expandafter\XINT_add_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_add_checklengths #1.#2.% +{% + \ifnum #2>#1 + \expandafter\XINT_add_exchange + \else + \expandafter\XINT_add_A + \fi + #1.#2.% }% -\def\XINT_addp_ABEA #1#2#3#4#5#6#7% +\def\XINT_add_exchange #1.#2.#3\Z!\Z!\Z!\Z!\Z!\W #4\Z {% - \XINT_addp_A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite + \XINT_add_A #2.#1.#4\Z!\Z!\Z!\Z!\Z!\W #3\Z }% -\def\XINT_addp_C #1#2#3#4#5% +\def\XINT_add_A #1.#2.% {% - \xint_gob_til_W - #5\xint_addp_cw - #4\xint_addp_cx - #3\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD {#5#4#3#2}{#1}% + \ifnum #1>\xint_c_vi % + \expandafter\XINT_add_aa + \else \expandafter\XINT_add_aa_small + \fi }% -\def\XINT_addp_CD #1% +%%%%%%%%%%%% +\def\XINT_add_out #1\Z #2\W% {% - \expandafter\XINT_addp_CC\the\numexpr 1+10#1\relax + \expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}% + #11\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_addp_CC #1#2#3#4#5#6#7% +\def\XINT_add_out_small #1\Z #2\W% {% - \XINT_addp_AC_checkcarry #2{#7#6#5#4#3}% + \XINT_smallunrevbyviii #11\Z!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\xint_addp_cw - #1\xint_addp_cx - #2\xint_addp_cy - #3\xint_addp_cz - \W\XINT_addp_CD +%%%%%%%%%%%% +\def\XINT_add_aa {\expandafter\XINT_add_out\the\numexpr\XINT_add_a \xint_c_ii}% +\def\XINT_add_aa_small + {\expandafter\XINT_add_out_small\the\numexpr\XINT_add_a \xint_c_ii}% +\def\XINT_add_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter\XINT_addp_CDw\the\numexpr \xint_c_i+10#1#2#3\relax + \XINT_add_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_addp_CDw #1#2#3#4#5#6% +\def\XINT_add_b #1#2!#3!% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDw_zeros - 0000\XINT_addp_endDw #2#3#4#5% + \xint_gob_til_Z #2\XINT_add_bi \Z + \expandafter\XINT_add_c\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_endDw_zeros 0000\XINT_addp_endDw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cx - #1\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD +\def\XINT_add_bi\Z + \expandafter\XINT_add_c + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6!#7!#8!#9!\Z !\W {% - \expandafter\XINT_addp_CDx\the\numexpr \xint_c_i+100#1#2\relax + \XINT_add_k #1#3!#5!#7!#9!% }% -\def\XINT_addp_CDx #1#2#3#4#5#6% +\def\XINT_add_c #1#2.% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDx_zeros - 0000\XINT_addp_endDx #2#3#4#5% + 1#2\expandafter!\the\numexpr\XINT_add_d #1% }% -\def\XINT_addp_endDx_zeros 0000\XINT_addp_endDx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cy #1\xint_addp_cz\W\XINT_addp_CD +\def\XINT_add_d #1#2!#3!% {% - \expandafter\XINT_addp_CDy\the\numexpr \xint_c_i+1000#1\relax + \xint_gob_til_Z #2\XINT_add_di \Z + \expandafter\XINT_add_e\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_CDy #1#2#3#4#5#6% +\def\XINT_add_di\Z\expandafter\XINT_add_e + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6!#7!#8\W {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDy_zeros - 0000\XINT_addp_endDy #2#3#4#5% + \XINT_add_k #1#3!#5!#7!% }% -\def\XINT_addp_endDy_zeros 0000\XINT_addp_endDy 0000#1\Z{ #1}% -\def\XINT_addp_endDy #1#2#3#4#5\Z{ #5#4#3#2#1}% -\def\xint_addp_cz\W\XINT_addp_CD #1#2{ #21000}% -\def\XINT_addp_F #1#2#3#4#5% +\def\XINT_add_e #1#2.% {% - \xint_gob_til_W - #5\xint_addp_Gw - #4\xint_addp_Gx - #3\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G {#2#3#4#5}{#1}% + 1#2\expandafter!\the\numexpr\XINT_add_f #1% }% -\def\XINT_addp_G #1#2% +\def\XINT_add_f #1#2!#3!% {% - \XINT_addp_F {#2#1}% + \xint_gob_til_Z #2\XINT_add_fi \Z + \expandafter\XINT_add_g\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\xint_addp_Gw - #1\xint_addp_Gx - #2\xint_addp_Gy - #3\xint_addp_Gz - \W\XINT_addp_G #4% +\def\XINT_add_fi\Z\expandafter\XINT_add_g + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6\W {% - \xint_gob_til_zeros_iv #3#2#10\XINT_addp_endGw_zeros - 0000\XINT_addp_endGw #3#2#10% + \XINT_add_k #1#3!#5!% }% -\def\XINT_addp_endGw_zeros 0000\XINT_addp_endGw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gx - #1\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G #3% +\def\XINT_add_g #1#2.% {% - \xint_gob_til_zeros_iv #2#100\XINT_addp_endGx_zeros - 0000\XINT_addp_endGx #2#100% + 1#2\expandafter!\the\numexpr\XINT_add_h #1% }% -\def\XINT_addp_endGx_zeros 0000\XINT_addp_endGx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gy - #1\xint_addp_Gz - \W\XINT_addp_G #2% +\def\XINT_add_h #1#2!#3!% {% - \xint_gob_til_zeros_iv #1000\XINT_addp_endGy_zeros - 0000\XINT_addp_endGy #1000% + \xint_gob_til_Z #2\XINT_add_hi \Z + \expandafter\XINT_add_i\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_endGy_zeros 0000\XINT_addp_endGy 0000#1\Z{ #1}% -\def\XINT_addp_endGy #1#2#3#4#5\Z{ #5#1#2#3#4}% -\def\xint_addp_Gz\W\XINT_addp_G #1#2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintiAdd}, \csh{xintiiAdd}} -% \lverb|ADDITION -% [algo plus efficace lorsque le premier argument plus long que le second] -% -% Note (octobre 2014, pendant la préparation de la sortie de 1.1) -% -% Je n'aurais pas dû l'appeler \xintAdd, mais seulement \xintiAdd. Le format -% de sortie de \xintAdd est modifié par xintfrac.sty, celui de \xintiAdd ne -% bouge pas, et \xintiiAdd reste la version stricte.| -% \begin{macrocode} -\def\xintiiAdd {\romannumeral0\xintiiadd }% -\def\xintiiadd #1{\expandafter\xint_iiadd\romannumeral-`0#1\Z }% -\def\xint_iiadd #1#2\Z #3% +\def\XINT_add_hi\Z + \expandafter\XINT_add_i\the\numexpr#1+#2+#3-\xint_c_ii.#4\W {% - \expandafter\XINT_add_fork\expandafter #1\romannumeral-`0#3\Z #2\Z + \XINT_add_k #1#3!% }% -\def\xintiAdd {\romannumeral0\xintiadd }% -\def\xintiadd #1% +\def\XINT_add_i #1#2.% {% - \expandafter\xint_add\romannumeral0\xintnum{#1}\Z + 1#2\expandafter!\the\numexpr\XINT_add_a #1% }% -\def\xint_add #1#2\Z #3% +%%%%%%%%%%%% +\def\XINT_add_k #1% + {\if #12\expandafter\XINT_add_ke\else\expandafter\XINT_add_l \fi}% +\def\XINT_add_ke #1% {% - \expandafter\XINT_add_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z + \xint_gob_til_Z #1\XINT_add_kf\Z 1% }% -\let\xintAdd\xintiAdd \let\xintadd\xintiadd -\def\XINT_add_fork #1#2% +\def\XINT_add_kf\Z 1{1}% +\def\XINT_add_l #1% {% - \xint_UDzerofork - #1\XINT_add_firstiszero - #2\XINT_add_secondiszero - 0{}% - \krof - \xint_UDsignsfork - #1#2\XINT_add_minusminus - #1-\XINT_add_minusplus - #2-\XINT_add_plusminus - --\XINT_add_plusplus - \krof #1#2% + \xint_gob_til_Z #1\XINT_add_lf\Z \XINT_add_m 1% }% -\def\XINT_add_firstiszero #1\krof #2#3\Z #4\Z { #3}% -\def\XINT_add_secondiszero #1\krof #2#3\Z #4\Z { #2#4}% -\def\XINT_add_plusplus #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#2#3}}% -\def\XINT_add_minusminus #1#2#3\Z #4\Z - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#3}}% -\def\XINT_add_minusplus #1#2#3\Z #4\Z {\XINT_sub_pre {#2#3}{#4}}% -\def\XINT_add_plusminus #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#3}}% -% \end{macrocode} -% \lverb|positive summands| -% \begin{macrocode} -\def\XINT_add_pre #1% +\def\XINT_add_lf\Z\XINT_add_m 1{100000001}% +\def\XINT_add_m #1!% {% - \expandafter\XINT_add_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \expandafter\XINT_add_n\the\numexpr\xint_c_i+#1.% }% -\def\XINT_add_pre_b #1#2% +\def\XINT_add_n #1#2.% {% - \expandafter\XINT_add_A - \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_add_o #1% }% +\def\XINT_add_o #1% + {\if #12\expandafter\XINT_add_l\else\expandafter\XINT_add_ke \fi}% % \end{macrocode} % \subsection{\csh{xintiSub}, \csh{xintiiSub}} -% \lverb|Release 1.09a has \xintnum added into \xintiSub.| +% \lverb|Entirely rewritten for v1.2.| % \begin{macrocode} \def\xintiiSub {\romannumeral0\xintiisub }% -\def\xintiisub #1{\expandafter\xint_iisub\romannumeral-`0#1\Z }% -\def\xint_iisub #1#2\Z #3% +\def\xintiisub #1{\expandafter\XINT_iisub\romannumeral-`0#1\Z }% +\def\XINT_iisub #1#2\Z #3% {% - \expandafter\XINT_sub_fork\expandafter #1\romannumeral-`0#3\Z #2\Z + \expandafter\XINT_sub_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xintiSub {\romannumeral0\xintisub }% -\def\xintisub #1% +\def\xintiSub {\romannumeral0\xintisub }% +\def\xintisub #1{\expandafter\XINT_isub\romannumeral0\xintnum{#1}\Z }% +\def\XINT_isub #1#2\Z #3% {% - \expandafter\xint_sub\romannumeral0\xintnum{#1}\Z + \expandafter\XINT_sub_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xint_sub #1#2\Z #3% -{% - \expandafter\XINT_sub_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z -}% -\let\xintSub\xintiSub \let\xintsub\xintisub -\def\XINT_sub_fork #1#2% +\def\XINT_sub_nfork #1#2% {% \xint_UDzerofork #1\XINT_sub_firstiszero @@ -16315,756 +16383,437 @@ $1$ or $-1$. --\XINT_sub_plusplus \krof #1#2% }% -\def\XINT_sub_firstiszero #1\krof #2#3\Z #4\Z {\XINT_opp #3}% -\def\XINT_sub_secondiszero #1\krof #2#3\Z #4\Z { #2#4}% -\def\XINT_sub_plusplus #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#2#3}}% -\def\XINT_sub_minusminus #1#2#3\Z #4\Z {\XINT_sub_pre {#3}{#4}}% -\def\XINT_sub_minusplus #1#2#3\Z #4\Z - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#2#3}}% -\def\XINT_sub_plusminus #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#3}}% -% \end{macrocode} -% \lverb|SOUSTRACTION A-B avec A premier argument, B second argument de -% \xintSub et ensuite \XINT_sub_pre ici| -% \begin{macrocode} -\def\XINT_sub_pre #1% -{% - \expandafter\XINT_sub_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_sub_pre_b #1#2% -{% - \expandafter\XINT_sub_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z -}% -% \end{macrocode} -% \lverb|& -% \romannumeral0\XINT_sub_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000.$\ output: N2 - N1$\ -% Elle donne le résultat dans le **bon ordre**, avec le bon signe, -% et sans zéros superflus.| -% \begin{macrocode} -\def\XINT_sub_A #1#2#3\W\X\Y\Z #4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_az - \W\XINT_sub_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z -}% -\def\XINT_sub_B #1#2#3#4#5#6#7% +\def\XINT_sub_firstiszero #1\krof 0#2#3\Z #4\Z {\XINT_opp #2#3}% +\def\XINT_sub_secondiszero #1\krof #20#3\Z #4\Z { #2#4}% +\def\XINT_sub_plusminus #1#2{\XINT_add_pp_a #1{}}% +\def\XINT_sub_plusplus #1#2% + {\expandafter\XINT_opp\romannumeral0\XINT_sub_mm_a #1#2}% +\def\XINT_sub_minusplus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pp_a {}#2}% +\def\XINT_sub_minusminus #1#2{\XINT_sub_mm_a {}{}}% +\def\XINT_sub_mm_a #1#2#3\Z +{% + \expandafter\XINT_sub_mm_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\def\XINT_sub_mm_b #1.#2\X #3\Z +{% + \expandafter\XINT_sub_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_sub_checklengths #1.#2.% {% - \xint_gob_til_W - #4\xint_sub_bz - \W\XINT_sub_onestep #1#2{#7#6#5#4}{#3}% -}% -% \end{macrocode} -% \lverb|& -% d'abord la branche principale -% #6 = 4 chiffres de N1, plus significatif en *premier*, -% #2#3#4#5 chiffres de N2, plus significatif en *dernier* -% On veut N2 - N1.| -% \begin{macrocode} -\def\XINT_sub_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_sub_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% + \ifnum #2>#1 + \expandafter\XINT_sub_exchange + \else + \expandafter\XINT_sub_aa + \fi }% -% \end{macrocode} -% \lverb|ON PRODUIT LE RÉSULTAT DANS LE BON ORDRE| -% \begin{macrocode} -\def\XINT_sub_backtoA #1#2#3.#4% +\def\XINT_sub_exchange #1\Z!\Z!\Z!\Z!\Z!\W #2\Z {% - \XINT_sub_A #2{#3#4}% + \expandafter\XINT_opp\romannumeral0\XINT_sub_aa + #2\Z!\Z!\Z!\Z!\Z!\W #1\Z }% -\def\xint_sub_bz - \W\XINT_sub_onestep #1#2#3#4#5#6#7% +%%%%%%%%%%%% +\def\XINT_sub_prepare_rescue #1\W {\relax\Z-\W}% +\def\XINT_sub_prepare_cuz #1\W {\relax\XINT_cuz_byviii!\Z 0\W\R}% +%%%%%%%%%%%% +\def\XINT_sub_aa {\expandafter\XINT_sub_out\the\numexpr\XINT_sub_a \xint_c_i }% +\def\XINT_sub_out #1\Z #2#3\W {% - \xint_UDzerofork - #1\XINT_sub_C % une retenue - 0\XINT_sub_D % pas de retenue - \krof - {#7}#2#3#4#5% + \if-#2\expandafter\XINT_sub_startrescue\fi + \expandafter\XINT_cuz_small + \romannumeral0\XINT_unrevbyviii {}#11\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_sub_D #1#2\W\X\Y\Z +\def\XINT_sub_startrescue\expandafter\XINT_cuz_small + \romannumeral0\XINT_unrevbyviii #1#2\Z!#3\W {% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% + \expandafter\XINT_sub_rescue_finish + \the\numexpr\XINT_sub_rescue_a #2!% + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W \R }% -\def\XINT_sub_C #1#2#3#4#5% +\def\XINT_sub_rescue_finish + {\expandafter-\romannumeral0\expandafter\XINT_cuz + \romannumeral0\XINT_unrevbyviii {}}% +\def\XINT_sub_rescue_a #1!% {% - \xint_gob_til_W - #2\xint_sub_cz - \W\XINT_sub_AC_onestep {#5#4#3#2}{#1}% + \expandafter\XINT_sub_rescue_c\the\numexpr \xint_c_xii_e_viii-#1.% }% -\def\XINT_sub_AC_onestep #1% +\def\XINT_sub_rescue_c 1#1#2.% {% - \expandafter\XINT_sub_backtoC\the\numexpr 11#1-\xint_c_i.% + 1#2\expandafter!\the\numexpr\XINT_sub_rescue_d #1% }% -\def\XINT_sub_backtoC #1#2#3.#4% +\def\XINT_sub_rescue_d #1#2#3!% {% - \XINT_sub_AC_checkcarry #2{#3#4}% la retenue va \^etre examin\'ee + \xint_gob_til_minus #2\XINT_sub_rescue_z -% + \expandafter\XINT_sub_rescue_c\the\numexpr \xint_c_xii_e_viii_mone-#2#3+#1.% }% -\def\XINT_sub_AC_checkcarry #1% +\def\XINT_sub_rescue_z #1.{1!}% +%%%%%%%%%%%% +\def\XINT_sub_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \xint_gob_til_one #1\xint_sub_AC_nocarry 1\XINT_sub_C + \XINT_sub_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\xint_sub_AC_nocarry 1\XINT_sub_C #1#2\W\X\Y\Z +\def\XINT_sub_b #1#2#3!#4!% {% - \expandafter - \XINT_cuz_loop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1\W\W\W\W\W\W\W\Z + \xint_gob_til_Z #2\XINT_sub_bi \Z + \expandafter\XINT_sub_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\xint_sub_cz\W\XINT_sub_AC_onestep #1% +\def\XINT_sub_c 1#1#2.% {% - \XINT_cuz + 1#2\expandafter!\the\numexpr\XINT_sub_d #1% }% -\def\xint_sub_az\W\XINT_sub_B #1#2#3#4#5#6#7% +\def\XINT_sub_d #1#2#3!#4!% {% - \xint_gob_til_W - #4\xint_sub_ez - \W\XINT_sub_Eenter #1{#3}#4#5#6#7% + \xint_gob_til_Z #2\XINT_sub_di \Z + \expandafter\XINT_sub_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -% \end{macrocode} -% \lverb|le premier nombre continue, le résultat sera < 0.| -% \begin{macrocode} -\def\XINT_sub_Eenter #1#2% +\def\XINT_sub_e 1#1#2.% {% - \expandafter - \XINT_sub_E\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - \W\X\Y\Z #1% + 1#2\expandafter!\the\numexpr\XINT_sub_f #1% }% -\def\XINT_sub_E #1#2#3#4#5#6% +\def\XINT_sub_f #1#2#3!#4!% {% - \xint_gob_til_W #3\xint_sub_F\W - \XINT_sub_Eonestep #1{#6#5#4#3}{#2}% + \xint_gob_til_Z #2\XINT_sub_fi \Z + \expandafter\XINT_sub_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_sub_Eonestep #1#2% +\def\XINT_sub_g 1#1#2.% {% - \expandafter\XINT_sub_backtoE\the\numexpr 109999-#2+#1.% + 1#2\expandafter!\the\numexpr\XINT_sub_h #1% }% -\def\XINT_sub_backtoE #1#2#3.#4% +\def\XINT_sub_h #1#2#3!#4!% {% - \XINT_sub_E #2{#3#4}% + \xint_gob_til_Z #2\XINT_sub_hi \Z + \expandafter\XINT_sub_i\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\xint_sub_F\W\XINT_sub_Eonestep #1#2#3#4% +\def\XINT_sub_i 1#1#2.% {% - \xint_UDonezerofork - #4#1{\XINT_sub_Fdec 0}% soustraire 1. Et faire signe - - #1#4{\XINT_sub_Finc 1}% additionner 1. Et faire signe - - 10\XINT_sub_DD % terminer. Mais avec signe - - \krof - {#3}% + 1#2\expandafter!\the\numexpr\XINT_sub_a #1% }% -\def\XINT_sub_DD {\expandafter\xint_minus_thenstop\romannumeral0\XINT_sub_D }% -\def\XINT_sub_Fdec #1#2#3#4#5#6% +\def\XINT_sub_bi\Z + \expandafter\XINT_sub_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W {% - \xint_gob_til_W #3\xint_sub_Fdec_finish\W - \XINT_sub_Fdec_onestep #1{#6#5#4#3}{#2}% + \XINT_sub_k #1#2!#5!#7!#9!% }% -\def\XINT_sub_Fdec_onestep #1#2% +\def\XINT_sub_di\Z + \expandafter\XINT_sub_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W {% - \expandafter\XINT_sub_backtoFdec\the\numexpr 11#2+#1-\xint_c_i.% + \XINT_sub_k #1#2!#5!#7!% }% -\def\XINT_sub_backtoFdec #1#2#3.#4% +\def\XINT_sub_fi\Z + \expandafter\XINT_sub_g\the\numexpr#1+1#2-#3.#4!#5!#6\W {% - \XINT_sub_Fdec #2{#3#4}% + \XINT_sub_k #1#2!#5!% }% -\def\xint_sub_Fdec_finish\W\XINT_sub_Fdec_onestep #1#2% +\def\XINT_sub_hi\Z + \expandafter\XINT_sub_i\the\numexpr#1+1#2-#3.#4\W {% - \expandafter\xint_minus_thenstop\romannumeral0\XINT_cuz + \XINT_sub_k #1#2!% }% -\def\XINT_sub_Finc #1#2#3#4#5#6% +%%%%%%%%%%%% +\def\XINT_sub_k #1#2% {% - \xint_gob_til_W #3\xint_sub_Finc_finish\W - \XINT_sub_Finc_onestep #1{#6#5#4#3}{#2}% + \xint_gob_til_Z #2\XINT_sub_p\Z \XINT_sub_l #1#2% }% -\def\XINT_sub_Finc_onestep #1#2% +\def\XINT_sub_l #1% {% - \expandafter\XINT_sub_backtoFinc\the\numexpr 10#2+#1.% + \xint_UDzerofork + #1\XINT_sub_m + 0{}% + \krof }% -\def\XINT_sub_backtoFinc #1#2#3.#4% +\def\XINT_sub_m #1!% {% - \XINT_sub_Finc #2{#3#4}% + \expandafter\XINT_sub_n\the\numexpr 1#1-\xint_c_i!% }% -\def\xint_sub_Finc_finish\W\XINT_sub_Finc_onestep #1#2#3% +\def\XINT_sub_n 1#1% {% \xint_UDzerofork - #1{\expandafter\expandafter\expandafter - \xint_minus_thenstop\xint_cleanupzeros_nostop}% - 0{ -1}% + #1{\XINT_sub_o}% + 0{\XINT_sub_n_checkzero}% \krof - #3% }% -\def\xint_sub_ez\W\XINT_sub_Eenter #1% +\def\XINT_sub_o #1!{1#1\expandafter!\the\numexpr\XINT_sub_m }% +\def\XINT_sub_n_checkzero #1!% {% - \xint_UDzerofork - #1\XINT_sub_K % il y a une retenue - 0\XINT_sub_L % pas de retenue - \krof + \xint_gob_til_eightzeroes #1\XINT_sub_n_prepare_cuz 00000000% + 1#1!% }% -\def\XINT_sub_L #1\W\X\Y\Z {\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z }% -\def\XINT_sub_K #1% +\def\XINT_sub_n_prepare_cuz 00000000100000000{1\XINT_sub_prepare_cuz}% +\def\XINT_sub_p\Z\XINT_sub_l #1\Z!% {% - \expandafter - \XINT_sub_KK\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_sub_KK #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_sub_KK_finish\W - \XINT_sub_KK_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_KK_onestep #1#2% -{% - \expandafter\XINT_sub_backtoKK\the\numexpr 109999-#2+#1.% -}% -\def\XINT_sub_backtoKK #1#2#3.#4% -{% - \XINT_sub_KK #2{#3#4}% -}% -\def\xint_sub_KK_finish\W\XINT_sub_KK_onestep #1#2#3% -{% - \expandafter\xint_minus_thenstop - \romannumeral0\XINT_cuz_loop #3\W\W\W\W\W\W\W\Z + \xint_UDzerofork + #1{-1\XINT_sub_prepare_rescue}% + 0{1\XINT_sub_prepare_cuz }% + \krof }% % \end{macrocode} % \subsection{\csh{xintiMul}, \csh{xintiiMul}} -% \lverb|1.09a adds \xintnum| +% \lverb|Completely rewritten for v1.2.| % \begin{macrocode} -\def\xintiiMul {\romannumeral0\xintiimul }% -\def\xintiimul #1% +\def\xintiMul {\romannumeral0\xintimul }% +\def\xintimul #1% {% - \expandafter\xint_iimul\expandafter {\romannumeral-`0#1}% + \expandafter\XINT_imul\romannumeral0\xintnum{#1}\Z }% -\def\xint_iimul #1#2% +\def\XINT_imul #1#2\Z #3% {% - \expandafter\XINT_mul_fork \romannumeral-`0#2\Z #1\Z + \expandafter\XINT_mul_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xintiMul {\romannumeral0\xintimul }% -\def\xintimul #1% +\def\xintiiMul {\romannumeral0\xintiimul }% +\def\xintiimul #1% {% - \expandafter\xint_mul\expandafter {\romannumeral0\xintnum{#1}}% + \expandafter\XINT_iimul\romannumeral-`0#1\Z }% -\def\xint_mul #1#2% +\def\XINT_iimul #1#2\Z #3% {% - \expandafter\XINT_mul_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_mul_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\let\xintMul\xintiMul \let\xintmul\xintimul -\def\XINT_Mul #1#2{\romannumeral0\XINT_mul_fork #2\Z #1\Z }% % \end{macrocode} -% \lverb|& -% MULTIPLICATION$\ -% Ici #1#2 = 2e input et #3#4 = 1er input $\ -% Release 1.03 adds some overhead to first compute and compare the -% lengths of the two inputs. The algorithm is asymmetrical and whether -% the first input is the longest or the shortest sometimes has a strong -% impact. 50 digits times 1000 digits used to be 5 times faster -% than 1000 digits times 50 digits. With the new code, the user input -% order does not matter as it is decided by the routine what is best. -% This is important for the extension to fractions, as there is no way -% then to generally control or guess the most frequent sizes of the -% inputs besides actually computing their lengths. | -% \begin{macrocode} -\def\XINT_mul_fork #1#2\Z #3#4\Z +% \lverb|I have changed the fork, and it complicates matters elsewhere.| +% \begin{macrocode} +\def\XINT_mul_fork #1#2\Z #3\Z{\XINT_mul_nfork #1#3\Z #2\Z}% +\def\XINT_mul_nfork #1#2% {% \xint_UDzerofork #1\XINT_mul_zero - #3\XINT_mul_zero + #2\XINT_mul_zero 0{}% \krof \xint_UDsignsfork - #1#3\XINT_mul_minusminus % #1 = #3 = - - #1-{\XINT_mul_minusplus #3}% % #1 = - - #3-{\XINT_mul_plusminus #1}% % #3 = - - --{\XINT_mul_plusplus #1#3}% - \krof - {#2}{#4}% -}% -\def\XINT_mul_zero #1\krof #2#3{ 0}% -\def\XINT_mul_minusminus #1#2% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2}}% - {\romannumeral0\xintlength {#1}}{#1}{#2}% -}% -\def\XINT_mul_minusplus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#1#3}}% - {\romannumeral0\xintlength {#2}}{#2}{#1#3}% -}% -\def\XINT_mul_plusminus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#3}}% - {\romannumeral0\xintlength {#1#2}}{#1#2}{#3}% -}% -\def\XINT_mul_plusplus #1#2#3#4% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2#4}}% - {\romannumeral0\xintlength {#1#3}}{#1#3}{#2#4}% -}% -\def\XINT_mul_choice_a #1#2% -{% - \expandafter\XINT_mul_choice_b\expandafter{#2}{#1}% -}% -\def\XINT_mul_choice_b #1#2% -{% - \ifnum #1<\xint_c_v - \expandafter\XINT_mul_choice_littlebyfirst - \else - \ifnum #2<\xint_c_v - \expandafter\expandafter\expandafter\XINT_mul_choice_littlebysecond - \else - \expandafter\expandafter\expandafter\XINT_mul_choice_compare - \fi - \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_littlebyfirst #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #3\expandafter}% - \romannumeral0\XINT_RQ {}#4\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z -}% -\def\XINT_mul_choice_littlebysecond #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #4\expandafter}% - \romannumeral0\XINT_RQ {}#3\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z + #1#2\XINT_mul_minusminus + #1-\XINT_mul_minusplus + #2-\XINT_mul_plusminus + --\XINT_mul_plusplus + \krof #1#2% }% -\def\XINT_mul_choice_compare #1#2% -{% - \ifnum #1>#2 - \expandafter \XINT_mul_choice_i +\def\XINT_mul_zero #1\krof #2#3\Z #4\Z { 0}% +\def\XINT_mul_minusminus #1#2{\XINT_mul_plusplus {}{}}% +\def\XINT_mul_minusplus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_mul_plusplus {}#2}% +\def\XINT_mul_plusminus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_mul_plusplus #1{}}% +\def\XINT_mul_plusplus #1#2#3\Z +{% + \expandafter\XINT_mul_pre_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \W #1% +}% +\def\XINT_mul_pre_b #1.#2\W #3\Z +{% + \expandafter\XINT_mul_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + 1\Z!\W #21\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +% \end{macrocode} +% \lverb|Cooking recipee, 2015/10/05.| +% \begin{macrocode} +\def\XINT_mul_checklengths #1.#2.% +{% + \ifnum #2=\xint_c_i\expandafter\XINT_mul_smallbyfirst\fi + \ifnum #1=\xint_c_i\expandafter\XINT_mul_smallbysecond\fi + \ifnum #2<#1 + \ifnum \numexpr (#2-\xint_c_i)*(#1-#2)<383 + \XINT_mul_exchange + \fi \else - \expandafter \XINT_mul_choice_ii + \ifnum \numexpr (#1-\xint_c_i)*(#2-#1)>383 + \XINT_mul_exchange + \fi \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_i #1#2% -{% - \ifnum #1<\numexpr\ifcase \numexpr (#2-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_same - \else - \expandafter\XINT_mul_choice_permute - \fi -}% -\def\XINT_mul_choice_ii #1#2% -{% - \ifnum #2<\numexpr\ifcase \numexpr (#1-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_permute - \else - \expandafter\XINT_mul_choice_same - \fi -}% -\def\XINT_mul_choice_same #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #2\W\W\W\W -}% -\def\XINT_mul_choice_permute #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W -}% -% \end{macrocode} -% \lverb|& -% Cette portion de routine d'addition se branche directement sur _addr_ -% lorsque -% le premier nombre est épuisé, ce qui est garanti arriver avant le second -% nombre. Elle produit son résultat toujours sur 4n, renversé. Ses deux inputs -% sont garantis sur 4n.| -% \begin{macrocode} -\def\XINT_mul_Ar #1#2#3#4#5#6% -{% - \xint_gob_til_Z #6\xint_mul_br\Z\XINT_mul_Br #1{#6#5#4#3}{#2}% -}% -\def\xint_mul_br\Z\XINT_mul_Br #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_mul_Br #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \expandafter\XINT_mul_ABEAr - \the\numexpr #1+10#2+#8#7#6#5.{#3}#4\W\X\Y\Z -}% -\def\XINT_mul_ABEAr #1#2#3#4#5#6.#7% -{% - \XINT_mul_Ar #2{#7#6#5#4#3}% -}% -% \end{macrocode} -% \subsubsection{``Small'' multiplication: \csh{XINT_mul_Mr}} -% \lverb|& -% << Petite >> multiplication. -% mul_Mr renvoie le résultat *à l'envers*, sur *4n*$\ -% \romannumeral0\XINT_mul_Mr {<n>}<N>\Z\Z\Z\Z$\ -% Fait la multiplication de <N> par <n>, qui est < 10000. -% <N> est présenté *à l'envers*, sur *4n*. Lorsque <n> vaut 0, donne 0000.| + \XINT_mul_start +}% +\def\XINT_mul_smallbyfirst #1\XINT_mul_start 1#2!1\Z!\W +{% + \ifnum#2=\xint_c_i\expandafter\XINT_mul_oneisone\fi + \ifnum#2<\xint_c_xxii\expandafter\XINT_mul_verysmall\fi + \expandafter\XINT_mul_out\the\numexpr\XINT_smallmul 1#2!% +}% +\def\XINT_mul_smallbysecond #1\XINT_mul_start #2\W 1#3!1\Z!% +{% + \ifnum#3=\xint_c_i\expandafter\XINT_mul_oneisone\fi + \ifnum#3<\xint_c_xxii\expandafter\XINT_mul_verysmall\fi + \expandafter\XINT_mul_out\the\numexpr\XINT_smallmul 1#3!#2% +}% +\def\XINT_mul_oneisone #1!{\XINT_mul_out }% +\def\XINT_mul_verysmall\expandafter\XINT_mul_out + \the\numexpr\XINT_smallmul 1#1!% + {\expandafter\XINT_mul_out\the\numexpr\XINT_verysmallmul 0.#1!}% +\def\XINT_mul_exchange #1\XINT_mul_start #2\W #31\Z!% + {\fi\fi\XINT_mul_start #31\Z!\W #2}% +\def\XINT_mul_start + {\expandafter\XINT_mul_out\the\numexpr\XINT_mul_loop 100000000!\Z\W}% +\def\XINT_mul_out + {\expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}}% +\def\XINT_mul_loop #1\Z #2\W #3\W 1#4!% +{% + \xint_gob_til_Z #4\XINT_mul_e \Z + \expandafter\XINT_mul_a\the\numexpr \XINT_smallmul 1#4!#3\W + #11!\W #3\W +}% +\def\XINT_mul_a #11\Z!\W #2!1!#3\W +{% + \expandafter\XINT_mul_b\the\numexpr + \XINT_add_a \xint_c_ii #2!\Z!\Z!\Z!\Z!\Z!\W #1\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_mul_b 1#1!{1#1\expandafter!\the\numexpr\XINT_mul_loop }% +\def\XINT_mul_e\Z #1\W #2!1!#3\W #4\W {#2!1\Z!}% +% \end{macrocode} +% \lverb|1.2 small and mini multiplication in base 10^8 with carry. On output +% the small multiplication suppresses ending zeroes. The situation is +% different with addition which may end up inserting a final 1!, thus +% multiplication is « cleaner » in that aspect. Used by the main +% multiplication routines. But division, float factorial, etc.. have their +% own variants as they need output with specific constraints. +% | % \begin{macrocode} -\def\XINT_mul_Mr #1% +\def\XINT_minimulwc_a 1#1.#2.#3!#4#5#6#7#8.% {% - \expandafter\XINT_mul_Mr_checkifzeroorone\expandafter{\the\numexpr #1}% + \expandafter\XINT_minimulwc_b + \the\numexpr \xint_c_x^ix+#1+#3*#8.#3*#4#5#6#7+#2*#8.#2*#4#5#6#7.% }% -\def\XINT_mul_Mr_checkifzeroorone #1% +\def\XINT_minimulwc_b 1#1#2#3#4#5#6.#7.% {% - \ifcase #1 - \expandafter\XINT_mul_Mr_zero - \or - \expandafter\XINT_mul_Mr_one - \else - \expandafter\XINT_mul_Nr - \fi - {0000}{}{#1}% + \expandafter\XINT_minimulwc_c + \the\numexpr \xint_c_x^ix+#1#2#3#4#5+#7.#6.% }% -\def\XINT_mul_Mr_zero #1\Z\Z\Z\Z { 0000}% -\def\XINT_mul_Mr_one #1#2#3#4\Z\Z\Z\Z { #4}% -\def\XINT_mul_Nr #1#2#3#4#5#6#7% +\def\XINT_minimulwc_c 1#1#2#3#4#5#6.#7.#8.% {% - \xint_gob_til_Z #4\xint_mul_pr\Z\XINT_mul_Pr {#1}{#3}{#7#6#5#4}{#2}{#3}% + 1#6#7\expandafter!% + \the\numexpr\expandafter\XINT_smallmul_a + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8.% }% -\def\XINT_mul_Pr #1#2#3% +\def\XINT_smallmul 1#1#2#3#4#5!{\XINT_smallmul_a 100000000.#1#2#3#4.#5!}% +\def\XINT_smallmul_a #1.#2.#3!1#4!% {% - \expandafter\XINT_mul_Lr\the\numexpr \xint_c_x^viii+#1+#2*#3\relax + \xint_gob_til_Z #4\XINT_smallmul_e\Z + \XINT_minimulwc_a #1.#2.#3!#4.#2.#3!% }% -\def\XINT_mul_Lr 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_Nr {#1#2#3#4}{#9#8#7#6#5}% -}% -\def\xint_mul_pr\Z\XINT_mul_Pr #1#2#3#4#5% -{% - \xint_gob_til_zeros_iv #1\XINT_mul_Mr_end_nocarry 0000% - \XINT_mul_Mr_end_carry #1{#4}% -}% -\def\XINT_mul_Mr_end_nocarry 0000\XINT_mul_Mr_end_carry 0000#1{ #1}% -\def\XINT_mul_Mr_end_carry #1#2#3#4#5{ #5#4#3#2#1}% +\def\XINT_smallmul_e\Z\XINT_minimulwc_a 1#1.#2\Z #3!% + {\xint_gob_til_eightzeroes #1\XINT_smallmul_f 000000001\relax #1!1\Z!}% +\def\XINT_smallmul_f 000000001\relax 00000000!1{1\relax}% % \end{macrocode} -% \subsubsection{``Small'' multiplication variant: \csh{XINT_mul_M}} -% \lverb|& -% << Petite >> multiplication. -% renvoie le résultat *à l'endroit*, avec *nettoyage des leading zéros*.$\ -% \romannumeral0\XINT_mul_M {<n>}<N>\Z\Z\Z\Z$\ -% Fait la multiplication de <N> par <n>, qui est < 10000. -% <N> est présenté *à l'envers*, sur *4n*. | +% \lverb|This is multiplication by 1 up to 21. Last time I checked it is never +% called with a wasteful multiplicand of 1.| % \begin{macrocode} -\def\XINT_mul_M #1% -{% - \expandafter\XINT_mul_M_checkifzeroorone\expandafter{\the\numexpr #1}% -}% -\def\XINT_mul_M_checkifzeroorone #1% -{% - \ifcase #1 - \expandafter\XINT_mul_M_zero - \or - \expandafter\XINT_mul_M_one - \else - \expandafter\XINT_mul_N - \fi - {0000}{}{#1}% -}% -\def\XINT_mul_M_zero #1\Z\Z\Z\Z { 0}% -\def\XINT_mul_M_one #1#2#3#4\Z\Z\Z\Z -{% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#4}% -}% -\def\XINT_mul_N #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\xint_mul_p\Z\XINT_mul_P {#1}{#3}{#7#6#5#4}{#2}{#3}% -}% -\def\XINT_mul_P #1#2#3% -{% - \expandafter\XINT_mul_L\the\numexpr \xint_c_x^viii+#1+#2*#3\relax -}% -\def\XINT_mul_L 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_N {#1#2#3#4}{#5#6#7#8#9}% -}% -\def\xint_mul_p\Z\XINT_mul_P #1#2#3#4#5% -{% - \XINT_mul_M_end #1#4% -}% -\edef\XINT_mul_M_end #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -% \end{macrocode} -% \subsubsection{Main routine: \csh{XINT_mul_enter}} -% \lverb|& -% Routine de multiplication principale -% (attention délimiteurs modifiés pour 1.08)$\ -% Le résultat partiel est toujours maintenu avec significatif à -% droite et il a un nombre multiple de 4 de chiffres$\ -% \romannumeral0\XINT_mul_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W$\ -% avec <N1> *renversé*, *longueur 4n* (zéros éventuellement ajoutés -% au-delà du chiffre le plus significatif) -% et <N2> dans l'ordre *normal*, et pas forcément longueur 4n. -% pas de signes.$\ -% Pour 1.08: dans \XINT_mul_enter et les modifs de 1.03 -% qui filtrent les courts, on pourrait croire que le -% second opérande a au moins quatre chiffres; mais le problème c'est que -% ceci est appelé par \XINT_sqr. Et de plus \XINT_sqr est utilisé dans -% la nouvelle routine d'extraction de racine carrée: je ne veux pas -% rajouter l'overhead à \XINT_sqr de voir si a longueur est au moins 4. -% Dilemme donc. Il ne semble pas y avoir d'autres accès -% directs (celui de big fac n'est pas un problème). J'ai presque été -% tenté de faire du 5x4, mais si on veut maintenir les résultats -% intermédiaires sur 4n, il y a des complications. Par ailleurs, -% je modifie aussi un petit peu la façon de coder la suite, compte tenu -% du style que j'ai développé ultérieurement. Attention terminaison -% modifiée pour le deuxième opérande.| -% \begin{macrocode} -\def\XINT_mul_enter #1\Z\Z\Z\Z #2#3#4#5% -{% - \xint_gob_til_W #5\XINT_mul_exit_a\W - \XINT_mul_start {#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_mul_exit_a\W\XINT_mul_start #1% -{% - \XINT_mul_exit_b #1% -}% -\def\XINT_mul_exit_b #1#2#3#4% +\def\XINT_verysmallmul #1.#2!1#3!% {% - \xint_gob_til_W - #2\XINT_mul_exit_ci - #3\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1#2#3#4% -}% -\def\XINT_mul_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_exit_cii\W\XINT_mul_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_exit_ci\W\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_start #1#2\Z\Z\Z\Z -{% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z -}% -\def\XINT_mul_main #1#2\Z\Z\Z\Z #3#4#5#6% -{% - \xint_gob_til_W #6\XINT_mul_finish_a\W - \XINT_mul_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z + \xint_gob_til_Z #3\XINT_verysmallmul_e\Z + \expandafter\XINT_verysmallmul_a + \the\numexpr #2*#3+#1.#2!% }% -\def\XINT_mul_compute #1#2#3\Z\Z\Z\Z +\def\XINT_verysmallmul_e\Z\expandafter\XINT_verysmallmul_a\the\numexpr + #1+#2#3.#4!% +{\xint_gob_til_zero #2\XINT_verysmallmul_f 0\xint_c_x^viii+#2#3!1\Z!}% +\def\XINT_verysmallmul_f #1!1{1\relax}% +\def\XINT_verysmallmul_a #1#2.% {% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z + \unless\ifnum #1#2<\xint_c_x^ix + \expandafter\XINT_verysmallmul_bi\else + \expandafter\XINT_verysmallmul_bj\fi + \the\numexpr \xint_c_x^ix+#1#2.% }% +\def\XINT_verysmallmul_bj{\expandafter\XINT_verysmallmul_cj }% +\def\XINT_verysmallmul_cj 1#1#2.% + {1#2\expandafter!\the\numexpr\XINT_verysmallmul #1.}% +\def\XINT_verysmallmul_bi\the\numexpr\xint_c_x^ix+#1#2#3.% + {1#3\expandafter!\the\numexpr\XINT_verysmallmul #1#2.}% % \end{macrocode} -% \lverb|& -% Ici, le deuxième nombre se termine. Fin du calcul. On utilise la variante -% \XINT_addm_A de l'addition car on sait que le deuxième terme est au moins -% aussi long que le premier. Lorsque le multiplicateur avait longueur 4n, la -% dernière addition a fourni le résultat à l'envers, il faut donc encore le -% renverser. | +% \lverb|Used by division and by squaring, not by multiplication itself.| % \begin{macrocode} -\def\XINT_mul_finish_a\W\XINT_mul_compute #1% -{% - \XINT_mul_finish_b #1% -}% -\def\XINT_mul_finish_b #1#2#3#4% -{% - \xint_gob_til_W - #1\XINT_mul_finish_c - #2\XINT_mul_finish_ci - #3\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1#2#3#4% -}% -\def\XINT_mul_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W +\def\XINT_minimul_a #1.#2!#3#4#5#6#7!% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z + \expandafter\XINT_minimul_b + \the\numexpr \xint_c_x^viii+#2*#7.#2*#3#4#5#6+#1*#7.#1*#3#4#5#6.% }% -\def\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W +\def\XINT_minimul_b 1#1#2#3#4#5.#6.% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z + \expandafter\XINT_minimul_c + \the\numexpr \xint_c_x^ix+#1#2#3#4+#6.#5.% }% -\def\XINT_mul_finish_ci #1\XINT_mul_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W +\def\XINT_minimul_c 1#1#2#3#4#5#6.#7.#8.% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z -}% -\def\XINT_mul_finish_c #1\XINT_mul_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z -{% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#2}% + 1#6#7\expandafter!\the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8!% }% % \end{macrocode} -% \subsubsection{Variant: \csh{XINT_mulr_enter}} -% \lverb|& -% \romannumeral0\XINT_mulr_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W $\ -% Ici <N1> est à l'envers sur 4n, et <N2> est à l'endroit, pas sur 4n, comme -% dans \XINT_mul_enter, mais le résultat est lui-même fourni *à l'envers*, sur -% *4n* (en faisant attention de ne pas avoir 0000 à la fin).$\ -% Utilisé par le calcul des puissances. J'ai modifié dans 1.08 sur le -% modèle de la nouvelle version de \XINT_mul_enter. Je pourrais économiser des -% macros et fusionner \XINT_mul_enter et \XINT_mulr_enter. Une autre fois.| +% \subsection{\csh{xintiSqr}, \csh{xintiiSqr}} +% \lverb|Rewritten for v1.2.| % \begin{macrocode} -\def\XINT_mulr_enter #1\Z\Z\Z\Z #2#3#4#5% -{% - \xint_gob_til_W #5\XINT_mulr_exit_a\W - \XINT_mulr_start {#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_a\W\XINT_mulr_start #1% -{% - \XINT_mulr_exit_b #1% -}% -\def\XINT_mulr_exit_b #1#2#3#4% -{% - \xint_gob_til_W - #2\XINT_mulr_exit_ci - #3\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1#2#3#4% -}% -\def\XINT_mulr_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_cii\W\XINT_mulr_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_ci\W\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_start #1#2\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_main #1#2\Z\Z\Z\Z #3#4#5#6% -{% - \xint_gob_til_W #6\XINT_mulr_finish_a\W - \XINT_mulr_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_compute #1#2#3\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z -}% -\def\XINT_mulr_finish_a\W\XINT_mulr_compute #1% -{% - \XINT_mulr_finish_b #1% -}% -\def\XINT_mulr_finish_b #1#2#3#4% +\def\xintiiSqr {\romannumeral0\xintiisqr }% +\def\xintiisqr #1% {% - \xint_gob_til_W - #1\XINT_mulr_finish_c - #2\XINT_mulr_finish_ci - #3\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1#2#3#4% + \expandafter\XINT_sqr\romannumeral0\xintiiabs{#1}\Z }% -\def\XINT_mulr_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W +\def\xintiSqr {\romannumeral0\xintisqr }% +\def\xintisqr #1% {% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z + \expandafter\XINT_sqr\romannumeral0\xintiabs{#1}\Z }% -\def\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W +\def\XINT_sqr #1\Z {% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z + \expandafter\XINT_sqr_a + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z }% -\def\XINT_mulr_finish_ci #1\XINT_mulr_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W -{% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z +\def\XINT_sqr_a #1.% +{% + \ifnum #1=\xint_c_i \expandafter\XINT_sqr_small + \else\expandafter\XINT_sqr_start\fi }% -\def\XINT_mulr_finish_c #1\XINT_mulr_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintiSqr}, \csh{xintiiSqr}} -% \begin{macrocode} -\def\xintiiSqr {\romannumeral0\xintiisqr }% -\def\xintiisqr #1% +\def\XINT_sqr_small 1#1#2#3#4#5!\Z {% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiiabs{#1}}% + \ifnum #1#2#3#4#5<46341 \expandafter\XINT_sqr_verysmall\fi + \expandafter\XINT_sqr_small_out + \the\numexpr\XINT_minimul_a #1#2#3#4.#5!#1#2#3#4#5!% }% -\def\xintiSqr {\romannumeral0\xintisqr }% -\def\xintisqr #1% +\edef\XINT_sqr_verysmall + \expandafter\XINT_sqr_small_out\the\numexpr\XINT_minimul_a #1!#2!% + {\noexpand\expandafter\space\noexpand\the\numexpr #2*#2\relax}% +\def\XINT_sqr_small_out 1#1!1#2!% {% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiabs{#1}}% + \XINT_cuz #2#1\R }% -\let\xintSqr\xintiSqr \let\xintsqr\xintisqr -\def\XINT_sqr #1% +\def\XINT_sqr_start #1\Z {% - \expandafter\XINT_mul_enter - \romannumeral0% - \XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W + \expandafter\XINT_mul_out + \the\numexpr\XINT_mul_loop 100000000!\Z\W #11\Z!\W #11\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% % \end{macrocode} % \subsection{\csh{xintiPow}, \csh{xintiiPow}} -% \lverb|1.02 modified the \XINT_posprod routine, the was renamed -% \XINT_pow_posprod and moved here, as it was well adapted for computing powers. -% Then 1.03 moved the special variants of multiplication (hence of addition) -% which were needed to earlier in this style file. -% -% Modified in 1.06, the exponent is given to a \numexpr rather than twice -% expanded. \xintnum added in 1.09a. -% -% \XINT_pow_posprod: Routine de produit servant pour le calcul des -% puissances. Chaque nouveau terme est plus grand que ce qui a déjà été calculé. -% Par conséquent on a intérêt à le conserver en second dans la routine de -% multiplication, donc le précédent calcul a intérêt à avoir été donné sur 4n, à -% l'envers. Il faut donc modifier la multiplication pour qu'elle fasse cela. Ce -% qui oblige à utiliser une version spéciale de l'addition également. -% -% 1.09j has reorganized the main loop, the described above \XINT_pow_posprod -% routine has been removed, intermediate multiplications are done -% immediately. Also, the maximal accepted exponent is now 100000 (no such -% restriction in \xintFloatPow, which accepts any exponent less than 2^31, and -% in \xintFloatPower which accepts long integers as exponent). -% -% 2^100000=9.990020930143845e30102 and multiplication of two numbers -% with 30000 digits would take hours on my laptop (seconds for 1000 digits).| +% \lverb|& +% The exponent is not limited but with current default settings of tex memory, +% with xint 1.2, the maximal exponent for 2^N is N = 2^17 = 131072.| % \begin{macrocode} \def\xintiiPow {\romannumeral0\xintiipow }% \def\xintiipow #1% @@ -17076,7 +16825,6 @@ $1$ or $-1$. {% \expandafter\xint_pow\romannumeral0\xintnum{#1}\Z% }% -\let\xintPow\xintiPow \let\xintpow\xintipow \def\xint_pow #1#2\Z {% \xint_UDsignfork @@ -17140,7 +16888,7 @@ $1$ or $-1$. \ifcase\XINT_cntSgn #1\Z \expandafter\XINT_pow_BisZero \or - \expandafter\XINT_pow_checkBsize + \expandafter\XINT_pow_I_in \else \expandafter\XINT_pow_BisNegative \fi @@ -17150,117 +16898,233 @@ $1$ or $-1$. {\noexpand\xintError:FractionRoundedToZero\space 0}% \def\XINT_pow_BisZero #1#2{ 1}% % \end{macrocode} -% \lverb|B = #1 > 0, A = #2 > 1. With 1.05, I replace \xintiLen{#1}>9 by -% direct use of \numexpr [to generate an error message if the exponent is too -% large] 1.06: \numexpr was already used above.| -% \begin{macrocode} -\def\XINT_pow_checkBsize #1% -{% - \ifnum #1>100000 - \expandafter\XINT_pow_BtooBig +% \lverb|B = #1 > 0, A = #2 > 1.| +% \begin{macrocode} +% \def\XINT_pow_checkBsize #1% +% {% +% \ifnum #1>131000 +% \expandafter\XINT_pow_BtooBig +% \else +% \expandafter\XINT_pow_I_in +% \fi +% {#1}% +% }% +% \edef\XINT_pow_BtooBig #1#2{\noexpand\xintError:ExponentTooBig\space 0}% +%%%%%%%%%%%% +\def\XINT_pow_I_in #1#2% +{% + \expandafter\XINT_pow_I_loop + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #2\R\R\R\R\R\R\R\R{10}0000001\W + #2\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W 1\Z!\W + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_pow_I_loop #1.% +{% + \ifnum #1 = \xint_c_i\expandafter\XINT_pow_I_exit\fi + \ifodd #1 + \expandafter\XINT_pow_II_in \else - \expandafter\XINT_pow_loopI - \fi - {#1}% + \expandafter\XINT_pow_I_squareit + \fi #1.% }% -\edef\XINT_pow_BtooBig #1#2{\noexpand\xintError:ExponentTooBig\space 0}% -\def\XINT_pow_loopI #1% +\def\XINT_pow_I_exit \ifodd #1\fi #2.#3\W {\XINT_mul_out #3}% +\def\XINT_pow_I_squareit #1.#2\W% {% - \ifnum #1=\xint_c_i\XINT_pow_Iend\fi + \expandafter\XINT_pow_I_loop + \the\numexpr #1/\xint_c_ii\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W +}% +%%%%%%%%%%%% +\def\XINT_pow_mulbutcheckifsmall #1!1#2% +{% + \xint_gob_til_Z #2\XINT_pow_mul_small\Z + \XINT_mul_loop 100000000!\Z\W #1!1#2% +}% +\def\XINT_pow_mul_small\Z\XINT_mul_loop 100000000!\Z\W 1#1!1\Z!\W +{% + \XINT_smallmul 1#1!% +}% +%%%%%%%%%%%% +\def\XINT_pow_II_in #1.#2\W +{% + \expandafter\XINT_pow_II_loop + \the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W #2\W +}% +\def\XINT_pow_II_loop #1.% +{% + \ifnum #1 = \xint_c_i\expandafter\XINT_pow_II_exit\fi \ifodd #1 - \expandafter\XINT_pow_loopI_odd + \expandafter\XINT_pow_II_odda \else - \expandafter\XINT_pow_loopI_even - \fi - {#1}% + \expandafter\XINT_pow_II_even + \fi #1.% }% -\edef\XINT_pow_Iend\fi #1\fi #2#3{\noexpand\fi\space #3}% -\def\XINT_pow_loopI_even #1#2% +\def\XINT_pow_II_exit\ifodd #1\fi #2.#3\W #4\W {% - \expandafter\XINT_pow_loopI\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% + \expandafter\XINT_mul_out + \the\numexpr\XINT_pow_mulbutcheckifsmall #4\W #3% }% -\def\XINT_pow_loopI_odd #1#2% +\def\XINT_pow_II_even #1.#2\W {% - \expandafter\XINT_pow_loopI_odda\expandafter - {\romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z }{#1}{#2}% + \expandafter\XINT_pow_II_loop + \the\numexpr #1/\xint_c_ii\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W }% -\def\XINT_pow_loopI_odda #1#2#3% +\def\XINT_pow_II_odda #1.#2\W #3\W {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% + \expandafter\XINT_pow_II_oddb + \the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #3\W #2\W #2\W }% -\def\XINT_pow_loopII #1% +\def\XINT_pow_II_oddb #1.#2\W #3\W {% - \ifnum #1 = \xint_c_i\XINT_pow_IIend\fi - \ifodd #1 - \expandafter\XINT_pow_loopII_odd + \expandafter\XINT_pow_II_loop + \the\numexpr #1\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #3\W #3\W #2\W +}% +% \end{macrocode} +% \subsection{\csh{xintiFac}, \csh{xintiiFac}} +% \lverb|Moved to xintcore.sty with release 1.2 (to be usable by \bnumexpr). +% The routine has been partially rewritten and there is an intrinsic limit at +% 9999. Anyhow with current default settings of the etex memory and the +% current 1.2 routine (last commit: eada1b1), the maximal possible computation +% is 5971! (which has 19956 digits). Also, I add \xintiiFac which does only +% \romannumeral-`0 and not \numexpr on its argument. This is for a silly +% slight optimization of the \xintiiexpr (and \bnumexpr) parsers. If the +% argument is >=2^31 an arithmetic overflow will occur in the \ifnum. This is +% not as good as in the \numexpr, but well.| +% \begin{macrocode} +\def\xintiFac {\romannumeral0\xintifac }% +\def\xintifac #1% +{% + \expandafter\XINT_fac_fork\expandafter {\the\numexpr#1}% +}% +\def\xintiiFac {\romannumeral0\xintiifac }% +\def\xintiifac #1% +{% + \expandafter\XINT_fac_fork\expandafter {\romannumeral-`0#1}% +}% +\let\xintFac\xintiFac \let\xintfac\xintifac +\def\XINT_fac_fork #1% +{% + \ifcase\XINT_cntSgn #1\Z + \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }% + \or + \expandafter\XINT_fac_checksize \else - \expandafter\XINT_pow_loopII_even + \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber + \expandafter\space\expandafter 1\xint_gobble_i }% \fi {#1}% }% -\def\XINT_pow_loopII_even #1#2% +\def\XINT_fac_checksize #1% {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% + \ifnum #1>9999 + \xint_dothis{\expandafter\xintError:FactorialOfTooBigNumber + \expandafter\space\expandafter 1\xint_gob_til_W }\fi + \ifnum #1>465 \xint_dothis{\XINT_fac_bigloop_a #1.}\fi + \ifnum #1>101 \xint_dothis{\XINT_fac_medloop_a #1.\XINT_mul_out}\fi + \xint_orthat{\XINT_fac_smallloop_a #1.\XINT_mul_out}% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_pow_loopII_odd #1#2#3% +\def\XINT_fac_bigloop_a #1.% {% - \expandafter\XINT_pow_loopII_odda\expandafter - {\romannumeral0\XINT_mulr_enter #3\Z\Z\Z\Z #2\W\W\W\W}{#1}{#2}% + \expandafter\XINT_fac_bigloop_b \the\numexpr + #1+\xint_c_i-\xint_c_ii*((#1-464)/\xint_c_ii).#1.% }% -\def\XINT_pow_loopII_odda #1#2#3% +\def\XINT_fac_bigloop_b #1.#2.% {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% + \expandafter\XINT_fac_medloop_a + \the\numexpr #1-\xint_c_i.{\XINT_fac_bigloop_loop #1.#2.}% +}% +\def\XINT_fac_bigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_fac_bigloop_exit\fi + \expandafter\XINT_fac_bigloop_loop + \the\numexpr #1+\xint_c_ii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_bigloop_mul #1!% +}% +\def\XINT_fac_bigloop_exit #1!{\XINT_mul_out}% +\def\XINT_fac_bigloop_mul #1!% +{% + \expandafter\XINT_smallmul + \the\numexpr \xint_c_x^viii+#1*(#1+\xint_c_i)!% +}% +\def\XINT_fac_medloop_a #1.% +{% + \expandafter\XINT_fac_medloop_b + \the\numexpr #1+\xint_c_i-\xint_c_iii*((#1-100)/\xint_c_iii).#1.% +}% +\def\XINT_fac_medloop_b #1.#2.% +{% + \expandafter\XINT_fac_smallloop_a + \the\numexpr #1-\xint_c_i.{\XINT_fac_medloop_loop #1.#2.}% +}% +\def\XINT_fac_medloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_fac_loop_exit\fi + \expandafter\XINT_fac_medloop_loop + \the\numexpr #1+\xint_c_iii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_medloop_mul #1!% +}% +\def\XINT_fac_medloop_mul #1!% +{% + \expandafter\XINT_smallmul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)!% +}% +\def\XINT_fac_smallloop_a #1.% +{% + \csname + XINT_fac_smallloop_\the\numexpr #1-\xint_c_iv*(#1/\xint_c_iv)\relax + \endcsname #1.% +}% +\expandafter\def\csname XINT_fac_smallloop_1\endcsname #1.% +{% + \XINT_fac_smallloop_loop 2.#1.100000001!1\Z!% +}% +\expandafter\def\csname XINT_fac_smallloop_-2\endcsname #1.% +{% + \XINT_fac_smallloop_loop 3.#1.100000002!1\Z!% +}% +\expandafter\def\csname XINT_fac_smallloop_-1\endcsname #1.% +{% + \XINT_fac_smallloop_loop 4.#1.100000006!1\Z!% +}% +\expandafter\def\csname XINT_fac_smallloop_0\endcsname #1.% +{% + \XINT_fac_smallloop_loop 5.#1.1000000024!1\Z!% +}% +\def\XINT_fac_smallloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_fac_loop_exit\fi + \expandafter\XINT_fac_smallloop_loop + \the\numexpr #1+\xint_c_iv\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_smallloop_mul #1!% }% -\def\XINT_pow_IIend\fi #1\fi #2#3#4% +\def\XINT_fac_smallloop_mul #1!% {% - \fi\XINT_mul_enter #4\Z\Z\Z\Z #3\W\W\W\W + \expandafter\XINT_smallmul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)*(#1+\xint_c_iii)!% }% +\def\XINT_fac_loop_exit #1!#2\Z!#3{#3#2\Z!}% % \end{macrocode} % \subsection{\csh{xintiDivision}, \csh{xintiQuo}, \csh{xintiRem}, % \csh{xintiiDivision}, \csh{xintiiQuo}, \csh{xintiiRem}} -% \lverb|The 1.09a release inserted the use of \xintnum. The \xintiiDivision -% etc... are the ones which do only \romannumeral-`0. -% -% January 5, 2014: Naturally, addition, subtraction, multiplication and division -% are the first things I did and since then I had left the division -% untouched. So in preparation of release 1.09j, I started revisiting the -% division, I did various minor improvements obtaining roughly -% 10$% efficiency gain. Then I decided I -% should deliberately impact the input save stack, with the hope to gain more -% speed from removing tokens and leaving them upstream. -% -% For this however I had to modify the underlying mathematical algorithm. The -% initial one is a bit unusual I guess, and, I trust, rather efficient, but it -% does not produce the quotient digits (in base 10000) one by one; at any given -% time it is possible that some correction will be made, which means it is not -% an appropriate algorithm for a TeX implementation which will abandon the -% quotient upstream. Thus I now have with 1.09j a new underlying mathematical -% algorithm, presumably much more standard. It is a bit complicated to implement -% expandably these things, but in the end I had regained the already mentioned -% 10$% efficiency and even more for -% small to medium sized inputs (up to 30$% perhaps). And in passing I did a -% special routine for divisors < 10000, which is 5 to 10 times faster still. -% -% But, I then tested a variant of my new implementation which again did -% not impact the input save stack and, for sizes of up to 200 digits, it -% is not much worse, indeed it is perhaps actually better than the one -% abandoning the quotient digits upstream (and in the end putting them -% in the correct order). So, finally, I re-incorporated the produced -% quotient digits within a tail recursion. Hence \xintiDivision, like all -% other routines in xint (except \xintSeq without optional parameter) -% does not impact the input save stack. One can have a produced -% quotient longer than 4x5000=20000 digits, and no need to worry about -% consequences propagating to \xintTrunc, \xintRound, \xintFloat, -% \xintFloatSqrt, etc... and all other places using the division. See -% also \xintXTrunc in this context.| +% \lverb|Completely rewritten for v1.2. +% WARNING: some comments below try to describe the flow of tokens but they +% date back from xint 1.09j and I updated them on the fly while doing the 1.2 +% version. As the new works in base 10^8, not 10^4 and "drops" the quotient +% digits,rather than store them upfront as the earlier code, I may well have +% not correctly converted all such comments. At the last minute some +% previously #1 became stuff like #1#2#3#4, then of course the old comments +% describing what the macro parameters stand for are necessarily wrong.| % \begin{macrocode} \def\xintiiQuo {\romannumeral0\xintiiquo }% \def\xintiiRem {\romannumeral0\xintiirem }% @@ -17270,16 +17134,15 @@ $1$ or $-1$. \def\xintiRem {\romannumeral0\xintirem }% \def\xintiquo {\expandafter\xint_firstoftwo_thenstop\romannumeral0\xintidivision }% \def\xintirem {\expandafter\xint_secondoftwo_thenstop\romannumeral0\xintidivision }% -\let\xintQuo\xintiQuo\let\xintquo\xintiquo % deprecated (1.1) -\let\xintRem\xintiRem\let\xintrem\xintirem % deprecated (1.1) +\let\xintQuo\xintiQuo\let\xintquo\xintiquo % deprecated +\let\xintRem\xintiRem\let\xintrem\xintirem % deprecated % \end{macrocode} % \lverb-#1 = A, #2 = B. On calcule le quotient et le reste dans la division % euclidienne de A par B: A=BQ+R, 0<= R < |B|.- % \begin{macrocode} \def\xintiDivision {\romannumeral0\xintidivision }% -\def\xintidivision #1{\expandafter\XINT_division\romannumeral0\xintnum{#1}\Z }% -\let\xintDivision\xintiDivision \let\xintdivision\xintidivision % deprecated -\def\XINT_division #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1% +\def\xintidivision #1{\expandafter\XINT_idivision\romannumeral0\xintnum{#1}\Z }% +\def\XINT_idivision #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1% \romannumeral0\xintnum{#3}\Z #2\Z }% \def\xintiiDivision {\romannumeral0\xintiidivision }% \def\xintiidivision #1{\expandafter\XINT_iidivision \romannumeral-`0#1\Z }% @@ -17293,10 +17156,10 @@ $1$ or $-1$. \romannumeral0\XINT_iidivision_bpos #1}\fi \xint_orthat{\XINT_iidivision_bpos #1#2}% }% -\def\XINT_iidivision_divbyzero #1\Z #2\Z {\xintError:DivisionByZero\space {0}{0}}% -\def\XINT_iidivision_aiszero #1\Z #2\Z { {0}{0}}% +\def\XINT_iidivision_divbyzero #1\Z #2\Z {\xintError:DivisionByZero{0}{0}}% +\def\XINT_iidivision_aiszero #1\Z #2\Z {{0}{0}}% \def\XINT_iidivision_bneg #1% q->-q, r unchanged - {\expandafter\space\expandafter{\romannumeral0\XINT_opp #1}}% + {\expandafter{\romannumeral0\XINT_opp #1}}% \def\XINT_iidivision_bpos #1% {% \xint_UDsignfork @@ -17313,7 +17176,7 @@ $1$ or $-1$. \else \expandafter\XINT_iidivision_aneg_rpos \fi {#1}{#2}}% -\def\XINT_iidivision_aneg_rzero #1#2#3{ {-#1}{0}}% necessarily q was >0 +\def\XINT_iidivision_aneg_rzero #1#2#3{{-#1}{0}}% necessarily q was >0 \def\XINT_iidivision_aneg_rpos #1% {% \expandafter\XINT_iidivision_aneg_end\expandafter @@ -17321,229 +17184,211 @@ $1$ or $-1$. }% \def\XINT_iidivision_aneg_end #1#2#3% {% - \expandafter\xint_exchangetwo_keepbraces_thenstop - \expandafter{\romannumeral0\XINT_sub_pre {#3}{#2}}{#1}% r-> b-r + \expandafter\xint_exchangetwo_keepbraces + \expandafter{\romannumeral0\XINT_sub_mm_a {}{}#3\Z #2\Z}{#1}% r-> b-r }% -% \end{macrocode} -% \lverb|& -% Pour la suite A et B sont > 0. -% #1 = B. Pour le moment à l'endroit. -% Calcul du plus petit K = 4n >= longueur de B| -% \begin{macrocode} +%%%%%%%%%%%% \def\XINT_div_prepare #1% {% - \expandafter \XINT_div_prepareB_aa \expandafter - {\romannumeral0\xintlength {#1}}{#1}% B > 0 ici + \XINT_div_prepare_a #1\R\R\R\R\R\R\R\R {10}0000001\W !{#1}% }% -\def\XINT_div_prepareB_aa #1% +\def\XINT_div_prepare_a #1#2#3#4#5#6#7#8#9% {% - \ifnum #1=\xint_c_i - \expandafter\XINT_div_prepareB_onedigit - \else - \expandafter\XINT_div_prepareB_a - \fi - {#1}% -}% -\def\XINT_div_prepareB_a #1% -{% - \expandafter\XINT_div_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \xint_gob_til_R #9\XINT_div_prepare_small\R + \XINT_div_prepare_b #9% }% -% \end{macrocode} -% \lverb|B=1 and B=2 treated specially.| -% \begin{macrocode} -\def\XINT_div_prepareB_onedigit #1#2% +%%%%%%%%%%%% +\def\XINT_div_prepare_small\R #1!#2% {% - \ifcase#2 + \ifcase #2 \or\expandafter\XINT_div_BisOne \or\expandafter\XINT_div_BisTwo - \else\expandafter\XINT_div_prepareB_e - \fi {000}{0}{4}{#2}% + \else\expandafter\XINT_div_small_a + \fi {#2}% }% -\def\XINT_div_BisOne #1#2#3#4#5{ {#5}{0}}% -\def\XINT_div_BisTwo #1#2#3#4#5% +\def\XINT_div_BisOne #1#2{{#2}{0}}% +\def\XINT_div_BisTwo #1#2% {% \expandafter\expandafter\expandafter\XINT_div_BisTwo_a - \ifodd\xintiiLDg{#5} \expandafter1\else \expandafter0\fi {#5}% + \ifodd\xintLDg{#2} \expandafter1\else \expandafter0\fi {#2}% }% -\edef\XINT_div_BisTwo_a #1#2% +\def\XINT_div_BisTwo_a #1#2% {% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\xinthalf {#2}}{#1}% + \expandafter{\romannumeral0\xinthalf {#2}}{#1}% }% -% \end{macrocode} -% \lverb|#1 = K. 1.09j uses \csname, earlier versions did it with -% \ifcase.| -% \begin{macrocode} -\def\XINT_div_prepareB_c #1#2% +\def\XINT_div_small_a #1#2% {% - \csname XINT_div_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% + \expandafter\XINT_div_small_b + \the\numexpr #1/\xint_c_ii\expandafter + .\the\numexpr \xint_c_x^viii+#1\expandafter!% + \romannumeral0% + \XINT_div_small_ba #2\R\R\R\R\R\R\R\R{10}0000001\W + #2\XINT_sepbyviii_Z_end 2345678\relax }% -\def\XINT_div_prepareB_d {\XINT_div_prepareB_e {}{0000}}% -\def\XINT_div_prepareB_di {\XINT_div_prepareB_e {0}{000}}% -\def\XINT_div_prepareB_dii {\XINT_div_prepareB_e {00}{00}}% -\def\XINT_div_prepareB_diii {\XINT_div_prepareB_e {000}{0}}% -\def\XINT_div_cleanR #10000.{{#1}}% -% \end{macrocode} -% \lverb|#1 = zéros à rajouter à B, #2=c [modifié dans 1.09j, ce sont maintenant -% des zéros explicites en nombre 4 - ancien c, et on utilisera -% \XINT_div_cleanR et non plus \XINT_dsh_checksignx pour nettoyer à la fin -% des zéros en excès dans le Reste; in all comments next, «c» stands now {0} or -% {00} or {000} or {0000} rather than a digit as in earlier versions], #3=K, #4 -% = B| -% \begin{macrocode} -\def\XINT_div_prepareB_e #1#2#3#4% +\def\XINT_div_small_b #1!#2{#2#1!}% +\def\XINT_div_small_ba #1#2#3#4#5#6#7#8#9% {% - \ifnum#3=\xint_c_iv\expandafter\XINT_div_prepareLittleB_f - \else\expandafter\XINT_div_prepareB_f - \fi - #4#1{#3}{#2}{#1}% + \xint_gob_til_R #9\XINT_div_smallsmall\R + \expandafter\XINT_div_dosmalldiv + \the\numexpr\expandafter\XINT_sepbyviii_Z + \romannumeral0\XINT_zeroes_forviii + #1#2#3#4#5#6#7#8#9% }% -% \end{macrocode} -% \lverb|x = #1#2#3#4 = 4 premiers chiffres de B. #1 est non nul. B is reversed. -% With 1.09j or latter x+1 and (x+1)/2 are pre-computed. Si K=4 on ne renverse -% pas B, et donc B=x dans la suite. De plus pour K=4 on ne travaille pas avec -% x+1 et (x+1)/2 mais avec x et x/2.| -% \begin{macrocode} -\def\XINT_div_prepareB_f #1#2#3#4#5#{% - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% +\def\XINT_div_smallsmall\R + \expandafter\XINT_div_dosmalldiv + \the\numexpr\expandafter\XINT_sepbyviii_Z + \romannumeral0\XINT_zeroes_forviii #1\R #2\relax + {{\XINT_div_dosmallsmall}{#1}}% +\def\XINT_div_dosmallsmall #1.1#2!#3% +{% + \expandafter\XINT_div_smallsmallend + \the\numexpr (#3+#1)/#2-\xint_c_i.#2.#3.% }% -\def\XINT_div_prepareLittleB_f #1#{% - \expandafter\XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% +\def\XINT_div_smallsmallend #1.#2.#3.{\expandafter + {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #3-#1*#2}}% +\def\XINT_div_dosmalldiv + {{\expandafter\XINT_sdiv_out\the\numexpr\XINT_smalldivx_a}}% +%%%%%%%%%%%% +\def\XINT_div_prepare_b + {\expandafter\XINT_div_prepare_c\romannumeral0\XINT_zeroes_forviii }% +\def\XINT_div_prepare_c #1!% +{% + \XINT_div_prepare_d #1.00000000!{#1}% }% -% \end{macrocode} -% \lverb|& -% #1 = x' = x+1= 1+quatre premiers chiffres de B, #2 = y = (x+1)/2 précalculé -% #3 = B préparé et maintenant renversé, #4=x, -% #5 = K, #6 = «c», #7= {} ou {0} ou {00} ou {000}, #8 = A initial -% On multiplie aussi A par 10^c. -> AK{x'yx}B«c». Par contre dans le -% cas little on a #1=y=(x/2), #2={}, #3={}, #4=x, donc cela donne -% ->AK{y{}x}{}«c», il n'y a pas de B.| -% \begin{macrocode} -\def\XINT_div_prepareB_g #1.#2.#3.#4#5#6#7#8% +\def\XINT_div_prepare_d #1#2#3#4#5#6#7#8#9% {% - \XINT_div_prepareA_a {#8#7}{#5}{{#1}{#2}{#4}}{#3}{#6}% + \expandafter\XINT_div_prepare_e\xint_gob_til_dot #1#2#3#4#5#6#7#8#9!% }% -% \end{macrocode} -% \lverb|A, K, {x'yx}, B«c» | -% \begin{macrocode} -\def\XINT_div_prepareA_a #1% +\def\XINT_div_prepare_e #1!#2!#3#4% {% - \expandafter\XINT_div_prepareA_b\expandafter - {\romannumeral0\xintlength {#1}}{#1}% + \XINT_div_prepare_f #4#3\X {#1}{#3}% }% -% \end{macrocode} -% \lverb|L0, A, K, {x'yx}, B«c»| -% \begin{macrocode} -\def\XINT_div_prepareA_b #1% +\def\XINT_div_prepare_f #1#2#3#4#5#6#7#8#9\X {% - \expandafter\XINT_div_prepareA_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \expandafter\XINT_div_prepare_g + \the\numexpr #1#2#3#4#5#6#7#8+\xint_c_i\expandafter + .\the\numexpr (#1#2#3#4#5#6#7#8+\xint_c_i)/\xint_c_ii\expandafter + .\the\numexpr #1#2#3#4#5#6#7#8\expandafter + .\romannumeral0\XINT_sepandrev_andcount + #1#2#3#4#5#6#7#8#9\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678% + \relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X }% -% \end{macrocode} -% \lverb|L, L0, A, K, {x'yx}, B, «c»| -% \begin{macrocode} -\def\XINT_div_prepareA_c #1#2% +\def\XINT_div_prepare_g #1.#2.#3.#4.#5\X #6#7#8% {% - \csname XINT_div_prepareA_d\romannumeral\numexpr #1-#2\endcsname - {#1}% + \expandafter\XINT_div_prepare_h + \the\numexpr\expandafter\XINT_sepbyviii_andcount + \romannumeral0\XINT_zeroes_forviii #8#7\R\R\R\R\R\R\R\R{10}0000001\W + #8#7\XINT_sepbyviii_end 2345678\relax + \xint_c_vii!\xint_c_vi!\xint_c_v!\xint_c_iv!% + \xint_c_iii!\xint_c_ii!\xint_c_i!\xint_c_\W + {#1}{#2}{#3}{#4}{#5}{#6}% }% -\def\XINT_div_prepareA_d {\XINT_div_prepareA_e {}}% -\def\XINT_div_prepareA_di {\XINT_div_prepareA_e {0}}% -\def\XINT_div_prepareA_dii {\XINT_div_prepareA_e {00}}% -\def\XINT_div_prepareA_diii {\XINT_div_prepareA_e {000}}% -% \end{macrocode} -% \lverb|#1#3 = A préparé, #2 = longueur de ce A préparé, #4=K, #5={x'yx}-> -% LKAx'yxB«c»| -% \begin{macrocode} -\def\XINT_div_prepareA_e #1#2#3#4#5% +\def\XINT_div_prepare_h #11.#2.#3#4#5#6%#7#8% {% - \XINT_div_start_a {#2}{#4}{#1#3}#5% + \XINT_div_start_a {#2}{#6}{#1}{#3}{#4}{#5}%{#7}{#8}% }% % \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c» (avec y{}x{} au lieu de x'yxB dans la -% variante little)| +% \lverb|L, K, A, x',y,x, B, «c». Attention que K est diminué de 1 plus loin. +% Comme xint 1.2 a déjà repéré K=1, on a ici au minimum K=2. Attention B est à +% l'envers, A est à l'endroit et les deux avec séparateurs. Attention que ce +% n'est pas ici qu'on boucle mais en \XINT_div_I_a.| % \begin{macrocode} \def\XINT_div_start_a #1#2% {% - \ifnum #2=\xint_c_iv \expandafter\XINT_div_little_b + \ifnum #1 < #2 + \expandafter\XINT_div_zeroQ \else - \ifnum #1 < #2 - \expandafter\expandafter\expandafter\XINT_div_III_aa - \else - \expandafter\expandafter\expandafter\XINT_div_start_b - \fi + \expandafter\XINT_div_start_b \fi {#1}{#2}% }% +\def\XINT_div_zeroQ #1#2#3#4#5#6#7% +{% + \expandafter\XINT_div_zeroQ_end + \romannumeral0\XINT_unsep_cuzsmall + #31\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W .% +}% +\def\XINT_div_zeroQ_end #1.#2% + {\expandafter{\expandafter0\expandafter}\XINT_div_cleanR #1#2.}% % \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c».| +% \lverb|L, K, A, x',y,x, B, «c»->K.A.x{LK{x'y}x}B«c»| % \begin{macrocode} -\def\XINT_div_III_aa #1#2#3#4#5#6#7% +\def\XINT_div_start_b #1#2#3#4#5#6% {% - \expandafter\expandafter\expandafter - \XINT_div_III_b\xint_cleanupzeros_nostop #3.{0000}% + \expandafter\XINT_div_finish\the\numexpr + \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}% }% +\def\XINT_div_finish +{% + \expandafter\XINT_div_finish_a \romannumeral-`0\XINT_div_unsepQ +}% +\def\XINT_div_finish_a #1\Z #2.{\XINT_div_finish_b #2.{#1}}% % \end{macrocode} -% \lverb|R.Q«c».| +% \lverb|Ici ce sont routines de fin. Le reste déjà nettoyé. R.Q«c».| % \begin{macrocode} -\def\XINT_div_III_b #1% +\def\XINT_div_finish_b #1% {% \if0#1% - \expandafter\XINT_div_III_bRzero + \expandafter\XINT_div_finish_bRzero \else - \expandafter\XINT_div_III_bRpos + \expandafter\XINT_div_finish_bRpos \fi #1% }% -\def\XINT_div_III_bRzero 0.#1#2% -{% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z}{0}% -}% -\def\XINT_div_III_bRpos #1.#2#3% -{% - \expandafter\XINT_div_III_c \XINT_div_cleanR #1#3.{#2}% -}% -\def\XINT_div_III_c #1#2% -{% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #2\W\W\W\W\W\W\W\Z}{#1}% +\def\XINT_div_finish_bRzero 0.#1#2{{#1}{0}}% +\def\XINT_div_finish_bRpos #1.#2#3% +{% + \expandafter\xint_exchangetwo_keepbraces\XINT_div_cleanR #1#3.{#2}% }% +\def\XINT_div_cleanR #100000000.{{#1}}% % \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c»->K.A.x{LK{x'y}x}B«c»| +% \lverb|Kalpha.A.x{LK{x'y}x}, B, «c», au début #2=alpha est vide. On fait une +% boucle pour prendre K unités de A (on a au moins L égal à K) et les mettre +% dans alpha.| % \begin{macrocode} -\def\XINT_div_start_b #1#2#3#4#5#6% +\def\XINT_div_start_c #1% {% - \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}% + \ifnum #1>\xint_c_vi + \expandafter\XINT_div_start_ca + \else + \expandafter\XINT_div_start_cb + \fi {#1}% }% -% \end{macrocode} -% \lverb|Kalpha.A.x{LK{x'y}x}, B, «c», au début #2=alpha est vide| -% \begin{macrocode} -\def\XINT_div_start_c #1#2.#3#4#5#6% +\def\XINT_div_start_ca #1#2.#3!#4!#5!#6!#7!#8!#9!% {% - \ifnum #1=\xint_c_iv\XINT_div_start_ca\fi \expandafter\XINT_div_start_c\expandafter - {\the\numexpr #1-\xint_c_iv}#2#3#4#5#6.% -}% -\def\XINT_div_start_ca\fi\expandafter\XINT_div_start_c\expandafter - #1#2#3#4#5{\fi\XINT_div_start_d {#2#3#4#5}#2#3#4#5}% + {\the\numexpr #1-\xint_c_vii}#2#3!#4!#5!#6!#7!#8!#9!.% +}% +\def\XINT_div_start_cb #1% + {\csname XINT_div_start_c_\romannumeral\numexpr#1\endcsname}% +\def\XINT_div_start_c_i #1.#2!% + {\XINT_div_start_c_ #1#2!.}% +\def\XINT_div_start_c_ii #1.#2!#3!% + {\XINT_div_start_c_ #1#2!#3!.}% +\def\XINT_div_start_c_iii #1.#2!#3!#4!% + {\XINT_div_start_c_ #1#2!#3!#4!.}% +\def\XINT_div_start_c_iv #1.#2!#3!#4!#5!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!.}% +\def\XINT_div_start_c_v #1.#2!#3!#4!#5!#6!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!#6!.}% +\def\XINT_div_start_c_vi #1.#2!#3!#4!#5!#6!#7!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!#6!#7!.}% % \end{macrocode} % \lverb|#1=a, #2=alpha (de longueur K, à l'endroit).#3=reste de A.#4=x, -% #5={LK{x'y}x},#6=B,«c» -> a, x, alpha, B, {0000}, L, K, {x'y},x, -% alpha'=reste de A, B{}«c». Pour K=4 on a en fait B=x, faudra revoir après.| +% #5={LK{x'y}x},#6=B,«c» -> a, x, alpha, B, {00000000}, L, K, {x'y},x, +% alpha'=reste de A, B«c».| % \begin{macrocode} -\def\XINT_div_start_d #1#2.#3.#4#5#6% +\def\XINT_div_start_c_ 1#1!#2.#3.#4#5#6% {% - \XINT_div_I_a {#1}{#4}{#2}{#6}{0000}#5{#3}{#6}{}% + \XINT_div_I_a {#1}{#4}{1#1!#2}{#6}{00000000}#5{#3}{#6}% }% % \end{macrocode} % \lverb|Ceci est le point de retour de la boucle principale. a, x, alpha, B, -% q0, L, K, {x'y}, x, alpha', BQ«c» | +% q0, L, K, {x'y}, x, alpha', B«c» | % \begin{macrocode} \def\XINT_div_I_a #1#2% {% @@ -17554,17 +17399,17 @@ $1$ or $-1$. \xint_gob_til_zero #1\XINT_div_I_czero 0\XINT_div_I_c #1% }% % \end{macrocode} -% \lverb|On intercepte quotient nul: #1=a, x, alpha, B, #5=q0, L, K, {x'y}, x, -% alpha', BQ«c» -> q{alpha} L, K, {x'y}, x, alpha', BQ«c»| +% \lverb|On intercepte petit quotient nul: #1=a, x, alpha, B, #5=q0, L, K, +% {x'y}, x, alpha', B«c» -> on lâche un q puis {alpha} L, K, {x'y}, x, +% alpha', B«c».| % \begin{macrocode} -\def\XINT_div_I_czero 0% - \XINT_div_I_c 0.#1#2#3#4#5{\XINT_div_I_g {#5}{#3}}% +\def\XINT_div_I_czero 0\XINT_div_I_c 0.#1#2#3#4#5{1#5\XINT_div_I_g {#3}}% \def\XINT_div_I_c #1.#2#3% {% - \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.% + \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.{#2}{#3}% }% % \end{macrocode} -% \lverb|r.q.alpha, B, q0, L, K, {x'y}, x, alpha', BQ«c»| +% \lverb|r.q.alpha, B, q0, L, K, {x'y}, x, alpha', B«c»| % \begin{macrocode} \def\XINT_div_I_da #1.% {% @@ -17578,413 +17423,451 @@ $1$ or $-1$. \fi \fi }% +% \end{macrocode} +% \lverb|attention très mauvaises notations avec _b et _db.| +% \begin{macrocode} \def\XINT_div_I_dN #1.% {% - \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.% + \expandafter\XINT_div_I_b\the\numexpr #1-\xint_c_i.% }% -\def\XINT_div_I_db #1.#2#3% #1=q=un chiffre, #2=alpha, #3=B +\def\XINT_div_I_db #1.#2#3#4#5% {% - \expandafter\XINT_div_I_dc\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% - #1{#2}{#3}% + \expandafter\XINT_div_I_dc\expandafter #1% + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#4\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_verysmallmul #1!#51\Z!}% + \Z {#4}{#5}% }% +% \end{macrocode} +% \lverb|La soustraction spéciale renvoie simplement - si le chiffre q est +% trop grand. On invoque dans ce cas I_dP.| +% \begin{macrocode} \def\XINT_div_I_dc #1#2% {% - \if-#1% s'arranger pour que si n\'egatif on ait renvoy\'e alpha=-. - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {\expandafter\XINT_div_I_dP\the\numexpr #2-\xint_c_i.}% - {\XINT_div_I_e {#1}#2}% + \if-#2\expandafter\XINT_div_I_dd\else\expandafter\XINT_div_I_de\fi + #1#2% }% -% \end{macrocode} -% \lverb|alpha,q,ancien alpha,B, q0->1nouveauq.alpha, L, K, {x'y},x, alpha', -% BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_e #1#2#3#4#5% +\def\XINT_div_I_dd #1-\Z {% - \expandafter\XINT_div_I_f \the\numexpr \xint_c_x^iv+#2+#5{#1}% + \if #11\expandafter\XINT_div_I_dz\fi + \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.XX% }% +\def\XINT_div_I_dz #1XX#2#3#4% +{% + 1#4\XINT_div_I_g {#2}% +}% +\def\XINT_div_I_de #1#2\Z #3#4#5{1#5+#1\XINT_div_I_g {#2}}% % \end{macrocode} -% \lverb|q.alpha, B, q0, L, K, {x'y},x, alpha'BQ«c» (intercepter q=0?) -% -> 1nouveauq.nouvel alpha, L, K, {x'y}, x, alpha',BQ«c»| +% \lverb|q.alpha, B, q0, L, K, {x'y},x, alpha'B«c» (q=0 has been intercepted) +% -> 1nouveauq.nouvel alpha, L, K, {x'y}, x, alpha',B«c»| % \begin{macrocode} -\def\XINT_div_I_dP #1.#2#3#4% +\def\XINT_div_I_dP #1.#2#3#4#5#6% {% - \expandafter \XINT_div_I_f \the\numexpr \xint_c_x^iv+#1+#4\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% + 1#6+#1\expandafter\XINT_div_I_g\expandafter + {\romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#4\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_verysmallmul #1!#51\Z!}% + }% }% % \end{macrocode} -% \lverb|1#1#2#3#4=nouveau q, nouvel alpha, L, K, {x'y},x,alpha', BQ«c»| +% \lverb|1#1=nouveau q. nouvel alpha, L, K, {x'y},x,alpha', BQ«c»| % \begin{macrocode} -\def\XINT_div_I_f 1#1#2#3#4{\XINT_div_I_g {#1#2#3#4}}% % \end{macrocode} % \lverb|#1=q,#2=nouvel alpha,#3=L, #4=K, #5={x'y}, #6=x, #7= alpha',#8=B, -% #9=Q«c» -> {x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c»| +% «c» -> on laisse q puis {x'y}alpha.alpha'.{{x'y}xKL}B«c»| % \begin{macrocode} -\def\XINT_div_I_g #1#2#3#4#5#6#7#8#9% +\def\XINT_div_I_g #1#2#3#4#5#6#7% {% - \ifnum#3=#4 - \expandafter\XINT_div_III_ab + \expandafter !\the\numexpr + \ifnum#2=#3 + \expandafter\XINT_div_exittofinish \else \expandafter\XINT_div_I_h \fi - {#5}#2.#7.{{#5}{#6}{#4}{#3}}{#8}{#9#1}% + {#4}#1.#6.{{#4}{#5}{#3}{#2}}{#7}% }% % \end{macrocode} -% \lverb|{x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c» -> R sans leading zeros.{Qq}«c»| +% \lverb|{x'y}alpha.alpha'.{{x'y}xKL}B«c» -> Attention retour à l'envoyeur ici +% par terminaison des \the\numexpr. On doit reprendre le Q déjà sorti, qui n'a +% plus de séparateurs, ni de leading 1. Ensuite R sans leading zeros.«c»| % \begin{macrocode} -\def\XINT_div_III_ab #1#2.#3.#4#5% +\def\XINT_div_exittofinish #1#2.#3.#4#5% {% - \expandafter\XINT_div_III_b - \romannumeral0\XINT_cuz_loop #2#3\W\W\W\W\W\W\W\Z.% + 1\expandafter\expandafter\expandafter!\expandafter\XINT_unsep_delim + \romannumeral0\XINT_div_unsepR #2#31\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W.% }% % \end{macrocode} -% \lverb|#1={x'y}alpha.#2#3#4#5#6=reste de A. -% #7={{x'y},x,K,L},#8=B,nouveauQ«c» devient {x'y},alpha sur K+4 chiffres.B, -% {{x'y},x,K,L}, #6= nouvel alpha',B,nouveauQ«c»| +% \lverb|#1={x'y}alpha.#2!#3=reste de A. +% #4={{x'y},x,K,L},#5=B,«c» devient {x'y},alpha sur K+4 chiffres.B, +% {{x'y},x,K,L}, #6= nouvel alpha',B,«c»| % \begin{macrocode} -\def\XINT_div_I_h #1.#2#3#4#5#6.#7#8% +\def\XINT_div_I_h #1.#2!#3.#4#5% {% - \XINT_div_II_b #1#2#3#4#5.{#8}{#7}{#6}{#8}% + \XINT_div_II_b #1#2!.{#5}{#4}{#3}{#5}% }% % \end{macrocode} -% \lverb|{x'y}alpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c» On -% intercepte la situation avec alpha débutant par 0000 qui est la seule qui -% pourrait donner un q1 nul. Donc q1 est non nul et la soustraction spéciale -% recevra un q1*B de longueur K ou K+4 et jamais 0000. Ensuite un q2 éventuel -% s'il est calculé est nécessairement non nul lui aussi. Comme dans la phase I -% on a aussi intercepté un q nul, la soustraction spéciale ne reçoit donc jamais -% un qB nul. Note: j'ai testé plusieurs fois que ma technique de gob_til_zeros -% est plus rapide que d'utiliser un \ifnum | +% \lverb|{x'y}alpha.B, {{x'y},x,K,L}, nouveau alpha',B,«c»| % \begin{macrocode} -\def\XINT_div_II_b #1#2#3#4#5#6#7#8#9% +\def\XINT_div_II_b #11#2!#3!% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_div_II_skipc 0000% - \XINT_div_II_c #1{#2#3#4#5}{#6#7#8#9}% + \xint_gob_til_eightzeroes #2\XINT_div_II_skipc 00000000% + \XINT_div_II_c #1{1#2}{#3}% }% % \end{macrocode} -% \lverb|x'y{0000}{4chiffres}reste de alpha.#6=B,#7={{x'y},x,K,L}, alpha',B, -% Q«c» -> {x'y}x,K,L (à diminuer de 4), {alpha sur -% K}B{q1=0000}{alpha'}B,Q«c»| +% \lverb|x'y{100000000}{1<8>}reste de alpha.#6=B,#7={{x'y},x,K,L}, alpha',B, +% «c» -> {x'y}x,K,L (à diminuer de 4), {alpha sur +% K}B{q1=00000000}{alpha'}B,«c»| % \begin{macrocode} -\def\XINT_div_II_skipc 0000\XINT_div_II_c #1#2#3#4#5.#6#7% +\def\XINT_div_II_skipc 00000000\XINT_div_II_c #1#2#3#4#5.#6#7% {% - \XINT_div_II_k #7{#4#5}{#6}{0000}% + \XINT_div_II_k #7{#4!#5}{#6}{00000000}% }% % \end{macrocode} -% \lverb|x'ya->1qx'yalpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c»| +% \lverb|x'ya->1qx'yalpha.B, {{x'y},x,K,L}, nouveau alpha',B, «c».| % \begin{macrocode} \def\XINT_div_II_c #1#2#3#4% {% - \expandafter\XINT_div_II_d\the\numexpr (#3#4+#2)/#1+\xint_c_ixixixix\relax - {#1}{#2}#3#4% + \expandafter\XINT_div_II_d\the\numexpr\XINT_div_mini + #1.#2!#3!#4!{#1}{#2}#3!#4!% }% % \end{macrocode} -% \lverb|1 suivi de q1 sur quatre chiffres, #5=x', #6=y, #7=alpha.#8=B, -% {{x'y},x,K,L}, alpha', B, Q«c» --> nouvel alpha.x',y,B,q1,{{x'y},x,K,L}, -% alpha', B, Q«c» | +% \lverb|1 suivi de q1 sur huit chiffres! #2=x', #3=y, #4=alpha.#5=B, +% {{x'y},x,K,L}, alpha', B, «c» --> nouvel alpha.x',y,B,q1,{{x'y},x,K,L}, +% alpha', B, «c» | % \begin{macrocode} -\def\XINT_div_II_d 1#1#2#3#4#5#6#7.#8% +\def\XINT_div_II_d 1#1#2#3#4#5!#6#7#8.#9% {% \expandafter\XINT_div_II_e - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#7}}% - {\romannumeral0\XINT_mul_Mr {#1#2#3#4}#8\Z\Z\Z\Z }.% - {#5}{#6}{#8}{#1#2#3#4}% + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#8\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_smallmul_a 100000000.#1#2#3#4.#5!#91\Z!}% + .{#6}{#7}{#9}{#1#2#3#4#5}% }% % \end{macrocode} -% \lverb|alpha.x',y,B,q1, {{x'y},x,K,L}, alpha', B, Q«c»| +% \lverb|alpha.x',y,B,q1, {{x'y},x,K,L}, alpha', B, «c». Attention la +% soustraction spéciale doit maintenir les blocs 1<8>!| % \begin{macrocode} -\def\XINT_div_II_e #1#2#3#4% +\def\XINT_div_II_e 1#1!% {% - \xint_gob_til_zeros_iv #1#2#3#4\XINT_div_II_skipf 0000% - \XINT_div_II_f #1#2#3#4% + \xint_gob_til_eightzeroes #1\XINT_div_II_skipf 00000000% + \XINT_div_II_f 1#1!% }% % \end{macrocode} -% \lverb|0000alpha sur K chiffres.#2=x',#3=y,#4=B,#5=q1, #6={{x'y},x,K,L}, -% #7=alpha',BQ«c» -> {x'y}x,K,L (à diminuer de 4), -% {alpha sur K}B{q1}{alpha'}BQ«c»| +% \lverb|100000000!alpha sur K chiffres.#2=x',#3=y,#4=B,#5=q1, #6={{x'y},x,K,L}, +% #7=alpha',B«c» -> {x'y}x,K,L (à diminuer de 1), +% {alpha sur K}B{q1}{alpha'}B«c»| % \begin{macrocode} -\def\XINT_div_II_skipf 0000\XINT_div_II_f 0000#1.#2#3#4#5#6% +\def\XINT_div_II_skipf 00000000\XINT_div_II_f 100000000!#1.#2#3#4#5#6% {% \XINT_div_II_k #6{#1}{#4}{#5}% }% % \end{macrocode} -% \lverb|a1 (huit chiffres), alpha (sur K+4), x', y, B, q1, {{x'y},x,K,L}, -% alpha', B,Q«c»| +% \lverb|1<a1>!1<a2>!, alpha (sur K+1 blocs de 8). x', y, B, q1, {{x'y},x,K,L}, +% alpha', B,«c».| % \begin{macrocode} -\def\XINT_div_II_f #1#2#3#4#5#6#7#8#9.% +\def\XINT_div_II_f #1!#2!#3.% {% - \XINT_div_II_fa {#1#2#3#4#5#6#7#8}{#1#2#3#4#5#6#7#8#9}% + \XINT_div_II_fa {#1!#2!}{#1!#2!#3}% }% \def\XINT_div_II_fa #1#2#3#4% {% - \expandafter\XINT_div_II_g\expandafter - {\the\numexpr (#1+#4)/#3-\xint_c_i}{#2}% + \expandafter\XINT_div_II_g \the\numexpr\XINT_div_mini #3.#4!#1{#2}% }% % \end{macrocode} % \lverb|#1=q, #2=alpha (K+4), #3=B, #4=q1, {{x'y},x,K,L}, alpha', BQ«c» -% -> 1 puis nouveau q sur 4 chiffres, nouvel alpha sur K chiffres, -% B, {{x'y},x,K,L}, alpha',BQ«c» | +% -> 1 puis nouveau q sur 8 chiffres. nouvel alpha sur K blocs, +% B, {{x'y},x,K,L}, alpha',B«c» | % \begin{macrocode} -\def\XINT_div_II_g #1#2#3#4% +\def\XINT_div_II_g 1#1#2#3#4#5!#6#7#8% {% \expandafter \XINT_div_II_h - \the\numexpr #4+#1+\xint_c_x^iv\expandafter\expandafter\expandafter - {\expandafter\xint_gobble_iv - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}{#3}% + \the\numexpr 1#1#2#3#4#5+#8\expandafter\expandafter\expandafter + .\expandafter\expandafter\expandafter + {\expandafter\xint_gob_til_exclam + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#6\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_smallmul_a 100000000.#1#2#3#4.#5!#71\Z!}}% + {#7}% }% % \end{macrocode} -% \lverb|1 puis nouveau q sur 4 chiffres, #5=nouvel alpha sur K chiffres, -% #6=B, #7={{x'y},x,K,L} avec L à ajuster, alpha', BQ«c» -% -> {x'y}x,K,L à diminuer de 4, {alpha}B{q}, alpha', BQ«c»| +% \lverb|1 puis nouveau q sur 8 chiffres, #2=nouvel alpha sur K blocs, +% #3=B, #4={{x'y},x,K,L} avec L à ajuster, alpha', BQ«c» +% -> {x'y}x,K,L à diminuer de 1, {alpha}B{q}, alpha', BQ«c»| % \begin{macrocode} -\def\XINT_div_II_h 1#1#2#3#4#5#6#7% +\def\XINT_div_II_h 1#1.#2#3#4% {% - \XINT_div_II_k #7{#5}{#6}{#1#2#3#4}% + \XINT_div_II_k #4{#2}{#3}{#1}% }% % \end{macrocode} -% \lverb|{x'y}x,K,L à diminuer de 4, alpha, B{q}alpha',BQ«c» -% ->nouveau L.K,x',y,x,alpha.B,q,alpha',B,Q«c» -% ->{LK{x'y}x},x,a,alpha.B,q,alpha',B,Q«c»| +% \lverb|{x'y}x,K,L à diminuer de 1, alpha, B{q}alpha',B«c» +% ->nouveau L.K,x',y,x,alpha.B,q,alpha',B,«c» +% ->{LK{x'y}x},x,a,alpha.B,q,alpha',B,«c»| % \begin{macrocode} \def\XINT_div_II_k #1#2#3#4#5% {% - \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_iv.{#3}#1{#2}#5.% + \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_i.{#3}#1{#2}#5.% }% -\def\XINT_div_II_l #1.#2#3#4#5#6#7#8#9% +\def\XINT_div_II_l #1.#2#3#4#51#6!% {% - \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6#7#8#9}#6#7#8#9% + \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6}1#6!% }% % \end{macrocode} -% \lverb|{LK{x'y}x},x,a,alpha.B{q}alpha'BQ -> a, x, alpha, B, q, -% L, K, {x'y}, x, alpha', BQ«c» | +% \lverb|{LK{x'y}x},x,a,alpha.B{q}alpha'B -> a, x, alpha, B, q, +% L, K, {x'y}, x, alpha', B«c» | % \begin{macrocode} \def\XINT_div_II_m #1#2#3#4.#5#6% {% \XINT_div_I_a {#3}{#2}{#4}{#5}{#6}#1% }% % \end{macrocode} -% \lverb|L, K, A, y,{},x, {},«c»->A.{yx}L{}«c» Comme ici K=4, dans -% la phase I on n'a pas besoin de alpha, car a = alpha. De plus on a maintenu B -% dans l'ordre qui est donc la même chose que x. Par ailleurs la phase I est -% simplifiée, il s'agit simplement de la division euclidienne de a par x, et de -% plus on n'a à la faire qu'une unique fois et ensuite la phase II peut boucler -% sur elle-même au lieu de revenir en phase I, par conséquent il n'y a pas non -% plus de q0 ici. Enfin, le y est (x/2) pas ((x+1)/2) il n'y a pas de x'=x+1| +% \lverb|This multiplication is exactly like \XINT_smallmul, but it always +% keeps the ending carry. For optimization I duplicated the whole code.| % \begin{macrocode} -\def\XINT_div_little_b #1#2#3#4#5#6#7% +\def\XINT_div_minimulwc_a 1#1.#2.#3!#4#5#6#7#8.% {% - \XINT_div_little_c #3.{{#4}{#6}}{#1}% + \expandafter\XINT_div_minimulwc_b + \the\numexpr \xint_c_x^ix+#1+#3*#8.#3*#4#5#6#7+#2*#8.#2*#4#5#6#7.% }% -% \end{macrocode} -% \lverb|#1#2#3#4=a, #5=alpha'=reste de A.#6={yx}, #7=L, «c» -> a, -% y, x, L, alpha'=reste de A, «c».| -% \begin{macrocode} -\def\XINT_div_little_c #1#2#3#4#5.#6#7% +\def\XINT_div_minimulwc_b 1#1#2#3#4#5#6.#7.% +{% + \expandafter\XINT_div_minimulwc_c + \the\numexpr \xint_c_x^ix+#1#2#3#4#5+#7.#6.% +}% +\def\XINT_div_minimulwc_c 1#1#2#3#4#5#6.#7.#8.% +{% + 1#6#7\expandafter!% + \the\numexpr\expandafter\XINT_div_smallmul_a + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8.% +}% +\def\XINT_div_smallmul_a #1.#2.#3!1#4!% {% - \XINT_div_littleI_a {#1#2#3#4}#6{#7}{#5}% + \xint_gob_til_Z #4\XINT_div_smallmul_e\Z + \XINT_div_minimulwc_a #1.#2.#3!#4.#2.#3!% }% +\def\XINT_div_smallmul_e\Z\XINT_div_minimulwc_a 1#1.#2\Z #3!{1\relax #1!}% % \end{macrocode} -% \lverb|a, y, x, L, alpha',«c» On calcule ici (contrairement à la -% phase I générale) le vrai quotient euclidien de a par x=B, c'est donc un -% chiffre de 0 à 9. De plus on n'a à faire cela qu'une unique fois.| +% \lverb|Special very small multiplication for division. We only need to cater +% for multiplicands from 1 to 9. The ending is different from standard +% verysmallmul, a zero carry is not suppressed. And no final 1\Z! is added. If +% #1=1 let's not forget to add the 100000000! at the end.| % \begin{macrocode} -\def\XINT_div_littleI_a #1#2#3% +\def\XINT_div_verysmallmul #1% + {\xint_gob_til_one #1\XINT_div_verysmallisone 1\XINT_div_verysmallmul_a 0.#1}% +\def\XINT_div_verysmallisone 1\XINT_div_verysmallmul_a 0.1!1#11\Z!% + {1\relax #1100000000!}% +\def\XINT_div_verysmallmul_a #1.#2!1#3!% {% - \expandafter\XINT_div_littleI_b - \the\numexpr (#1+#2)/#3-\xint_c_i{#1}{#2}{#3}% + \xint_gob_til_Z #3\XINT_div_verysmallmul_e\Z + \expandafter\XINT_div_verysmallmul_b + \the\numexpr \xint_c_x^ix+#2*#3+#1.#2!% }% +\def\XINT_div_verysmallmul_b 1#1#2.% + {1#2\expandafter!\the\numexpr\XINT_div_verysmallmul_a #1.}% +\def\XINT_div_verysmallmul_e\Z #1\Z +#2#3!{1\relax 0000000#2!}% % \end{macrocode} -% \lverb|On intercepte quotient nul: [est-ce vraiment utile? ou n'est-ce pas -% plutôt une perte de temps en moyenne? il faudrait tester] q=0#1=a, -% #2=y, x, L, alpha', «c» -> -% II_a avec L{alpha}alpha'.{yx}{0000}«c». Et en cas de quotient non nul on -% procède avec littleI_c avec #1=q, #2=a, #3=y, #4=x -> {nouvel alpha sur 4 -% chiffres}q{yx},L,alpha',«c».| +% \lverb|Special subtraction for division purposes.| % \begin{macrocode} -\def\XINT_div_littleI_b #1% +\def\XINT_div_sub #1#2% {% - \xint_gob_til_zero #1\XINT_div_littleI_skip 0\XINT_div_littleI_c #1% + \expandafter\XINT_div_sub_clean + \the\numexpr\expandafter\XINT_div_sub_a\expandafter + 1#2\Z!\Z!\Z!\Z!\Z!\W #1\Z!\Z!\Z!\Z!\Z!\W }% -\def\XINT_div_littleI_skip 0\XINT_div_littleI_c 0#1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{{#2}{#3}}{0000}}% -\def\XINT_div_littleI_c #1#2#3#4% +\def\XINT_div_sub_clean #1-#2#3\W {% - \expandafter\expandafter\expandafter\XINT_div_littleI_e - \expandafter\expandafter\expandafter - {\expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4}#1{{#3}{#4}}% + \if1#2\expandafter\XINT_rev_nounsep\else\expandafter\XINT_div_sub_neg\fi + {}#1\R!\R!\R!\R!\R!\R!\R!\R!\W }% -% \end{macrocode} -% \lverb|#1=nouvel alpha sur 4 chiffres#2=q,#3={yx}, #4=L, #5=alpha',«c» -> -% L{alpha}alpha'.{yx}{000q}«c» point d'entrée de la boucle principale| -% \begin{macrocode} -\def\XINT_div_littleI_e #1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{#3}{000#2}}% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c» et c'est là qu'on boucle| -% \begin{macrocode} -\def\XINT_div_littleII_a #1% +\def\XINT_div_sub_neg #1\W { -}% +\def\XINT_div_sub_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \ifnum#1=\xint_c_iv - \expandafter\XINT_div_littleIII_ab - \else - \expandafter\XINT_div_littleII_b - \fi {#1}% + \XINT_div_sub_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c» -> (en fait #3 est vide normalement ici) R -% sans leading zeros.Q«c»| -% \begin{macrocode} -\def\XINT_div_littleIII_ab #1#2#3.#4% +\def\XINT_div_sub_b #1#2#3!#4!% {% - \expandafter\XINT_div_III_b\the\numexpr #2#3.% + \xint_gob_til_Z #4\XINT_div_sub_bi \Z + \expandafter\XINT_div_sub_c\the\numexpr#1-#3+1#4-\xint_c_i.% }% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c». On diminue L de quatre, comme cela c'est -% fait.| -% \begin{macrocode} -\def\XINT_div_littleII_b #1% +\def\XINT_div_sub_c 1#1#2.% {% - \expandafter\XINT_div_littleII_c\expandafter {\the\numexpr #1-\xint_c_iv}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_d #1% }% -% \end{macrocode} -% \lverb|{nouveauL}{alpha}alpha'.{yx}Q«c». On prélève 4 chiffres de alpha' -> -% {nouvel alpha sur huit chiffres}yx{nouveau L}{nouvel alpha'}Q«c». Regarder -% si l'ancien alpha était 0000 n'avancerait à rien car obligerait à refaire une -% chose comme la phase I, donc on ne perd pas de temps avec ça, on reste en -% permanence en phase II.| -% \begin{macrocode} -\def\XINT_div_littleII_c #1#2#3#4#5#6#7.#8% +\def\XINT_div_sub_d #1#2#3!#4!% {% - \XINT_div_littleII_d {#2#3#4#5#6}#8{#1}{#7}% + \xint_gob_til_Z #4\XINT_div_sub_di \Z + \expandafter\XINT_div_sub_e\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_littleII_d #1#2#3% +\def\XINT_div_sub_e 1#1#2.% {% - \expandafter\XINT_div_littleII_e\the\numexpr (#1+#2)/#3+\xint_c_ixixixix.% - {#1}{#2}{#3}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_f #1% }% -% \end{macrocode} -% \lverb|1 suivi de #1=q1 sur quatre chiffres.#2=alpha, #3=y, #4=x, -% L, alpha', Q«c» --> nouvel alpha sur 4.{q1}{yx},L,alpha', Q«c» | -% \begin{macrocode} -\def\XINT_div_littleII_e 1#1.#2#3#4% +\def\XINT_div_sub_f #1#2#3!#4!% {% - \expandafter\expandafter\expandafter\XINT_div_littleII_f - \expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4.% - {#1}{{#3}{#4}}% + \xint_gob_til_Z #4\XINT_div_sub_fi \Z + \expandafter\XINT_div_sub_g\the\numexpr#1-#3+1#4-\xint_c_i.% }% -% \end{macrocode} -% \lverb|alpha.q,{yx},L,alpha',Q«c»->L{alpha}alpha'.{yx}{Qq}«c»| -% \begin{macrocode} -\def\XINT_div_littleII_f #1.#2#3#4#5#6% +\def\XINT_div_sub_g 1#1#2.% {% - \XINT_div_littleII_a {#4}{#1}#5.{#3}{#6#2}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_h #1% }% -% \end{macrocode} -% \lverb|La soustraction spéciale. Dans 1.09j, elle fait A-qB, pour A (en fait -% alpha dans mes dénominations des commentaires du code) et qB chacun de -% longueur K ou K+4, avec K au moins huit multiple de quatre, qB a ses quatre -% chiffres significatifs (qui sont à droite) non nuls. Si A-qB<0 il suffit de -% renvoyer -, le résultat n'importe pas. On est sûr que qB est non nul. On le -% met dans cette version en premier pour tester plus facilement le cas avec qB -% de longueur K+4 et A de longueur seulement K. Lorsque la longueur de qB est -% inférieure ou égale à celle de A, on va jusqu'à la fin de A et donc c'est la -% retenue finale qui décide du cas négatif éventuel. Le résultat non négatif est -% toujours donc renvoyé avec la même longueur que A, et il est dans l'ordre. -% J'ai fait une implémentation des phases I et II en maintenant alpha toujours à -% l'envers afin d'éviter le reverse order systématique fait sur A (ou plutôt -% alpha), mais alors il fallait que la soustraction ici s'arrange pour repérer -% les huit chiffres les plus significatifs, au final ce n'était pas plus rapide, -% et même pénalisant pour de gros inputs. Dans les versions 1.09i et antérieures -% (en fait je pense qu'ici rien quasiment n'avait bougé depuis la première -% implémentation), la soustraction spéciale n'était pratiquée que dans des cas -% avec certainement A-qB positif ou nul. De plus on n'excluait pas q=0, donc il -% fallait aussi faire un éventuel reverseorder sur ce qui était encore non -% traité. Les cas avec q=0 sont maintenant interceptés en amont et comme A et qB -% ont toujours quasiment la même longueur on ne s'embarrasse pas de -% complications pour la fin.| -% \begin{macrocode} -\def\XINT_div_sub_xpxp #1#2% #1=alpha d\'ej\`a renvers\'e, #2 se d\'eveloppe en qB +\def\XINT_div_sub_h #1#2#3!#4!% {% - \expandafter\XINT_div_sub_xpxp_b #2\W\X\Y\Z #1\W\X\Y\Z + \xint_gob_til_Z #4\XINT_div_sub_hi \Z + \expandafter\XINT_div_sub_i\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_sub_xpxp_b +\def\XINT_div_sub_i 1#1#2.% {% - \XINT_div_sub_A 1{}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_a #1% }% -\def\XINT_div_sub_A #1#2#3#4#5#6% +\def\XINT_div_sub_bi\Z + \expandafter\XINT_div_sub_c\the\numexpr#1-#2+#3.#4!#5!#6!#7!#8!#9!\Z !\W {% - \xint_gob_til_W #3\xint_div_sub_az\W - \XINT_div_sub_B #1{#3#4#5#6}{#2}% + \XINT_div_sub_l #1#2!#5!#7!#9!% }% -\def\XINT_div_sub_B #1#2#3#4\W\X\Y\Z #5#6#7#8% +\def\XINT_div_sub_di\Z + \expandafter\XINT_div_sub_e\the\numexpr#1-#2+#3.#4!#5!#6!#7!#8\W {% - \xint_gob_til_W #5\xint_div_sub_bz\W - \XINT_div_sub_onestep #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z + \XINT_div_sub_l #1#2!#5!#7!% }% -\def\XINT_div_sub_onestep #1#2#3#4#5#6% +\def\XINT_div_sub_fi\Z + \expandafter\XINT_div_sub_g\the\numexpr#1-#2+#3.#4!#5!#6\W {% - \expandafter\XINT_div_sub_backtoA - \the\numexpr 11#6-#5#4#3#2+#1-\xint_c_i.% + \XINT_div_sub_l #1#2!#5!% }% -\def\XINT_div_sub_backtoA #1#2#3.#4% +\def\XINT_div_sub_hi\Z + \expandafter\XINT_div_sub_i\the\numexpr#1-#2+#3.#4\W {% - \XINT_div_sub_A #2{#3#4}% + \XINT_div_sub_l #1#2!% }% -% \end{macrocode} -% \lverb|si on arrive en sub_bz c'est que qB était de longueur K+4 et A -% seulement de longueur K, le résultat est donc < 0, renvoyer juste -| -% \begin{macrocode} -\def\xint_div_sub_bz\W\XINT_div_sub_onestep #1\Z { -}% -% \end{macrocode} -% \lverb|si on arrive en sub_az c'est que qB était de longueur inférieure ou -% égale à celle de A, donc on continue jusqu'à la fin de A, et on vérifiera la -% retenue à la fin.| -% \begin{macrocode} -\def\xint_div_sub_az\W\XINT_div_sub_B #1#2{\XINT_div_sub_C #1}% -\def\XINT_div_sub_C #1#2#3#4#5#6% +\def\XINT_div_sub_l #1% {% - \xint_gob_til_W #3\xint_div_sub_cz\W - \XINT_div_sub_C_onestep #1{#6#5#4#3}{#2}% + \xint_UDzerofork + #1{-2\relax}% + 0\XINT_div_sub_r + \krof }% -\def\XINT_div_sub_C_onestep #1#2% +\def\XINT_div_sub_r #1!% {% - \expandafter\XINT_div_sub_backtoC \the\numexpr 11#2+#1-\xint_c_i.% + -\ifnum 0#1=\xint_c_ 1\else2\fi\relax }% -\def\XINT_div_sub_backtoC #1#2#3.#4% +%%%%%%%%%%%% +\def\XINT_sdiv_out #1\Z #2\W% + {\expandafter + {\romannumeral0\XINT_unsep_cuzsmall#11\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W}% + {#2}}% +\def\XINT_smalldivx_a #1.1#2!1#3!% {% - \XINT_div_sub_C #2{#3#4}% + \expandafter\XINT_smalldivx_b + \the\numexpr (#3+#1)/#2-\xint_c_i!#1.#2!#3!% +}% +\def\XINT_smalldivx_b #1!% +{% + \if0#1\else + \xint_c_x^viii+#1\xint_afterfi{\expandafter!\the\numexpr}\fi + \XINT_smalldiv_c #1!% +}% +\def\XINT_smalldiv_c #1!#2.#3!#4!% +{% + \expandafter\XINT_smalldiv_d\the\numexpr #4-#1*#3!#2.#3!% +}% +\def\XINT_smalldiv_d #1!#2!#3#4!% +{% + \xint_gob_til_Z #4\XINT_smalldiv_end \Z + \XINT_smalldiv_e #1!#2!#3#4!% +}% +\def\XINT_smalldiv_end\Z\XINT_smalldiv_e #1!#2!1\Z!{1!\Z #1\W }% +\def\XINT_smalldiv_e #1!#2.#3!% +{% + \expandafter\XINT_smalldiv_f\the\numexpr + \xint_c_xi_e_viii_mone+#1*\xint_c_x^viii/#3!#2.#3!#1!% +}% +\def\XINT_smalldiv_f 1#1#2#3#4#5#6!#7.#8!% +{% + \xint_gob_til_zero #1\XINT_smalldiv_fz 0% + \expandafter\XINT_smalldiv_g + \the\numexpr\XINT_minimul_a #2#3#4#5.#6!#8!#2#3#4#5#6!#7.#8!% +}% +\def\XINT_smalldiv_fz 0% + \expandafter\XINT_smalldiv_g\the\numexpr\XINT_minimul_a + 9999.9999!#1!99999999!#2!0!1#3!% +{% + \XINT_smalldiv_i .#3!\xint_c_!#2!% +}% +\def\XINT_smalldiv_g 1#1!1#2!#3!#4!#5!#6!% +{% + \expandafter\XINT_smalldiv_h + \the\numexpr 1#6-#1.#2!#5!#3!#4!% +}% +\def\XINT_smalldiv_h 1#1#2.#3!#4!% +{% + \expandafter\XINT_smalldiv_i + \the\numexpr #4-#3+#1-\xint_c_i.#2!% +}% +\def\XINT_smalldiv_i #1.#2!#3!#4.#5!% +{% + \expandafter\XINT_smalldiv_j + \the\numexpr (#1#2+#4)/#5-\xint_c_i!#3!#1#2!#4.#5!% +}% +\def\XINT_smalldiv_j #1!#2!% +{% + \xint_c_x^viii+#1+#2\expandafter!\the\numexpr\XINT_smalldiv_k + #1!% +}% +\def\XINT_smalldiv_k #1!#2!#3.#4!% +{% + \expandafter\XINT_smalldiv_d\the\numexpr #2-#1*#4!#3.#4!% +}% +%%%%%%%%%%%% +\def\XINT_div_mini #1.#2!1#3!% +{% + \expandafter\XINT_div_mini_a\the\numexpr + \xint_c_xi_e_viii_mone+#3*\xint_c_x^viii/#1!#1.#2!#3!% }% % \end{macrocode} -% \lverb|une fois arrivé en sub_cz on teste la retenue pour voir si le résultat -% final est en fait négatif, dans ce cas on renvoie seulement -| +% \lverb|Note (2015/10/08). Attention à la différence dans l'ordre des +% arguments avec ce que je vois en comparaison avec \XINT_smalldiv_f. Je ne me +% souviens plus du tout s'il y a une raison quelconque.| % \begin{macrocode} -\def\xint_div_sub_cz\W\XINT_div_sub_C_onestep #1#2% +\def\XINT_div_mini_a 1#1#2#3#4#5#6!#7.#8!% {% - \if#10% retenue - \expandafter\xint_div_sub_neg - \else\expandafter\xint_div_sub_ok - \fi + \xint_gob_til_zero #1\XINT_div_mini_w 0% + \expandafter\XINT_div_mini_b + \the\numexpr\XINT_minimul_a #2#3#4#5.#6!#7!#2#3#4#5#6!#7.#8!% +}% +\def\XINT_div_mini_w 0% + \expandafter\XINT_div_mini_b\the\numexpr\XINT_minimul_a + 9999.9999!#1!99999999!#2.#3!00000000!#4!% +{% + \xint_c_x^viii_mone+(#4+#3)/#2!% +}% +\def\XINT_div_mini_b 1#1!1#2!#3!#4!#5!#6!% +{% + \expandafter\XINT_div_mini_c + \the\numexpr 1#6-#1.#2!#5!#3!#4!% +}% +\def\XINT_div_mini_c 1#1#2.#3!#4!% +{% + \expandafter\XINT_div_mini_d + \the\numexpr #4-#3+#1-\xint_c_i.#2!% +}% +\def\XINT_div_mini_d #1.#2!#3!#4.#5!% +{% + \xint_c_x^viii_mone+#3+(#1#2+#5)/#4!% }% -\def\xint_div_sub_neg #1{ -}% -\def\xint_div_sub_ok #1{ #1}% % \end{macrocode} % \subsection{\csh{xintiDivRound}, \csh{xintiiDivRound}} -% \lverb|v1.1, transferred from first release of bnumexpr.| +% \lverb|v1.1, transferred from first release of bnumexpr. Rewritten for v1.2.| % \begin{macrocode} \def\xintiDivRound {\romannumeral0\xintidivround }% -\def\xintidivround #1{\expandafter\XINT_iidivround\romannumeral0\xintnum{#1}\Z }% +\def\xintidivround #1% + {\expandafter\XINT_idivround\romannumeral0\xintnum{#1}\Z }% \def\xintiiDivRound {\romannumeral0\xintiidivround }% \def\xintiidivround #1{\expandafter\XINT_iidivround \romannumeral-`0#1\Z }% -\def\XINT_iidivround #1#2\Z #3{\expandafter\XINT_iidivround_a\expandafter #1% - \romannumeral-`0#3\Z #2\Z }% +\def\XINT_idivround #1#2\Z #3% + {\expandafter\XINT_iidivround_a\expandafter #1% + \romannumeral0\xintnum{#3}\Z #2\Z }% +\def\XINT_iidivround #1#2\Z #3% + {\expandafter\XINT_iidivround_a\expandafter #1\romannumeral-`0#3\Z #2\Z }% \def\XINT_iidivround_a #1#2% #1 de A, #2 de B. {% \if0#2\xint_dothis\XINT_iidivround_divbyzero\fi @@ -18008,19 +17891,50 @@ $1$ or $-1$. -{\xintiiopp\XINT_iidivround_pos #1}% \krof }% -\def\XINT_iidivround_pos #1#2\Z #3\Z{\expandafter\XINT_iidivround_pos_a - \romannumeral0\XINT_div_prepare {#2}{#1#30}}% -\def\XINT_iidivround_pos_a #1#2{\xintReverseOrder {#1\XINT_iidivround_pos_b}\Z }% -\def\XINT_iidivround_pos_b #1#2{\xint_gob_til_Z #2\XINT_iidivround_pos_small\Z - \XINT_iidivround_pos_c #1#2}% -\def\XINT_iidivround_pos_c #1#2\Z {\ifnum #1>\xint_c_iv - \expandafter\XINT_iidivround_pos_up - \else \expandafter\xintreverseorder - \fi {#2}}% -\def\XINT_iidivround_pos_up #1{\xintinc {\xintReverseOrder{#1}}}% -\def\XINT_iidivround_pos_small\Z\XINT_iidivround_pos_c #1#2% - {\ifnum #1>\xint_c_iv\expandafter\xint_secondoftwo\else\expandafter - \xint_firstoftwo\fi { 0}{ 1}}% +\def\XINT_iidivround_pos #1#2\Z #3\Z +{% + \expandafter\XINT_iidivround_pos_a + \romannumeral0\XINT_div_prepare {#2}{#1#30}% +}% +\def\XINT_iidivround_pos_a #1#2% +{% + \expandafter\XINT_iidivround_pos_b + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678\XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_iidivround_pos_b 1#1#2#3#4#5#6#7#8!#9% +{% + \xint_gob_til_Z #9\XINT_iidivround_small\Z + \ifnum #8>\xint_c_iv + \expandafter\XINT_iidivround_pos_up + \else \expandafter\XINT_iidivround_pos_finish + \fi + 1#1#2#3#4#5#6#70!#9% +}% +\def\XINT_iidivround_pos_up +{% + \expandafter\XINT_iidivround_pos_finish + \the\numexpr\XINT_add_a\xint_c_ii 100000010!\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_iidivround_pos_finish #10!#2\Z #3\W +{% + \expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}% + #1!#21\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_iidivround_small\Z\ifnum #1>#2\fi 1#30!#4\W +{% + \ifnum #1>\xint_c_iv + \expandafter\XINT_iidivround_small_up + \else \expandafter\XINT_iidivround_small_trunc + \fi {#3}% +}% +\edef\XINT_iidivround_small_up #1% + {\noexpand\expandafter\space\noexpand\the\numexpr #1+\xint_c_i\relax }% +\edef\XINT_iidivround_small_trunc #1% + {\noexpand\expandafter\space\noexpand\the\numexpr #1\relax }% % \end{macrocode} % \subsection{\csh{xintiDivTrunc}, \csh{xintiiDivTrunc}} % \begin{macrocode} @@ -18052,7 +17966,8 @@ $1$ or $-1$. \krof }% \def\XINT_iidivtrunc_pos #1#2\Z #3\Z% - {\expandafter\xint_firstoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}% + {\expandafter\xint_firstoftwo_thenstop + \romannumeral0\XINT_div_prepare {#2}{#1#3}}% % \end{macrocode} % \subsection{\csh{xintiMod}, \csh{xintiiMod}} % \begin{macrocode} @@ -18084,7 +17999,20 @@ $1$ or $-1$. \krof }% \def\XINT_iimod_pos #1#2\Z #3\Z% - {\expandafter\xint_secondoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}% + {\expandafter\xint_secondoftwo_thenstop\romannumeral0\XINT_div_prepare + {#2}{#1#3}}% +% \end{macrocode} +% \subsection{``Load \xintfracnameimp'' macros} +% \lverb|Originally was used in \xintiiexpr. Transferred from xintfrac for 1.1.| +% \begin{macrocode} +\catcode`! 11 +\def\xintAbs {\Did_you_mean_iiAbs?or_load_xintfrac!}% +\def\xintOpp {\Did_you_mean_iiOpp?or_load_xintfrac!}% +\def\xintAdd {\Did_you_mean_iiAdd?or_load_xintfrac!}% +\def\xintSub {\Did_you_mean_iiSub?or_load_xintfrac!}% +\def\xintMul {\Did_you_mean_iiMul?or_load_xintfrac!}% +\def\xintPow {\Did_you_mean_iiPow?or_load_xintfrac!}% +\def\xintSqr {\Did_you_mean_iiSqr?or_load_xintfrac!}% \XINT_restorecatcodes_endinput% % \end{macrocode} %\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 @@ -18099,9 +18027,10 @@ $1$ or $-1$. % % \localtableofcontents % -% The basic arithmetic routines |\xintiiAdd|, |\xintiiSub|, -% |\xintiiMul|, |\xintiiQuo| and |\xintiiPow| have been moved to new -% package \xintcorenameimp. +% With release |1.1| the core arithmetic routines |\xintiiAdd|, +% |\xintiiSub|, |\xintiiMul|, |\xintiiQuo|, |\xintiiPow| were separated to be +% the main component of the then new +% \xintcorenameimp. % \begin{macrocode} \begingroup\catcode61\catcode48\catcode32=10\relax% \catcode13=5 % ^^M @@ -18151,7 +18080,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xint}% - [2015/09/12 v1.1c Expandable operations on big integers (jfB)]% + [2015/10/10 v1.2 Expandable operations on big integers (jfB)]% % \end{macrocode} % \subsection{More token management} % \begin{macrocode} @@ -18161,6 +18090,10 @@ $1$ or $-1$. \long\def\xint_firstofthree_thenstop #1#2#3{ #1}% 1.09i \long\def\xint_secondofthree_thenstop #1#2#3{ #2}% \long\def\xint_thirdofthree_thenstop #1#2#3{ #3}% +\edef\xint_cleanupzeros_andstop #1#2#3#4% +{% + \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax +}% % \end{macrocode} % \subsection{\csh{xintSgnFork}} % \lverb|Expandable three-way fork added in 1.07. The argument #1 must expand @@ -18201,24 +18134,12 @@ $1$ or $-1$. % However this last aspect does not appear like a very useful thing. And despite % the fact that a special check is made for a sign, actually the input is not % given to \xintnum, contrarily to \xintLen. This is all a bit incoherent. -% Should be fixed.| +% Should be fixed. +% +% 1.2 has \xintReverseDigits and I thus make \xintRev an alias. Remarks above +% not addressed.| % \begin{macrocode} -\def\xintRev {\romannumeral0\xintrev }% -\def\xintrev #1% -{% - \expandafter\XINT_rev_fork - \romannumeral-`0#1\xint_relax % empty #1 ok, \xint_relax stops expansion - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_rev_fork #1% -{% - \xint_UDsignfork - #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_rord_main {}}% - -{\XINT_rord_main {}#1}% - \krof -}% +\let\xintRev\xintReverseDigits % \end{macrocode} % \subsection{\csh{xintLen}} % \lverb|\xintLen is ONLY for (possibly long) integers. Gets extended to @@ -18463,143 +18384,183 @@ $1$ or $-1$. }% % \end{macrocode} % \subsection{\csh{xintCmp}, \csh{xintiiCmp}} -% \lverb|Release 1.09a has \xintnum inserted into \xintCmp. Unnecessary -% \xintiCmp suppressed in 1.09f. And 1.1a does \xintiiCmp, for -% optimization in \xintiiexpr. (not needed before, because \XINT_cmp_fork was -% directly used, or \XINT_Cmp)| +% \lverb|Faster than doing the full subtraction.| % \begin{macrocode} -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% +\def\xintCmp {\romannumeral0\xintcmp }% +\def\xintcmp #1{\expandafter\XINT_icmp\romannumeral0\xintnum{#1}\Z }% +\def\xintiiCmp {\romannumeral0\xintiicmp }% +\def\xintiicmp #1{\expandafter\XINT_iicmp\romannumeral-`0#1\Z }% +\def\XINT_iicmp #1#2\Z #3% {% - \expandafter\xint_cmp\expandafter{\romannumeral0\xintnum{#1}}% + \expandafter\XINT_cmp_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xint_cmp #1#2% +% \end{macrocode} +% \lverb|New fork of 1.2 makes it less convenient here for \XINT_cmp_pre and +% \XINT_Cmp, which just avoided the \romannumeral-`0. Nanosecond loss ? I +% vaguely recalled that for \xintNewExpr things, I did need another name such +% as \XINT_cmp for \xintiiCmp.| +% \begin{macrocode} +\let\XINT_Cmp \xintiiCmp +\def\XINT_icmp #1#2\Z #3% {% - \expandafter\XINT_cmp_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_cmp_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xintiiCmp {\romannumeral0\xintiicmp }% -\def\xintiicmp #1% +\def\XINT_cmp_nfork #1#2% {% - \expandafter\xint_iicmp\expandafter{\romannumeral-`0#1}% + \xint_UDzerofork + #1\XINT_cmp_firstiszero + #2\XINT_cmp_secondiszero + 0{}% + \krof + \xint_UDsignsfork + #1#2\XINT_cmp_minusminus + #1-\XINT_cmp_minusplus + #2-\XINT_cmp_plusminus + --\XINT_cmp_plusplus + \krof #1#2% }% -\def\xint_iicmp #1#2% +\def\XINT_cmp_firstiszero #1\krof 0#2#3\Z #4\Z {% - \expandafter\XINT_cmp_fork \romannumeral-`0#2\Z #1\Z -}% -\def\XINT_Cmp #1#2{\romannumeral0\XINT_cmp_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|& -% COMPARAISON $\ -% 1 si #3#4>#1#2, 0 si #3#4=#1#2, -1 si #3#4<#1#2$\ -% #3#4 vient du *premier*,$ -% #1#2 vient du *second*| -% \begin{macrocode} -\def\XINT_cmp_fork #1#2\Z #3#4\Z + \xint_UDzerominusfork + #2-{ 0}% + 0#2{ 1}% + 0-{ -1}% + \krof +}% +\def\XINT_cmp_secondiszero #1\krof #20#3\Z #4\Z {% - \xint_UDsignsfork - #1#3\XINT_cmp_minusminus - #1-\XINT_cmp_minusplus - #3-\XINT_cmp_plusminus - --{\xint_UDzerosfork - #1#3\XINT_cmp_zerozero - #10\XINT_cmp_zeroplus - #30\XINT_cmp_pluszero - 00\XINT_cmp_plusplus - \krof }% + \xint_UDzerominusfork + #2-{ 0}% + 0#2{ -1}% + 0-{ 1}% \krof - {#2}{#4}#1#3% +}% +\def\XINT_cmp_plusminus #1\Z #2\Z{ 1}% +\def\XINT_cmp_minusplus #1\Z #2\Z{ -1}% +\def\XINT_cmp_minusminus + --{\expandafter\XINT_opp\romannumeral0\XINT_cmp_plusplus {}{}}% +\def\XINT_cmp_plusplus #1#2#3\Z +{% + \expandafter\XINT_cmp_pp + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\def\XINT_cmp_pp #1.#2\X #3\Z +{% + \expandafter\XINT_cmp_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_cmp_checklengths #1.#2.% +{% + \ifnum #1=#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + \XINT_cmp_aa {\XINT_cmp_distinctlengths {#1}{#2}}% }% -\def\XINT_cmp_minusplus #1#2#3#4{ 1}% -\def\XINT_cmp_plusminus #1#2#3#4{ -1}% -\def\XINT_cmp_zerozero #1#2#3#4{ 0}% -\def\XINT_cmp_zeroplus #1#2#3#4{ 1}% -\def\XINT_cmp_pluszero #1#2#3#4{ -1}% -\def\XINT_cmp_plusplus #1#2#3#4% +\def\XINT_cmp_distinctlengths #1#2#3\W #4\W {% - \XINT_cmp_pre {#4#2}{#3#1}% + \ifnum #1>#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + { -1}{ 1}% }% -\def\XINT_cmp_minusminus #1#2#3#4% +%%%%%%%%%%%% +\def\XINT_cmp_aa {\expandafter\XINT_cmp_w\the\numexpr\XINT_cmp_a \xint_c_i }% +%%%%%%%%%%%% +\def\XINT_cmp_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \XINT_cmp_pre {#1}{#2}% + \XINT_cmp_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_cmp_pre #1% +\def\XINT_cmp_b #1#2#3!#4!% {% - \expandafter\XINT_cmp_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \xint_gob_til_Z #2\XINT_cmp_bi \Z + \expandafter\XINT_cmp_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_pre_b #1#2% +\def\XINT_cmp_c 1#1#2.% {% - \expandafter\XINT_cmp_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_cmp_d #1% }% -% \end{macrocode} -% \lverb|& -% COMPARAISON$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEUR LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000. -% routine appelée via$\ -% \XINT_cmp_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% ATTENTION RENVOIE 1 SI N1 < N2, 0 si N1 = N2, -1 si N1 > N2| -% \begin{macrocode} -\def\XINT_cmp_A #1#2#3\W\X\Y\Z #4#5#6#7% +\def\XINT_cmp_d #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_cmp_az\W - \XINT_cmp_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z + \xint_gob_til_Z #2\XINT_cmp_di \Z + \expandafter\XINT_cmp_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_B #1#2#3#4#5#6#7% +\def\XINT_cmp_e 1#1#2.% {% - \xint_gob_til_W#4\xint_cmp_bz\W - \XINT_cmp_onestep #1#2{#7#6#5#4}{#3}% + 1#2\expandafter!\the\numexpr\XINT_cmp_f #1% }% -\def\XINT_cmp_onestep #1#2#3#4#5#6% +\def\XINT_cmp_f #1#2#3!#4!% {% - \expandafter\XINT_cmp_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% + \xint_gob_til_Z #2\XINT_cmp_fi \Z + \expandafter\XINT_cmp_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_backtoA #1#2#3.#4% +\def\XINT_cmp_g 1#1#2.% {% - \XINT_cmp_A #2{#3#4}% + 1#2\expandafter!\the\numexpr\XINT_cmp_h #1% }% -\def\xint_cmp_bz\W\XINT_cmp_onestep #1\Z { 1}% -\def\xint_cmp_az\W\XINT_cmp_B #1#2#3#4#5#6#7% +\def\XINT_cmp_h #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_cmp_ez\W - \XINT_cmp_Eenter #1{#3}#4#5#6#7% + \xint_gob_til_Z #2\XINT_cmp_hi \Z + \expandafter\XINT_cmp_i\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_Eenter #1\Z { -1}% -\def\xint_cmp_ez\W\XINT_cmp_Eenter #1% +\def\XINT_cmp_i 1#1#2.% {% - \xint_UDzerofork - #1\XINT_cmp_K % il y a une retenue - 0\XINT_cmp_L % pas de retenue - \krof + 1#2\expandafter!\the\numexpr\XINT_cmp_a #1% +}% +\def\XINT_cmp_bi\Z + \expandafter\XINT_cmp_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W +{% + \XINT_cmp_k #1#2!#5!#7!#9!% }% -\def\XINT_cmp_K #1\Z { -1}% -\def\XINT_cmp_L #1{\XINT_OneIfPositive_main #1}% -\def\XINT_OneIfPositive #1% +\def\XINT_cmp_di\Z + \expandafter\XINT_cmp_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W {% - \XINT_OneIfPositive_main #1\W\X\Y\Z% + \XINT_cmp_k #1#2!#5!#7!% }% -\def\XINT_OneIfPositive_main #1#2#3#4% +\def\XINT_cmp_fi\Z + \expandafter\XINT_cmp_g\the\numexpr#1+1#2-#3.#4!#5!#6\W {% - \xint_gob_til_Z #4\xint_OneIfPositive_terminated\Z - \XINT_OneIfPositive_onestep #1#2#3#4% + \XINT_cmp_k #1#2!#5!% }% -\def\xint_OneIfPositive_terminated\Z\XINT_OneIfPositive_onestep\W\X\Y\Z { 0}% -\def\XINT_OneIfPositive_onestep #1#2#3#4% +\def\XINT_cmp_hi\Z + \expandafter\XINT_cmp_i\the\numexpr#1+1#2-#3.#4\W {% - \expandafter\XINT_OneIfPositive_check\the\numexpr #1#2#3#4\relax + \XINT_cmp_k #1#2!% }% -\def\XINT_OneIfPositive_check #1% +%%%%%%%%%%%% +\def\XINT_cmp_k #1#2\W {% - \xint_gob_til_zero #1\xint_OneIfPositive_backtomain 0% - \XINT_OneIfPositive_finish #1% + \xint_UDzerofork + #1{-1\relax \XINT_cmp_greater}% + 0{-1\relax \XINT_cmp_lessorequal}% + \krof }% -\def\XINT_OneIfPositive_finish #1\W\X\Y\Z{ 1}% -\def\xint_OneIfPositive_backtomain 0\XINT_OneIfPositive_finish 0% - {\XINT_OneIfPositive_main }% +\def\XINT_cmp_w #1-1#2{#2#11\Z!\W}% +\def\XINT_cmp_greater #1\Z!\W{ 1}% +\def\XINT_cmp_lessorequal 1#1!% + {\xint_gob_til_Z #1\XINT_cmp_equal\Z + \xint_gob_til_eightzeroes #1\XINT_cmp_continue 00000000% + \XINT_cmp_less }% +\def\XINT_cmp_less #1\W { -1}% +\def\XINT_cmp_continue 00000000\XINT_cmp_less {\XINT_cmp_lessorequal }% +\def\XINT_cmp_equal\Z\xint_gob_til_eightzeroes\Z\XINT_cmp_continue + 00000000\XINT_cmp_less\W { 0}% % \end{macrocode} % \subsection{\csh{xintEq}, \csh{xintGt}, \csh{xintLt}} % \lverb|1.09a.| @@ -18711,101 +18672,159 @@ $1$ or $-1$. }% \def\XINT_xorof_e #1\Z #2{ #2}% % \end{macrocode} -% \subsection{\csh{xintGeq}} +% \subsection{\csh{xintGeq}, \csh{xintiiGeq}} % \lverb|& -% Release 1.09a has \xintnum added into \xintGeq. % PLUS GRAND OU ÉGAL % attention compare les **valeurs absolues**| % \begin{macrocode} -\def\xintGeq {\romannumeral0\xintgeq }% -\def\xintgeq #1% +\def\xintGeq {\romannumeral0\xintgeq }% +\def\xintgeq #1{\expandafter\XINT_geq\romannumeral0\xintnum{#1}\Z }% +\def\xintiiGeq {\romannumeral0\xintiigeq }% +\def\xintiigeq #1{\expandafter\XINT_iigeq\romannumeral-`0#1\Z }% +\def\XINT_iigeq #1#2\Z #3% {% - \expandafter\xint_geq\expandafter {\romannumeral0\xintnum{#1}}% + \expandafter\XINT_geq_fork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xint_geq #1#2% +\let\XINT_geq_pre \xintiigeq % TEMPORAIRE +\let\XINT_Geq \xintGeq % TEMPORAIRE ATTENTION FAIT xintNum +\def\XINT_geq #1#2\Z #3% {% - \expandafter\XINT_geq_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_geq_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\XINT_Geq #1#2{\romannumeral0\XINT_geq_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|& -% PLUS GRAND OU ÉGAL -% ATTENTION, TESTE les VALEURS ABSOLUES| -% \begin{macrocode} -\def\XINT_geq_fork #1#2\Z #3#4\Z +\def\XINT_geq_fork #1#2% {% \xint_UDzerofork - #1\XINT_geq_secondiszero % |#1#2|=0 - #3\XINT_geq_firstiszero % |#1#2|>0 - 0{\xint_UDsignsfork - #1#3\XINT_geq_minusminus - #1-\XINT_geq_minusplus - #3-\XINT_geq_plusminus - --\XINT_geq_plusplus - \krof }% + #1\XINT_geq_firstiszero + #2\XINT_geq_secondiszero + 0{}% \krof - {#2}{#4}#1#3% + \xint_UDsignsfork + #1#2\XINT_geq_minusminus + #1-\XINT_geq_minusplus + #2-\XINT_geq_plusminus + --\XINT_geq_plusplus + \krof #1#2% +}% +\def\XINT_geq_firstiszero #1\krof 0#2#3\Z #4\Z + {\xint_UDzerofork #2{ 1}0{ 0}\krof }% +\def\XINT_geq_secondiszero #1\krof #20#3\Z #4\Z { 1}% +\def\XINT_geq_plusminus #1-{\XINT_geq_plusplus #1{}}% +\def\XINT_geq_minusplus -#1{\XINT_geq_plusplus {}#1}% +\def\XINT_geq_minusminus --{\XINT_geq_plusplus {}{}}% +\def\XINT_geq_plusplus #1#2#3\Z #4\Z {\XINT_geq_pp #1#4\Z #2#3\Z }% +\def\XINT_geq_pp #1\Z +{% + \expandafter\XINT_geq_pp_a + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X +}% +\def\XINT_geq_pp_a #1.#2\X #3\Z +{% + \expandafter\XINT_geq_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_geq_checklengths #1.#2.% +{% + \ifnum #1=#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + \XINT_geq_aa {\XINT_geq_distinctlengths {#1}{#2}} +}% +\def\XINT_geq_distinctlengths #1#2#3\W #4\W +{% + \ifnum #1>#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + { 1}{ 0}% }% -\def\XINT_geq_secondiszero #1#2#3#4{ 1}% -\def\XINT_geq_firstiszero #1#2#3#4{ 0}% -\def\XINT_geq_plusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#3#1}}% -\def\XINT_geq_minusminus #1#2#3#4{\XINT_geq_pre {#2}{#1}}% -\def\XINT_geq_minusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#1}}% -\def\XINT_geq_plusminus #1#2#3#4{\XINT_geq_pre {#2}{#3#1}}% -\def\XINT_geq_pre #1% +%%%%%%%%%%%% +\def\XINT_geq_aa {\expandafter\XINT_geq_w\the\numexpr\XINT_geq_a \xint_c_i }% +%%%%%%%%%%%% +\def\XINT_geq_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter\XINT_geq_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \XINT_geq_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_geq_pre_b #1#2% +\def\XINT_geq_b #1#2#3!#4!% {% - \expandafter\XINT_geq_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z + \xint_gob_til_Z #2\XINT_geq_bi \Z + \expandafter\XINT_geq_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -% \end{macrocode} -% \lverb|& -% PLUS GRAND OU ÉGAL$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000$\ -% routine appelée via$\ -% \romannumeral0\XINT_geq_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% ATTENTION RENVOIE 1 SI N1 < N2 ou N1 = N2 et 0 si N1 > N2| -% \begin{macrocode} -\def\XINT_geq_A #1#2#3\W\X\Y\Z #4#5#6#7% +\def\XINT_geq_c 1#1#2.% {% - \xint_gob_til_W #4\xint_geq_az\W - \XINT_geq_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_geq_d #1% }% -\def\XINT_geq_B #1#2#3#4#5#6#7% +\def\XINT_geq_d #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_geq_bz\W - \XINT_geq_onestep #1#2{#7#6#5#4}{#3}% + \xint_gob_til_Z #2\XINT_geq_di \Z + \expandafter\XINT_geq_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_geq_onestep #1#2#3#4#5#6% +\def\XINT_geq_e 1#1#2.% {% - \expandafter\XINT_geq_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% + 1#2\expandafter!\the\numexpr\XINT_geq_f #1% }% -\def\XINT_geq_backtoA #1#2#3.#4% +\def\XINT_geq_f #1#2#3!#4!% {% - \XINT_geq_A #2{#3#4}% + \xint_gob_til_Z #2\XINT_geq_fi \Z + \expandafter\XINT_geq_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\xint_geq_bz\W\XINT_geq_onestep #1\W\X\Y\Z { 1}% -\def\xint_geq_az\W\XINT_geq_B #1#2#3#4#5#6#7% +\def\XINT_geq_g 1#1#2.% {% - \xint_gob_til_W #4\xint_geq_ez\W - \XINT_geq_Eenter #1% + 1#2\expandafter!\the\numexpr\XINT_geq_h #1% }% -\def\XINT_geq_Eenter #1\W\X\Y\Z { 0}% -\def\xint_geq_ez\W\XINT_geq_Eenter #1% +\def\XINT_geq_h #1#2#3!#4!% {% - \xint_UDzerofork - #1{ 0} % il y a une retenue - 0{ 1} % pas de retenue - \krof + \xint_gob_til_Z #2\XINT_geq_hi \Z + \expandafter\XINT_geq_i\the\numexpr#1+1#4-#3-\xint_c_i.% +}% +\def\XINT_geq_i 1#1#2.% +{% + 1#2\expandafter!\the\numexpr\XINT_geq_a #1% +}% +\def\XINT_geq_bi\Z + \expandafter\XINT_geq_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W +{% + \XINT_geq_k #1#2!#5!#7!#9!% +}% +\def\XINT_geq_di\Z + \expandafter\XINT_geq_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W +{% + \XINT_geq_k #1#2!#5!#7!% }% +\def\XINT_geq_fi\Z + \expandafter\XINT_geq_g\the\numexpr#1+1#2-#3.#4!#5!#6\W +{% + \XINT_geq_k #1#2!#5!% +}% +\def\XINT_geq_hi\Z + \expandafter\XINT_geq_i\the\numexpr#1+1#2-#3.#4\W +{% + \XINT_geq_k #1#2!% +}% +%%%%%%%%%%%% +\def\XINT_geq_k #1#2\W +{% + \xint_UDzerofork + #1{-1\relax { 0}}% + 0{-1\relax { 1}}% + \krof +}% +\def\XINT_geq_w #1-1#2{#2}% % \end{macrocode} % \subsection{\csh{xintiMax}, \csh{xintiiMax}} % \lverb|& @@ -18816,7 +18835,9 @@ $1$ or $-1$. % 1.09a has \xintnum added into \xintiMax. % % 1.1 adds the missing \xintiiMax. Using \xintMax and not \xintiMax in xint is -% deprecated.| +% deprecated. +% +% 1.2 REMOVES \xintMax, \xintMin, \xintMaxof, \xintMinof.| % \begin{macrocode} \def\xintiMax {\romannumeral0\xintimax }% \def\xintimax #1% @@ -18836,7 +18857,6 @@ $1$ or $-1$. {% \expandafter\XINT_max_pre\expandafter {\romannumeral-`0#2}{#1}% }% -\let\xintMax\xintiMax \let\xintmax\xintimax % deprecated, should be only with xintfrac \def\XINT_max_pre #1#2{\XINT_max_fork #1\Z #2\Z {#2}{#1}}% \def\XINT_Max #1#2{\romannumeral0\XINT_max_fork #2\Z #1\Z {#1}{#2}}% % \end{macrocode} @@ -18888,8 +18908,8 @@ $1$ or $-1$. \fi }% % \end{macrocode} -% \subsection{\csh{xintMaxof}} -% \lverb|New with 1.09a.| +% \subsection{\csh{xintiMaxof}} +% \lverb|New with 1.09a. 1.2 has NO MORE \xintMaxof, requires \xintfracname.| % \begin{macrocode} \def\xintiMaxof {\romannumeral0\xintimaxof }% \def\xintimaxof #1{\expandafter\XINT_imaxof_a\romannumeral-`0#1\relax }% @@ -18901,11 +18921,11 @@ $1$ or $-1$. \def\XINT_imaxof_d #1\Z {\expandafter\XINT_imaxof_b\romannumeral0\xintimax {#1}}% \def\XINT_imaxof_e #1\Z #2\Z { #2}% -\let\xintMaxof\xintiMaxof \let\xintmaxof\xintimaxof % \end{macrocode} % \subsection{\csh{xintiMin}, \csh{xintiiMin}} % \lverb|\xintnum added New with 1.09a. I add \xintiiMin in 1.1 and mark as -% deprecated \xintMin, renamed \xintiMin.| +% deprecated \xintMin, renamed \xintiMin. \xintMin NOW REMOVED (1.2, as +% \xintMax, \xintMaxof), only provided by \xintfracnameimp.| % \begin{macrocode} \def\xintiMin {\romannumeral0\xintimin }% \def\xintimin #1% @@ -18925,7 +18945,6 @@ $1$ or $-1$. {% \expandafter\XINT_min_pre\expandafter {\romannumeral-`0#2}{#1}% }% -\let\xintMin\xintiMin \let\xintmin\xintimin % deprecated \def\XINT_min_pre #1#2{\XINT_min_fork #1\Z #2\Z {#2}{#1}}% \def\XINT_Min #1#2{\romannumeral0\XINT_min_fork #2\Z #1\Z {#1}{#2}}% % \end{macrocode} @@ -18990,12 +19009,11 @@ $1$ or $-1$. \def\XINT_iminof_d #1\Z {\expandafter\XINT_iminof_b\romannumeral0\xintimin {#1}}% \def\XINT_iminof_e #1\Z #2\Z { #2}% -\let\xintMinof\xintiMinof \let\xintminof\xintiminof % \end{macrocode} % \subsection{\csh{xintiiSum}} % \lverb|& -% \xintSum {{a}{b}...{z}}$\ -% \xintSumExpr {a}{b}...{z}\relax$\ +% \xintiiSum {{a}{b}...{z}}$\ +% \xintiiSumExpr {a}{b}...{z}\relax$\ % 1.03 (drastically) simplifies and makes the routines more efficient (for big % computations). Also the way \xintSum and \xintSumExpr ...\relax are related. % has been modified. Now \xintSumExpr \z \relax is accepted input when @@ -19003,55 +19021,36 @@ $1$ or $-1$. % was possible). % % 1.09a does NOT add the \xintnum overhead. 1.09h renames \xintiSum to -% \xintiiSum to correctly reflect this.| +% \xintiiSum to correctly reflect this. +% +% The xint 1.0x routine could benefit from the fact that addition and +% subtraction did not check the lengths of the arguments and were able to do +% their job independently of the order (but not at equal speed). Thus it was +% possible to add separately positive and negative summands and do one big +% subtraction at the end, keeping during all that time the intermediate result +% in reverse order suitable for both addition and subtraction. The lazy +% programmer being a bit tired after the 95$% rewrite of xintcore has not +% tried to do the same with the new model. Thus we just do stupidly repeated +% additions. The code is thus much shorter... and in fact I just copied the +% routine for products and changed products to sums.| % \begin{macrocode} \def\xintiiSum {\romannumeral0\xintiisum }% \def\xintiisum #1{\xintiisumexpr #1\relax }% \def\xintiiSumExpr {\romannumeral0\xintiisumexpr }% \def\xintiisumexpr {\expandafter\XINT_sumexpr\romannumeral-`0}% -\let\xintSum\xintiiSum \let\xintsum\xintiisum -\let\xintSumExpr\xintiiSumExpr \let\xintsumexpr\xintiisumexpr -\def\XINT_sumexpr {\XINT_sum_loop {0000}{0000}}% -\def\XINT_sum_loop #1#2#3% -{% - \expandafter\XINT_sum_checksign\romannumeral-`0#3\Z {#1}{#2}% -}% -\def\XINT_sum_checksign #1% -{% - \xint_gob_til_relax #1\XINT_sum_finished\relax - \xint_gob_til_zero #1\XINT_sum_skipzeroinput0% - \xint_UDsignfork - #1\XINT_sum_N - -{\XINT_sum_P #1}% - \krof -}% -\def\XINT_sum_finished #1\Z #2#3% -{% - \XINT_sub_A 1{}#3\W\X\Y\Z #2\W\X\Y\Z -}% -\def\XINT_sum_skipzeroinput #1\krof #2\Z {\XINT_sum_loop }% -\def\XINT_sum_P #1\Z #2% -{% - \expandafter\XINT_sum_loop\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT_sum_N #1\Z #2#3% -{% - \expandafter\XINT_sum_NN\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3\W\X\Y\Z }{#2}% -}% -\def\XINT_sum_NN #1#2{\XINT_sum_loop {#2}{#1}}% +\def\XINT_sumexpr {\XINT_sum_loop_a 0\Z }% +\def\XINT_sum_loop_a #1\Z #2% + {\expandafter\XINT_sum_loop_b \romannumeral-`0#2\Z #1\Z \Z}% +\def\XINT_sum_loop_b #1% + {\xint_gob_til_relax #1\XINT_sum_finished\relax\XINT_sum_loop_c #1}% +\def\XINT_sum_loop_c + {\expandafter\XINT_sum_loop_a\romannumeral0\XINT_add_fork }% +\def\XINT_sum_finished #1\Z #2\Z \Z { #2}% % \end{macrocode} % \subsection{\csh{xintiiPrd}} % \lverb|& -% \xintPrd {{a}...{z}}$\ -% \xintPrdExpr {a}...{z}\relax$\ +% \xintiiPrd {{a}...{z}}$\ +% \xintiiPrdExpr {a}...{z}\relax$\ % Release 1.02 modified the product routine. The earlier version was faster in % situations where each new term is bigger than the product of all previous % terms, a situation which arises in the algorithm for computing powers. The @@ -19077,119 +19076,16 @@ $1$ or $-1$. % \begin{macrocode} \def\xintiiPrd {\romannumeral0\xintiiprd }% \def\xintiiprd #1{\xintiiprdexpr #1\relax }% -\let\xintPrd\xintiiPrd -\let\xintprd\xintiiprd \def\xintiiPrdExpr {\romannumeral0\xintiiprdexpr }% \def\xintiiprdexpr {\expandafter\XINT_prdexpr\romannumeral-`0}% -\let\xintPrdExpr\xintiiPrdExpr -\let\xintprdexpr\xintiiprdexpr \def\XINT_prdexpr {\XINT_prod_loop_a 1\Z }% \def\XINT_prod_loop_a #1\Z #2% - {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}% + {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}% \def\XINT_prod_loop_b #1% {\xint_gob_til_relax #1\XINT_prod_finished\relax\XINT_prod_loop_c #1}% \def\XINT_prod_loop_c {\expandafter\XINT_prod_loop_a\romannumeral0\XINT_mul_fork }% -\def\XINT_prod_finished #1\Z #2\Z \Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintFac}} -% \lverb|& -% Modified with 1.02 and again in 1.03 for greater efficiency. I am -% tempted, -% here and elsewhere, to use \ifcase\XINT_Geq {#1}{1000000000} rather than -% \ifnum\xintLength {#1}>9 but for the time being I leave things as they stand. -% With release 1.05, rather than using \xintLength I opt finally for direct use -% of \numexpr (which will throw a suitable number too big message), and to raise -% the \xintError:$\ FactorialOfTooBigNumber for argument larger than 1000000 -% (rather than 1000000000). With 1.09a, \xintFac uses \xintnum. -% -% 1.09j for no special reason, I lower the maximal number from 999999 to 100000. -% Any how this computation would need more memory than TL2013 standard allows to -% TeX. And I don't even mention time... | -% \begin{macrocode} -\def\xintiFac {\romannumeral0\xintifac }% -\def\xintifac #1% -{% - \expandafter\XINT_fac_fork\expandafter{\the\numexpr #1}% -}% -\let\xintFac\xintiFac \let\xintfac\xintifac -\def\XINT_fac_fork #1% -{% - \ifcase\XINT_cntSgn #1\Z - \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }% - \or - \expandafter\XINT_fac_checklength - \else - \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \fi - {#1}% -}% -\def\XINT_fac_checklength #1% -{% - \ifnum #1>100000 - \xint_afterfi{\expandafter\xintError:FactorialOfTooBigNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \else - \xint_afterfi{\ifnum #1>\xint_c_ixixixix - \expandafter\XINT_fac_big_loop - \else - \expandafter\XINT_fac_loop - \fi }% - \fi - {#1}% -}% -\def\XINT_fac_big_loop #1{\XINT_fac_big_loop_main {10000}{#1}{}}% -\def\XINT_fac_big_loop_main #1#2#3% -{% - \ifnum #1<#2 - \expandafter - \XINT_fac_big_loop_main - \expandafter - {\the\numexpr #1+1\expandafter }% - \else - \expandafter\XINT_fac_big_docomputation - \fi - {#2}{#3{#1}}% -}% -\def\XINT_fac_big_docomputation #1#2% -{% - \expandafter \XINT_fac_bigcompute_loop \expandafter - {\romannumeral0\XINT_fac_loop {9999}}#2\relax -}% -\def\XINT_fac_bigcompute_loop #1#2% -{% - \xint_gob_til_relax #2\XINT_fac_bigcompute_end\relax - \expandafter\XINT_fac_bigcompute_loop\expandafter - {\expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W }% -}% -\def\XINT_fac_bigcompute_end #1#2#3#4#5% -{% - \XINT_fac_bigcompute_end_ #5% -}% -\def\XINT_fac_bigcompute_end_ #1\R #2\Z \W\X\Y\Z #3\W\X\Y\Z { #3}% -\def\XINT_fac_loop #1{\XINT_fac_loop_main 1{1000}{#1}}% -\def\XINT_fac_loop_main #1#2#3% -{% - \ifnum #3>#1 - \else - \expandafter\XINT_fac_loop_exit - \fi - \expandafter\XINT_fac_loop_main\expandafter - {\the\numexpr #1+1\expandafter }\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z }% - {#3}% -}% -\def\XINT_fac_loop_exit #1#2#3#4#5#6#7% -{% - \XINT_fac_loop_exit_ #6% -}% -\def\XINT_fac_loop_exit_ #1#2#3% -{% - \XINT_mul_M -}% +\def\XINT_prod_finished\relax\XINT_prod_loop_c #1\Z #2\Z \Z { #2}% % \end{macrocode} % \lverb|& % & @@ -19735,15 +19631,15 @@ $1$ or $-1$. \def\xintisqrt {\expandafter\XINT_sqrt_post\romannumeral0\xintisquareroot }% \def\xintiisqrt {\expandafter\XINT_sqrt_post\romannumeral0\xintiisquareroot }% \def\xintiisqrtr {\expandafter\XINT_sqrtr_post\romannumeral0\xintiisquareroot }% -\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }% +\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\Z }% % \end{macrocode} % \lverb|N = (#1)^2 - #2 avec #1 le plus petit possible et #2>0 (hence #2<2*#1). % (#1-.5)^2=#1^2-#1+.25=N+#2-#1+.25. Si 0<#2<#1, <= N-0.75<N, donc rounded->#1 % si #2>=#1, (#1-.5)^2>=N+.25>N, donc rounded->#1-1.| % \begin{macrocode} -\def\XINT_sqrtr_post #1#2{\xintiiifLt {#2}{#1}% Lt <-> a<b - { #1}{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }}% -\def\xintisquareroot #1{\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z }% +\def\XINT_sqrtr_post #1#2{\xintiiifLt {#2}{#1}{ #1}{\XINT_dec_pos #1\Z}}% +\def\xintisquareroot #1% + {\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z }% \def\xintiisquareroot #1{\expandafter\XINT_sqrt_checkin\romannumeral-`0#1\Z }% \def\XINT_sqrt_checkin #1% {% @@ -19889,8 +19785,7 @@ $1$ or $-1$. \def\XINT_sqrt_big_g #1#2% {% \expandafter\XINT_sqrt_big_j - \romannumeral0\xintiidivision{#1}% - {\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% + \romannumeral0\xintiidivision{#1}{\romannumeral0\XINT_dbl_pos #2\Z}{#2}% }% \def\XINT_sqrt_big_j #1% {% @@ -19920,6 +19815,19 @@ $1$ or $-1$. {\expandafter\XINT_iie\the\numexpr #2\expandafter.\expandafter{\romannumeral-`0#1}}% \def\XINT_iie #1.#2{\ifnum#1>\xint_c_ \xint_dothis{\xint_dsh {#2}{-#1}}\fi \xint_orthat{ #2}}% +% \end{macrocode} +% \subsection{``Load \xintfracnameimp'' macros} +% \lverb|Originally was used in \xintiiexpr. Transferred from xintfrac for 1.1.| +% \begin{macrocode} +\catcode`! 11 +\def\xintMax {\Did_you_mean_iiMax?or_load_xintfrac!}% +\def\xintMin {\Did_you_mean_iiMin?or_load_xintfrac!}% +\def\xintMaxof {\Did_you_mean_iMaxof?or_load_xintfrac!}% +\def\xintMinof {\Did_you_mean_iMinof?or_load_xintfrac!}% +\def\xintSum {\Did_you_mean_iiSum?or_load_xintfrac!}% +\def\xintPrd {\Did_you_mean_iiPrd?or_load_xintfrac!}% +\def\xintPrdExpr {\Did_you_mean_iiPrdExpr?or_load_xintfrac!}% +\def\xintSumExpr {\Did_you_mean_iiSumExpr?or_load_xintfrac!}% \XINT_restorecatcodes_endinput% % \end{macrocode} %\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 @@ -19994,21 +19902,14 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintbinhex}% - [2015/09/12 v1.1c Expandable binary and hexadecimal conversions (jfB)]% + [2015/10/10 v1.2 Expandable binary and hexadecimal conversions (jfB)]% % \end{macrocode} % \subsection{Constants, etc...} % \lverb!v1.08! % \begin{macrocode} -\chardef\xint_c_xvi 16 -% \chardef\xint_c_ii^v 32 % already in xint.sty -% \chardef\xint_c_ii^vi 64 % already in xint.sty -\chardef\xint_c_ii^vii 128 -\mathchardef\xint_c_ii^viii 256 -\mathchardef\xint_c_ii^xii 4096 -\newcount\xint_c_ii^xv \xint_c_ii^xv 32768 -\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536 +\newcount\xint_c_ii^xv \xint_c_ii^xv 32768 +\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536 \newcount\xint_c_x^v \xint_c_x^v 100000 -\newcount\xint_c_x^ix \xint_c_x^ix 1000000000 \def\XINT_tmpa #1{\ifx\relax#1\else \expandafter\edef\csname XINT_sdth_#1\endcsname {\ifcase #1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or @@ -20103,6 +20004,40 @@ $1$ or $-1$. \csname XINT_sdtb_\the\numexpr #2-\xint_c_xvi*#1\endcsname }% % \end{macrocode} +% \subsection{\csh{XINT_OQ}} +% \lverb|Moved with release 1.2 from xintcore 1.1 as it is used only here. +% Will be probably suppressed once I review the code of xintbinhex.| +% \begin{macrocode} +\def\XINT_OQ #1#2#3#4#5#6#7#8#9% +{% + \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}% +}% +\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z +{% + \XINT_OQ_end_b #1\Z +}% +\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8% +{% + \xint_gob_til_R + #8\XINT_OQ_end_viii + #7\XINT_OQ_end_vii + #6\XINT_OQ_end_vi + #5\XINT_OQ_end_v + #4\XINT_OQ_end_iv + #3\XINT_OQ_end_iii + #2\XINT_OQ_end_ii + \R\XINT_OQ_end_i + \Z #2#3#4#5#6#7#8% +}% +\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% +\def\XINT_OQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}% +\def\XINT_OQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}% +\def\XINT_OQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}% +\def\XINT_OQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}% +\def\XINT_OQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% +\def\XINT_OQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% +\def\XINT_OQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% +% \end{macrocode} % \subsection{\csh{xintDecToHex}, \csh{xintDecToBin}} % \lverb!v1.08! % \begin{macrocode} @@ -20668,7 +20603,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintgcd}% - [2015/09/12 v1.1c Euclide algorithm with xint package (jfB)]% + [2015/10/10 v1.2 Euclide algorithm with xint package (jfB)]% % \end{macrocode} % \subsection{\csh{xintGCD}, \csh{xintiiGCD}} % \lverb|The macros of 1.09a benefits from the \xintnum which has been inserted @@ -20918,8 +20853,8 @@ $1$ or $-1$. \def\XINT_bezout_loop_b #1#2#3#4#5#6#7#8% {% \expandafter \XINT_bezout_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#2}}{#7}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#6}{#2}}{#8}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #5\Z #2\Z}{#7}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #6\Z #2\Z}{#8}}% {#1}{#3}{#4}{#5}{#6}% }% % \end{macrocode} @@ -21370,8 +21305,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintfrac}% - [2015/09/12 v1.1c Expandable operations on fractions (jfB)]% -\chardef\xint_c_xviii 18 + [2015/10/10 v1.2 Expandable operations on fractions (jfB)]% % \end{macrocode} % \subsection{\csh{XINT_cntSgnFork}} % \lverb|1.09i. Used internally, #1 must expand to \m@ne, \z@, or \@ne or @@ -21481,137 +21415,168 @@ $1$ or $-1$. % e only. The \xintexpr parser does accept uppercase E also. Ah, by the way, % perhaps I should at least say what this macro does? (belated addition % 2014/10/22...), before I forget! It prepares the fraction in the internal -% format {exponent}{Numerator}{Denominator} where Denominator is at least 1.| +% format {exponent}{Numerator}{Denominator} where Denominator is at least 1. +% +% 2015/10/09: this venerable macro from the early days (1.03, 2013/04/14) has +% gotten a lifting for release 1.2. There were two kinds of issues:$newline +% +% 1) use of \W, \Z, \T delimiters was very poor choice as this could clash with +% user input, +% +% 2) the new \XINT_frac_gen handles macros (possibly empty) in the input as +% general as \A.\Be\C/\D.\Ee\F. The earlier version would not have expanded +% the \B for example (only \A, \D, \C, \F). +% +% I wanted to make stricter the restricted A/B[N] case, doing no expansion of +% B, but this clashed with some established uses in the documentation like +% 1/\xintiiSqr{...}[0] for example. Thus I maintained it despite overhead of +% having to go over A one more time. Careful also here about potential brace +% removals if one does stuff like #1/#2#3[#4] regarding the #3. And while I +% was at it I added \numexpr parsing of the N, which earlier was restricted to +% be only explicit digits, and I even allowed [] with an empty N. +% +% This little event makes me think I should read again other remaining +% portions my early code, as I was still learning TeX coding at that time.| % \begin{macrocode} \def\XINT_inFrac {\romannumeral0\XINT_infrac }% \def\XINT_infrac #1% {% - \expandafter\XINT_infrac_ \romannumeral-`0#1[\W]\Z\T + \expandafter\XINT_infrac_fork\romannumeral-`0#1/\XINT_W[\XINT_W\XINT_T }% -\def\XINT_infrac_ #1[#2#3]#4\Z +\def\XINT_infrac_fork #1[#2% {% - \xint_UDwfork - #2\XINT_infrac_A - \W\XINT_infrac_B + \xint_UDXINTWfork + #2\XINT_frac_gen + \XINT_W\XINT_infrac_res_a % strict A[N] or A/B[N] input \krof - #1[#2#3]#4% + #1[#2% }% -\def\XINT_infrac_A #1[\W]\T +\def\XINT_infrac_res_a #1% {% - \XINT_frac #1/\W\Z + \xint_gob_til_zero #1\XINT_infrac_res_zero 0\XINT_infrac_res_b #1% }% -\def\XINT_infrac_B #1% +\def\XINT_infrac_res_zero 0\XINT_infrac_res_b #1\XINT_T {{0}{0}{1}}% +\def\XINT_infrac_res_b #1/#2% {% - \xint_gob_til_zero #1\XINT_infrac_Zero0\XINT_infrac_BB #1% -}% -\def\XINT_infrac_BB #1[\W]\T {\XINT_infrac_BC #1/\W\Z }% -\def\XINT_infrac_BC #1/#2#3\Z -{% - \xint_UDwfork - #2\XINT_infrac_BCa - \W{\expandafter\XINT_infrac_BCb \romannumeral-`0#2}% + \xint_UDXINTWfork + #2\XINT_infrac_res_ca + \XINT_W\XINT_infrac_res_cb \krof - #3\Z #1\Z + #1/#2% }% -\def\XINT_infrac_BCa \Z #1[#2]#3\Z { {#2}{#1}{1}}% -\def\XINT_infrac_BCb #1[#2]/\W\Z #3\Z { {#2}{#3}{#1}}% -\def\XINT_infrac_Zero #1\T { {0}{0}{1}}% +\def\XINT_infrac_res_ca #1[#2]/\XINT_W[\XINT_W\XINT_T + {\expandafter{\the\numexpr 0#2}{#1}{1}}% +\def\XINT_infrac_res_cb #1/#2[% + {\expandafter\XINT_infrac_res_cc\romannumeral-`0#2~#1[}% +\def\XINT_infrac_res_cc #1~#2[#3]/\XINT_W[\XINT_W\XINT_T + {\expandafter{\the\numexpr 0#3}{#2}{#1}}% % \end{macrocode} -% \subsection{\csh{XINT_frac}} +% \subsection{\csh{XINT_frac_gen}} % \lverb|Extended in 1.07 to recognize and accept scientific notation both at % the numerator and (possible) denominator. Only a lowercase e will do here, but -% uppercase E is possible within an \xintexpr..\relax | +% uppercase E is possible within an \xintexpr..\relax +% +% Completely rewritten for 1.2 2015/10/10. It now is able to handles inputs +% such as \A.\Be\C/\D.\Ee\F where each of \A, \B, \D, and \E may need +% \fexpan sion and \C and \F will end up in \numexpr.| % \begin{macrocode} -\def\XINT_frac #1/#2#3\Z +\def\XINT_frac_gen #1/#2% {% - \xint_UDwfork - #2\XINT_frac_A - \W{\expandafter\XINT_frac_U \romannumeral-`0#2}% + \xint_UDXINTWfork + #2\XINT_frac_gen_A + \XINT_W\XINT_frac_gen_B \krof - #3e\W\Z #1e\W\Z + #1/#2% }% -\def\XINT_frac_U #1e#2#3\Z +\def\XINT_frac_gen_A #1/\XINT_W [\XINT_W {\XINT_frac_gen_C 0~1!#1ee.\XINT_W }% +\def\XINT_frac_gen_B #1/#2/\XINT_W[%\XINT_W {% - \xint_UDwfork - #2\XINT_frac_Ua - \W{\XINT_frac_Ub #2}% - \krof - #3\Z #1\Z + \expandafter\XINT_frac_gen_Ba + \romannumeral-`0#2ee.\XINT_W\XINT_Z #1ee.%\XINT_W }% -\def\XINT_frac_Ua \Z #1/\W\Z {\XINT_frac_B #1.\W\Z {0}}% -\def\XINT_frac_Ub #1/\W e\W\Z #2\Z {\XINT_frac_B #2.\W\Z {#1}}% -\def\XINT_frac_B #1.#2#3\Z +\def\XINT_frac_gen_Ba #1.#2% {% - \xint_UDwfork - #2\XINT_frac_Ba - \W{\XINT_frac_Bb #2}% + \xint_UDXINTWfork + #2\XINT_frac_gen_Bb + \XINT_W\XINT_frac_gen_Bc \krof - #3\Z #1\Z + #1.#2% }% -\def\XINT_frac_Ba \Z #1\Z {\XINT_frac_T {0}{#1}}% -\def\XINT_frac_Bb #1.\W\Z #2\Z +\def\XINT_frac_gen_Bb #1e#2e#3\XINT_Z + {\expandafter\XINT_frac_gen_C\the\numexpr 0#2~#1!}% +\def\XINT_frac_gen_Bc #1.#2e% {% - \expandafter \XINT_frac_T \expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% + \expandafter\XINT_frac_gen_Bd\romannumeral-`0#2.#1e% }% -\def\XINT_frac_A e\W\Z {\XINT_frac_T {0}{1}{0}}% -\def\XINT_frac_T #1#2#3#4e#5#6\Z +\def\XINT_frac_gen_Bd #1.#2e#3e#4\XINT_Z {% - \xint_UDwfork - #5\XINT_frac_Ta - \W{\XINT_frac_Tb #5}% - \krof - #6\Z #4\Z {#1}{#2}{#3}% + \expandafter\XINT_frac_gen_C\the\numexpr 0#3-\romannumeral0\expandafter + \XINT_length_loop + 0.#1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye~#2#1!% }% -\def\XINT_frac_Ta \Z #1\Z {\XINT_frac_C #1.\W\Z {0}}% -\def\XINT_frac_Tb #1e\W\Z #2\Z {\XINT_frac_C #2.\W\Z {#1}}% -\def\XINT_frac_C #1.#2#3\Z +\def\XINT_frac_gen_C #1!#2.#3% {% - \xint_UDwfork - #2\XINT_frac_Ca - \W{\XINT_frac_Cb #2}% + \xint_UDXINTWfork + #3\XINT_frac_gen_Ca + \XINT_W\XINT_frac_gen_Cb \krof - #3\Z #1\Z + #1!#2.#3% }% -\def\XINT_frac_Ca \Z #1\Z {\XINT_frac_D {0}{#1}}% -\def\XINT_frac_Cb #1.\W\Z #2\Z +\def\XINT_frac_gen_Ca #1~#2!#3e#4e#5\XINT_T {% - \expandafter\XINT_frac_D\expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% + \expandafter\XINT_frac_gen_F\the\numexpr #4-#1\expandafter + ~\romannumeral0\XINT_num_loop + #2\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#3~% }% -\def\XINT_frac_D #1#2#3#4#5#6% +\def\XINT_frac_gen_Cb #1.#2e% {% - \expandafter \XINT_frac_E \expandafter - {\the\numexpr -#1+#3+#4-#6\expandafter}\expandafter - {\romannumeral0\XINT_num_loop #2% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% - {\romannumeral0\XINT_num_loop #5% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% + \expandafter\XINT_frac_gen_Cc\romannumeral-`0#2.#1e% }% -\def\XINT_frac_E #1#2#3% +\def\XINT_frac_gen_Cc #1.#2~#3!#4e#5e#6\XINT_T {% - \expandafter \XINT_frac_F #3\Z {#2}{#1}% + \expandafter\XINT_frac_gen_F\the\numexpr #5-#2-% + \romannumeral0\XINT_length_loop + 0.#1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye\expandafter + ~\romannumeral0\XINT_num_loop + #3\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z + ~#4#1~% }% -\def\XINT_frac_F #1% +\def\XINT_frac_gen_F #1~#2% {% \xint_UDzerominusfork - #1-\XINT_frac_Gdivisionbyzero - 0#1\XINT_frac_Gneg - 0-{\XINT_frac_Gpos #1}% - \krof + #2-\XINT_frac_gen_Gdivbyzero + 0#2{\XINT_frac_gen_G -{}}% + 0-{\XINT_frac_gen_G {}#2}% + \krof #1~% +}% +\def\XINT_frac_gen_Gdivbyzero #1~~#2~% +{% + \expandafter\XINT_frac_gen_Gdivbyzero_a + \romannumeral0\XINT_num_loop + #2\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#1~% }% -\edef\XINT_frac_Gdivisionbyzero #1\Z #2#3% +\def\XINT_frac_gen_Gdivbyzero_a #1~#2~% {% - \noexpand\xintError:DivisionByZero\space {0}{#2}{0}% + \xintError:DivisionByZero {#2}{#1}{0}% }% -\def\XINT_frac_Gneg #1\Z #2#3% +\def\XINT_frac_gen_G #1#2#3~#4~#5~% {% - \expandafter\XINT_frac_H \expandafter{\romannumeral0\XINT_opp #2}{#3}{#1}% + \expandafter\XINT_frac_gen_Ga + \romannumeral0\XINT_num_loop + #1#5\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#3~{#2#4}% }% -\def\XINT_frac_H #1#2{ {#2}{#1}}% -\def\XINT_frac_Gpos #1\Z #2#3{ {#3}{#2}{#1}}% +\def\XINT_frac_gen_Ga #1#2~#3~% +{% + \xint_gob_til_zero #1\XINT_frac_gen_zero 0% + {#3}{#1#2}% +}% +\def\XINT_frac_gen_zero 0#1#2#3{{0}{0}{1}}% % \end{macrocode} % \subsection{\csh{XINT_factortens}, \csh{XINT_cuz_cnt}} % \begin{macrocode} @@ -21709,6 +21674,102 @@ $1$ or $-1$. \xint_relax }{#1}% }% % \end{macrocode} +% \subsection{\csh{XINT_addm_A}} +% \lverb|This is a routine from xintcore 1.0x, which is needed by \xintFloat, +% \XINTinFloat and \xintRound, for the time being. I should moved it here, now +% that xintcore has been entirely rewritten with release 1.2.| +% \begin{macrocode} +\def\XINT_addm_A #1#2#3#4#5#6% +{% + \xint_gob_til_W #3\xint_addm_az\W + \XINT_addm_AB #1{#3#4#5#6}{#2}% +}% +\def\xint_addm_az\W\XINT_addm_AB #1#2% +{% + \XINT_addm_AC_checkcarry #1% +}% +\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT_addm_ABE #1#2#3#4#5#6% +{% + \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.% +}% +\def\XINT_addm_ABEA #1#2#3.#4% +{% + \XINT_addm_A #2{#3#4}% +}% +\def\XINT_addm_AC_checkcarry #1% +{% + \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C +}% +\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z +{% + \expandafter + \xint_cleanupzeros_andstop + \romannumeral0% + \XINT_rord_main {}#2% + \xint_relax + \xint_bye\xint_bye\xint_bye\xint_bye + \xint_bye\xint_bye\xint_bye\xint_bye + \xint_relax + #1% +}% +\def\XINT_addm_C #1#2#3#4#5% +{% + \xint_gob_til_W + #5\xint_addm_cw + #4\xint_addm_cx + #3\xint_addm_cy + #2\xint_addm_cz + \W\XINT_addm_CD {#5#4#3#2}{#1}% +}% +\def\XINT_addm_CD #1% +{% + \expandafter\XINT_addm_CC\the\numexpr 1+10#1.% +}% +\def\XINT_addm_CC #1#2#3.#4% +{% + \XINT_addm_AC_checkcarry #2{#3#4}% +}% +\def\xint_addm_cw + #1\xint_addm_cx + #2\xint_addm_cy + #3\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.% +}% +\def\XINT_addm_CDw #1.#2#3\X\Y\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cx + #1\xint_addm_cy + #2\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.% +}% +\def\XINT_addm_CDx #1.#2#3\Y\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cy + #1\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDy\the\numexpr 1+#1.% +}% +\def\XINT_addm_CDy #1.#2#3\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}% +\edef\XINT_addm_end #1#2#3#4#5% + {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}% +% \end{macrocode} % \subsection{\csh{xintRaw}} % \lverb|& % 1.07: this macro simply prints in a user readable form the fraction after its @@ -22175,8 +22236,8 @@ $1$ or $-1$. \def\XINT_jrr_loop_b #1#2#3#4#5#6#7% {% \expandafter \XINT_jrr_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#4}{#1}}{#6}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#1}}{#7}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #4\Z #1\Z}{#6}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #5\Z #1\Z}{#7}}% {#2}{#3}{#4}{#5}% }% \def\XINT_jrr_loop_c #1#2% @@ -22489,7 +22550,15 @@ $1$ or $-1$. % D-|N| last digits from Q, etc.. we compare D-|N| with the length M of Q etc... % (well in this last, very uncommon, branch, I stopped trying to optimize things % and I even do an \xintnum to ensure a 0 if something comes out empty from -% \xintDecSplit).@ +% \xintDecSplit). +% +% [2015/10/04] Although the explanations above are extremely clear, there are +% just too complicated for me to be now able to understand them fully. I +% miraculously managed to do the minimal changes (all happens between +% \XINT_xtrunc_Q and \XINT_xtrunc_Pa) in order for \xintXTrunc to use the 1.2 +% division routine. Seems to work. But some thought should be given to how to +% adapt \xintXTrunc for it to better use the abilities and characteristics of +% the new division routines in xincore.@ % \begin{macrocode} \def\xintXTrunc #1#2% {% @@ -22615,33 +22684,7 @@ $1$ or $-1$. \expandafter\XINT_xtrunc_negNC\expandafter {\the\numexpr\xintLength {#1}-#2}{#1}% }% -\def\XINT_xtrunc_Q #1% -{% - \expandafter\XINT_xtrunc_prepare_I - \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z -}% -\def\XINT_xtrunc_prepare_I #1.#2#3% -{% - \expandafter\XINT_xtrunc_prepareB_aa\expandafter - {\romannumeral0\xintlength {#2}}{#2}{#1}% -}% -\def\XINT_xtrunc_prepareB_aa #1% -{% - \ifnum #1=\xint_c_i - \expandafter\XINT_xtrunc_prepareB_onedigit - \else - \expandafter\XINT_xtrunc_prepareB_PaBa - \fi - {#1}% -}% -\def\XINT_xtrunc_prepareB_onedigit #1#2% -{% - \ifcase#2 - \or\expandafter\XINT_xtrunc_BisOne - \or\expandafter\XINT_xtrunc_BisTwo - \else\expandafter\XINT_xtrunc_prepareB_PaBe - \fi {000}{0}{4}{#2}% -}% +%%%%%%%%%%%% \def\XINT_xtrunc_BisOne #1#2#3#4#5#6#7% {% #5.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter @@ -22661,48 +22704,76 @@ $1$ or $-1$. 0000000000000000000000000000000000000000000000000000000000000000% \repeat }% -\def\XINT_xtrunc_prepareB_PaBa #1#2% +%%%%%%%%%%%% +\def\XINT_xtrunc_Q #1% +{% + \expandafter\XINT_xtrunc_prepare + \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z +}% +\def\XINT_xtrunc_prepare #1.#2#3% {% \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_a {#1}{#2}}% + {\romannumeral0% + \XINT_xtrunc_prepare_a #2\R\R\R\R\R\R\R\R {10}0000001\W !{#2}}{#1}% }% -\def\XINT_xtrunc_prepareB_a #1% +%%%%%%%%%%%% +\def\XINT_xtrunc_prepare_a #1#2#3#4#5#6#7#8#9% {% - \expandafter\XINT_xtrunc_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \xint_gob_til_R #9\XINT_xtrunc_prepare_small\R + \XINT_xtrunc_prepare_b #9% }% -\def\XINT_xtrunc_prepareB_c #1#2% +\def\XINT_xtrunc_prepare_small\R #1!#2% {% - \csname XINT_xtrunc_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% + \ifcase #2 + \or\xint_afterfi{ \XINT_div_BisOne}% + \or\xint_afterfi{ \XINT_div_BisTwo}% + \else\expandafter\XINT_xtrunc_small_aa + \fi {#2}% }% -\def\XINT_xtrunc_prepareB_d {\XINT_xtrunc_prepareB_e {}{0000}}% -\def\XINT_xtrunc_prepareB_di {\XINT_xtrunc_prepareB_e {0}{000}}% -\def\XINT_xtrunc_prepareB_dii {\XINT_xtrunc_prepareB_e {00}{00}}% -\def\XINT_xtrunc_prepareB_diii {\XINT_xtrunc_prepareB_e {000}{0}}% -\def\XINT_xtrunc_prepareB_PaBe #1#2#3#4% +\def\XINT_xtrunc_small_aa #1% {% - \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_e {#1}{#2}{#3}{#4}}% + \expandafter\space\expandafter\XINT_xtrunc_small_a + \the\numexpr #1/\xint_c_ii\expandafter + .\the\numexpr \xint_c_x^viii+#1!% }% -\def\XINT_xtrunc_prepareB_e #1#2#3#4% +%%%%%%%%%%%% +\def\XINT_xtrunc_small_a #1.#2!#3% {% - \ifnum#3=\xint_c_iv\expandafter\XINT_xtrunc_prepareLittleB_f - \else\expandafter\XINT_xtrunc_prepareB_f - \fi - #4#1{#3}{#2}{#1}% + \expandafter\XINT_div_small_b\the\numexpr #1\expandafter + .\the\numexpr #2\expandafter!% + \romannumeral0\XINT_div_small_ba #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_sepbyviii_Z_end 2345678\relax }% -\def\XINT_xtrunc_prepareB_f #1#2#3#4#5#{% - \expandafter\space - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% +%%%%%%%%%%%% +\def\XINT_xtrunc_prepare_b + {\expandafter\XINT_xtrunc_prepare_c\romannumeral0\XINT_zeroes_forviii }% +\def\XINT_xtrunc_prepare_c #1!% +{% + \XINT_xtrunc_prepare_d #1.00000000!{#1}% }% -\def\XINT_xtrunc_prepareLittleB_f #1#{% - \expandafter\space\expandafter - \XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% +\def\XINT_xtrunc_prepare_d #1#2#3#4#5#6#7#8#9% +{% + \expandafter\XINT_xtrunc_prepare_e\xint_gob_til_dot #1#2#3#4#5#6#7#8#9!% +}% +\def\XINT_xtrunc_prepare_e #1!#2!#3#4% +{% + \XINT_xtrunc_prepare_f #4#3\X {#1}{#3}% }% +\def\XINT_xtrunc_prepare_f #1#2#3#4#5#6#7#8#9\X +{% + \expandafter\space\expandafter\XINT_div_prepare_g + \the\numexpr #1#2#3#4#5#6#7#8+\xint_c_i\expandafter + .\the\numexpr (#1#2#3#4#5#6#7#8+\xint_c_i)/\xint_c_ii\expandafter + .\the\numexpr #1#2#3#4#5#6#7#8\expandafter + .\romannumeral0\XINT_sepandrev_andcount + #1#2#3#4#5#6#7#8#9\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678% + \relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X +}% +%%%%%%%%%%%% \def\XINT_xtrunc_Pa #1#2% {% \expandafter\XINT_xtrunc_Pb\romannumeral0#1{#2}{#1}% @@ -22758,10 +22829,7 @@ $1$ or $-1$. % gains. The earlier version was seriously silly when dealing with % inputs having a big power of ten. Again some modifications in 1.08b % for a better treatment of cases with long explicit numerators or -% denominators. -% -% Here again some inner macros used the \xintiquo with extra \xintnum overhead -% in 1.09a, 1.09f reinstalled use of \xintiiquo without this overhead.| +% denominators.| % \begin{macrocode} \def\xintFloat {\romannumeral0\xintfloat }% \def\xintfloat #1{\XINT_float_chkopt #1\xint_relax }% @@ -23147,7 +23215,7 @@ $1$ or $-1$. }% \def\XINT_fadd_C #1#2#3% {% - \ifcase\romannumeral0\XINT_cmp_pre {#2}{#3} %<- intentional space here. + \ifcase\romannumeral0\xintiicmp {#2}{#3} %<- intentional space here. \expandafter\XINT_fadd_eq \or\expandafter\XINT_fadd_D \else\expandafter\XINT_fadd_Da @@ -23636,8 +23704,8 @@ $1$ or $-1$. \def\XINT_minof_e #1\Z #2\Z { #2}% % \end{macrocode} % \subsection{\csh{xintCmp}} -% \lverb|Rewritten completely in 1.08a to be less dumb when comparing fractions having -% big powers of tens.| +% \lverb|Rewritten completely in 1.08a to be less dumb when comparing +% fractions having big powers of tens.| % \begin{macrocode} %\def\xintCmp {\romannumeral0\xintcmp }% \def\xintcmp #1% @@ -23708,32 +23776,42 @@ $1$ or $-1$. \expandafter\XINT_fcmp_Fe\expandafter {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}% }% -\def\XINT_fcmp_Fe #1#2{\XINT_cmp_pre {#2}{#1}}% +\def\XINT_fcmp_Fe #1#2{\xintiicmp {#2}{#1}}% \def\XINT_fcmp_Fn #1\Z #2#3% {% - \expandafter\XINT_cmp_pre\expandafter + \expandafter\xintiicmp\expandafter {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}% }% % \end{macrocode} % \subsection{\csh{xintAbs}} -% \lverb|Simplified in 1.09i. (original macro had been written before \xintRaw)| % \begin{macrocode} \def\xintAbs {\romannumeral0\xintabs }% \def\xintabs #1{\expandafter\XINT_abs\romannumeral0\xintraw {#1}}% % \end{macrocode} % \subsection{\csh{xintOpp}} -% \lverb|caution that -#1 would not be ok if #1 has [n] -% stuff. Simplified in 1.09i. (original macro had been written before \xintRaw)| % \begin{macrocode} \def\xintOpp {\romannumeral0\xintopp }% \def\xintopp #1{\expandafter\XINT_opp\romannumeral0\xintraw {#1}}% % \end{macrocode} % \subsection{\csh{xintSgn}} -% \lverb|Simplified in 1.09i. (original macro had been written before \xintRaw)| % \begin{macrocode} \def\xintSgn {\romannumeral0\xintsgn }% \def\xintsgn #1{\expandafter\XINT_sgn\romannumeral0\xintraw {#1}\Z }% % \end{macrocode} +% \subsection{Floating point macros} +% \begin{framed} +% 1.2 release has not touched the floating point routines apart from adding +% the new \csh{xintFloatFac}. The others should be revised for some +% optimizations related to the underlying model of the new core routines. +% This is particularly the case for \csh{xintFloatPow} and +% \csh{xintFloatPower} which should keep intermediate results in a suitable +% format, like \csh{xintiiPow} does. +% +% The switch to 1.2 was smooth (apart from the writing up of the new +% \csh{xintFloatFac}), as I didn't have to change a single line of code +% anywhere here ! +% \end{framed} +% % \subsection{\csh{xintFloatAdd}, \csh{XINTinFloatAdd}} % \lverb|1.07; 1.09ka improves a bit the efficieny of the coding of % \XINT_FL_Add_d.| @@ -23810,6 +23888,10 @@ $1$ or $-1$. }% % \end{macrocode} % \subsection{\csh{xintFloatMul}, \csh{XINTinFloatMul}} +% \begin{framed} +% It is a long-standing issue here that I must at some point revise the code +% and avoid compute with 2P digits the exact intermediate result. +% \end{framed} % \lverb|1.07| % \begin{macrocode} \def\xintFloatMul {\romannumeral0\xintfloatmul}% @@ -23874,6 +23956,11 @@ $1$ or $-1$. \def\XINT_FL_Div_b #1[#2]#3[#4]{\xintE{#3/#1}{#4-#2}}% % \end{macrocode} % \subsection{\csh{xintFloatPow}, \csh{XINTinFloatPow}} +% \begin{framed} +% This definitely should be revised to better take into account the new +% multiplication to maintain through intermediate states a suitable internal +% format, optimized for calls to \csh{XINT_mul_loop}. +% \end{framed} % \lverb|1.07. Release 1.09j has re-organized the core loop, and % \XINT_flpow_prd sub-routine has been removed.| % \begin{macrocode} @@ -24144,6 +24231,215 @@ $1$ or $-1$. #4{#3}{#5}% }% % \end{macrocode} +% \subsection{\csh{xintFloatFac}, \csh{XINTFloatFac}} +% \lverb|1.2. Je dois documenter le raisonnement sur la précision à imposer +% pour les calculs par blocs de huit faits en sous-main. Par ailleurs j'ai été +% amené à une routine smallmul spéciale.| +% \begin{macrocode} +\def\xintFloatFac {\romannumeral0\xintfloatfac}% +\def\xintfloatfac #1{\XINT_flfac_chkopt \xintfloat #1\xint_relax }% +\def\XINTinFloatFac {\romannumeral0\XINTinfloatfac }% +\def\XINTinfloatfac #1{\XINT_flfac_chkopt \XINTinfloat #1\xint_relax }% +\def\XINT_flfac_chkopt #1#2% +{% + \ifx [#2\expandafter\XINT_flfac_opt + \else\expandafter\XINT_flfac_noopt + \fi + #1#2% +}% +\def\XINT_flfac_noopt #1#2\xint_relax +{% + \expandafter\XINT_FL_fac_start\expandafter + {\the\numexpr #2}{\XINTdigits}{#1[\XINTdigits]}% +}% +\def\XINT_flfac_opt #1[\xint_relax #2]#3% +{% + \expandafter\XINT_FL_fac_start\expandafter + {\the\numexpr #3\expandafter}\expandafter{\the\numexpr#2}{#1[#2]}% +}% +\def\XINT_FL_fac_start #1% +{% + \ifcase\XINT_cntSgn #1\Z + \expandafter\XINT_FL_fac_iszero + \or + \expandafter\XINT_FL_fac_increaseP + \else + \expandafter\XINT_FL_fac_isneg + \fi {#1}% +}% +\def\XINT_FL_fac_iszero #1#2#3{#3{1/1[0]}}% +\def\XINT_FL_fac_isneg #1#2#3% + {\expandafter\xintError:FactorialOfNegativeNumber #3{1/1[0]}}% +\def\XINT_FL_fac_increaseP #1#2% +{% + \expandafter\XINT_FL_fac_fork + \the\numexpr \xint_c_viii*% + ((\xint_c_v+#2+\XINT_FL_fac_extradigits #187654321\Z)/\xint_c_viii).% + #1.% +}% +\def\XINT_FL_fac_extradigits #1#2#3#4#5#6#7#8{\XINT_FL_fac_extra_a }% +\def\XINT_FL_fac_extra_a #1#2\Z {#1}% +\def\XINT_FL_fac_fork #1.#2.#3% +{% + \ifnum #2>99999999 \xint_dothis{\XINT_FL_fac_toobig }\fi + \ifnum #2>9999 \xint_dothis{\XINT_FL_fac_vbigloop_a }\fi + \ifnum #2>465 \xint_dothis{\XINT_FL_fac_bigloop_a }\fi + \ifnum #2>101 \xint_dothis{\XINT_FL_fac_medloop_a }\fi + \xint_orthat{\XINT_FL_fac_smallloop_a }% + #2.#1.{\XINT_FL_fac_out}{#3}% +}% +\def\XINT_FL_fac_toobig #1.#2.#3#4% + {\expandafter\xintError:FactorialOfTooBigNumber #4{1/1[0]}}% +\def\XINT_FL_fac_out #1\Z![#2]#3{#3{\romannumeral0\XINT_mul_out + #1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W [#2]}}% +\def\XINT_FL_fac_vbigloop_a #1.#2.% +{% + \XINT_FL_fac_bigloop_a 9999.#2.% + {\expandafter\XINT_FL_fac_vbigloop_loop\the\numexpr 100010000\expandafter.% + \the\numexpr \xint_c_x^viii+#1.}% +}% +\def\XINT_FL_fac_vbigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_vbigloop_loop + \the\numexpr #1+\xint_c_i\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_mul #1!% +}% +\def\XINT_FL_fac_bigloop_a #1.% +{% + \expandafter\XINT_FL_fac_bigloop_b \the\numexpr + #1+\xint_c_i-\xint_c_ii*((#1-464)/\xint_c_ii).#1.% +}% +\def\XINT_FL_fac_bigloop_b #1.#2.#3.% +{% + \expandafter\XINT_FL_fac_medloop_a + \the\numexpr #1-\xint_c_i.#3.{\XINT_FL_fac_bigloop_loop #1.#2.}% +}% +\def\XINT_FL_fac_bigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_bigloop_loop + \the\numexpr #1+\xint_c_ii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_bigloop_mul #1!% +}% +\def\XINT_FL_fac_bigloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr \xint_c_x^viii+#1*(#1+\xint_c_i)!% +}% +\def\XINT_FL_fac_medloop_a #1.% +{% + \expandafter\XINT_FL_fac_medloop_b + \the\numexpr #1+\xint_c_i-\xint_c_iii*((#1-100)/\xint_c_iii).#1.% +}% +\def\XINT_FL_fac_medloop_b #1.#2.#3.% +{% + \expandafter\XINT_FL_fac_smallloop_a + \the\numexpr #1-\xint_c_i.#3.{\XINT_FL_fac_medloop_loop #1.#2.}% +}% +\def\XINT_FL_fac_medloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_medloop_loop + \the\numexpr #1+\xint_c_iii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_medloop_mul #1!% +}% +\def\XINT_FL_fac_medloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)!% +}% +\def\XINT_FL_fac_smallloop_a #1.% +{% + \csname + XINT_FL_fac_smallloop_\the\numexpr #1-\xint_c_iv*(#1/\xint_c_iv)\relax + \endcsname #1.% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_1\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000001!.{2.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_-2\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000002!.{3.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_-1\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000006!.{4.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_0\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000024!.{5.#1.}{#2}% +}% +\def\XINT_FL_fac_addzeros #1.% +{% + \ifnum #1=\xint_c_viii \expandafter\XINT_FL_fac_addzeros_exit\fi + \expandafter\XINT_FL_fac_addzeros\the\numexpr #1-\xint_c_viii.100000000!% +}% +\def\XINT_FL_fac_addzeros_exit #1.#2.#3#4% + {\XINT_FL_fac_smallloop_loop #3#21\Z![-#4]}% +\def\XINT_FL_fac_smallloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_smallloop_loop + \the\numexpr #1+\xint_c_iv\expandafter.% + \the\numexpr #2\expandafter.\romannumeral0\XINT_FL_fac_smallloop_mul #1!% +}% +\def\XINT_FL_fac_smallloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)*(#1+\xint_c_iii)!% +}%[[ +\def\XINT_FL_fac_loop_exit #1!#2]#3{#3#2]}% +\def\XINT_FL_fac_mul 1#1!% + {\expandafter\XINT_FL_fac_mul_a\the\numexpr\XINT_FL_fac_smallmul 10!{#1}}% +\def\XINT_FL_fac_mul_a #1-#2% +{% + \if#21\xint_afterfi{\expandafter\space\xint_gob_til_exclam}\else + \expandafter\space\fi #11\Z!% +}% +\def\XINT_FL_fac_minimulwc_a #1#2#3#4#5!#6#7#8#9% +{% + \XINT_FL_fac_minimulwc_b {#1#2#3#4}{#5}{#6#7#8#9}% +}% +\def\XINT_FL_fac_minimulwc_b #1#2#3#4!#5% +{% + \expandafter\XINT_FL_fac_minimulwc_c + \the\numexpr \xint_c_x^ix+#5+#2*#4.{{#1}{#2}{#3}{#4}}% +}% +\def\XINT_FL_fac_minimulwc_c 1#1#2#3#4#5#6.#7% +{% + \expandafter\XINT_FL_fac_minimulwc_d {#1#2#3#4#5}#7{#6}% +}% +\def\XINT_FL_fac_minimulwc_d #1#2#3#4#5% +{% + \expandafter\XINT_FL_fac_minimulwc_e + \the\numexpr \xint_c_x^ix+#1+#2*#5+#3*#4.{#2}{#4}% +}% +\def\XINT_FL_fac_minimulwc_e 1#1#2#3#4#5#6.#7#8#9% +{% + 1#6#9\expandafter!% + \the\numexpr\expandafter\XINT_FL_fac_smallmul + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#7*#8!% +}% +\def\XINT_FL_fac_smallmul 1#1!#21#3!% +{% + \xint_gob_til_Z #3\XINT_FL_fac_smallmul_end\Z + \XINT_FL_fac_minimulwc_a #2!#3!{#1}{#2}% +}% +\def\XINT_FL_fac_smallmul_end\Z\XINT_FL_fac_minimulwc_a #1!\Z!#2#3[#4]% +{% + \ifnum #2=\xint_c_ + \expandafter\xint_firstoftwo\else + \expandafter\xint_secondoftwo + \fi + {-2\relax[#4]}% + {1#2\expandafter!\expandafter-\expandafter1\expandafter + [\the\numexpr #4+\xint_c_viii]}% +}% +% \end{macrocode} % \subsection{\csh{xintFloatSqrt}, \csh{XINTinFloatSqrt}} % \lverb|1.08| % \begin{macrocode} @@ -24295,7 +24591,7 @@ $1$ or $-1$. {% \expandafter\XINT_flsqrt_big_j \romannumeral0\xintiidivision - {#1}{\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% + {#1}{\romannumeral0\XINT_dbl_pos #2\Z}{#2}% }% \def\XINT_flsqrt_big_j #1% {% @@ -24307,8 +24603,8 @@ $1$ or $-1$. \def\XINT_flsqrt_big_k #1#2#3% {% \expandafter\XINT_flsqrt_big_l\expandafter - {\romannumeral0\XINT_sub_pre {#3}{#1}}% - {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr {#1}}}% + {\romannumeral0\xintiisub {#3}{#1}}% + {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr #1\Z}}% }% \def\XINT_flsqrt_big_l #1#2% {% @@ -24398,7 +24694,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintseries}% - [2015/09/12 v1.1c Expandable partial sums with xint package (jfB)]% + [2015/10/10 v1.2 Expandable partial sums with xint package (jfB)]% % \end{macrocode} % \subsection{\csh{xintSeries}} % \lverb|& @@ -24903,7 +25199,7 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintcfrac}% - [2015/09/12 v1.1c Expandable continued fractions with xint package (jfB)]% + [2015/10/10 v1.2 Expandable continued fractions with xint package (jfB)]% % \end{macrocode} % \subsection{\csh{xintCFrac}} % \begin{macrocode} @@ -25362,8 +25658,8 @@ $1$ or $-1$. \expandafter\XINT_ctf_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_ctf_loop_c #1#2% {% @@ -25399,8 +25695,8 @@ $1$ or $-1$. \def\XINT_icstf_loop_b #1.#2#3#4#5% {% \expandafter\XINT_icstf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {#2}{#3}% }% \def\XINT_icstf_loop_c #1#2% @@ -25430,8 +25726,8 @@ $1$ or $-1$. \expandafter\XINT_gctf_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_gctf_loop_c #1#2% {% @@ -25492,8 +25788,8 @@ $1$ or $-1$. \def\XINT_igctf_loop_b #1.#2#3#4#5% {% \expandafter\XINT_igctf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {#2}{#3}% }% \def\XINT_igctf_loop_c #1#2% @@ -25553,8 +25849,8 @@ $1$ or $-1$. \expandafter\XINT_ctcv_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_ctcv_loop_c #1#2% {% @@ -25596,8 +25892,8 @@ $1$ or $-1$. \def\XINT_icstcv_loop_b #1.#2#3#4#5% {% \expandafter\XINT_icstcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {{#2}{#3}}% }% \def\XINT_icstcv_loop_c #1#2% @@ -25633,8 +25929,8 @@ $1$ or $-1$. \expandafter\XINT_gctcv_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_gctcv_loop_c #1#2% {% @@ -25705,8 +26001,8 @@ $1$ or $-1$. \def\XINT_igctcv_loop_b #1.#2#3#4#5% {% \expandafter\XINT_igctcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {{#2}{#3}}% }% \def\XINT_igctcv_loop_c #1#2% @@ -26029,6 +26325,32 @@ $1$ or $-1$. % retrieving data expandably as \emph{names} of control sequences. Intermediate % computation results are stored as control sequences |\.=a/b[n]|. % +% Release |1.2| |[2015/10/10]| has the following changes: +% \begin{description} +% \item[not anymore limited to 5000 +% digits:] |1.2| replaces chains of |\romannumeral-`0| used earlier to +% gather digits by |\csname| governed expansions. The use of +% |\csname.=A/B[N]\endcsname| storage has been part of the design from the +% start, hence it was very natural and not too hard to gather the number +% directly inside |\csname|. With the chains of |\romannumeral-`0| gone, +% there is no more a limit at about 5000 (with the standard settings of the +% maximal expansion depth at 10000) on the maximal number of digits for each +% gathered number. +% \item[faster gathering of digits:] the previous item and some other changes +% have accelerated the building up of numbers. +% \item[optional accelerated parsing:] the new functions |qint|, |qfrac|, +% |qfloat| allow to skip entirely the digit by digit parsing and hand over +% directly responsability to \csa{xintiNum}, \csa{xintRaw}, or +% \csa{xintFloat} respectively. +% \item[float factorial:] the factorial operator |!| maps to the new macro +% \csa{xintFloatFac} inside \csa{xintfloatexpr}. +% \item[isolated dot now illegal:] the decimal mark must have digits either +% before or after it, an isolated |.| is now illegal input. +% \item[more recognized tokens:] |\ht|, |\dp|, |\wd|, |\fontcharht|, +% |\fontcharwd|, |\fontchardp| and |\fontcharit| are recognized and prefixed +% with |\number| automatically. +% \end{description} +% % Release |1.1| |[2014/10/28]| has made many extensions, some bug fixes, and % some breaking changes: % \begin{description} @@ -26302,18 +26624,52 @@ $1$ or $-1$. % \begin{macrocode} \XINT_providespackage \ProvidesPackage{xintexpr}% - [2015/09/12 v1.1c Expandable expression parser (jfB)]% + [2015/10/10 v1.2 Expandable expression parser (jfB)]% +\catcode`! 11 % \end{macrocode} % \subsection{Locking and unlocking} -% je dois réfléchir si je dois bloquer expansion après |unlock_a|, à -% cause de nil. -% \begin{macrocode} -\def\xint_gob_til_! #1!{}% this ! has catcode 11 -\edef\XINT_expr_lockscan#1!{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% -\edef\XINT_expr_lockit #1{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% -\def\XINT_expr_inintpart #1!{\XINT_num{#1}}% -\def\XINT_expr_infracpart #1e#2!{#1![\the\numexpr#2-\xintLength{#1}]!}% -\def\XINT_expr_inexppart e#1!{![\the\numexpr #1]!}% +% \lverb|Some renaming and modifications here with release 1.2 to switch from +% using chains of \romannumeral-`0 in order to gather numbers, possibly +% hexadecimals, to using a \csname governed expansion. In this way no more +% limit at 5000 digits, and besides this is a logical move because the +% \xintexpr parser is already based on \csname...\endcsname storage of numbers +% as one token. +% +% The limitation at 5000 digits didn't worry me too much because it was not +% very realistic to launch computations with thousands of digits... such +% computations are still slow with 1.2 but less so now. Chains or +% \romannumeral are still used for the gathering of function names and other +% stuff which I have half-forgotten because the parser does many things. +% +% In the earlier versions we used the lockscan macro after a chain of +% \romannumeral-`0 had ended gathering digits; this uses has been replaced by +% direct processing inside a \csname...\endcsname and the macro is kept only +% for matters of dummy variables. +% +% Currently, the parsing of hexadecimal numbers needs two nested +% \csname...\endcsname, first to gather the letters (possibly with a hexadecimal +% fractional part), and in a second stage to apply \xintHexToDec to do the +% actual conversion. This should be faster than updating on the fly the number +% (which would be hard for the fraction part...). The macro \xintHexToDec +% could probably be made faster by using techniques similar as the ones v1.2 +% uses in xintcore.sty.| +% \begin{macrocode} +\def\xint_gob_til_! #1!{}% catcode 11 ! default in xintexpr.sty code. +\edef\XINT_expr_lockscan#1!% not used for decimal numbers in xintexpr 1.2 + {\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% +\edef\XINT_expr_lockit + #1{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% +\def\XINT_expr_unlock_hex_in #1% expanded inside \csname..\endcsname + {\expandafter\XINT_expr_inhex\romannumeral-`0\XINT_expr_unlock#1;}% +\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname +{% + \if#2>\xintHexToDec{#1}% + \else + \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}% + [\the\numexpr-4*\xintLength{#3}]% + \fi +}% +%%%%%%%%%%%% \def\XINT_expr_unlock {\expandafter\XINT_expr_unlock_a\string }% \def\XINT_expr_unlock_a #1.={}% \def\XINT_expr_unexpectedtoken {\xintError:ignored }% @@ -26395,7 +26751,7 @@ $1$ or $-1$. \def\xintiieval {\expandafter\XINT_iiexpr_wrap\romannumeral0\xintbareiieval }% % \end{macrocode} % \subsection{\csh{xintieval}, \csh{XINT_iexpr_wrap}} -% \lverb|Optional argument since 1.1| +% \lverb|Optional argument since 1.1.| % \begin{macrocode} \def\xintieval #1% {\ifx [#1\expandafter\XINT_iexpr_withopt\else\expandafter\XINT_iexpr_noopt \fi #1}% @@ -26592,16 +26948,23 @@ $1$ or $-1$. #1% }% \def\XINT_expr_subexpr !#1\fi !{\expandafter\XINT_expr_getop\xint_gobble_iii }% +% \end{macrocode} +% \lverb|1.2 adds \ht, \dp, \wd and the eTeX font things.| +% \begin{macrocode} \def\XINT_expr_countetc #1% {% - \ifx\count#1\else\ifx#1\dimen\else\ifx#1\numexpr\else\ifx#1\dimexpr\else - \ifx\skip#1\else\ifx\glueexpr#1\else\ifx\fontdimen#1\else + \ifx\count#1\else\ifx\dimen#1\else\ifx\numexpr#1\else\ifx\dimexpr#1\else + \ifx\skip#1\else\ifx\glueexpr#1\else\ifx\fontdimen#1\else\ifx\ht#1\else + \ifx\dp#1\else\ifx\wd#1\else\ifx\fontcharht#1\else\ifx\fontcharwd#1\else + \ifx\fontchardp#1\else\ifx\fontcharic#1\else \XINT_expr_unpackvar - \fi\fi\fi\fi\fi\fi\fi + \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi \expandafter\XINT_expr_getnext\number #1% }% -\def\XINT_expr_unpackvar\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getnext\number #1% - {\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getop\csname .=\number#1\endcsname }% +\def\XINT_expr_unpackvar\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi + \expandafter\XINT_expr_getnext\number #1% + {\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi + \expandafter\XINT_expr_getop\csname .=\number#1\endcsname }% \begingroup \lccode`*=`# \lowercase{\endgroup @@ -26609,7 +26972,7 @@ $1$ or $-1$. \if#1*\xint_dothis {\XINT_expr_scan_macropar *}\fi \if#1[\xint_dothis {\xint_c_xviii ({}}\fi \if#1+\xint_dothis \XINT_expr_getnext \fi - \if#1.\xint_dothis {\XINT_expr_scandec_II\XINT_expr_infracpart}\fi + \if#1.\xint_dothis {\XINT_expr_startdec}\fi \if#1-\xint_dothis -\fi \if#1(\xint_dothis {\xint_c_xviii ({}}\fi \xint_orthat {\XINT_expr_scan_nbr_or_func #1}% @@ -26618,158 +26981,220 @@ $1$ or $-1$. % \end{macrocode} % \subsection{The integer or decimal number or hexa-decimal number or % function name or variable name or special hacky things big parser} +% \lverb|1.2 release has replaced chains of \romannumeral-`0 by \csname +% governed expansion. Thus there is no more the limit at about 5000 digits for +% parsed numbers. +% +% In order to avoid having to lock and unlock in succession to handle the +% scientific part and adjust the exponent according to the number of digits of +% the decimal part, the parsing of this decimal part counts on the fly the +% number of digits it encounters. +% +% There is some slight annoyance with \xintiiexpr which should never be given +% a [n] inside its \csname.=<digits>\endcsname storage of numbers (because its +% arithmetic uses the ii macros which know nothing about the [N] notation). +% Hence if the parser has only seen digits when hitting something else than +% the dot or e (or E), it will not insert a [0]. Thus we very slightly +% compromise the efficiency of \xintexpr and \xintfloatexpr in order to be +% able to share the same code with \xintiiexpr. +% +% Indeed, the parser at this location is completely common to all, it does not +% know if it is working inside \xintexpr or \xintiiexpr. On the other hand if +% a dot or a e (or E) is met, then the (common) parser has no scrupules ending +% this number with a [n], this will provoke an error later if that was within +% an \xintiiexpr, as soon as an arithmetic macro is used. +% +% As the gathered numbers have no spaces, no pluses, no minuses, the only +% remaining issue is with leading zeroes, which are discarded on the fly. The +% hexadecimal numbers leading zeroes are stripped in a second stage by the +% \xintHexToDec macro. +% +% With v1.2, \xinttheexpr . \relax does not work anymore (it did in earlier +% releases). There must be digits either before or after the decimal mark. Thus +% both \xinttheexpr 1.\relax and \xinttheexpr .1\relax are legal.| % \begin{macrocode} \catcode96 11 % ` \def\XINT_expr_scan_nbr_or_func #1% this #1 has necessarily here catcode 12 {% \if "#1\xint_dothis \XINT_expr_scanhex_I\fi \if `#1\xint_dothis {\XINT_expr_onlitteral_`}\fi - \ifnum \xint_c_ix<1#1 \xint_dothis \XINT_expr_scandec_I\fi + \ifnum \xint_c_ix<1#1 \xint_dothis \XINT_expr_startint\fi \xint_orthat \XINT_expr_scanfunc #1% }% \catcode96 12 % ` -\def\XINT_expr_scandec_I -{% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart - \romannumeral-`0\XINT_expr_scanintpart_b -}% -\def\XINT_expr_scandec_II +\def\XINT_expr_startint #1% {% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart - \romannumeral-`0\XINT_expr_scanfracpart_b + \if #10\expandafter\XINT_expr_gobz_a\else\XINT_expr_scanint_a\fi #1% }% +\def\XINT_expr_scanint_a #1#2% + {\expandafter\XINT_expr_getop\csname.=#1% + \expandafter\XINT_expr_scanint_b\romannumeral-`0#2}% +\def\XINT_expr_gobz_a #1% + {\expandafter\XINT_expr_getop\csname.=% + \expandafter\XINT_expr_gobz_scanint_b\romannumeral-`0#1}% +\def\XINT_expr_startdec #1% + {\expandafter\XINT_expr_getop\csname.=% + \expandafter\XINT_expr_scandec_a\romannumeral-`0#1}% % \end{macrocode} -% \subsubsection{Integral part} -% \begin{macrocode} -\def\XINT_expr_scanintpart_a #1% -{% careful that ! has catcode letter here - \ifcat \relax #1\xint_dothis{!!#1}\fi % stops the scan - \if e#1\xint_dothis{\expandafter\XINT_expr_inexppart - \romannumeral-`0\XINT_expr_scanexppart_a e}\fi - \if E#1\xint_dothis{\expandafter\XINT_expr_inexppart - \romannumeral-`0\XINT_expr_scanexppart_a e}\fi -% \end{macrocode} -% \lverb|\if @#1\xint_dothis{!*#1}\fi % tacit multiplication later| +% \subsubsection{Integral part (skipping zeroes)} +% \lverb|Sub-expressions are recognized as startaing with catcode 11 +% exclamation mark, which is treated together with variable names composed of +% letters below, hence induces a tacit multiplication if encountered while +% gathering a number. % -% \lverb|\if _#1\xint_dothis{!*#1}\fi % tacit multiplication for variables| +% 1.2 has modified the code to give highest priority to digits, the impact is +% non-negligeable. I don't think the doubled \string is a serious penalty.| % \begin{macrocode} - \ifcat a#1\xint_dothis{!!*#1}\fi % includes subexpressions (#1=! letter) - \xint_orthat {\expandafter\XINT_expr_scanintpart_aa\string #1}% +\def\XINT_expr_scanint_b #1% +{% + \ifcat \relax #1\expandafter\XINT_expr_scanint_endbycs\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanint_c\fi + \string#1\XINT_expr_scanint_d +}% +\def\XINT_expr_scanint_d #1% +{% + \expandafter\XINT_expr_scanint_b\romannumeral-`0#1% +}% +\def\XINT_expr_scanint_endbycs#1#2\XINT_expr_scanint_d{\endcsname #1}% +\def\XINT_expr_scanint_c\string #1\XINT_expr_scanint_d +{% + \if e#1\xint_dothis{[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{\endcsname*#1}\fi + \if .#1\xint_dothis{\XINT_expr_startdec_a .}\fi + \xint_orthat {\expandafter\endcsname \string#1}% +}% +\def\XINT_expr_startdec_a .#1% +{% + \expandafter\XINT_expr_scandec_a\romannumeral-`0#1% +}% +\def\XINT_expr_scandec_a #1% +{% + \if .#1\xint_dothis{\endcsname..}\fi + \xint_orthat {\XINT_expr_scandec_b 0.#1}% +}% +\def\XINT_expr_gobz_scanint_b #1% +{% + \ifcat \relax #1\expandafter\XINT_expr_gobz_scanint_endbycs\expandafter #1\fi + \ifnum\xint_c_x<1\string#1 \else\expandafter\XINT_expr_gobz_scanint_c\fi + \string#1\XINT_expr_scanint_d }% -\def\XINT_expr_scanintpart_aa #1% +\def\XINT_expr_gobz_scanint_endbycs#1#2\XINT_expr_scanint_d{0\endcsname #1}% +\def\XINT_expr_gobz_scanint_c\string #1\XINT_expr_scanint_d {% - \if .#1\xint_dothis\XINT_expr_scandec_transition\fi - \ifnum \xint_c_ix<1#1 \xint_dothis\XINT_expr_scanintpart_b\fi - \xint_orthat {!!}#1% + \if e#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{0\endcsname*#1}\fi + \if .#1\xint_dothis{\XINT_expr_gobz_startdec_a .}\fi + \if 0#1\xint_dothis\XINT_expr_gobz_scanint_d\fi + \xint_orthat {0\expandafter\endcsname \string#1}% }% -\def\XINT_expr_scanintpart_b #1#2% +\def\XINT_expr_gobz_scanint_d #1% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanintpart_a\romannumeral-`0#2% + \expandafter\XINT_expr_gobz_scanint_b\romannumeral-`0#1% }% -\def\XINT_expr_scandec_transition .#1% +\def\XINT_expr_gobz_startdec_a .#1% {% - \expandafter\XINT_expr_scandec_trans_a\romannumeral-`0#1% + \expandafter\XINT_expr_gobz_scandec_a\romannumeral-`0#1% }% -\def\XINT_expr_scandec_trans_a #1% +\def\XINT_expr_gobz_scandec_a #1% {% - \if .#1\xint_dothis{!!..}\fi - \xint_orthat {\expandafter\XINT_expr_infracpart - \romannumeral-`0\XINT_expr_scanfracpart_a #1}% + \if .#1\xint_dothis{0\endcsname..}\fi + \xint_orthat {\XINT_expr_gobz_scandec_b 0.#1}% }% % \end{macrocode} % \subsubsection{Fractional part} +% \lverb|Annoying duplication of code to allow 0. as input.| % \begin{macrocode} -\def\XINT_expr_scanfracpart_a #1% +\def\XINT_expr_scandec_b #1.#2% {% - \ifcat \relax #1\xint_dothis{e!#1}\fi % stops the scan - \if e#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi - \if E#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi - \ifcat a#1\xint_dothis{e!*#1}\fi % and also the case of subexpressions (!) - \xint_orthat {\expandafter\XINT_expr_scanfracpart_aa\string #1}% + \ifcat \relax #2\expandafter\XINT_expr_scandec_endbycs\expandafter#2\fi + \ifnum\xint_c_ix<1\string#2 \else\expandafter\XINT_expr_scandec_c\fi + \string#2\expandafter\XINT_expr_scandec_d\the\numexpr #1-\xint_c_i.% }% -\def\XINT_expr_scanfracpart_aa #1% +\def\XINT_expr_scandec_endbycs #1#2\XINT_expr_scandec_d + \the\numexpr#3-\xint_c_i.{[#3]\endcsname #1}% +\def\XINT_expr_scandec_d #1.#2% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanfracpart_b - \else - \xint_afterfi {e!}% - \fi - #1% + \expandafter\XINT_expr_scandec_b + \the\numexpr #1\expandafter.\romannumeral-`0#2% }% -\def\XINT_expr_scanfracpart_b #1#2% +\def\XINT_expr_scandec_c\string #1#2\the\numexpr#3-\xint_c_i.% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanfracpart_a\romannumeral-`0#2% + \if e#1\xint_dothis{[\the\numexpr#3\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{[\the\numexpr#3\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{[#3]\endcsname *#1}\fi + \xint_orthat {[#3]\expandafter\endcsname \string#1}% +}% +\def\XINT_expr_gobz_scandec_b 0.#1% +{% + \ifcat \relax #1\expandafter\XINT_expr_gobz_scandec_endbycs\expandafter#1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_gobz_scandec_c\fi + \string#1\expandafter\XINT_expr_scandec_d\the\numexpr\xint_c_mone.% +}% +\def\XINT_expr_gobz_scandec_endbycs #1#2\xint_c_mone.{0[0]\endcsname #1}% +\def\XINT_expr_gobz_scandec_c\string #1#2\xint_c_mone.% +{% + \if e#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{0[0]\endcsname *#1}\fi + \xint_orthat {0[0]\expandafter\endcsname \string#1}% }% % \end{macrocode} % \subsubsection{Scientific notation} +% \lverb|Some pluses and minuses are allowed at the start of the scientific +% part, however not later, and no parenthesis.| % \begin{macrocode} -\def\XINT_expr_scanexppart_a #1#2% +\def\XINT_expr_scanexp_a #1#2% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanexppart_b\romannumeral-`0#2% + #1\expandafter\XINT_expr_scanexp_b\romannumeral-`0#2% }% -\def\XINT_expr_scanexppart_b #1% +\def\XINT_expr_scanexp_b #1% {% - \ifcat \relax #1\xint_dothis{0!#1}\fi % stops the scan (incorrect syntax) - \ifcat a#1\xint_dothis{0!*#1}\fi % idem - \if +#1\xint_dothis {\XINT_expr_scanexppart_a +}\fi - \if -#1\xint_dothis {\XINT_expr_scanexppart_a -}\fi - \xint_orthat {\expandafter\XINT_expr_scanexppart_c\string #1}% + \ifcat \relax #1\expandafter\XINT_expr_scanexp_endbycs\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanexp_c\fi + \string#1\XINT_expr_scanexp_d }% -\def\XINT_expr_scanexppart_c #1% +\def\XINT_expr_scanexpr_endbycs#1#2\XINT_expr_scanexp_d {]\endcsname #1}% +\def\XINT_expr_scanexp_d #1% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanexppart_d - \else - \expandafter !% - \fi - #1% + \expandafter\XINT_expr_scanexp_bb\romannumeral-`0#1% }% -\def\XINT_expr_scanexppart_d #1#2% +\def\XINT_expr_scanexp_c\string #1\XINT_expr_scanexp_d {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanexppart_e\romannumeral-`0#2% + \ifcat a#1\xint_dothis {]\endcsname *#1}\fi + \if +#1\xint_dothis {\XINT_expr_scanexp_a +}\fi + \if -#1\xint_dothis {\XINT_expr_scanexp_a -}\fi + \xint_orthat {]\expandafter\endcsname\string #1}% }% -\def\XINT_expr_scanexppart_e #1% +\def\XINT_expr_scanexp_bb #1% {% - \ifcat \relax #1\xint_dothis{!#1}\fi % stops the scan - \ifcat a#1\xint_dothis{!*#1}\fi % idem - \xint_orthat {\expandafter\XINT_expr_scanexppart_f\string #1}% + \ifcat \relax #1\expandafter\XINT_expr_scanexp_endbycs_b\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanexp_cb\fi + \string#1\XINT_expr_scanexp_db }% -\def\XINT_expr_scanexppart_f #1% +\def\XINT_expr_scanexp_endbycs_b#1#2\XINT_expr_scanexp_db {]\endcsname #1}% +\def\XINT_expr_scanexp_db #1% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanexppart_d - \else - \expandafter !% - \fi - #1% + \expandafter\XINT_expr_scanexp_bb\romannumeral-`0#1% +}% +\def\XINT_expr_scanexp_cb\string #1\XINT_expr_scanexp_db +{% + \ifcat a#1\xint_dothis {]\endcsname *#1}\fi + \xint_orthat {]\expandafter\endcsname\string #1}% }% % \end{macrocode} % \subsubsection{Hexadecimal numbers} % \begin{macrocode} -\def\XINT_expr_scanhex_I #1% -{% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\expandafter\XINT_expr_inhex - \romannumeral-`0\XINT_expr_scanhexI_a -}% -\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname +\def\XINT_expr_scanhex_I #1% #1=" {% - \if#2I\xintHexToDec{#1}% - \else - \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}% - [\the\numexpr-4*\xintLength{#3}]% - \fi + \expandafter\XINT_expr_getop\csname.=\expandafter + \XINT_expr_unlock_hex_in\csname.=\XINT_expr_scanhexI_a }% \def\XINT_expr_scanhexI_a #1% {% - \ifcat #1\relax\xint_dothis{.I;!#1}\fi - \ifx !#1\xint_dothis{.I;!*!}\fi % tacit multiplication + \ifcat #1\relax\xint_dothis{.>\endcsname\endcsname #1}\fi + \ifx !#1\xint_dothis{.>\endcsname\endcsname*!}\fi % tacit multiplication \xint_orthat {\expandafter\XINT_expr_scanhexI_aa\string #1}% }% \def\XINT_expr_scanhexI_aa #1% @@ -26787,24 +27212,23 @@ $1$ or $-1$. \expandafter\xint_secondoftwo \fi {\expandafter\XINT_expr_scanhex_transition}% - {\xint_afterfi {.I;!}}% + {\xint_afterfi {.>\endcsname\endcsname}}% \fi #1% }% \def\XINT_expr_scanhexI_b #1#2% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexI_a\romannumeral-`0#2% + #1\expandafter\XINT_expr_scanhexI_a\romannumeral-`0#2% }% \def\XINT_expr_scanhex_transition .#1% {% - \expandafter.\expandafter.\romannumeral-`0\expandafter + \expandafter.\expandafter.\expandafter \XINT_expr_scanhexII_a\romannumeral-`0#1% }% \def\XINT_expr_scanhexII_a #1% {% - \ifcat #1\relax\xint_dothis{;!#1}\fi - \ifx !#1\xint_dothis{;!*!}\fi % tacit multiplication + \ifcat #1\relax\xint_dothis{\endcsname\endcsname#1}\fi + \ifx !#1\xint_dothis{\endcsname\endcsname*!}\fi % tacit multiplication \xint_orthat {\expandafter\XINT_expr_scanhexII_aa\string #1}% }% \def\XINT_expr_scanhexII_aa #1% @@ -26816,14 +27240,13 @@ $1$ or $-1$. 0\else1\fi\else0\fi\else1\fi\else0\fi 1% \expandafter\XINT_expr_scanhexII_b \else - \xint_afterfi {;!}% + \xint_afterfi {\endcsname\endcsname}% \fi #1% }% \def\XINT_expr_scanhexII_b #1#2% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexII_a\romannumeral-`0#2% + #1\expandafter\XINT_expr_scanhexII_a\romannumeral-`0#2% }% % \end{macrocode} % \subsubsection{Function and variable names} @@ -27684,22 +28107,25 @@ $1$ or $-1$. }% % \end{macrocode} % \subsection{! as postfix factorial operator} -% \lverb|As of 2014/11/07, not yet a float version of factorial. I must do it!| +% \lverb|Float version was at last done 2015/10/06. As xint does not have yet +% exp/log, Stirling is no go.| % \begin{macrocode} \let\XINT_expr_precedence_! \xint_c_x \def\XINT_expr_op_! #1{\expandafter\XINT_expr_getop \csname .=\xintFac{\XINT_expr_unlock #1}\endcsname }% -\let\XINT_flexpr_op_!\XINT_expr_op_! +\def\XINT_flexpr_op_! #1{\expandafter\XINT_expr_getop + \csname .=\XINTinFloatFac{\XINT_expr_unlock #1}\endcsname }% \def\XINT_iiexpr_op_! #1{\expandafter\XINT_expr_getop - \csname .=\xintiFac{\XINT_expr_unlock #1}\endcsname }% + \csname .=\xintiiFac{\XINT_expr_unlock #1}\endcsname }% % \end{macrocode} % \subsection{The A/B[N] mechanism} % \lverb|Releases earlier than 1.1 required the use of braces around A/B[N] -% input. The [N] is now implemented directly. *BUT* uses a delimited macro! +% input. The [N] is now implemented directly. *BUT* this uses a delimited macro! % thus N is not allowed to be itself an expression (I could add it...). -% \xintE, \xintiiE, and \XINTinFloatE all put #2 in a \numexpr. BUT ATTENTION -% TO CRAZYNESS OF NUMEXPR: \the\numexpr 3 + 7 9 \relax !! Hence we have to do -% the job ourselves.| +% \xintE, \xintiiE, and \XINTinFloatE all put #2 in a \numexpr. But attention +% to the fact that \numexpr stops at spaces separating digits: +% \the\numexpr 3 + 7 9\relax gives 109\relax !! Hence we have to be +% careful.| % \begin{macrocode} \catcode`[ 11 \catcode`* 11 @@ -27742,7 +28168,7 @@ $1$ or $-1$. % all high punctuation ?, !, :, ;. % % It is not recommended to overwrite single Latin letters which are -% pre-defined to serve as dummy variables. Variable names may contains +% pre-defined to serve as dummy variables. Variable names may contain % letters, digits, underscores, and must not start with a digit.| % \begin{macrocode} \catcode`: 12 @@ -27848,7 +28274,7 @@ $1$ or $-1$. }% \expandafter\def\csname XINT_expr_onlitteral_`\endcsname #1#2#3({\xint_c_xviii `{#2}}% % \end{macrocode} -% \subsection{The bool, togl, protect, unknown, and break "functions"} +% \subsection{The bool, togl, protect, unknown, and break ``functions''} % \lverb|bool, togl and protect use delimited macros. Only unknown and break % are true functions with a more flexible parsing of the opening and closing % parentheses, which may possibly arise from expansion itself.| @@ -27859,12 +28285,24 @@ $1$ or $-1$. {\expandafter\XINT_expr_getop\csname .=\xintToggle{#1}\endcsname }% \def\XINT_expr_onlitteral_protect #1)% {\expandafter\XINT_expr_getop\csname .=\detokenize{#1}\endcsname }% -\def\XINT_expr_func_unknown #1#2#3{\expandafter #1\expandafter #2\csname .=0\endcsname }% +\def\XINT_expr_func_unknown #1#2#3% + {\expandafter #1\expandafter #2\csname .=0\endcsname }% \def\XINT_expr_func_break #1#2#3% -{\expandafter #1\expandafter #2\csname.=?\romannumeral-`0\XINT_expr_unlock #3\endcsname }% + {\expandafter #1\expandafter #2\csname.=?\romannumeral-`0\XINT_expr_unlock #3\endcsname }% \let\XINT_flexpr_func_break \XINT_expr_func_break \let\XINT_iiexpr_func_break \XINT_expr_func_break % \end{macrocode} +% \subsection{The qint, qfrac, qfloat ``functions''} +% \lverb|New with 1.2. Allows the user to hand over quickly a big number to the +% parser, spaces not immediately removed but should be harmless in general.| +% \begin{macrocode} +\def\XINT_expr_onlitteral_qint #1)% + {\expandafter\XINT_expr_getop\csname .=\xintiNum{#1}\endcsname }% +\def\XINT_expr_onlitteral_qfrac #1)% + {\expandafter\XINT_expr_getop\csname .=\xintRaw{#1}\endcsname }% +\def\XINT_expr_onlitteral_qfloat #1)% + {\expandafter\XINT_expr_getop\csname .=\XINTinFloatdigits{#1}\endcsname }% +% \end{macrocode} % \subsection{seq and the implementation of dummy variables} % \lverb|All of seq, add, mul, rseq, etc... (actually all of the extensive % changes from xintexpr 1.09n to 1.1) was done around June 15-25th 2014, but the @@ -27921,9 +28359,9 @@ $1$ or $-1$. \def\XINT_expr_onlitteral_seq_e #1#2{\XINT_expr_onlitteral_seq_d {#1}{#2)}}% % \end{macrocode} % \subsubsection{\csh{XINT_isbalanced_a} for \csh{XINT_expr_onlitteral_seq_a}} -%\lverb|Expands to \m@ne in case a closing ) had no opening ( matching it, to -% \@ne if opening ) had no closing ) matching it, to \z@ if expression was -% balanced.| +% \lverb|Expands to \xint_c_mone in case a closing ) had no opening ( matching +% it, to \@ne if opening ) had no closing ) matching it, to \z@ if expression +% was balanced.| % \begin{macrocode} % use as \XINT_isbalanced_a \relax #1(\xint_bye)\xint_bye \def\XINT_isbalanced_a #1({\XINT_isbalanced_b #1)\xint_bye }% @@ -27932,7 +28370,7 @@ $1$ or $-1$. % \end{macrocode} % \lverb|if #2 is not \xint_bye, a ) was found, but there was no (. Hence error -> -1| % \begin{macrocode} -\def\XINT_isbalanced_error #1)\xint_bye {\m@ne}% +\def\XINT_isbalanced_error #1)\xint_bye {\xint_c_mone}% % \end{macrocode} % \lverb|#2 was \xint_bye, was there a ) in original #1?| % \begin{macrocode} @@ -28978,10 +29416,11 @@ $1$ or $-1$. \romannumeral-`0\expandafter\XINT_xptwo_getab_b \romannumeral-`0####2!{####1}{~xint#1}{xint#1}}% }% -}% +}% cela aurait-il un sens d'ajouter Raw et iNum (à cause de qint, qfrac, + % qfloat?). Pas le temps d'y réfléchir. Je ne fais rien. \xintFor #1 in {Num,Irr,Abs,iiAbs,Sgn,iiSgn,TFrac,Floor,iFloor,Ceil,iCeil,% Sqr,iiSqr,iiSqrt,iiSqrtR,iiIsZero,iiIsNotZero,iiifNotZero,iiifSgn,% - Odd,Even,iiOdd,iiEven,Opp,iiOpp,iiifZero,Fac,iFac,Bool,Toggle}\do + Odd,Even,iiOdd,iiEven,Opp,iiOpp,iiifZero,Fac,iiFac,Bool,Toggle}\do {\toks0 \expandafter{\the\toks0% \expandafter\let\csname xint#1NE\expandafter\endcsname\csname xint#1\expandafter @@ -28989,6 +29428,13 @@ $1$ or $-1$. \expandafter\XINT_NEfork_one\romannumeral-`0####1!{~xint#1}{xint#1}{}{}}% }% }% +\toks0 + \expandafter{\the\toks0 + \let\XINTinFloatFacNE\XINTinFloatFac + \def\XINTinFloatFac ##1{% + \expandafter\XINT_NEfork_one + \romannumeral-`0##1!{~XINTinFloatFac}{XINTinFloatFac}{}{}}% + }% \xintFor #1 in {Add,Sub,Mul,Div,Power,E,Mod,SeqA::csv}\do {\toks0 \expandafter{\the\toks0% @@ -29189,12 +29635,41 @@ $1$ or $-1$. \def\mymacroaux #1#2{#2}% % \parbox[t]{10cm}{Total number of code lines: - \dtt{\the\numexpr - \xintListWithSep+{\xintApply\mymacro\storedlinecounts}\relax }. Each - package starts - with circa \dtt{50} lines dealing with catcodes, package identification - and reloading management, also for Plain \TeX\strut. Version - {\xintbndlversion} of {\xintbndldate}.\par} + \dtt{\the\numexpr + \xintListWithSep+{\xintApply\mymacro\storedlinecounts}\relax }. + Among those, release 1.2 has about 3000 lines starting with either + \{\% or \}\%.% en fait 3013 mais je devrais automatiser. + + Each package starts with circa \dtt{50} lines dealing with catcodes, + package identification and reloading management, also for Plain + \TeX\strut. Version {\xintbndlversion} of {\xintbndldate}.\par +} + +% il faut que je patche doc.sty pour faire ça automatiquement: +% +% TEMP$ grep -c -e "^{%" *sty +% xint.sty:170 +% xintbinhex.sty:69 +% xintcfrac.sty:183 +% xintcore.sty:296 +% xintexpr.sty:133 +% xintfrac.sty:415 +% xintgcd.sty:59 +% xintkernel.sty:7 +% xintseries.sty:48 +% xinttools.sty:112 +% +% TEMP$ grep -c -e "^}%" *sty +% xint.sty:170 +% xintbinhex.sty:69 +% xintcfrac.sty:183 +% xintcore.sty:296 +% xintexpr.sty:163 +% xintfrac.sty:415 +% xintgcd.sty:61 +% xintkernel.sty:8 +% xintseries.sty:48 +% xinttools.sty:112 \CharacterTable {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z @@ -29211,7 +29686,7 @@ $1$ or $-1$. Right bracket \] Circumflex \^ Underscore \_ Grave accent \` Left brace \{ Vertical bar \| Right brace \} Tilde \~} -\CheckSum {25543} +\CheckSum {26711}% \makeatletter\check@checksum\makeatother \Finale %% End of file xint.dtx diff --git a/Master/texmf-dist/source/generic/xint/xint.ins b/Master/texmf-dist/source/generic/xint/xint.ins index 5244bbf6c17..2be4ee69a10 100644 --- a/Master/texmf-dist/source/generic/xint/xint.ins +++ b/Master/texmf-dist/source/generic/xint/xint.ins @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% --------------------------------------------------------------- %% diff --git a/Master/texmf-dist/tex/generic/xint/xint.sty b/Master/texmf-dist/tex/generic/xint/xint.sty index b07e96b1cbe..b06c32ff687 100644 --- a/Master/texmf-dist/tex/generic/xint/xint.sty +++ b/Master/texmf-dist/tex/generic/xint/xint.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xint: Expandable operations on big integers %% --------------------------------------------------------------- @@ -70,13 +70,17 @@ \XINTsetupcatcodes% defined in xintkernel.sty (loaded by xintcore.sty) \XINT_providespackage \ProvidesPackage{xint}% - [2015/09/12 v1.1c Expandable operations on big integers (jfB)]% + [2015/10/10 v1.2 Expandable operations on big integers (jfB)]% \long\def\xint_firstofthree #1#2#3{#1}% \long\def\xint_secondofthree #1#2#3{#2}% \long\def\xint_thirdofthree #1#2#3{#3}% \long\def\xint_firstofthree_thenstop #1#2#3{ #1}% 1.09i \long\def\xint_secondofthree_thenstop #1#2#3{ #2}% \long\def\xint_thirdofthree_thenstop #1#2#3{ #3}% +\edef\xint_cleanupzeros_andstop #1#2#3#4% +{% + \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax +}% \def\xintSgnFork {\romannumeral0\xintsgnfork }% \def\xintsgnfork #1% {% @@ -101,22 +105,7 @@ \expandafter\space\expandafter 0\xint_gob_til_Z }% \def\XINT_isone_yes #1\Z { 1}% -\def\xintRev {\romannumeral0\xintrev }% -\def\xintrev #1% -{% - \expandafter\XINT_rev_fork - \romannumeral-`0#1\xint_relax % empty #1 ok, \xint_relax stops expansion - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_rev_fork #1% -{% - \xint_UDsignfork - #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_rord_main {}}% - -{\XINT_rord_main {}#1}% - \krof -}% +\let\xintRev\xintReverseDigits \def\xintLen {\romannumeral0\xintlen }% \def\xintlen #1% {% @@ -301,121 +290,175 @@ \expandafter\xint_secondoftwo_thenstop \fi }% -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% +\def\xintCmp {\romannumeral0\xintcmp }% +\def\xintcmp #1{\expandafter\XINT_icmp\romannumeral0\xintnum{#1}\Z }% +\def\xintiiCmp {\romannumeral0\xintiicmp }% +\def\xintiicmp #1{\expandafter\XINT_iicmp\romannumeral-`0#1\Z }% +\def\XINT_iicmp #1#2\Z #3% {% - \expandafter\xint_cmp\expandafter{\romannumeral0\xintnum{#1}}% + \expandafter\XINT_cmp_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xint_cmp #1#2% +\let\XINT_Cmp \xintiiCmp +\def\XINT_icmp #1#2\Z #3% {% - \expandafter\XINT_cmp_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_cmp_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xintiiCmp {\romannumeral0\xintiicmp }% -\def\xintiicmp #1% +\def\XINT_cmp_nfork #1#2% {% - \expandafter\xint_iicmp\expandafter{\romannumeral-`0#1}% + \xint_UDzerofork + #1\XINT_cmp_firstiszero + #2\XINT_cmp_secondiszero + 0{}% + \krof + \xint_UDsignsfork + #1#2\XINT_cmp_minusminus + #1-\XINT_cmp_minusplus + #2-\XINT_cmp_plusminus + --\XINT_cmp_plusplus + \krof #1#2% }% -\def\xint_iicmp #1#2% +\def\XINT_cmp_firstiszero #1\krof 0#2#3\Z #4\Z {% - \expandafter\XINT_cmp_fork \romannumeral-`0#2\Z #1\Z + \xint_UDzerominusfork + #2-{ 0}% + 0#2{ 1}% + 0-{ -1}% + \krof }% -\def\XINT_Cmp #1#2{\romannumeral0\XINT_cmp_fork #2\Z #1\Z }% -\def\XINT_cmp_fork #1#2\Z #3#4\Z +\def\XINT_cmp_secondiszero #1\krof #20#3\Z #4\Z {% - \xint_UDsignsfork - #1#3\XINT_cmp_minusminus - #1-\XINT_cmp_minusplus - #3-\XINT_cmp_plusminus - --{\xint_UDzerosfork - #1#3\XINT_cmp_zerozero - #10\XINT_cmp_zeroplus - #30\XINT_cmp_pluszero - 00\XINT_cmp_plusplus - \krof }% + \xint_UDzerominusfork + #2-{ 0}% + 0#2{ -1}% + 0-{ 1}% \krof - {#2}{#4}#1#3% }% -\def\XINT_cmp_minusplus #1#2#3#4{ 1}% -\def\XINT_cmp_plusminus #1#2#3#4{ -1}% -\def\XINT_cmp_zerozero #1#2#3#4{ 0}% -\def\XINT_cmp_zeroplus #1#2#3#4{ 1}% -\def\XINT_cmp_pluszero #1#2#3#4{ -1}% -\def\XINT_cmp_plusplus #1#2#3#4% +\def\XINT_cmp_plusminus #1\Z #2\Z{ 1}% +\def\XINT_cmp_minusplus #1\Z #2\Z{ -1}% +\def\XINT_cmp_minusminus + --{\expandafter\XINT_opp\romannumeral0\XINT_cmp_plusplus {}{}}% +\def\XINT_cmp_plusplus #1#2#3\Z +{% + \expandafter\XINT_cmp_pp + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\def\XINT_cmp_pp #1.#2\X #3\Z +{% + \expandafter\XINT_cmp_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_cmp_checklengths #1.#2.% +{% + \ifnum #1=#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + \XINT_cmp_aa {\XINT_cmp_distinctlengths {#1}{#2}}% +}% +\def\XINT_cmp_distinctlengths #1#2#3\W #4\W {% - \XINT_cmp_pre {#4#2}{#3#1}% + \ifnum #1>#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + { -1}{ 1}% }% -\def\XINT_cmp_minusminus #1#2#3#4% +%%%%%%%%%%%% +\def\XINT_cmp_aa {\expandafter\XINT_cmp_w\the\numexpr\XINT_cmp_a \xint_c_i }% +%%%%%%%%%%%% +\def\XINT_cmp_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \XINT_cmp_pre {#1}{#2}% + \XINT_cmp_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_cmp_pre #1% +\def\XINT_cmp_b #1#2#3!#4!% {% - \expandafter\XINT_cmp_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \xint_gob_til_Z #2\XINT_cmp_bi \Z + \expandafter\XINT_cmp_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_pre_b #1#2% +\def\XINT_cmp_c 1#1#2.% {% - \expandafter\XINT_cmp_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_cmp_d #1% }% -\def\XINT_cmp_A #1#2#3\W\X\Y\Z #4#5#6#7% +\def\XINT_cmp_d #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_cmp_az\W - \XINT_cmp_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z + \xint_gob_til_Z #2\XINT_cmp_di \Z + \expandafter\XINT_cmp_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_B #1#2#3#4#5#6#7% +\def\XINT_cmp_e 1#1#2.% {% - \xint_gob_til_W#4\xint_cmp_bz\W - \XINT_cmp_onestep #1#2{#7#6#5#4}{#3}% + 1#2\expandafter!\the\numexpr\XINT_cmp_f #1% }% -\def\XINT_cmp_onestep #1#2#3#4#5#6% +\def\XINT_cmp_f #1#2#3!#4!% {% - \expandafter\XINT_cmp_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% + \xint_gob_til_Z #2\XINT_cmp_fi \Z + \expandafter\XINT_cmp_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_backtoA #1#2#3.#4% +\def\XINT_cmp_g 1#1#2.% {% - \XINT_cmp_A #2{#3#4}% + 1#2\expandafter!\the\numexpr\XINT_cmp_h #1% }% -\def\xint_cmp_bz\W\XINT_cmp_onestep #1\Z { 1}% -\def\xint_cmp_az\W\XINT_cmp_B #1#2#3#4#5#6#7% +\def\XINT_cmp_h #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_cmp_ez\W - \XINT_cmp_Eenter #1{#3}#4#5#6#7% + \xint_gob_til_Z #2\XINT_cmp_hi \Z + \expandafter\XINT_cmp_i\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_cmp_Eenter #1\Z { -1}% -\def\xint_cmp_ez\W\XINT_cmp_Eenter #1% +\def\XINT_cmp_i 1#1#2.% {% - \xint_UDzerofork - #1\XINT_cmp_K % il y a une retenue - 0\XINT_cmp_L % pas de retenue - \krof + 1#2\expandafter!\the\numexpr\XINT_cmp_a #1% }% -\def\XINT_cmp_K #1\Z { -1}% -\def\XINT_cmp_L #1{\XINT_OneIfPositive_main #1}% -\def\XINT_OneIfPositive #1% +\def\XINT_cmp_bi\Z + \expandafter\XINT_cmp_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W {% - \XINT_OneIfPositive_main #1\W\X\Y\Z% + \XINT_cmp_k #1#2!#5!#7!#9!% }% -\def\XINT_OneIfPositive_main #1#2#3#4% +\def\XINT_cmp_di\Z + \expandafter\XINT_cmp_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W {% - \xint_gob_til_Z #4\xint_OneIfPositive_terminated\Z - \XINT_OneIfPositive_onestep #1#2#3#4% + \XINT_cmp_k #1#2!#5!#7!% }% -\def\xint_OneIfPositive_terminated\Z\XINT_OneIfPositive_onestep\W\X\Y\Z { 0}% -\def\XINT_OneIfPositive_onestep #1#2#3#4% +\def\XINT_cmp_fi\Z + \expandafter\XINT_cmp_g\the\numexpr#1+1#2-#3.#4!#5!#6\W {% - \expandafter\XINT_OneIfPositive_check\the\numexpr #1#2#3#4\relax + \XINT_cmp_k #1#2!#5!% }% -\def\XINT_OneIfPositive_check #1% +\def\XINT_cmp_hi\Z + \expandafter\XINT_cmp_i\the\numexpr#1+1#2-#3.#4\W {% - \xint_gob_til_zero #1\xint_OneIfPositive_backtomain 0% - \XINT_OneIfPositive_finish #1% + \XINT_cmp_k #1#2!% }% -\def\XINT_OneIfPositive_finish #1\W\X\Y\Z{ 1}% -\def\xint_OneIfPositive_backtomain 0\XINT_OneIfPositive_finish 0% - {\XINT_OneIfPositive_main }% +%%%%%%%%%%%% +\def\XINT_cmp_k #1#2\W +{% + \xint_UDzerofork + #1{-1\relax \XINT_cmp_greater}% + 0{-1\relax \XINT_cmp_lessorequal}% + \krof +}% +\def\XINT_cmp_w #1-1#2{#2#11\Z!\W}% +\def\XINT_cmp_greater #1\Z!\W{ 1}% +\def\XINT_cmp_lessorequal 1#1!% + {\xint_gob_til_Z #1\XINT_cmp_equal\Z + \xint_gob_til_eightzeroes #1\XINT_cmp_continue 00000000% + \XINT_cmp_less }% +\def\XINT_cmp_less #1\W { -1}% +\def\XINT_cmp_continue 00000000\XINT_cmp_less {\XINT_cmp_lessorequal }% +\def\XINT_cmp_equal\Z\xint_gob_til_eightzeroes\Z\XINT_cmp_continue + 00000000\XINT_cmp_less\W { 0}% \def\xintEq {\romannumeral0\xinteq }\def\xinteq #1#2{\xintifeq{#1}{#2}{1}{0}}% \def\xintGt {\romannumeral0\xintgt }\def\xintgt #1#2{\xintifgt{#1}{#2}{1}{0}}% \def\xintLt {\romannumeral0\xintlt }\def\xintlt #1#2{\xintiflt{#1}{#2}{1}{0}}% @@ -482,80 +525,154 @@ {\XINT_xorof_a #2}% }% \def\XINT_xorof_e #1\Z #2{ #2}% -\def\xintGeq {\romannumeral0\xintgeq }% -\def\xintgeq #1% +\def\xintGeq {\romannumeral0\xintgeq }% +\def\xintgeq #1{\expandafter\XINT_geq\romannumeral0\xintnum{#1}\Z }% +\def\xintiiGeq {\romannumeral0\xintiigeq }% +\def\xintiigeq #1{\expandafter\XINT_iigeq\romannumeral-`0#1\Z }% +\def\XINT_iigeq #1#2\Z #3% {% - \expandafter\xint_geq\expandafter {\romannumeral0\xintnum{#1}}% + \expandafter\XINT_geq_fork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xint_geq #1#2% +\let\XINT_geq_pre \xintiigeq % TEMPORAIRE +\let\XINT_Geq \xintGeq % TEMPORAIRE ATTENTION FAIT xintNum +\def\XINT_geq #1#2\Z #3% {% - \expandafter\XINT_geq_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_geq_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\XINT_Geq #1#2{\romannumeral0\XINT_geq_fork #2\Z #1\Z }% -\def\XINT_geq_fork #1#2\Z #3#4\Z +\def\XINT_geq_fork #1#2% {% \xint_UDzerofork - #1\XINT_geq_secondiszero % |#1#2|=0 - #3\XINT_geq_firstiszero % |#1#2|>0 - 0{\xint_UDsignsfork - #1#3\XINT_geq_minusminus - #1-\XINT_geq_minusplus - #3-\XINT_geq_plusminus - --\XINT_geq_plusplus - \krof }% + #1\XINT_geq_firstiszero + #2\XINT_geq_secondiszero + 0{}% \krof - {#2}{#4}#1#3% + \xint_UDsignsfork + #1#2\XINT_geq_minusminus + #1-\XINT_geq_minusplus + #2-\XINT_geq_plusminus + --\XINT_geq_plusplus + \krof #1#2% +}% +\def\XINT_geq_firstiszero #1\krof 0#2#3\Z #4\Z + {\xint_UDzerofork #2{ 1}0{ 0}\krof }% +\def\XINT_geq_secondiszero #1\krof #20#3\Z #4\Z { 1}% +\def\XINT_geq_plusminus #1-{\XINT_geq_plusplus #1{}}% +\def\XINT_geq_minusplus -#1{\XINT_geq_plusplus {}#1}% +\def\XINT_geq_minusminus --{\XINT_geq_plusplus {}{}}% +\def\XINT_geq_plusplus #1#2#3\Z #4\Z {\XINT_geq_pp #1#4\Z #2#3\Z }% +\def\XINT_geq_pp #1\Z +{% + \expandafter\XINT_geq_pp_a + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X +}% +\def\XINT_geq_pp_a #1.#2\X #3\Z +{% + \expandafter\XINT_geq_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_geq_checklengths #1.#2.% +{% + \ifnum #1=#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + \XINT_geq_aa {\XINT_geq_distinctlengths {#1}{#2}} }% -\def\XINT_geq_secondiszero #1#2#3#4{ 1}% -\def\XINT_geq_firstiszero #1#2#3#4{ 0}% -\def\XINT_geq_plusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#3#1}}% -\def\XINT_geq_minusminus #1#2#3#4{\XINT_geq_pre {#2}{#1}}% -\def\XINT_geq_minusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#1}}% -\def\XINT_geq_plusminus #1#2#3#4{\XINT_geq_pre {#2}{#3#1}}% -\def\XINT_geq_pre #1% +\def\XINT_geq_distinctlengths #1#2#3\W #4\W {% - \expandafter\XINT_geq_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \ifnum #1>#2 + \expandafter\xint_firstoftwo + \else + \expandafter\xint_secondoftwo + \fi + { 1}{ 0}% }% -\def\XINT_geq_pre_b #1#2% +%%%%%%%%%%%% +\def\XINT_geq_aa {\expandafter\XINT_geq_w\the\numexpr\XINT_geq_a \xint_c_i }% +%%%%%%%%%%%% +\def\XINT_geq_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter\XINT_geq_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z + \XINT_geq_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_geq_A #1#2#3\W\X\Y\Z #4#5#6#7% +\def\XINT_geq_b #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_geq_az\W - \XINT_geq_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z + \xint_gob_til_Z #2\XINT_geq_bi \Z + \expandafter\XINT_geq_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_geq_B #1#2#3#4#5#6#7% +\def\XINT_geq_c 1#1#2.% {% - \xint_gob_til_W #4\xint_geq_bz\W - \XINT_geq_onestep #1#2{#7#6#5#4}{#3}% + 1#2\expandafter!\the\numexpr\XINT_geq_d #1% }% -\def\XINT_geq_onestep #1#2#3#4#5#6% +\def\XINT_geq_d #1#2#3!#4!% {% - \expandafter\XINT_geq_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% + \xint_gob_til_Z #2\XINT_geq_di \Z + \expandafter\XINT_geq_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_geq_backtoA #1#2#3.#4% +\def\XINT_geq_e 1#1#2.% {% - \XINT_geq_A #2{#3#4}% + 1#2\expandafter!\the\numexpr\XINT_geq_f #1% }% -\def\xint_geq_bz\W\XINT_geq_onestep #1\W\X\Y\Z { 1}% -\def\xint_geq_az\W\XINT_geq_B #1#2#3#4#5#6#7% +\def\XINT_geq_f #1#2#3!#4!% {% - \xint_gob_til_W #4\xint_geq_ez\W - \XINT_geq_Eenter #1% + \xint_gob_til_Z #2\XINT_geq_fi \Z + \expandafter\XINT_geq_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_geq_Eenter #1\W\X\Y\Z { 0}% -\def\xint_geq_ez\W\XINT_geq_Eenter #1% +\def\XINT_geq_g 1#1#2.% {% - \xint_UDzerofork - #1{ 0} % il y a une retenue - 0{ 1} % pas de retenue - \krof + 1#2\expandafter!\the\numexpr\XINT_geq_h #1% +}% +\def\XINT_geq_h #1#2#3!#4!% +{% + \xint_gob_til_Z #2\XINT_geq_hi \Z + \expandafter\XINT_geq_i\the\numexpr#1+1#4-#3-\xint_c_i.% +}% +\def\XINT_geq_i 1#1#2.% +{% + 1#2\expandafter!\the\numexpr\XINT_geq_a #1% +}% +\def\XINT_geq_bi\Z + \expandafter\XINT_geq_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W +{% + \XINT_geq_k #1#2!#5!#7!#9!% +}% +\def\XINT_geq_di\Z + \expandafter\XINT_geq_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W +{% + \XINT_geq_k #1#2!#5!#7!% +}% +\def\XINT_geq_fi\Z + \expandafter\XINT_geq_g\the\numexpr#1+1#2-#3.#4!#5!#6\W +{% + \XINT_geq_k #1#2!#5!% }% +\def\XINT_geq_hi\Z + \expandafter\XINT_geq_i\the\numexpr#1+1#2-#3.#4\W +{% + \XINT_geq_k #1#2!% +}% +%%%%%%%%%%%% +\def\XINT_geq_k #1#2\W +{% + \xint_UDzerofork + #1{-1\relax { 0}}% + 0{-1\relax { 1}}% + \krof +}% +\def\XINT_geq_w #1-1#2{#2}% \def\xintiMax {\romannumeral0\xintimax }% \def\xintimax #1% {% @@ -574,7 +691,6 @@ {% \expandafter\XINT_max_pre\expandafter {\romannumeral-`0#2}{#1}% }% -\let\xintMax\xintiMax \let\xintmax\xintimax % deprecated, should be only with xintfrac \def\XINT_max_pre #1#2{\XINT_max_fork #1\Z #2\Z {#2}{#1}}% \def\XINT_Max #1#2{\romannumeral0\XINT_max_fork #2\Z #1\Z {#1}{#2}}% \def\XINT_max_fork #1#2\Z #3#4\Z @@ -623,7 +739,6 @@ \def\XINT_imaxof_d #1\Z {\expandafter\XINT_imaxof_b\romannumeral0\xintimax {#1}}% \def\XINT_imaxof_e #1\Z #2\Z { #2}% -\let\xintMaxof\xintiMaxof \let\xintmaxof\xintimaxof \def\xintiMin {\romannumeral0\xintimin }% \def\xintimin #1% {% @@ -642,7 +757,6 @@ {% \expandafter\XINT_min_pre\expandafter {\romannumeral-`0#2}{#1}% }% -\let\xintMin\xintiMin \let\xintmin\xintimin % deprecated \def\XINT_min_pre #1#2{\XINT_min_fork #1\Z #2\Z {#2}{#1}}% \def\XINT_Min #1#2{\romannumeral0\XINT_min_fork #2\Z #1\Z {#1}{#2}}% \def\XINT_min_fork #1#2\Z #3#4\Z @@ -691,148 +805,30 @@ \def\XINT_iminof_d #1\Z {\expandafter\XINT_iminof_b\romannumeral0\xintimin {#1}}% \def\XINT_iminof_e #1\Z #2\Z { #2}% -\let\xintMinof\xintiMinof \let\xintminof\xintiminof \def\xintiiSum {\romannumeral0\xintiisum }% \def\xintiisum #1{\xintiisumexpr #1\relax }% \def\xintiiSumExpr {\romannumeral0\xintiisumexpr }% \def\xintiisumexpr {\expandafter\XINT_sumexpr\romannumeral-`0}% -\let\xintSum\xintiiSum \let\xintsum\xintiisum -\let\xintSumExpr\xintiiSumExpr \let\xintsumexpr\xintiisumexpr -\def\XINT_sumexpr {\XINT_sum_loop {0000}{0000}}% -\def\XINT_sum_loop #1#2#3% -{% - \expandafter\XINT_sum_checksign\romannumeral-`0#3\Z {#1}{#2}% -}% -\def\XINT_sum_checksign #1% -{% - \xint_gob_til_relax #1\XINT_sum_finished\relax - \xint_gob_til_zero #1\XINT_sum_skipzeroinput0% - \xint_UDsignfork - #1\XINT_sum_N - -{\XINT_sum_P #1}% - \krof -}% -\def\XINT_sum_finished #1\Z #2#3% -{% - \XINT_sub_A 1{}#3\W\X\Y\Z #2\W\X\Y\Z -}% -\def\XINT_sum_skipzeroinput #1\krof #2\Z {\XINT_sum_loop }% -\def\XINT_sum_P #1\Z #2% -{% - \expandafter\XINT_sum_loop\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT_sum_N #1\Z #2#3% -{% - \expandafter\XINT_sum_NN\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3\W\X\Y\Z }{#2}% -}% -\def\XINT_sum_NN #1#2{\XINT_sum_loop {#2}{#1}}% +\def\XINT_sumexpr {\XINT_sum_loop_a 0\Z }% +\def\XINT_sum_loop_a #1\Z #2% + {\expandafter\XINT_sum_loop_b \romannumeral-`0#2\Z #1\Z \Z}% +\def\XINT_sum_loop_b #1% + {\xint_gob_til_relax #1\XINT_sum_finished\relax\XINT_sum_loop_c #1}% +\def\XINT_sum_loop_c + {\expandafter\XINT_sum_loop_a\romannumeral0\XINT_add_fork }% +\def\XINT_sum_finished #1\Z #2\Z \Z { #2}% \def\xintiiPrd {\romannumeral0\xintiiprd }% \def\xintiiprd #1{\xintiiprdexpr #1\relax }% -\let\xintPrd\xintiiPrd -\let\xintprd\xintiiprd \def\xintiiPrdExpr {\romannumeral0\xintiiprdexpr }% \def\xintiiprdexpr {\expandafter\XINT_prdexpr\romannumeral-`0}% -\let\xintPrdExpr\xintiiPrdExpr -\let\xintprdexpr\xintiiprdexpr \def\XINT_prdexpr {\XINT_prod_loop_a 1\Z }% \def\XINT_prod_loop_a #1\Z #2% - {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}% + {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}% \def\XINT_prod_loop_b #1% {\xint_gob_til_relax #1\XINT_prod_finished\relax\XINT_prod_loop_c #1}% \def\XINT_prod_loop_c {\expandafter\XINT_prod_loop_a\romannumeral0\XINT_mul_fork }% -\def\XINT_prod_finished #1\Z #2\Z \Z { #2}% -\def\xintiFac {\romannumeral0\xintifac }% -\def\xintifac #1% -{% - \expandafter\XINT_fac_fork\expandafter{\the\numexpr #1}% -}% -\let\xintFac\xintiFac \let\xintfac\xintifac -\def\XINT_fac_fork #1% -{% - \ifcase\XINT_cntSgn #1\Z - \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }% - \or - \expandafter\XINT_fac_checklength - \else - \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \fi - {#1}% -}% -\def\XINT_fac_checklength #1% -{% - \ifnum #1>100000 - \xint_afterfi{\expandafter\xintError:FactorialOfTooBigNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \else - \xint_afterfi{\ifnum #1>\xint_c_ixixixix - \expandafter\XINT_fac_big_loop - \else - \expandafter\XINT_fac_loop - \fi }% - \fi - {#1}% -}% -\def\XINT_fac_big_loop #1{\XINT_fac_big_loop_main {10000}{#1}{}}% -\def\XINT_fac_big_loop_main #1#2#3% -{% - \ifnum #1<#2 - \expandafter - \XINT_fac_big_loop_main - \expandafter - {\the\numexpr #1+1\expandafter }% - \else - \expandafter\XINT_fac_big_docomputation - \fi - {#2}{#3{#1}}% -}% -\def\XINT_fac_big_docomputation #1#2% -{% - \expandafter \XINT_fac_bigcompute_loop \expandafter - {\romannumeral0\XINT_fac_loop {9999}}#2\relax -}% -\def\XINT_fac_bigcompute_loop #1#2% -{% - \xint_gob_til_relax #2\XINT_fac_bigcompute_end\relax - \expandafter\XINT_fac_bigcompute_loop\expandafter - {\expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W }% -}% -\def\XINT_fac_bigcompute_end #1#2#3#4#5% -{% - \XINT_fac_bigcompute_end_ #5% -}% -\def\XINT_fac_bigcompute_end_ #1\R #2\Z \W\X\Y\Z #3\W\X\Y\Z { #3}% -\def\XINT_fac_loop #1{\XINT_fac_loop_main 1{1000}{#1}}% -\def\XINT_fac_loop_main #1#2#3% -{% - \ifnum #3>#1 - \else - \expandafter\XINT_fac_loop_exit - \fi - \expandafter\XINT_fac_loop_main\expandafter - {\the\numexpr #1+1\expandafter }\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z }% - {#3}% -}% -\def\XINT_fac_loop_exit #1#2#3#4#5#6#7% -{% - \XINT_fac_loop_exit_ #6% -}% -\def\XINT_fac_loop_exit_ #1#2#3% -{% - \XINT_mul_M -}% +\def\XINT_prod_finished\relax\XINT_prod_loop_c #1\Z #2\Z \Z { #2}% \def\xintiiMON {\romannumeral0\xintiimon }% \def\xintiimon #1% {% @@ -1251,10 +1247,10 @@ \def\xintisqrt {\expandafter\XINT_sqrt_post\romannumeral0\xintisquareroot }% \def\xintiisqrt {\expandafter\XINT_sqrt_post\romannumeral0\xintiisquareroot }% \def\xintiisqrtr {\expandafter\XINT_sqrtr_post\romannumeral0\xintiisquareroot }% -\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }% -\def\XINT_sqrtr_post #1#2{\xintiiifLt {#2}{#1}% Lt <-> a<b - { #1}{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }}% -\def\xintisquareroot #1{\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z }% +\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\Z }% +\def\XINT_sqrtr_post #1#2{\xintiiifLt {#2}{#1}{ #1}{\XINT_dec_pos #1\Z}}% +\def\xintisquareroot #1% + {\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z }% \def\xintiisquareroot #1{\expandafter\XINT_sqrt_checkin\romannumeral-`0#1\Z }% \def\XINT_sqrt_checkin #1% {% @@ -1400,8 +1396,7 @@ \def\XINT_sqrt_big_g #1#2% {% \expandafter\XINT_sqrt_big_j - \romannumeral0\xintiidivision{#1}% - {\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% + \romannumeral0\xintiidivision{#1}{\romannumeral0\XINT_dbl_pos #2\Z}{#2}% }% \def\XINT_sqrt_big_j #1% {% @@ -1427,6 +1422,15 @@ {\expandafter\XINT_iie\the\numexpr #2\expandafter.\expandafter{\romannumeral-`0#1}}% \def\XINT_iie #1.#2{\ifnum#1>\xint_c_ \xint_dothis{\xint_dsh {#2}{-#1}}\fi \xint_orthat{ #2}}% +\catcode`! 11 +\def\xintMax {\Did_you_mean_iiMax?or_load_xintfrac!}% +\def\xintMin {\Did_you_mean_iiMin?or_load_xintfrac!}% +\def\xintMaxof {\Did_you_mean_iMaxof?or_load_xintfrac!}% +\def\xintMinof {\Did_you_mean_iMinof?or_load_xintfrac!}% +\def\xintSum {\Did_you_mean_iiSum?or_load_xintfrac!}% +\def\xintPrd {\Did_you_mean_iiPrd?or_load_xintfrac!}% +\def\xintPrdExpr {\Did_you_mean_iiPrdExpr?or_load_xintfrac!}% +\def\xintSumExpr {\Did_you_mean_iiSumExpr?or_load_xintfrac!}% \XINT_restorecatcodes_endinput% \endinput %% diff --git a/Master/texmf-dist/tex/generic/xint/xintbinhex.sty b/Master/texmf-dist/tex/generic/xint/xintbinhex.sty index 3876c864057..1fb50a91f74 100644 --- a/Master/texmf-dist/tex/generic/xint/xintbinhex.sty +++ b/Master/texmf-dist/tex/generic/xint/xintbinhex.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintbinhex: Expandable binary and hexadecimal conversions %% --------------------------------------------------------------- @@ -70,15 +70,10 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintbinhex}% - [2015/09/12 v1.1c Expandable binary and hexadecimal conversions (jfB)]% -\chardef\xint_c_xvi 16 -\chardef\xint_c_ii^vii 128 -\mathchardef\xint_c_ii^viii 256 -\mathchardef\xint_c_ii^xii 4096 -\newcount\xint_c_ii^xv \xint_c_ii^xv 32768 -\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536 + [2015/10/10 v1.2 Expandable binary and hexadecimal conversions (jfB)]% +\newcount\xint_c_ii^xv \xint_c_ii^xv 32768 +\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536 \newcount\xint_c_x^v \xint_c_x^v 100000 -\newcount\xint_c_x^ix \xint_c_x^ix 1000000000 \def\XINT_tmpa #1{\ifx\relax#1\else \expandafter\edef\csname XINT_sdth_#1\endcsname {\ifcase #1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or @@ -172,6 +167,35 @@ \csname XINT_sdtb_#1\expandafter\expandafter\expandafter\endcsname \csname XINT_sdtb_\the\numexpr #2-\xint_c_xvi*#1\endcsname }% +\def\XINT_OQ #1#2#3#4#5#6#7#8#9% +{% + \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}% +}% +\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z +{% + \XINT_OQ_end_b #1\Z +}% +\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8% +{% + \xint_gob_til_R + #8\XINT_OQ_end_viii + #7\XINT_OQ_end_vii + #6\XINT_OQ_end_vi + #5\XINT_OQ_end_v + #4\XINT_OQ_end_iv + #3\XINT_OQ_end_iii + #2\XINT_OQ_end_ii + \R\XINT_OQ_end_i + \Z #2#3#4#5#6#7#8% +}% +\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% +\def\XINT_OQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}% +\def\XINT_OQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}% +\def\XINT_OQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}% +\def\XINT_OQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}% +\def\XINT_OQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% +\def\XINT_OQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% +\def\XINT_OQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% \def\xintDecToHex {\romannumeral0\xintdectohex }% \def\xintdectohex #1% {\expandafter\XINT_dth_checkin\romannumeral-`0#1\W\W\W\W \T}% diff --git a/Master/texmf-dist/tex/generic/xint/xintcfrac.sty b/Master/texmf-dist/tex/generic/xint/xintcfrac.sty index f3585235d2f..4a1656c4cb0 100644 --- a/Master/texmf-dist/tex/generic/xint/xintcfrac.sty +++ b/Master/texmf-dist/tex/generic/xint/xintcfrac.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintcfrac: Expandable continued fractions with xint package %% --------------------------------------------------------------- @@ -70,7 +70,7 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintcfrac}% - [2015/09/12 v1.1c Expandable continued fractions with xint package (jfB)]% + [2015/10/10 v1.2 Expandable continued fractions with xint package (jfB)]% \def\xintCFrac {\romannumeral0\xintcfrac }% \def\xintcfrac #1% {% @@ -483,8 +483,8 @@ \expandafter\XINT_ctf_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_ctf_loop_c #1#2% {% @@ -517,8 +517,8 @@ \def\XINT_icstf_loop_b #1.#2#3#4#5% {% \expandafter\XINT_icstf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {#2}{#3}% }% \def\XINT_icstf_loop_c #1#2% @@ -545,8 +545,8 @@ \expandafter\XINT_gctf_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_gctf_loop_c #1#2% {% @@ -604,8 +604,8 @@ \def\XINT_igctf_loop_b #1.#2#3#4#5% {% \expandafter\XINT_igctf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {#2}{#3}% }% \def\XINT_igctf_loop_c #1#2% @@ -658,8 +658,8 @@ \expandafter\XINT_ctcv_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_ctcv_loop_c #1#2% {% @@ -698,8 +698,8 @@ \def\XINT_icstcv_loop_b #1.#2#3#4#5% {% \expandafter\XINT_icstcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {{#2}{#3}}% }% \def\XINT_icstcv_loop_c #1#2% @@ -732,8 +732,8 @@ \expandafter\XINT_gctcv_loop_c\expandafter {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #6\Z}{\XINT_mul_fork #1\Z #4\Z}}% + {\romannumeral0\xintiiadd {\XINT_mul_fork #2\Z #5\Z}{\XINT_mul_fork #1\Z #3\Z}}% }% \def\XINT_gctcv_loop_c #1#2% {% @@ -801,8 +801,8 @@ \def\XINT_igctcv_loop_b #1.#2#3#4#5% {% \expandafter\XINT_igctcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% + {\romannumeral0\xintiiadd {#5}{\XINT_mul_fork #1\Z #3\Z}}% + {\romannumeral0\xintiiadd {#4}{\XINT_mul_fork #1\Z #2\Z}}% {{#2}{#3}}% }% \def\XINT_igctcv_loop_c #1#2% diff --git a/Master/texmf-dist/tex/generic/xint/xintcore.sty b/Master/texmf-dist/tex/generic/xint/xintcore.sty index d9bbb3779f8..3352af4f189 100644 --- a/Master/texmf-dist/tex/generic/xint/xintcore.sty +++ b/Master/texmf-dist/tex/generic/xint/xintcore.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintcore: Expandable arithmetic on big integers %% --------------------------------------------------------------- @@ -70,160 +70,15 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintcore}% - [2015/09/12 v1.1c Expandable arithmetic on big integers (jfB)]% -\def\xint_minus_thenstop { -}% -\def\xint_gob_til_zeros_iii #1000{}% -\def\xint_gob_til_zeros_iv #10000{}% -\def\xint_gob_til_one #11{}% -\def\xint_gob_til_G #1G{}% -\def\xint_gob_til_minus #1-{}% -\def\xint_gob_til_relax #1\relax {}% -\def\xint_exchangetwo_keepbraces #1#2{{#2}{#1}}% -\def\xint_exchangetwo_keepbraces_thenstop #1#2{ {#2}{#1}}% -\def\xint_UDzerofork #10#2#3\krof {#2}% -\def\xint_UDsignfork #1-#2#3\krof {#2}% -\def\xint_UDwfork #1\W#2#3\krof {#2}% -\def\xint_UDzerosfork #100#2#3\krof {#2}% -\def\xint_UDonezerofork #110#2#3\krof {#2}% -\def\xint_UDsignsfork #1--#2#3\krof {#2}% -\chardef\xint_c_ix 9 -\chardef\xint_c_x 10 -\chardef\xint_c_ii^v 32 % not used in xint, common to xintfrac and xintbinhex -\chardef\xint_c_ii^vi 64 -\mathchardef\xint_c_ixixixix 9999 -\mathchardef\xint_c_x^iv 10000 -\newcount\xint_c_x^viii \xint_c_x^viii 100000000 -\def\XINT_RQ #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #9\XINT_RQ_end_a\R\XINT_RQ {#9#8#7#6#5#4#3#2#1}% -}% -\def\XINT_RQ_end_a\R\XINT_RQ #1#2\Z -{% - \XINT_RQ_end_b #1\Z -}% -\def\XINT_RQ_end_b #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R - #8\XINT_RQ_end_viii - #7\XINT_RQ_end_vii - #6\XINT_RQ_end_vi - #5\XINT_RQ_end_v - #4\XINT_RQ_end_iv - #3\XINT_RQ_end_iii - #2\XINT_RQ_end_ii - \R\XINT_RQ_end_i - \Z #2#3#4#5#6#7#8% -}% -\def\XINT_RQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_RQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#9000}% -\def\XINT_RQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#900}% -\def\XINT_RQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#90}% -\def\XINT_RQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#9}% -\def\XINT_RQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_RQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_RQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% -\def\XINT_OQ #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}% -}% -\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z -{% - \XINT_OQ_end_b #1\Z -}% -\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R - #8\XINT_OQ_end_viii - #7\XINT_OQ_end_vii - #6\XINT_OQ_end_vi - #5\XINT_OQ_end_v - #4\XINT_OQ_end_iv - #3\XINT_OQ_end_iii - #2\XINT_OQ_end_ii - \R\XINT_OQ_end_i - \Z #2#3#4#5#6#7#8% -}% -\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_OQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}% -\def\XINT_OQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}% -\def\XINT_OQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}% -\def\XINT_OQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}% -\def\XINT_OQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_OQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_OQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% -\def\XINT_SQ #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R #8\XINT_SQ_end_a\R\XINT_SQ {#8#7#6#5#4#3#2#1}% -}% -\def\XINT_SQ_end_a\R\XINT_SQ #1#2\Z -{% - \XINT_SQ_end_b #1\Z -}% -\def\XINT_SQ_end_b #1#2#3#4#5#6#7% -{% - \xint_gob_til_R - #7\XINT_SQ_end_vii - #6\XINT_SQ_end_vi - #5\XINT_SQ_end_v - #4\XINT_SQ_end_iv - #3\XINT_SQ_end_iii - #2\XINT_SQ_end_ii - \R\XINT_SQ_end_i - \Z #2#3#4#5#6#7% -}% -\def\XINT_SQ_end_vii #1\Z #2#3#4#5#6#7#8\Z { #8}% -\def\XINT_SQ_end_vi #1\Z #2#3#4#5#6#7#8\Z { #7#8000000}% -\def\XINT_SQ_end_v #1\Z #2#3#4#5#6#7#8\Z { #6#7#800000}% -\def\XINT_SQ_end_iv #1\Z #2#3#4#5#6#7#8\Z { #5#6#7#80000}% -\def\XINT_SQ_end_iii #1\Z #2#3#4#5#6#7#8\Z { #4#5#6#7#8000}% -\def\XINT_SQ_end_ii #1\Z #2#3#4#5#6#7#8\Z { #3#4#5#6#7#800}% -\def\XINT_SQ_end_i \Z #1#2#3#4#5#6#7\Z { #1#2#3#4#5#6#70}% -\edef\xint_cleanupzeros_andstop #1#2#3#4% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax -}% -\def\xint_cleanupzeros_nostop #1#2#3#4% -{% - \the\numexpr #1#2#3#4\relax -}% -\def\XINT_rev_andcuz #1% -{% - \expandafter\xint_cleanupzeros_andstop - \romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_cuz #1% -{% - \XINT_cuz_loop #1\W\W\W\W\W\W\W\Z% -}% -\def\XINT_cuz_loop #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W #8\xint_cuz_end_a\W - \xint_gob_til_Z #8\xint_cuz_end_A\Z - \XINT_cuz_check_a {#1#2#3#4#5#6#7#8}% -}% -\def\xint_cuz_end_a #1\XINT_cuz_check_a #2% -{% - \xint_cuz_end_b #2% -}% -\edef\xint_cuz_end_b #1#2#3#4#5\Z -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax -}% -\def\xint_cuz_end_A \Z\XINT_cuz_check_a #1{ 0}% -\def\XINT_cuz_check_a #1% -{% - \expandafter\XINT_cuz_check_b\the\numexpr #1\relax -}% -\def\XINT_cuz_check_b #1% -{% - \xint_gob_til_zero #1\xint_cuz_backtoloop 0\XINT_cuz_stop #1% -}% -\def\XINT_cuz_stop #1\W #2\Z{ #1}% -\def\xint_cuz_backtoloop 0\XINT_cuz_stop 0{\XINT_cuz_loop }% + [2015/10/10 v1.2 Expandable arithmetic on big integers (jfB)]% +\ifdefined\m@ne\let\xint_c_mone\m@ne + \else\csname newcount\endcsname\xint_c_mone \xint_c_mone -1 \fi +\newcount\xint_c_x^viii \xint_c_x^viii 100000000 +\newcount\xint_c_x^ix \xint_c_x^ix 1000000000 +\newcount\xint_c_x^viii_mone \xint_c_x^viii_mone 99999999 +\newcount\xint_c_xii_e_viii \xint_c_xii_e_viii 1200000000 +\newcount\xint_c_xi_e_viii_mone \xint_c_xi_e_viii_mone 1099999999 +\newcount\xint_c_xii_e_viii_mone\xint_c_xii_e_viii_mone 1199999999 \def\xintiNum {\romannumeral0\xintinum }% \def\xintinum #1% {% @@ -263,6 +118,195 @@ \def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }% \def\XINT_num_keepsign_b #1{\XINT_num_loop -}% \def\XINT_num_finish #1\xint_relax #2\Z { #1}% +\edef\XINT_cuz_small #1#2#3#4#5#6#7#8% +{% + \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax +}% +%%%%%%%%%%%% +\def\XINT_cuz #1#2#3#4#5#6#7#8#9% +{% + \xint_gob_til_R #9\XINT_cuz_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_cuz_z 00000000% + \XINT_cuz_clean #1#2#3#4#5#6#7#8#9% +}% +\edef\XINT_cuz_clean #1#2#3#4#5#6#7#8#9\R + {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax #9}% +\edef\XINT_cuz_e\R #1\XINT_cuz_clean #2\R + {\noexpand\expandafter\space\noexpand\the\numexpr #2\relax }% +\def\XINT_cuz_z 00000000\XINT_cuz_clean 00000000{\XINT_cuz }% +%%%%%%%%%%%% +\def\XINT_cuz_byviii #1#2#3#4#5#6#7#8#9% +{% + \xint_gob_til_R #9\XINT_cuz_byviii_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_cuz_byviii_z 00000000% + \XINT_cuz_byviii_clean #1#2#3#4#5#6#7#8#9% +}% +\def\XINT_cuz_byviii_clean #1\R { #1}% +\def\XINT_cuz_byviii_e\R #1\XINT_cuz_byviii_clean #2\R{ #2}% +\def\XINT_cuz_byviii_z 00000000\XINT_cuz_byviii_clean 00000000{\XINT_cuz_byviii}% +\def\XINT_zeroes_forviii #1#2#3#4#5#6#7#8% +{% + \xint_gob_til_R #8\XINT_zeroes_forviii_end\R\XINT_zeroes_forviii +}% +\edef\XINT_zeroes_forviii_end\R\XINT_zeroes_forviii #1#2#3#4#5#6#7#8#9\W +{% + \noexpand\expandafter\space\noexpand\xint_gob_til_one #2#3#4#5#6#7#8% +}% +%%%%%%%%%%%% +\def\XINT_rsepbyviii #1#2#3#4#5#6#7#8% +{% + \XINT_rsepbyviii_b {#1#2#3#4#5#6#7#8}% +}% +\def\XINT_rsepbyviii_b #1#2#3#4#5#6#7#8#9% +{% + #2#3#4#5#6#7#8#9\expandafter!\the\numexpr + 1#1\expandafter.\the\numexpr 1\XINT_rsepbyviii +}% +\def\XINT_rsepbyviii_end_B #1\relax #2#3{#2.}% +\def\XINT_rsepbyviii_end_A #11#2\expandafter #3\relax #4#5{#2.1#5.}% +%%%%%%%%%%%% +\def\XINT_sepandrev +{% + \expandafter\XINT_sepandrev_a\the\numexpr 1\XINT_rsepbyviii +}% +\def\XINT_sepandrev_a {\XINT_sepandrev_b {}}% +\def\XINT_sepandrev_b #1#2.#3.#4.#5.#6.#7.#8.#9.% +{% + \xint_gob_til_R #9\XINT_sepandrev_end\R + \XINT_sepandrev_b {#9!#8!#7!#6!#5!#4!#3!#2!#1}% +}% +\def\XINT_sepandrev_end\R\XINT_sepandrev_b #1#2\W {\XINT_sepandrev_done #1}% +\def\XINT_sepandrev_done #11#2!{ }% +%%%%%%%%%%%% +\def\XINT_sepandrev_andcount +{% + \expandafter\XINT_sepandrev_andcount_a\the\numexpr 1\XINT_rsepbyviii +}% +\def\XINT_sepandrev_andcount_a {\XINT_sepandrev_andcount_b 0.{}}% +\def\XINT_sepandrev_andcount_b #1.#2#3.#4.#5.#6.#7.#8.#9.% +{% + \xint_gob_til_R #9\XINT_sepandrev_andcount_end\R + \expandafter\XINT_sepandrev_andcount_b \the\numexpr #1+\xint_c_xiv.% + {#9!#8!#7!#6!#5!#4!#3!#2}% +}% +\def\XINT_sepandrev_andcount_end\R + \expandafter\XINT_sepandrev_andcount_b\the\numexpr #1+\xint_c_xiv.#2#3#4\W +{\expandafter\XINT_sepandrev_andcount_done\the\numexpr \xint_c_ii*#3+#1.#2}% +\edef\XINT_sepandrev_andcount_done #1.#21#3!% + {\noexpand\expandafter\space\noexpand\the\numexpr #1-#3.}% +\def\XINT_sepbyviii_andcount +{% + \expandafter\XINT_sepbyviii_andcount_a\the\numexpr\XINT_sepbyviii +}% +\def\XINT_sepbyviii #1#2#3#4#5#6#7#8% +{% + 1#1#2#3#4#5#6#7#8\expandafter!\the\numexpr\XINT_sepbyviii +}% +\def\XINT_sepbyviii_end #1\relax {\relax\XINT_sepbyviii_andcount_end!}% +\def\XINT_sepbyviii_andcount_a {\XINT_sepbyviii_andcount_b \xint_c_.}% +\def\XINT_sepbyviii_andcount_b #1.#2!#3!#4!#5!#6!#7!#8!#9!% +{% + #2\expandafter!\the\numexpr#3\expandafter!\the\numexpr#4\expandafter + !\the\numexpr#5\expandafter!\the\numexpr#6\expandafter!\the\numexpr + #7\expandafter + !\the\numexpr#8\expandafter!\the\numexpr#9\expandafter!\the\numexpr + \expandafter\XINT_sepbyviii_andcount_b\the\numexpr #1+\xint_c_viii.% +}% +\def\XINT_sepbyviii_andcount_end #1\XINT_sepbyviii_andcount_b\the\numexpr + #2+\xint_c_viii.#3#4\W {\expandafter.\the\numexpr #2+#3.}% +%%%%%%%%%%%% +\def\XINT_rev_nounsep #1#2!#3!#4!#5!#6!#7!#8!#9!% +{% + \xint_gob_til_R #9\XINT_rev_nounsep_end\R + \XINT_rev_nounsep {#9!#8!#7!#6!#5!#4!#3!#2!#1}% +}% +\def\XINT_rev_nounsep_end\R\XINT_rev_nounsep #1#2\W {\XINT_rev_nounsep_done #1}% +\def\XINT_rev_nounsep_done #11{ 1}% +%%%%%%%%%%%% +\def\XINT_sepbyviii_Z #1#2#3#4#5#6#7#8% +{% + 1#1#2#3#4#5#6#7#8\expandafter!\the\numexpr\XINT_sepbyviii_Z +}% +\def\XINT_sepbyviii_Z_end #1\relax {\relax\Z!}% +%%%%%%%%%%%% +\def\XINT_unsep_cuzsmall #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_unsep_cuzsmall_end\R + \XINT_unsep_cuzsmall {#1#2#3#4#5#6#7#8#9}% +}% +\def\XINT_unsep_cuzsmall_end\R + \XINT_unsep_cuzsmall #1{\XINT_unsep_cuzsmall_done #1}% +\def\XINT_unsep_cuzsmall_done #1\R #2\W{\XINT_cuz_small #1}% +\def\XINT_unsep_delim {1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W}% +%%%%%%%%%%%% +\def\XINT_div_unsepQ #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_div_unsepQ_end\R + \XINT_div_unsepQ {#1#2#3#4#5#6#7#8#9}% +}% +\def\XINT_div_unsepQ_end\R\XINT_div_unsepQ #1{\XINT_div_unsepQ_x #1}% +\def\XINT_div_unsepQ_x #1#2#3#4#5#6#7#8#9% +{% + \xint_gob_til_R #9\XINT_div_unsepQ_e \R + \xint_gob_til_eightzeroes #1#2#3#4#5#6#7#8\XINT_div_unsepQ_y 00000000% + \expandafter\XINT_div_unsepQ_done \the\numexpr #1#2#3#4#5#6#7#8.#9% +}% +\def\XINT_div_unsepQ_e\R\xint_gob_til_eightzeroes #1\XINT_div_unsepQ_y #2\W + {\the\numexpr #1\relax \Z}% +\def\XINT_div_unsepQ_y #1.#2\R #3\W{\XINT_cuz_small #2\Z}% +\def\XINT_div_unsepQ_done #1.#2\R #3\W { #1#2\Z}% +%%%%%%%%%%%% +\def\XINT_div_unsepR #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_div_unsepR_end\R + \XINT_div_unsepR {#1#2#3#4#5#6#7#8#9}% +}% +\def\XINT_div_unsepR_end\R\XINT_div_unsepR #1{\XINT_div_unsepR_done #1}% +\def\XINT_div_unsepR_done #1\R #2\W {\XINT_cuz #1\R}% +%%%%%%%%%%%% +\def\XINT_unrevbyviii #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_unrevbyviii_a\R + \XINT_unrevbyviii {#9#8#7#6#5#4#3#2#1}% +}% +\edef\XINT_unrevbyviii_a\R\XINT_unrevbyviii #1#2\W + {\noexpand\expandafter\space + \noexpand\romannumeral-`0\noexpand\xint_gob_til_Z #1}% +\def\XINT_smallunrevbyviii 1#1!1#2!1#3!1#4!1#5!1#6!1#7!1#8!#9\W% +{% + \expandafter\XINT_cuz_small\xint_gob_til_Z #8#7#6#5#4#3#2#1% +}% +\def\XINT_microrevsep #1#2#3#4#5#6#7#8% +{% + 1#8#7#6#5#4#3#2#1\expandafter!\the\numexpr\XINT_microrevsep +}% +\def\XINT_microrevsep_end #1\W #2\expandafter #3\Z{#2!}% +\def\xintReverseDigits {\romannumeral0\xintreversedigits }% +\def\xintreversedigits #1{\expandafter\XINT_reversedigits\romannumeral-`0#1\Z}% +\def\XINT_reversedigits #1% +{% + \xint_UDsignfork + #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_reversedigits_a}% + -{\XINT_reversedigits_a #1}% + \krof +}% +\def\XINT_reversedigits_a #1\Z +{% + \expandafter\XINT_revdigits_a\the\numexpr\expandafter\XINT_microrevsep + \romannumeral-`0#1{\XINT_microrevsep_end\W}\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end\Z + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_revdigits_a {\XINT_revdigits_b {}}% +\def\XINT_revdigits_b #11#2!1#3!1#4!1#5!1#6!1#7!1#8!1#9!% +{% + \xint_gob_til_R #9\XINT_revdigits_end\R + \XINT_revdigits_b {#9#8#7#6#5#4#3#2#1}% +}% +\edef\XINT_revdigits_end\R\XINT_revdigits_b #1#2\W + {\noexpand\expandafter\space\noexpand\xint_gob_til_Z #1}% \def\xintiiSgn {\romannumeral0\xintiisgn }% \def\xintiisgn #1% {% @@ -293,7 +337,7 @@ {% \xint_UDzerominusfork #1-\xint_c_ - 0#1\m@ne % I will not allocate a count only for -1? + 0#1\xint_c_mone 0-\xint_c_i \krof }% @@ -307,7 +351,6 @@ {% \expandafter\XINT_opp \romannumeral0\xintnum{#1}% }% -\let\xintOpp\xintiOpp \let\xintopp\xintiopp \def\XINT_Opp #1{\romannumeral0\XINT_opp #1}% \def\XINT_opp #1% {% @@ -327,7 +370,6 @@ {% \expandafter\XINT_abs \romannumeral0\xintnum{#1}% }% -\let\xintAbs\xintiAbs \let\xintabs\xintiabs \def\XINT_Abs #1{\romannumeral0\XINT_abs #1}% \def\XINT_abs #1% {% @@ -355,27 +397,26 @@ 0-{ #1}% positive \krof }% +\def\xintLDg {\romannumeral0\xintldg }% +\def\xintldg #1{\xintiildg {\xintNum{#1}}}% \def\xintiiLDg {\romannumeral0\xintiildg }% \def\xintiildg #1% {% - \expandafter\XINT_ldg\expandafter {\romannumeral-`0#1}% -}% -\def\xintLDg {\romannumeral0\xintldg }% -\def\xintldg #1% -{% - \expandafter\XINT_ldg\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\XINT_LDg #1{\romannumeral0\XINT_ldg {#1}}% -\def\XINT_ldg #1% -{% - \expandafter\XINT_ldg_\romannumeral0\xintreverseorder {#1}\Z -}% -\def\XINT_ldg_ #1#2\Z{ #1}% + \expandafter\XINT_ldg_done\romannumeral0% + \expandafter\XINT_revdigits_a\the\numexpr\expandafter\XINT_microrevsep + \romannumeral0\expandafter\XINT_abs + \romannumeral-`0#1{\XINT_microrevsep_end\W}\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end + \XINT_microrevsep_end\XINT_microrevsep_end\Z + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W + \Z +}% +\def\XINT_ldg_done #1#2\Z { #1}% \def\xintDouble {\romannumeral0\xintdouble }% \def\xintdouble #1% {% - \expandafter\XINT_dbl\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W + \expandafter\XINT_dbl\romannumeral-`0#1\Z }% \def\XINT_dbl #1% {% @@ -385,37 +426,27 @@ 0-{\XINT_dbl_pos #1}% \krof }% -\def\XINT_dbl_zero #1\Z \W\W\W\W\W\W\W { 0}% +\def\XINT_dbl_zero #1\Z { 0}% \def\XINT_dbl_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dbl_pos }% -\def\XINT_dbl_pos -{% - \expandafter\XINT_dbl_a \expandafter{\expandafter}\expandafter 0% - \romannumeral0\XINT_SQ {}% -}% -\def\XINT_dbl_a #1#2#3#4#5#6#7#8#9% +\def\XINT_dbl_pos #1\Z {% - \xint_gob_til_W #9\XINT_dbl_end_a\W - \expandafter\XINT_dbl_b - \the\numexpr \xint_c_x^viii+#2+\xint_c_ii*#9#8#7#6#5#4#3\relax {#1}% + \expandafter\XINT_dbl_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W 1\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_dbl_b 1#1#2#3#4#5#6#7#8#9% +\def\XINT_dbl_pos_aa {% - \XINT_dbl_a {#2#3#4#5#6#7#8#9}{#1}% -}% -\def\XINT_dbl_end_a #1+#2+#3\relax #4% -{% - \expandafter\XINT_dbl_end_b #2#4% -}% -\edef\XINT_dbl_end_b #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax + \expandafter\XINT_mul_out\the\numexpr\XINT_verysmallmul 0.2!% }% \def\xintHalf {\romannumeral0\xinthalf }% \def\xinthalf #1% {% - \expandafter\XINT_half\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W + \expandafter\XINT_half\romannumeral-`0#1\Z }% \def\XINT_half #1% {% @@ -425,48 +456,34 @@ 0-{\XINT_half_pos #1}% \krof }% -\def\XINT_half_zero #1\Z \W\W\W\W\W\W\W { 0}% +\def\XINT_half_zero #1\Z { 0}% \def\XINT_half_neg {\expandafter\XINT_opp\romannumeral0\XINT_half_pos }% -\def\XINT_half_pos {\expandafter\XINT_half_a\romannumeral0\XINT_SQ {}}% -\def\XINT_half_a #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W #8\XINT_half_dont\W - \expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#7#6#5#4#3#2#1\relax #8% -}% -\edef\XINT_half_dont\W\expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#1#2#3#4#5#6#7\relax \W\W\W\W\W\W\W -{% - \noexpand\expandafter\space - \noexpand\the\numexpr (#1#2#3#4#5#6#7+\xint_c_i)/\xint_c_ii-\xint_c_i \relax -}% -\def\XINT_half_b 1#1#2#3#4#5#6#7#8% -{% - \XINT_half_c {#2#3#4#5#6#7}{#1}% -}% -\def\XINT_half_c #1#2#3#4#5#6#7#8#9% +\def\XINT_half_pos #1\Z {% - \xint_gob_til_W #3\XINT_half_end_a #2\W - \expandafter\XINT_half_d - \the\numexpr \xint_c_x^viii+\xint_c_v*#9#8#7#6#5#4#3+#2\relax {#1}% + \expandafter\XINT_half_pos_a + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + 1\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_half_d 1#1#2#3#4#5#6#7#8#9% +\def\XINT_half_pos_a + {\expandafter\XINT_half_pos_b\the\numexpr\XINT_verysmallmul 0.5!}% +\def\XINT_half_pos_b 1#1#2#3#4#5#6#7#8!1#9% {% - \XINT_half_c {#2#3#4#5#6#7#8#9}{#1}% + \xint_gob_til_Z #9\XINT_half_small \Z + \XINT_mul_out 1#1#2#3#4#5#6#7!1#9% }% -\def\XINT_half_end_a #1\W #2\relax #3% +\edef\XINT_half_small \Z\XINT_mul_out 1#1!#2\W {% - \xint_gob_til_zero #1\XINT_half_end_b 0\space #1#3% -}% -\edef\XINT_half_end_b 0\space 0#1#2#3#4#5#6#7% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7\relax + \noexpand\expandafter\space\noexpand\the\numexpr #1\relax }% \def\xintDec {\romannumeral0\xintdec }% \def\xintdec #1% {% - \expandafter\XINT_dec\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W + \expandafter\XINT_dec\romannumeral-`0#1\Z }% \def\XINT_dec #1% {% @@ -476,43 +493,24 @@ 0-{\XINT_dec_pos #1}% \krof }% -\def\XINT_dec_zero #1\W\W\W\W\W\W\W\W { -1}% +\def\XINT_dec_zero #1\Z { -1}% \def\XINT_dec_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_inc_pos }% -\def\XINT_dec_pos -{% - \expandafter\XINT_dec_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_dec_a #1#2#3#4#5#6#7#8#9% -{% - \expandafter\XINT_dec_b - \the\numexpr 11#9#8#7#6#5#4#3#2-\xint_c_i\relax {#1}% -}% -\def\XINT_dec_b 1#1% -{% - \xint_gob_til_one #1\XINT_dec_A 1\XINT_dec_c -}% -\def\XINT_dec_c #1#2#3#4#5#6#7#8#9{\XINT_dec_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_A 1\XINT_dec_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_B #1#2\W\W\W\W\W\W\W\W +\def\XINT_dec_pos #1\Z {% - \expandafter\XINT_dec_cleanup - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% + \expandafter\XINT_dec_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W }% -\edef\XINT_dec_cleanup #1#2#3#4#5#6#7#8% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax }% +\def\XINT_dec_pos_aa {\XINT_sub_aa 100000001!\Z!\Z!\Z!\Z!\Z!\W }% \def\xintInc {\romannumeral0\xintinc }% \def\xintinc #1% {% - \expandafter\XINT_inc\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W + \expandafter\XINT_inc\romannumeral-`0#1\Z }% \def\XINT_inc #1% {% @@ -522,415 +520,208 @@ 0-{\XINT_inc_pos #1}% \krof }% -\def\XINT_inc_zero #1\W\W\W\W\W\W\W\W { 1}% +\def\XINT_inc_zero #1\Z { 1}% \def\XINT_inc_neg {\expandafter\XINT_opp\romannumeral0\XINT_dec_pos }% -\def\XINT_inc_pos -{% - \expandafter\XINT_inc_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_inc_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_inc_end\W - \expandafter\XINT_inc_b - \the\numexpr 10#9#8#7#6#5#4#3#2+\xint_c_i\relax {#1}% -}% -\def\XINT_inc_b 1#1% -{% - \xint_gob_til_zero #1\XINT_inc_A 0\XINT_inc_c -}% -\def\XINT_inc_c #1#2#3#4#5#6#7#8#9{\XINT_inc_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_A 0\XINT_inc_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_end\W #1\relax #2{ 1#2}% -\def\XINT_add_A #1#2#3#4#5#6% +\def\XINT_inc_pos #1\Z {% - \xint_gob_til_W #3\xint_add_az\W - \XINT_add_AB #1{#3#4#5#6}{#2}% + \expandafter\XINT_inc_pos_aa + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W }% -\def\xint_add_az\W\XINT_add_AB #1#2% +\def\XINT_inc_pos_aa {\XINT_add_aa 100000001!\Z!\Z!\Z!\Z!\Z!\W }% +\def\xintiAdd {\romannumeral0\xintiadd }% +\def\xintiadd #1{\expandafter\XINT_iadd\romannumeral0\xintnum{#1}\Z }% +\def\xintiiAdd {\romannumeral0\xintiiadd }% +\def\xintiiadd #1{\expandafter\XINT_iiadd\romannumeral-`0#1\Z }% +\def\XINT_iiadd #1#2\Z #3% {% - \XINT_add_AC_checkcarry #1% -}% -\def\XINT_add_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_add_bz\W - \XINT_add_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_add_ABEA #1#2#3.#4% -{% - \XINT_add_A #2{#3#4}% -}% -\def\xint_add_bz\W\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_CC\the\numexpr #1+10#5#4#3#2.% -}% -\def\XINT_add_CC #1#2#3.#4% -{% - \XINT_add_AC_checkcarry #2{#3#4}% on va examiner et \'eliminer #2 -}% -\def\XINT_add_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_add_AC_nocarry 0\XINT_add_C -}% -\def\xint_add_AC_nocarry 0\XINT_add_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% + \expandafter\XINT_add_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\XINT_add_C #1#2#3#4#5% +\def\XINT_iadd #1#2\Z #3% {% - \xint_gob_til_W #2\xint_add_cz\W - \XINT_add_CD {#5#4#3#2}{#1}% + \expandafter\XINT_add_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\XINT_add_CD #1% +\def\XINT_add_fork #1#2\Z #3\Z {\XINT_add_nfork #1#3\Z #2\Z}% +\def\XINT_add_nfork #1#2% {% - \expandafter\XINT_add_CC\the\numexpr 1+10#1.% -}% -\def\xint_add_cz\W\XINT_add_CD #1#2{ 1#2}% -\def\XINT_addr_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addr_az\W - \XINT_addr_B #1{#3#4#5#6}{#2}% -}% -\def\xint_addr_az\W\XINT_addr_B #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_addr_B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_addr_bz\W - \XINT_addr_E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT_addr_ABEA #1#2#3#4#5#6#7% -{% - \XINT_addr_A #2{#7#6#5#4#3}% -}% -\def\xint_addr_bz\W\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_CC\the\numexpr #1+10#5#4#3#2\relax -}% -\def\XINT_addr_CC #1#2#3#4#5#6#7% -{% - \XINT_addr_AC_checkcarry #2{#7#6#5#4#3}% -}% -\def\XINT_addr_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addr_AC_nocarry 0\XINT_addr_C -}% -\def\xint_addr_AC_nocarry 0\XINT_addr_C #1#2\W\X\Y\Z { #1#2}% -\def\XINT_addr_C #1#2#3#4#5% -{% - \xint_gob_til_W #2\xint_addr_cz\W - \XINT_addr_D {#5#4#3#2}{#1}% -}% -\def\XINT_addr_D #1% -{% - \expandafter\XINT_addr_CC\the\numexpr 1+10#1\relax -}% -\def\xint_addr_cz\W\XINT_addr_D #1#2{ #21000}% -\def\XINT_addm_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addm_az\W - \XINT_addm_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addm_az\W\XINT_addm_AB #1#2% -{% - \XINT_addm_AC_checkcarry #1% -}% -\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addm_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_addm_ABEA #1#2#3.#4% -{% - \XINT_addm_A #2{#3#4}% -}% -\def\XINT_addm_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C -}% -\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_addm_C #1#2#3#4#5% -{% - \xint_gob_til_W - #5\xint_addm_cw - #4\xint_addm_cx - #3\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD {#5#4#3#2}{#1}% -}% -\def\XINT_addm_CD #1% -{% - \expandafter\XINT_addm_CC\the\numexpr 1+10#1.% -}% -\def\XINT_addm_CC #1#2#3.#4% -{% - \XINT_addm_AC_checkcarry #2{#3#4}% -}% -\def\xint_addm_cw - #1\xint_addm_cx - #2\xint_addm_cy - #3\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.% -}% -\def\XINT_addm_CDw #1.#2#3\X\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cx - #1\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.% -}% -\def\XINT_addm_CDx #1.#2#3\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cy - #1\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDy\the\numexpr 1+#1.% -}% -\def\XINT_addm_CDy #1.#2#3\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}% -\edef\XINT_addm_end #1#2#3#4#5% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}% -\def\XINT_addp_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addp_az\W - \XINT_addp_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addp_az\W\XINT_addp_AB #1#2% -{% - \XINT_addp_AC_checkcarry #1% -}% -\def\XINT_addp_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addp_AC_nocarry 0\XINT_addp_C -}% -\def\xint_addp_AC_nocarry 0\XINT_addp_C -{% - \XINT_addp_F -}% -\def\XINT_addp_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addp_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z + \xint_UDzerofork + #1\XINT_add_firstiszero + #2\XINT_add_secondiszero + 0{}% + \krof + \xint_UDsignsfork + #1#2\XINT_add_minusminus + #1-\XINT_add_minusplus + #2-\XINT_add_plusminus + --\XINT_add_plusplus + \krof #1#2% }% -\def\XINT_addp_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addp_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax +\def\XINT_add_firstiszero #1\krof 0#2#3\Z #4\Z { #2#3}% +\def\XINT_add_secondiszero #1\krof #20#3\Z #4\Z { #2#4}% +\def\XINT_add_minusminus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pp_a {}{}}% +\def\XINT_add_minusplus #1#2{\XINT_sub_mm_a {}#2}% +\def\XINT_add_plusminus #1#2% + {\expandafter\XINT_opp\romannumeral0\XINT_sub_mm_a #1{}}% +\def\XINT_add_pp_a #1#2#3\Z +{% + \expandafter\XINT_add_pp_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\let\XINT_add_plusplus \XINT_add_pp_a +\def\XINT_add_pp_b #1.#2\X #3\Z +{% + \expandafter\XINT_add_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_add_checklengths #1.#2.% +{% + \ifnum #2>#1 + \expandafter\XINT_add_exchange + \else + \expandafter\XINT_add_A + \fi + #1.#2.% }% -\def\XINT_addp_ABEA #1#2#3#4#5#6#7% +\def\XINT_add_exchange #1.#2.#3\Z!\Z!\Z!\Z!\Z!\W #4\Z {% - \XINT_addp_A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite + \XINT_add_A #2.#1.#4\Z!\Z!\Z!\Z!\Z!\W #3\Z }% -\def\XINT_addp_C #1#2#3#4#5% +\def\XINT_add_A #1.#2.% {% - \xint_gob_til_W - #5\xint_addp_cw - #4\xint_addp_cx - #3\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD {#5#4#3#2}{#1}% + \ifnum #1>\xint_c_vi % + \expandafter\XINT_add_aa + \else \expandafter\XINT_add_aa_small + \fi }% -\def\XINT_addp_CD #1% +%%%%%%%%%%%% +\def\XINT_add_out #1\Z #2\W% {% - \expandafter\XINT_addp_CC\the\numexpr 1+10#1\relax + \expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}% + #11\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_addp_CC #1#2#3#4#5#6#7% +\def\XINT_add_out_small #1\Z #2\W% {% - \XINT_addp_AC_checkcarry #2{#7#6#5#4#3}% + \XINT_smallunrevbyviii #11\Z!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\xint_addp_cw - #1\xint_addp_cx - #2\xint_addp_cy - #3\xint_addp_cz - \W\XINT_addp_CD +%%%%%%%%%%%% +\def\XINT_add_aa {\expandafter\XINT_add_out\the\numexpr\XINT_add_a \xint_c_ii}% +\def\XINT_add_aa_small + {\expandafter\XINT_add_out_small\the\numexpr\XINT_add_a \xint_c_ii}% +\def\XINT_add_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter\XINT_addp_CDw\the\numexpr \xint_c_i+10#1#2#3\relax + \XINT_add_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_addp_CDw #1#2#3#4#5#6% +\def\XINT_add_b #1#2!#3!% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDw_zeros - 0000\XINT_addp_endDw #2#3#4#5% + \xint_gob_til_Z #2\XINT_add_bi \Z + \expandafter\XINT_add_c\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_endDw_zeros 0000\XINT_addp_endDw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cx - #1\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD +\def\XINT_add_bi\Z + \expandafter\XINT_add_c + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6!#7!#8!#9!\Z !\W {% - \expandafter\XINT_addp_CDx\the\numexpr \xint_c_i+100#1#2\relax + \XINT_add_k #1#3!#5!#7!#9!% }% -\def\XINT_addp_CDx #1#2#3#4#5#6% +\def\XINT_add_c #1#2.% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDx_zeros - 0000\XINT_addp_endDx #2#3#4#5% + 1#2\expandafter!\the\numexpr\XINT_add_d #1% }% -\def\XINT_addp_endDx_zeros 0000\XINT_addp_endDx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cy #1\xint_addp_cz\W\XINT_addp_CD +\def\XINT_add_d #1#2!#3!% {% - \expandafter\XINT_addp_CDy\the\numexpr \xint_c_i+1000#1\relax + \xint_gob_til_Z #2\XINT_add_di \Z + \expandafter\XINT_add_e\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_CDy #1#2#3#4#5#6% +\def\XINT_add_di\Z\expandafter\XINT_add_e + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6!#7!#8\W {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDy_zeros - 0000\XINT_addp_endDy #2#3#4#5% + \XINT_add_k #1#3!#5!#7!% }% -\def\XINT_addp_endDy_zeros 0000\XINT_addp_endDy 0000#1\Z{ #1}% -\def\XINT_addp_endDy #1#2#3#4#5\Z{ #5#4#3#2#1}% -\def\xint_addp_cz\W\XINT_addp_CD #1#2{ #21000}% -\def\XINT_addp_F #1#2#3#4#5% +\def\XINT_add_e #1#2.% {% - \xint_gob_til_W - #5\xint_addp_Gw - #4\xint_addp_Gx - #3\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G {#2#3#4#5}{#1}% + 1#2\expandafter!\the\numexpr\XINT_add_f #1% }% -\def\XINT_addp_G #1#2% +\def\XINT_add_f #1#2!#3!% {% - \XINT_addp_F {#2#1}% + \xint_gob_til_Z #2\XINT_add_fi \Z + \expandafter\XINT_add_g\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\xint_addp_Gw - #1\xint_addp_Gx - #2\xint_addp_Gy - #3\xint_addp_Gz - \W\XINT_addp_G #4% +\def\XINT_add_fi\Z\expandafter\XINT_add_g + \the\numexpr#1+#2+#3-\xint_c_ii.#4!#5!#6\W {% - \xint_gob_til_zeros_iv #3#2#10\XINT_addp_endGw_zeros - 0000\XINT_addp_endGw #3#2#10% + \XINT_add_k #1#3!#5!% }% -\def\XINT_addp_endGw_zeros 0000\XINT_addp_endGw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gx - #1\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G #3% +\def\XINT_add_g #1#2.% {% - \xint_gob_til_zeros_iv #2#100\XINT_addp_endGx_zeros - 0000\XINT_addp_endGx #2#100% + 1#2\expandafter!\the\numexpr\XINT_add_h #1% }% -\def\XINT_addp_endGx_zeros 0000\XINT_addp_endGx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gy - #1\xint_addp_Gz - \W\XINT_addp_G #2% +\def\XINT_add_h #1#2!#3!% {% - \xint_gob_til_zeros_iv #1000\XINT_addp_endGy_zeros - 0000\XINT_addp_endGy #1000% + \xint_gob_til_Z #2\XINT_add_hi \Z + \expandafter\XINT_add_i\the\numexpr#1+#2+#3-\xint_c_ii.% }% -\def\XINT_addp_endGy_zeros 0000\XINT_addp_endGy 0000#1\Z{ #1}% -\def\XINT_addp_endGy #1#2#3#4#5\Z{ #5#1#2#3#4}% -\def\xint_addp_Gz\W\XINT_addp_G #1#2{ #2}% -\def\xintiiAdd {\romannumeral0\xintiiadd }% -\def\xintiiadd #1{\expandafter\xint_iiadd\romannumeral-`0#1\Z }% -\def\xint_iiadd #1#2\Z #3% +\def\XINT_add_hi\Z + \expandafter\XINT_add_i\the\numexpr#1+#2+#3-\xint_c_ii.#4\W {% - \expandafter\XINT_add_fork\expandafter #1\romannumeral-`0#3\Z #2\Z + \XINT_add_k #1#3!% }% -\def\xintiAdd {\romannumeral0\xintiadd }% -\def\xintiadd #1% +\def\XINT_add_i #1#2.% {% - \expandafter\xint_add\romannumeral0\xintnum{#1}\Z + 1#2\expandafter!\the\numexpr\XINT_add_a #1% }% -\def\xint_add #1#2\Z #3% +%%%%%%%%%%%% +\def\XINT_add_k #1% + {\if #12\expandafter\XINT_add_ke\else\expandafter\XINT_add_l \fi}% +\def\XINT_add_ke #1% {% - \expandafter\XINT_add_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z + \xint_gob_til_Z #1\XINT_add_kf\Z 1% }% -\let\xintAdd\xintiAdd \let\xintadd\xintiadd -\def\XINT_add_fork #1#2% +\def\XINT_add_kf\Z 1{1}% +\def\XINT_add_l #1% {% - \xint_UDzerofork - #1\XINT_add_firstiszero - #2\XINT_add_secondiszero - 0{}% - \krof - \xint_UDsignsfork - #1#2\XINT_add_minusminus - #1-\XINT_add_minusplus - #2-\XINT_add_plusminus - --\XINT_add_plusplus - \krof #1#2% + \xint_gob_til_Z #1\XINT_add_lf\Z \XINT_add_m 1% }% -\def\XINT_add_firstiszero #1\krof #2#3\Z #4\Z { #3}% -\def\XINT_add_secondiszero #1\krof #2#3\Z #4\Z { #2#4}% -\def\XINT_add_plusplus #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#2#3}}% -\def\XINT_add_minusminus #1#2#3\Z #4\Z - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#3}}% -\def\XINT_add_minusplus #1#2#3\Z #4\Z {\XINT_sub_pre {#2#3}{#4}}% -\def\XINT_add_plusminus #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#3}}% -\def\XINT_add_pre #1% +\def\XINT_add_lf\Z\XINT_add_m 1{100000001}% +\def\XINT_add_m #1!% {% - \expandafter\XINT_add_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \expandafter\XINT_add_n\the\numexpr\xint_c_i+#1.% }% -\def\XINT_add_pre_b #1#2% +\def\XINT_add_n #1#2.% {% - \expandafter\XINT_add_A - \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_add_o #1% }% +\def\XINT_add_o #1% + {\if #12\expandafter\XINT_add_l\else\expandafter\XINT_add_ke \fi}% \def\xintiiSub {\romannumeral0\xintiisub }% -\def\xintiisub #1{\expandafter\xint_iisub\romannumeral-`0#1\Z }% -\def\xint_iisub #1#2\Z #3% +\def\xintiisub #1{\expandafter\XINT_iisub\romannumeral-`0#1\Z }% +\def\XINT_iisub #1#2\Z #3% {% - \expandafter\XINT_sub_fork\expandafter #1\romannumeral-`0#3\Z #2\Z + \expandafter\XINT_sub_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\def\xintiSub {\romannumeral0\xintisub }% -\def\xintisub #1% +\def\xintiSub {\romannumeral0\xintisub }% +\def\xintisub #1{\expandafter\XINT_isub\romannumeral0\xintnum{#1}\Z }% +\def\XINT_isub #1#2\Z #3% {% - \expandafter\xint_sub\romannumeral0\xintnum{#1}\Z + \expandafter\XINT_sub_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% -\def\xint_sub #1#2\Z #3% -{% - \expandafter\XINT_sub_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z -}% -\let\xintSub\xintiSub \let\xintsub\xintisub -\def\XINT_sub_fork #1#2% +\def\XINT_sub_nfork #1#2% {% \xint_UDzerofork #1\XINT_sub_firstiszero @@ -944,612 +735,401 @@ --\XINT_sub_plusplus \krof #1#2% }% -\def\XINT_sub_firstiszero #1\krof #2#3\Z #4\Z {\XINT_opp #3}% -\def\XINT_sub_secondiszero #1\krof #2#3\Z #4\Z { #2#4}% -\def\XINT_sub_plusplus #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#2#3}}% -\def\XINT_sub_minusminus #1#2#3\Z #4\Z {\XINT_sub_pre {#3}{#4}}% -\def\XINT_sub_minusplus #1#2#3\Z #4\Z - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#2#3}}% -\def\XINT_sub_plusminus #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#3}}% -\def\XINT_sub_pre #1% -{% - \expandafter\XINT_sub_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_sub_pre_b #1#2% -{% - \expandafter\XINT_sub_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z -}% -\def\XINT_sub_A #1#2#3\W\X\Y\Z #4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_az - \W\XINT_sub_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z -}% -\def\XINT_sub_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_bz - \W\XINT_sub_onestep #1#2{#7#6#5#4}{#3}% -}% -\def\XINT_sub_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_sub_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% +\def\XINT_sub_firstiszero #1\krof 0#2#3\Z #4\Z {\XINT_opp #2#3}% +\def\XINT_sub_secondiszero #1\krof #20#3\Z #4\Z { #2#4}% +\def\XINT_sub_plusminus #1#2{\XINT_add_pp_a #1{}}% +\def\XINT_sub_plusplus #1#2% + {\expandafter\XINT_opp\romannumeral0\XINT_sub_mm_a #1#2}% +\def\XINT_sub_minusplus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pp_a {}#2}% +\def\XINT_sub_minusminus #1#2{\XINT_sub_mm_a {}{}}% +\def\XINT_sub_mm_a #1#2#3\Z +{% + \expandafter\XINT_sub_mm_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X #1% +}% +\def\XINT_sub_mm_b #1.#2\X #3\Z +{% + \expandafter\XINT_sub_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax \xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z!\Z!\Z!\Z!\Z!\W #2\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_sub_checklengths #1.#2.% +{% + \ifnum #2>#1 + \expandafter\XINT_sub_exchange + \else + \expandafter\XINT_sub_aa + \fi }% -\def\XINT_sub_backtoA #1#2#3.#4% +\def\XINT_sub_exchange #1\Z!\Z!\Z!\Z!\Z!\W #2\Z {% - \XINT_sub_A #2{#3#4}% + \expandafter\XINT_opp\romannumeral0\XINT_sub_aa + #2\Z!\Z!\Z!\Z!\Z!\W #1\Z }% -\def\xint_sub_bz - \W\XINT_sub_onestep #1#2#3#4#5#6#7% +%%%%%%%%%%%% +\def\XINT_sub_prepare_rescue #1\W {\relax\Z-\W}% +\def\XINT_sub_prepare_cuz #1\W {\relax\XINT_cuz_byviii!\Z 0\W\R}% +%%%%%%%%%%%% +\def\XINT_sub_aa {\expandafter\XINT_sub_out\the\numexpr\XINT_sub_a \xint_c_i }% +\def\XINT_sub_out #1\Z #2#3\W {% - \xint_UDzerofork - #1\XINT_sub_C % une retenue - 0\XINT_sub_D % pas de retenue - \krof - {#7}#2#3#4#5% -}% -\def\XINT_sub_D #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% + \if-#2\expandafter\XINT_sub_startrescue\fi + \expandafter\XINT_cuz_small + \romannumeral0\XINT_unrevbyviii {}#11\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\def\XINT_sub_C #1#2#3#4#5% +\def\XINT_sub_startrescue\expandafter\XINT_cuz_small + \romannumeral0\XINT_unrevbyviii #1#2\Z!#3\W {% - \xint_gob_til_W - #2\xint_sub_cz - \W\XINT_sub_AC_onestep {#5#4#3#2}{#1}% + \expandafter\XINT_sub_rescue_finish + \the\numexpr\XINT_sub_rescue_a #2!% + 1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W \R }% -\def\XINT_sub_AC_onestep #1% +\def\XINT_sub_rescue_finish + {\expandafter-\romannumeral0\expandafter\XINT_cuz + \romannumeral0\XINT_unrevbyviii {}}% +\def\XINT_sub_rescue_a #1!% {% - \expandafter\XINT_sub_backtoC\the\numexpr 11#1-\xint_c_i.% + \expandafter\XINT_sub_rescue_c\the\numexpr \xint_c_xii_e_viii-#1.% }% -\def\XINT_sub_backtoC #1#2#3.#4% +\def\XINT_sub_rescue_c 1#1#2.% {% - \XINT_sub_AC_checkcarry #2{#3#4}% la retenue va \^etre examin\'ee + 1#2\expandafter!\the\numexpr\XINT_sub_rescue_d #1% }% -\def\XINT_sub_AC_checkcarry #1% +\def\XINT_sub_rescue_d #1#2#3!% {% - \xint_gob_til_one #1\xint_sub_AC_nocarry 1\XINT_sub_C + \xint_gob_til_minus #2\XINT_sub_rescue_z -% + \expandafter\XINT_sub_rescue_c\the\numexpr \xint_c_xii_e_viii_mone-#2#3+#1.% }% -\def\xint_sub_AC_nocarry 1\XINT_sub_C #1#2\W\X\Y\Z +\def\XINT_sub_rescue_z #1.{1!}% +%%%%%%%%%%%% +\def\XINT_sub_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter - \XINT_cuz_loop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1\W\W\W\W\W\W\W\Z + \XINT_sub_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\xint_sub_cz\W\XINT_sub_AC_onestep #1% +\def\XINT_sub_b #1#2#3!#4!% {% - \XINT_cuz + \xint_gob_til_Z #2\XINT_sub_bi \Z + \expandafter\XINT_sub_c\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\xint_sub_az\W\XINT_sub_B #1#2#3#4#5#6#7% +\def\XINT_sub_c 1#1#2.% {% - \xint_gob_til_W - #4\xint_sub_ez - \W\XINT_sub_Eenter #1{#3}#4#5#6#7% + 1#2\expandafter!\the\numexpr\XINT_sub_d #1% }% -\def\XINT_sub_Eenter #1#2% +\def\XINT_sub_d #1#2#3!#4!% {% - \expandafter - \XINT_sub_E\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - \W\X\Y\Z #1% + \xint_gob_til_Z #2\XINT_sub_di \Z + \expandafter\XINT_sub_e\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_sub_E #1#2#3#4#5#6% +\def\XINT_sub_e 1#1#2.% {% - \xint_gob_til_W #3\xint_sub_F\W - \XINT_sub_Eonestep #1{#6#5#4#3}{#2}% + 1#2\expandafter!\the\numexpr\XINT_sub_f #1% }% -\def\XINT_sub_Eonestep #1#2% +\def\XINT_sub_f #1#2#3!#4!% {% - \expandafter\XINT_sub_backtoE\the\numexpr 109999-#2+#1.% + \xint_gob_til_Z #2\XINT_sub_fi \Z + \expandafter\XINT_sub_g\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_sub_backtoE #1#2#3.#4% +\def\XINT_sub_g 1#1#2.% {% - \XINT_sub_E #2{#3#4}% + 1#2\expandafter!\the\numexpr\XINT_sub_h #1% }% -\def\xint_sub_F\W\XINT_sub_Eonestep #1#2#3#4% +\def\XINT_sub_h #1#2#3!#4!% {% - \xint_UDonezerofork - #4#1{\XINT_sub_Fdec 0}% soustraire 1. Et faire signe - - #1#4{\XINT_sub_Finc 1}% additionner 1. Et faire signe - - 10\XINT_sub_DD % terminer. Mais avec signe - - \krof - {#3}% + \xint_gob_til_Z #2\XINT_sub_hi \Z + \expandafter\XINT_sub_i\the\numexpr#1+1#4-#3-\xint_c_i.% }% -\def\XINT_sub_DD {\expandafter\xint_minus_thenstop\romannumeral0\XINT_sub_D }% -\def\XINT_sub_Fdec #1#2#3#4#5#6% +\def\XINT_sub_i 1#1#2.% {% - \xint_gob_til_W #3\xint_sub_Fdec_finish\W - \XINT_sub_Fdec_onestep #1{#6#5#4#3}{#2}% + 1#2\expandafter!\the\numexpr\XINT_sub_a #1% }% -\def\XINT_sub_Fdec_onestep #1#2% +\def\XINT_sub_bi\Z + \expandafter\XINT_sub_c\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8!#9!\Z !\W {% - \expandafter\XINT_sub_backtoFdec\the\numexpr 11#2+#1-\xint_c_i.% + \XINT_sub_k #1#2!#5!#7!#9!% }% -\def\XINT_sub_backtoFdec #1#2#3.#4% +\def\XINT_sub_di\Z + \expandafter\XINT_sub_e\the\numexpr#1+1#2-#3.#4!#5!#6!#7!#8\W {% - \XINT_sub_Fdec #2{#3#4}% + \XINT_sub_k #1#2!#5!#7!% }% -\def\xint_sub_Fdec_finish\W\XINT_sub_Fdec_onestep #1#2% +\def\XINT_sub_fi\Z + \expandafter\XINT_sub_g\the\numexpr#1+1#2-#3.#4!#5!#6\W {% - \expandafter\xint_minus_thenstop\romannumeral0\XINT_cuz + \XINT_sub_k #1#2!#5!% }% -\def\XINT_sub_Finc #1#2#3#4#5#6% +\def\XINT_sub_hi\Z + \expandafter\XINT_sub_i\the\numexpr#1+1#2-#3.#4\W {% - \xint_gob_til_W #3\xint_sub_Finc_finish\W - \XINT_sub_Finc_onestep #1{#6#5#4#3}{#2}% + \XINT_sub_k #1#2!% }% -\def\XINT_sub_Finc_onestep #1#2% +%%%%%%%%%%%% +\def\XINT_sub_k #1#2% {% - \expandafter\XINT_sub_backtoFinc\the\numexpr 10#2+#1.% + \xint_gob_til_Z #2\XINT_sub_p\Z \XINT_sub_l #1#2% }% -\def\XINT_sub_backtoFinc #1#2#3.#4% +\def\XINT_sub_l #1% {% - \XINT_sub_Finc #2{#3#4}% + \xint_UDzerofork + #1\XINT_sub_m + 0{}% + \krof }% -\def\xint_sub_Finc_finish\W\XINT_sub_Finc_onestep #1#2#3% +\def\XINT_sub_m #1!% {% - \xint_UDzerofork - #1{\expandafter\expandafter\expandafter - \xint_minus_thenstop\xint_cleanupzeros_nostop}% - 0{ -1}% - \krof - #3% + \expandafter\XINT_sub_n\the\numexpr 1#1-\xint_c_i!% }% -\def\xint_sub_ez\W\XINT_sub_Eenter #1% +\def\XINT_sub_n 1#1% {% \xint_UDzerofork - #1\XINT_sub_K % il y a une retenue - 0\XINT_sub_L % pas de retenue + #1{\XINT_sub_o}% + 0{\XINT_sub_n_checkzero}% \krof }% -\def\XINT_sub_L #1\W\X\Y\Z {\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z }% -\def\XINT_sub_K #1% +\def\XINT_sub_o #1!{1#1\expandafter!\the\numexpr\XINT_sub_m }% +\def\XINT_sub_n_checkzero #1!% {% - \expandafter - \XINT_sub_KK\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax + \xint_gob_til_eightzeroes #1\XINT_sub_n_prepare_cuz 00000000% + 1#1!% }% -\def\XINT_sub_KK #1#2#3#4#5#6% +\def\XINT_sub_n_prepare_cuz 00000000100000000{1\XINT_sub_prepare_cuz}% +\def\XINT_sub_p\Z\XINT_sub_l #1\Z!% {% - \xint_gob_til_W #3\xint_sub_KK_finish\W - \XINT_sub_KK_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_KK_onestep #1#2% -{% - \expandafter\XINT_sub_backtoKK\the\numexpr 109999-#2+#1.% + \xint_UDzerofork + #1{-1\XINT_sub_prepare_rescue}% + 0{1\XINT_sub_prepare_cuz }% + \krof }% -\def\XINT_sub_backtoKK #1#2#3.#4% +\def\xintiMul {\romannumeral0\xintimul }% +\def\xintimul #1% {% - \XINT_sub_KK #2{#3#4}% + \expandafter\XINT_imul\romannumeral0\xintnum{#1}\Z }% -\def\xint_sub_KK_finish\W\XINT_sub_KK_onestep #1#2#3% +\def\XINT_imul #1#2\Z #3% {% - \expandafter\xint_minus_thenstop - \romannumeral0\XINT_cuz_loop #3\W\W\W\W\W\W\W\Z + \expandafter\XINT_mul_nfork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z }% \def\xintiiMul {\romannumeral0\xintiimul }% \def\xintiimul #1% {% - \expandafter\xint_iimul\expandafter {\romannumeral-`0#1}% -}% -\def\xint_iimul #1#2% -{% - \expandafter\XINT_mul_fork \romannumeral-`0#2\Z #1\Z -}% -\def\xintiMul {\romannumeral0\xintimul }% -\def\xintimul #1% -{% - \expandafter\xint_mul\expandafter {\romannumeral0\xintnum{#1}}% + \expandafter\XINT_iimul\romannumeral-`0#1\Z }% -\def\xint_mul #1#2% +\def\XINT_iimul #1#2\Z #3% {% - \expandafter\XINT_mul_fork \romannumeral0\xintnum{#2}\Z #1\Z + \expandafter\XINT_mul_nfork\expandafter #1\romannumeral-`0#3\Z #2\Z }% -\let\xintMul\xintiMul \let\xintmul\xintimul -\def\XINT_Mul #1#2{\romannumeral0\XINT_mul_fork #2\Z #1\Z }% -\def\XINT_mul_fork #1#2\Z #3#4\Z +\def\XINT_mul_fork #1#2\Z #3\Z{\XINT_mul_nfork #1#3\Z #2\Z}% +\def\XINT_mul_nfork #1#2% {% \xint_UDzerofork #1\XINT_mul_zero - #3\XINT_mul_zero + #2\XINT_mul_zero 0{}% \krof \xint_UDsignsfork - #1#3\XINT_mul_minusminus % #1 = #3 = - - #1-{\XINT_mul_minusplus #3}% % #1 = - - #3-{\XINT_mul_plusminus #1}% % #3 = - - --{\XINT_mul_plusplus #1#3}% - \krof - {#2}{#4}% -}% -\def\XINT_mul_zero #1\krof #2#3{ 0}% -\def\XINT_mul_minusminus #1#2% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2}}% - {\romannumeral0\xintlength {#1}}{#1}{#2}% -}% -\def\XINT_mul_minusplus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#1#3}}% - {\romannumeral0\xintlength {#2}}{#2}{#1#3}% -}% -\def\XINT_mul_plusminus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#3}}% - {\romannumeral0\xintlength {#1#2}}{#1#2}{#3}% -}% -\def\XINT_mul_plusplus #1#2#3#4% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2#4}}% - {\romannumeral0\xintlength {#1#3}}{#1#3}{#2#4}% -}% -\def\XINT_mul_choice_a #1#2% -{% - \expandafter\XINT_mul_choice_b\expandafter{#2}{#1}% -}% -\def\XINT_mul_choice_b #1#2% -{% - \ifnum #1<\xint_c_v - \expandafter\XINT_mul_choice_littlebyfirst - \else - \ifnum #2<\xint_c_v - \expandafter\expandafter\expandafter\XINT_mul_choice_littlebysecond - \else - \expandafter\expandafter\expandafter\XINT_mul_choice_compare - \fi - \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_littlebyfirst #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #3\expandafter}% - \romannumeral0\XINT_RQ {}#4\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z -}% -\def\XINT_mul_choice_littlebysecond #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #4\expandafter}% - \romannumeral0\XINT_RQ {}#3\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z -}% -\def\XINT_mul_choice_compare #1#2% -{% - \ifnum #1>#2 - \expandafter \XINT_mul_choice_i - \else - \expandafter \XINT_mul_choice_ii - \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_i #1#2% -{% - \ifnum #1<\numexpr\ifcase \numexpr (#2-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_same - \else - \expandafter\XINT_mul_choice_permute - \fi -}% -\def\XINT_mul_choice_ii #1#2% -{% - \ifnum #2<\numexpr\ifcase \numexpr (#1-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_permute - \else - \expandafter\XINT_mul_choice_same - \fi -}% -\def\XINT_mul_choice_same #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #2\W\W\W\W -}% -\def\XINT_mul_choice_permute #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W -}% -\def\XINT_mul_Ar #1#2#3#4#5#6% -{% - \xint_gob_til_Z #6\xint_mul_br\Z\XINT_mul_Br #1{#6#5#4#3}{#2}% -}% -\def\xint_mul_br\Z\XINT_mul_Br #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_mul_Br #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \expandafter\XINT_mul_ABEAr - \the\numexpr #1+10#2+#8#7#6#5.{#3}#4\W\X\Y\Z -}% -\def\XINT_mul_ABEAr #1#2#3#4#5#6.#7% -{% - \XINT_mul_Ar #2{#7#6#5#4#3}% -}% -\def\XINT_mul_Mr #1% -{% - \expandafter\XINT_mul_Mr_checkifzeroorone\expandafter{\the\numexpr #1}% -}% -\def\XINT_mul_Mr_checkifzeroorone #1% -{% - \ifcase #1 - \expandafter\XINT_mul_Mr_zero - \or - \expandafter\XINT_mul_Mr_one - \else - \expandafter\XINT_mul_Nr - \fi - {0000}{}{#1}% -}% -\def\XINT_mul_Mr_zero #1\Z\Z\Z\Z { 0000}% -\def\XINT_mul_Mr_one #1#2#3#4\Z\Z\Z\Z { #4}% -\def\XINT_mul_Nr #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\xint_mul_pr\Z\XINT_mul_Pr {#1}{#3}{#7#6#5#4}{#2}{#3}% -}% -\def\XINT_mul_Pr #1#2#3% -{% - \expandafter\XINT_mul_Lr\the\numexpr \xint_c_x^viii+#1+#2*#3\relax -}% -\def\XINT_mul_Lr 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_Nr {#1#2#3#4}{#9#8#7#6#5}% -}% -\def\xint_mul_pr\Z\XINT_mul_Pr #1#2#3#4#5% -{% - \xint_gob_til_zeros_iv #1\XINT_mul_Mr_end_nocarry 0000% - \XINT_mul_Mr_end_carry #1{#4}% -}% -\def\XINT_mul_Mr_end_nocarry 0000\XINT_mul_Mr_end_carry 0000#1{ #1}% -\def\XINT_mul_Mr_end_carry #1#2#3#4#5{ #5#4#3#2#1}% -\def\XINT_mul_M #1% -{% - \expandafter\XINT_mul_M_checkifzeroorone\expandafter{\the\numexpr #1}% + #1#2\XINT_mul_minusminus + #1-\XINT_mul_minusplus + #2-\XINT_mul_plusminus + --\XINT_mul_plusplus + \krof #1#2% }% -\def\XINT_mul_M_checkifzeroorone #1% -{% - \ifcase #1 - \expandafter\XINT_mul_M_zero - \or - \expandafter\XINT_mul_M_one +\def\XINT_mul_zero #1\krof #2#3\Z #4\Z { 0}% +\def\XINT_mul_minusminus #1#2{\XINT_mul_plusplus {}{}}% +\def\XINT_mul_minusplus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_mul_plusplus {}#2}% +\def\XINT_mul_plusminus #1#2% + {\expandafter\xint_minus_thenstop\romannumeral0\XINT_mul_plusplus #1{}}% +\def\XINT_mul_plusplus #1#2#3\Z +{% + \expandafter\XINT_mul_pre_b + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #2#3\R\R\R\R\R\R\R\R{10}0000001\W + #2#3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \W #1% +}% +\def\XINT_mul_pre_b #1.#2\W #3\Z +{% + \expandafter\XINT_mul_checklengths + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + 1\Z!\W #21\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_mul_checklengths #1.#2.% +{% + \ifnum #2=\xint_c_i\expandafter\XINT_mul_smallbyfirst\fi + \ifnum #1=\xint_c_i\expandafter\XINT_mul_smallbysecond\fi + \ifnum #2<#1 + \ifnum \numexpr (#2-\xint_c_i)*(#1-#2)<383 + \XINT_mul_exchange + \fi \else - \expandafter\XINT_mul_N + \ifnum \numexpr (#1-\xint_c_i)*(#2-#1)>383 + \XINT_mul_exchange + \fi \fi - {0000}{}{#1}% -}% -\def\XINT_mul_M_zero #1\Z\Z\Z\Z { 0}% -\def\XINT_mul_M_one #1#2#3#4\Z\Z\Z\Z -{% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#4}% -}% -\def\XINT_mul_N #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\xint_mul_p\Z\XINT_mul_P {#1}{#3}{#7#6#5#4}{#2}{#3}% -}% -\def\XINT_mul_P #1#2#3% -{% - \expandafter\XINT_mul_L\the\numexpr \xint_c_x^viii+#1+#2*#3\relax -}% -\def\XINT_mul_L 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_N {#1#2#3#4}{#5#6#7#8#9}% -}% -\def\xint_mul_p\Z\XINT_mul_P #1#2#3#4#5% -{% - \XINT_mul_M_end #1#4% -}% -\edef\XINT_mul_M_end #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -\def\XINT_mul_enter #1\Z\Z\Z\Z #2#3#4#5% -{% - \xint_gob_til_W #5\XINT_mul_exit_a\W - \XINT_mul_start {#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_mul_exit_a\W\XINT_mul_start #1% -{% - \XINT_mul_exit_b #1% -}% -\def\XINT_mul_exit_b #1#2#3#4% -{% - \xint_gob_til_W - #2\XINT_mul_exit_ci - #3\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1#2#3#4% -}% -\def\XINT_mul_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z + \XINT_mul_start }% -\def\XINT_mul_exit_cii\W\XINT_mul_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W +\def\XINT_mul_smallbyfirst #1\XINT_mul_start 1#2!1\Z!\W {% - \XINT_mul_M {#1}#2\Z\Z\Z\Z + \ifnum#2=\xint_c_i\expandafter\XINT_mul_oneisone\fi + \ifnum#2<\xint_c_xxii\expandafter\XINT_mul_verysmall\fi + \expandafter\XINT_mul_out\the\numexpr\XINT_smallmul 1#2!% }% -\def\XINT_mul_exit_ci\W\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W +\def\XINT_mul_smallbysecond #1\XINT_mul_start #2\W 1#3!1\Z!% {% - \XINT_mul_M {#1}#2\Z\Z\Z\Z + \ifnum#3=\xint_c_i\expandafter\XINT_mul_oneisone\fi + \ifnum#3<\xint_c_xxii\expandafter\XINT_mul_verysmall\fi + \expandafter\XINT_mul_out\the\numexpr\XINT_smallmul 1#3!#2% }% -\def\XINT_mul_start #1#2\Z\Z\Z\Z +\def\XINT_mul_oneisone #1!{\XINT_mul_out }% +\def\XINT_mul_verysmall\expandafter\XINT_mul_out + \the\numexpr\XINT_smallmul 1#1!% + {\expandafter\XINT_mul_out\the\numexpr\XINT_verysmallmul 0.#1!}% +\def\XINT_mul_exchange #1\XINT_mul_start #2\W #31\Z!% + {\fi\fi\XINT_mul_start #31\Z!\W #2}% +\def\XINT_mul_start + {\expandafter\XINT_mul_out\the\numexpr\XINT_mul_loop 100000000!\Z\W}% +\def\XINT_mul_out + {\expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}}% +\def\XINT_mul_loop #1\Z #2\W #3\W 1#4!% {% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z + \xint_gob_til_Z #4\XINT_mul_e \Z + \expandafter\XINT_mul_a\the\numexpr \XINT_smallmul 1#4!#3\W + #11!\W #3\W }% -\def\XINT_mul_main #1#2\Z\Z\Z\Z #3#4#5#6% +\def\XINT_mul_a #11\Z!\W #2!1!#3\W {% - \xint_gob_til_W #6\XINT_mul_finish_a\W - \XINT_mul_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z + \expandafter\XINT_mul_b\the\numexpr + \XINT_add_a \xint_c_ii #2!\Z!\Z!\Z!\Z!\Z!\W #1\Z!\Z!\Z!\Z!\Z!\W }% -\def\XINT_mul_compute #1#2#3\Z\Z\Z\Z +\def\XINT_mul_b 1#1!{1#1\expandafter!\the\numexpr\XINT_mul_loop }% +\def\XINT_mul_e\Z #1\W #2!1!#3\W #4\W {#2!1\Z!}% +\def\XINT_minimulwc_a 1#1.#2.#3!#4#5#6#7#8.% {% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z + \expandafter\XINT_minimulwc_b + \the\numexpr \xint_c_x^ix+#1+#3*#8.#3*#4#5#6#7+#2*#8.#2*#4#5#6#7.% }% -\def\XINT_mul_finish_a\W\XINT_mul_compute #1% +\def\XINT_minimulwc_b 1#1#2#3#4#5#6.#7.% {% - \XINT_mul_finish_b #1% + \expandafter\XINT_minimulwc_c + \the\numexpr \xint_c_x^ix+#1#2#3#4#5+#7.#6.% }% -\def\XINT_mul_finish_b #1#2#3#4% +\def\XINT_minimulwc_c 1#1#2#3#4#5#6.#7.#8.% {% - \xint_gob_til_W - #1\XINT_mul_finish_c - #2\XINT_mul_finish_ci - #3\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1#2#3#4% + 1#6#7\expandafter!% + \the\numexpr\expandafter\XINT_smallmul_a + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8.% }% -\def\XINT_mul_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W +\def\XINT_smallmul 1#1#2#3#4#5!{\XINT_smallmul_a 100000000.#1#2#3#4.#5!}% +\def\XINT_smallmul_a #1.#2.#3!1#4!% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z + \xint_gob_til_Z #4\XINT_smallmul_e\Z + \XINT_minimulwc_a #1.#2.#3!#4.#2.#3!% }% -\def\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W +\def\XINT_smallmul_e\Z\XINT_minimulwc_a 1#1.#2\Z #3!% + {\xint_gob_til_eightzeroes #1\XINT_smallmul_f 000000001\relax #1!1\Z!}% +\def\XINT_smallmul_f 000000001\relax 00000000!1{1\relax}% +\def\XINT_verysmallmul #1.#2!1#3!% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z + \xint_gob_til_Z #3\XINT_verysmallmul_e\Z + \expandafter\XINT_verysmallmul_a + \the\numexpr #2*#3+#1.#2!% }% -\def\XINT_mul_finish_ci #1\XINT_mul_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W +\def\XINT_verysmallmul_e\Z\expandafter\XINT_verysmallmul_a\the\numexpr + #1+#2#3.#4!% +{\xint_gob_til_zero #2\XINT_verysmallmul_f 0\xint_c_x^viii+#2#3!1\Z!}% +\def\XINT_verysmallmul_f #1!1{1\relax}% +\def\XINT_verysmallmul_a #1#2.% {% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z + \unless\ifnum #1#2<\xint_c_x^ix + \expandafter\XINT_verysmallmul_bi\else + \expandafter\XINT_verysmallmul_bj\fi + \the\numexpr \xint_c_x^ix+#1#2.% }% -\def\XINT_mul_finish_c #1\XINT_mul_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z +\def\XINT_verysmallmul_bj{\expandafter\XINT_verysmallmul_cj }% +\def\XINT_verysmallmul_cj 1#1#2.% + {1#2\expandafter!\the\numexpr\XINT_verysmallmul #1.}% +\def\XINT_verysmallmul_bi\the\numexpr\xint_c_x^ix+#1#2#3.% + {1#3\expandafter!\the\numexpr\XINT_verysmallmul #1#2.}% +\def\XINT_minimul_a #1.#2!#3#4#5#6#7!% {% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#2}% + \expandafter\XINT_minimul_b + \the\numexpr \xint_c_x^viii+#2*#7.#2*#3#4#5#6+#1*#7.#1*#3#4#5#6.% }% -\def\XINT_mulr_enter #1\Z\Z\Z\Z #2#3#4#5% +\def\XINT_minimul_b 1#1#2#3#4#5.#6.% {% - \xint_gob_til_W #5\XINT_mulr_exit_a\W - \XINT_mulr_start {#2#3#4#5}#1\Z\Z\Z\Z + \expandafter\XINT_minimul_c + \the\numexpr \xint_c_x^ix+#1#2#3#4+#6.#5.% }% -\def\XINT_mulr_exit_a\W\XINT_mulr_start #1% +\def\XINT_minimul_c 1#1#2#3#4#5#6.#7.#8.% {% - \XINT_mulr_exit_b #1% + 1#6#7\expandafter!\the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8!% }% -\def\XINT_mulr_exit_b #1#2#3#4% -{% - \xint_gob_til_W - #2\XINT_mulr_exit_ci - #3\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1#2#3#4% -}% -\def\XINT_mulr_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_cii\W\XINT_mulr_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_ci\W\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_start #1#2\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_main #1#2\Z\Z\Z\Z #3#4#5#6% -{% - \xint_gob_til_W #6\XINT_mulr_finish_a\W - \XINT_mulr_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_compute #1#2#3\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z -}% -\def\XINT_mulr_finish_a\W\XINT_mulr_compute #1% -{% - \XINT_mulr_finish_b #1% -}% -\def\XINT_mulr_finish_b #1#2#3#4% +\def\xintiiSqr {\romannumeral0\xintiisqr }% +\def\xintiisqr #1% {% - \xint_gob_til_W - #1\XINT_mulr_finish_c - #2\XINT_mulr_finish_ci - #3\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1#2#3#4% + \expandafter\XINT_sqr\romannumeral0\xintiiabs{#1}\Z }% -\def\XINT_mulr_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W +\def\xintiSqr {\romannumeral0\xintisqr }% +\def\xintisqr #1% {% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z + \expandafter\XINT_sqr\romannumeral0\xintiabs{#1}\Z }% -\def\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W +\def\XINT_sqr #1\Z {% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z + \expandafter\XINT_sqr_a + \romannumeral0\expandafter\XINT_sepandrev_andcount + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \Z }% -\def\XINT_mulr_finish_ci #1\XINT_mulr_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W +\def\XINT_sqr_a #1.% {% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z + \ifnum #1=\xint_c_i \expandafter\XINT_sqr_small + \else\expandafter\XINT_sqr_start\fi }% -\def\XINT_mulr_finish_c #1\XINT_mulr_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z { #2}% -\def\xintiiSqr {\romannumeral0\xintiisqr }% -\def\xintiisqr #1% +\def\XINT_sqr_small 1#1#2#3#4#5!\Z {% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiiabs{#1}}% + \ifnum #1#2#3#4#5<46341 \expandafter\XINT_sqr_verysmall\fi + \expandafter\XINT_sqr_small_out + \the\numexpr\XINT_minimul_a #1#2#3#4.#5!#1#2#3#4#5!% }% -\def\xintiSqr {\romannumeral0\xintisqr }% -\def\xintisqr #1% +\edef\XINT_sqr_verysmall + \expandafter\XINT_sqr_small_out\the\numexpr\XINT_minimul_a #1!#2!% + {\noexpand\expandafter\space\noexpand\the\numexpr #2*#2\relax}% +\def\XINT_sqr_small_out 1#1!1#2!% {% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiabs{#1}}% + \XINT_cuz #2#1\R }% -\let\xintSqr\xintiSqr \let\xintsqr\xintisqr -\def\XINT_sqr #1% +\def\XINT_sqr_start #1\Z {% - \expandafter\XINT_mul_enter - \romannumeral0% - \XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W + \expandafter\XINT_mul_out + \the\numexpr\XINT_mul_loop 100000000!\Z\W #11\Z!\W #11\Z!% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% \def\xintiiPow {\romannumeral0\xintiipow }% \def\xintiipow #1% @@ -1561,7 +1141,6 @@ {% \expandafter\xint_pow\romannumeral0\xintnum{#1}\Z% }% -\let\xintPow\xintiPow \let\xintpow\xintipow \def\xint_pow #1#2\Z {% \xint_UDsignfork @@ -1614,7 +1193,7 @@ \ifcase\XINT_cntSgn #1\Z \expandafter\XINT_pow_BisZero \or - \expandafter\XINT_pow_checkBsize + \expandafter\XINT_pow_I_in \else \expandafter\XINT_pow_BisNegative \fi @@ -1623,75 +1202,198 @@ \edef\XINT_pow_BisNegative #1#2% {\noexpand\xintError:FractionRoundedToZero\space 0}% \def\XINT_pow_BisZero #1#2{ 1}% -\def\XINT_pow_checkBsize #1% +%%%%%%%%%%%% +\def\XINT_pow_I_in #1#2% {% - \ifnum #1>100000 - \expandafter\XINT_pow_BtooBig - \else - \expandafter\XINT_pow_loopI - \fi - {#1}% + \expandafter\XINT_pow_I_loop + \the\numexpr #1\expandafter.% + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #2\R\R\R\R\R\R\R\R{10}0000001\W + #2\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W 1\Z!\W + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W }% -\edef\XINT_pow_BtooBig #1#2{\noexpand\xintError:ExponentTooBig\space 0}% -\def\XINT_pow_loopI #1% +\def\XINT_pow_I_loop #1.% {% - \ifnum #1=\xint_c_i\XINT_pow_Iend\fi + \ifnum #1 = \xint_c_i\expandafter\XINT_pow_I_exit\fi \ifodd #1 - \expandafter\XINT_pow_loopI_odd + \expandafter\XINT_pow_II_in \else - \expandafter\XINT_pow_loopI_even - \fi - {#1}% + \expandafter\XINT_pow_I_squareit + \fi #1.% +}% +\def\XINT_pow_I_exit \ifodd #1\fi #2.#3\W {\XINT_mul_out #3}% +\def\XINT_pow_I_squareit #1.#2\W% +{% + \expandafter\XINT_pow_I_loop + \the\numexpr #1/\xint_c_ii\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W }% -\edef\XINT_pow_Iend\fi #1\fi #2#3{\noexpand\fi\space #3}% -\def\XINT_pow_loopI_even #1#2% +%%%%%%%%%%%% +\def\XINT_pow_mulbutcheckifsmall #1!1#2% {% - \expandafter\XINT_pow_loopI\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% + \xint_gob_til_Z #2\XINT_pow_mul_small\Z + \XINT_mul_loop 100000000!\Z\W #1!1#2% }% -\def\XINT_pow_loopI_odd #1#2% +\def\XINT_pow_mul_small\Z\XINT_mul_loop 100000000!\Z\W 1#1!1\Z!\W {% - \expandafter\XINT_pow_loopI_odda\expandafter - {\romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z }{#1}{#2}% + \XINT_smallmul 1#1!% }% -\def\XINT_pow_loopI_odda #1#2#3% +%%%%%%%%%%%% +\def\XINT_pow_II_in #1.#2\W {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% + \expandafter\XINT_pow_II_loop + \the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W #2\W }% -\def\XINT_pow_loopII #1% +\def\XINT_pow_II_loop #1.% {% - \ifnum #1 = \xint_c_i\XINT_pow_IIend\fi + \ifnum #1 = \xint_c_i\expandafter\XINT_pow_II_exit\fi \ifodd #1 - \expandafter\XINT_pow_loopII_odd + \expandafter\XINT_pow_II_odda \else - \expandafter\XINT_pow_loopII_even + \expandafter\XINT_pow_II_even + \fi #1.% +}% +\def\XINT_pow_II_exit\ifodd #1\fi #2.#3\W #4\W +{% + \expandafter\XINT_mul_out + \the\numexpr\XINT_pow_mulbutcheckifsmall #4\W #3% +}% +\def\XINT_pow_II_even #1.#2\W +{% + \expandafter\XINT_pow_II_loop + \the\numexpr #1/\xint_c_ii\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #2\W #2\W +}% +\def\XINT_pow_II_odda #1.#2\W #3\W +{% + \expandafter\XINT_pow_II_oddb + \the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #3\W #2\W #2\W +}% +\def\XINT_pow_II_oddb #1.#2\W #3\W +{% + \expandafter\XINT_pow_II_loop + \the\numexpr #1\expandafter.% + \the\numexpr\XINT_pow_mulbutcheckifsmall #3\W #3\W #2\W +}% +\def\xintiFac {\romannumeral0\xintifac }% +\def\xintifac #1% +{% + \expandafter\XINT_fac_fork\expandafter {\the\numexpr#1}% +}% +\def\xintiiFac {\romannumeral0\xintiifac }% +\def\xintiifac #1% +{% + \expandafter\XINT_fac_fork\expandafter {\romannumeral-`0#1}% +}% +\let\xintFac\xintiFac \let\xintfac\xintifac +\def\XINT_fac_fork #1% +{% + \ifcase\XINT_cntSgn #1\Z + \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }% + \or + \expandafter\XINT_fac_checksize + \else + \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber + \expandafter\space\expandafter 1\xint_gobble_i }% \fi {#1}% }% -\def\XINT_pow_loopII_even #1#2% +\def\XINT_fac_checksize #1% +{% + \ifnum #1>9999 + \xint_dothis{\expandafter\xintError:FactorialOfTooBigNumber + \expandafter\space\expandafter 1\xint_gob_til_W }\fi + \ifnum #1>465 \xint_dothis{\XINT_fac_bigloop_a #1.}\fi + \ifnum #1>101 \xint_dothis{\XINT_fac_medloop_a #1.\XINT_mul_out}\fi + \xint_orthat{\XINT_fac_smallloop_a #1.\XINT_mul_out}% + 1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_fac_bigloop_a #1.% +{% + \expandafter\XINT_fac_bigloop_b \the\numexpr + #1+\xint_c_i-\xint_c_ii*((#1-464)/\xint_c_ii).#1.% +}% +\def\XINT_fac_bigloop_b #1.#2.% +{% + \expandafter\XINT_fac_medloop_a + \the\numexpr #1-\xint_c_i.{\XINT_fac_bigloop_loop #1.#2.}% +}% +\def\XINT_fac_bigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_fac_bigloop_exit\fi + \expandafter\XINT_fac_bigloop_loop + \the\numexpr #1+\xint_c_ii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_bigloop_mul #1!% +}% +\def\XINT_fac_bigloop_exit #1!{\XINT_mul_out}% +\def\XINT_fac_bigloop_mul #1!% +{% + \expandafter\XINT_smallmul + \the\numexpr \xint_c_x^viii+#1*(#1+\xint_c_i)!% +}% +\def\XINT_fac_medloop_a #1.% +{% + \expandafter\XINT_fac_medloop_b + \the\numexpr #1+\xint_c_i-\xint_c_iii*((#1-100)/\xint_c_iii).#1.% +}% +\def\XINT_fac_medloop_b #1.#2.% +{% + \expandafter\XINT_fac_smallloop_a + \the\numexpr #1-\xint_c_i.{\XINT_fac_medloop_loop #1.#2.}% +}% +\def\XINT_fac_medloop_loop #1.#2.% {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% + \ifnum #1>#2 \expandafter\XINT_fac_loop_exit\fi + \expandafter\XINT_fac_medloop_loop + \the\numexpr #1+\xint_c_iii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_medloop_mul #1!% }% -\def\XINT_pow_loopII_odd #1#2#3% +\def\XINT_fac_medloop_mul #1!% {% - \expandafter\XINT_pow_loopII_odda\expandafter - {\romannumeral0\XINT_mulr_enter #3\Z\Z\Z\Z #2\W\W\W\W}{#1}{#2}% + \expandafter\XINT_smallmul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)!% }% -\def\XINT_pow_loopII_odda #1#2#3% +\def\XINT_fac_smallloop_a #1.% {% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% + \csname + XINT_fac_smallloop_\the\numexpr #1-\xint_c_iv*(#1/\xint_c_iv)\relax + \endcsname #1.% }% -\def\XINT_pow_IIend\fi #1\fi #2#3#4% +\expandafter\def\csname XINT_fac_smallloop_1\endcsname #1.% {% - \fi\XINT_mul_enter #4\Z\Z\Z\Z #3\W\W\W\W + \XINT_fac_smallloop_loop 2.#1.100000001!1\Z!% }% +\expandafter\def\csname XINT_fac_smallloop_-2\endcsname #1.% +{% + \XINT_fac_smallloop_loop 3.#1.100000002!1\Z!% +}% +\expandafter\def\csname XINT_fac_smallloop_-1\endcsname #1.% +{% + \XINT_fac_smallloop_loop 4.#1.100000006!1\Z!% +}% +\expandafter\def\csname XINT_fac_smallloop_0\endcsname #1.% +{% + \XINT_fac_smallloop_loop 5.#1.1000000024!1\Z!% +}% +\def\XINT_fac_smallloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_fac_loop_exit\fi + \expandafter\XINT_fac_smallloop_loop + \the\numexpr #1+\xint_c_iv\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_fac_smallloop_mul #1!% +}% +\def\XINT_fac_smallloop_mul #1!% +{% + \expandafter\XINT_smallmul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)*(#1+\xint_c_iii)!% +}% +\def\XINT_fac_loop_exit #1!#2\Z!#3{#3#2\Z!}% \def\xintiiQuo {\romannumeral0\xintiiquo }% \def\xintiiRem {\romannumeral0\xintiirem }% \def\xintiiquo {\expandafter\xint_firstoftwo_thenstop\romannumeral0\xintiidivision }% @@ -1700,12 +1402,11 @@ \def\xintiRem {\romannumeral0\xintirem }% \def\xintiquo {\expandafter\xint_firstoftwo_thenstop\romannumeral0\xintidivision }% \def\xintirem {\expandafter\xint_secondoftwo_thenstop\romannumeral0\xintidivision }% -\let\xintQuo\xintiQuo\let\xintquo\xintiquo % deprecated (1.1) -\let\xintRem\xintiRem\let\xintrem\xintirem % deprecated (1.1) +\let\xintQuo\xintiQuo\let\xintquo\xintiquo % deprecated +\let\xintRem\xintiRem\let\xintrem\xintirem % deprecated \def\xintiDivision {\romannumeral0\xintidivision }% -\def\xintidivision #1{\expandafter\XINT_division\romannumeral0\xintnum{#1}\Z }% -\let\xintDivision\xintiDivision \let\xintdivision\xintidivision % deprecated -\def\XINT_division #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1% +\def\xintidivision #1{\expandafter\XINT_idivision\romannumeral0\xintnum{#1}\Z }% +\def\XINT_idivision #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1% \romannumeral0\xintnum{#3}\Z #2\Z }% \def\xintiiDivision {\romannumeral0\xintiidivision }% \def\xintiidivision #1{\expandafter\XINT_iidivision \romannumeral-`0#1\Z }% @@ -1719,10 +1420,10 @@ \romannumeral0\XINT_iidivision_bpos #1}\fi \xint_orthat{\XINT_iidivision_bpos #1#2}% }% -\def\XINT_iidivision_divbyzero #1\Z #2\Z {\xintError:DivisionByZero\space {0}{0}}% -\def\XINT_iidivision_aiszero #1\Z #2\Z { {0}{0}}% +\def\XINT_iidivision_divbyzero #1\Z #2\Z {\xintError:DivisionByZero{0}{0}}% +\def\XINT_iidivision_aiszero #1\Z #2\Z {{0}{0}}% \def\XINT_iidivision_bneg #1% q->-q, r unchanged - {\expandafter\space\expandafter{\romannumeral0\XINT_opp #1}}% + {\expandafter{\romannumeral0\XINT_opp #1}}% \def\XINT_iidivision_bpos #1% {% \xint_UDsignfork @@ -1739,7 +1440,7 @@ \else \expandafter\XINT_iidivision_aneg_rpos \fi {#1}{#2}}% -\def\XINT_iidivision_aneg_rzero #1#2#3{ {-#1}{0}}% necessarily q was >0 +\def\XINT_iidivision_aneg_rzero #1#2#3{{-#1}{0}}% necessarily q was >0 \def\XINT_iidivision_aneg_rpos #1% {% \expandafter\XINT_iidivision_aneg_end\expandafter @@ -1747,155 +1448,185 @@ }% \def\XINT_iidivision_aneg_end #1#2#3% {% - \expandafter\xint_exchangetwo_keepbraces_thenstop - \expandafter{\romannumeral0\XINT_sub_pre {#3}{#2}}{#1}% r-> b-r + \expandafter\xint_exchangetwo_keepbraces + \expandafter{\romannumeral0\XINT_sub_mm_a {}{}#3\Z #2\Z}{#1}% r-> b-r }% +%%%%%%%%%%%% \def\XINT_div_prepare #1% {% - \expandafter \XINT_div_prepareB_aa \expandafter - {\romannumeral0\xintlength {#1}}{#1}% B > 0 ici -}% -\def\XINT_div_prepareB_aa #1% -{% - \ifnum #1=\xint_c_i - \expandafter\XINT_div_prepareB_onedigit - \else - \expandafter\XINT_div_prepareB_a - \fi - {#1}% + \XINT_div_prepare_a #1\R\R\R\R\R\R\R\R {10}0000001\W !{#1}% }% -\def\XINT_div_prepareB_a #1% +\def\XINT_div_prepare_a #1#2#3#4#5#6#7#8#9% {% - \expandafter\XINT_div_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \xint_gob_til_R #9\XINT_div_prepare_small\R + \XINT_div_prepare_b #9% }% -\def\XINT_div_prepareB_onedigit #1#2% +%%%%%%%%%%%% +\def\XINT_div_prepare_small\R #1!#2% {% - \ifcase#2 + \ifcase #2 \or\expandafter\XINT_div_BisOne \or\expandafter\XINT_div_BisTwo - \else\expandafter\XINT_div_prepareB_e - \fi {000}{0}{4}{#2}% + \else\expandafter\XINT_div_small_a + \fi {#2}% }% -\def\XINT_div_BisOne #1#2#3#4#5{ {#5}{0}}% -\def\XINT_div_BisTwo #1#2#3#4#5% +\def\XINT_div_BisOne #1#2{{#2}{0}}% +\def\XINT_div_BisTwo #1#2% {% \expandafter\expandafter\expandafter\XINT_div_BisTwo_a - \ifodd\xintiiLDg{#5} \expandafter1\else \expandafter0\fi {#5}% + \ifodd\xintLDg{#2} \expandafter1\else \expandafter0\fi {#2}% }% -\edef\XINT_div_BisTwo_a #1#2% +\def\XINT_div_BisTwo_a #1#2% {% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\xinthalf {#2}}{#1}% + \expandafter{\romannumeral0\xinthalf {#2}}{#1}% }% -\def\XINT_div_prepareB_c #1#2% +\def\XINT_div_small_a #1#2% {% - \csname XINT_div_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% + \expandafter\XINT_div_small_b + \the\numexpr #1/\xint_c_ii\expandafter + .\the\numexpr \xint_c_x^viii+#1\expandafter!% + \romannumeral0% + \XINT_div_small_ba #2\R\R\R\R\R\R\R\R{10}0000001\W + #2\XINT_sepbyviii_Z_end 2345678\relax }% -\def\XINT_div_prepareB_d {\XINT_div_prepareB_e {}{0000}}% -\def\XINT_div_prepareB_di {\XINT_div_prepareB_e {0}{000}}% -\def\XINT_div_prepareB_dii {\XINT_div_prepareB_e {00}{00}}% -\def\XINT_div_prepareB_diii {\XINT_div_prepareB_e {000}{0}}% -\def\XINT_div_cleanR #10000.{{#1}}% -\def\XINT_div_prepareB_e #1#2#3#4% +\def\XINT_div_small_b #1!#2{#2#1!}% +\def\XINT_div_small_ba #1#2#3#4#5#6#7#8#9% {% - \ifnum#3=\xint_c_iv\expandafter\XINT_div_prepareLittleB_f - \else\expandafter\XINT_div_prepareB_f - \fi - #4#1{#3}{#2}{#1}% + \xint_gob_til_R #9\XINT_div_smallsmall\R + \expandafter\XINT_div_dosmalldiv + \the\numexpr\expandafter\XINT_sepbyviii_Z + \romannumeral0\XINT_zeroes_forviii + #1#2#3#4#5#6#7#8#9% }% -\def\XINT_div_prepareB_f #1#2#3#4#5#{% - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% +\def\XINT_div_smallsmall\R + \expandafter\XINT_div_dosmalldiv + \the\numexpr\expandafter\XINT_sepbyviii_Z + \romannumeral0\XINT_zeroes_forviii #1\R #2\relax + {{\XINT_div_dosmallsmall}{#1}}% +\def\XINT_div_dosmallsmall #1.1#2!#3% +{% + \expandafter\XINT_div_smallsmallend + \the\numexpr (#3+#1)/#2-\xint_c_i.#2.#3.% }% -\def\XINT_div_prepareLittleB_f #1#{% - \expandafter\XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% +\def\XINT_div_smallsmallend #1.#2.#3.{\expandafter + {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #3-#1*#2}}% +\def\XINT_div_dosmalldiv + {{\expandafter\XINT_sdiv_out\the\numexpr\XINT_smalldivx_a}}% +%%%%%%%%%%%% +\def\XINT_div_prepare_b + {\expandafter\XINT_div_prepare_c\romannumeral0\XINT_zeroes_forviii }% +\def\XINT_div_prepare_c #1!% +{% + \XINT_div_prepare_d #1.00000000!{#1}% }% -\def\XINT_div_prepareB_g #1.#2.#3.#4#5#6#7#8% +\def\XINT_div_prepare_d #1#2#3#4#5#6#7#8#9% {% - \XINT_div_prepareA_a {#8#7}{#5}{{#1}{#2}{#4}}{#3}{#6}% + \expandafter\XINT_div_prepare_e\xint_gob_til_dot #1#2#3#4#5#6#7#8#9!% }% -\def\XINT_div_prepareA_a #1% +\def\XINT_div_prepare_e #1!#2!#3#4% {% - \expandafter\XINT_div_prepareA_b\expandafter - {\romannumeral0\xintlength {#1}}{#1}% + \XINT_div_prepare_f #4#3\X {#1}{#3}% }% -\def\XINT_div_prepareA_b #1% +\def\XINT_div_prepare_f #1#2#3#4#5#6#7#8#9\X {% - \expandafter\XINT_div_prepareA_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \expandafter\XINT_div_prepare_g + \the\numexpr #1#2#3#4#5#6#7#8+\xint_c_i\expandafter + .\the\numexpr (#1#2#3#4#5#6#7#8+\xint_c_i)/\xint_c_ii\expandafter + .\the\numexpr #1#2#3#4#5#6#7#8\expandafter + .\romannumeral0\XINT_sepandrev_andcount + #1#2#3#4#5#6#7#8#9\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678% + \relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X }% -\def\XINT_div_prepareA_c #1#2% +\def\XINT_div_prepare_g #1.#2.#3.#4.#5\X #6#7#8% {% - \csname XINT_div_prepareA_d\romannumeral\numexpr #1-#2\endcsname - {#1}% + \expandafter\XINT_div_prepare_h + \the\numexpr\expandafter\XINT_sepbyviii_andcount + \romannumeral0\XINT_zeroes_forviii #8#7\R\R\R\R\R\R\R\R{10}0000001\W + #8#7\XINT_sepbyviii_end 2345678\relax + \xint_c_vii!\xint_c_vi!\xint_c_v!\xint_c_iv!% + \xint_c_iii!\xint_c_ii!\xint_c_i!\xint_c_\W + {#1}{#2}{#3}{#4}{#5}{#6}% }% -\def\XINT_div_prepareA_d {\XINT_div_prepareA_e {}}% -\def\XINT_div_prepareA_di {\XINT_div_prepareA_e {0}}% -\def\XINT_div_prepareA_dii {\XINT_div_prepareA_e {00}}% -\def\XINT_div_prepareA_diii {\XINT_div_prepareA_e {000}}% -\def\XINT_div_prepareA_e #1#2#3#4#5% +\def\XINT_div_prepare_h #11.#2.#3#4#5#6%#7#8% {% - \XINT_div_start_a {#2}{#4}{#1#3}#5% + \XINT_div_start_a {#2}{#6}{#1}{#3}{#4}{#5}%{#7}{#8}% }% \def\XINT_div_start_a #1#2% {% - \ifnum #2=\xint_c_iv \expandafter\XINT_div_little_b + \ifnum #1 < #2 + \expandafter\XINT_div_zeroQ \else - \ifnum #1 < #2 - \expandafter\expandafter\expandafter\XINT_div_III_aa - \else - \expandafter\expandafter\expandafter\XINT_div_start_b - \fi + \expandafter\XINT_div_start_b \fi {#1}{#2}% }% -\def\XINT_div_III_aa #1#2#3#4#5#6#7% +\def\XINT_div_zeroQ #1#2#3#4#5#6#7% {% - \expandafter\expandafter\expandafter - \XINT_div_III_b\xint_cleanupzeros_nostop #3.{0000}% + \expandafter\XINT_div_zeroQ_end + \romannumeral0\XINT_unsep_cuzsmall + #31\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W .% }% -\def\XINT_div_III_b #1% +\def\XINT_div_zeroQ_end #1.#2% + {\expandafter{\expandafter0\expandafter}\XINT_div_cleanR #1#2.}% +\def\XINT_div_start_b #1#2#3#4#5#6% {% - \if0#1% - \expandafter\XINT_div_III_bRzero - \else - \expandafter\XINT_div_III_bRpos - \fi - #1% + \expandafter\XINT_div_finish\the\numexpr + \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}% }% -\def\XINT_div_III_bRzero 0.#1#2% +\def\XINT_div_finish {% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z}{0}% + \expandafter\XINT_div_finish_a \romannumeral-`0\XINT_div_unsepQ }% -\def\XINT_div_III_bRpos #1.#2#3% +\def\XINT_div_finish_a #1\Z #2.{\XINT_div_finish_b #2.{#1}}% +\def\XINT_div_finish_b #1% {% - \expandafter\XINT_div_III_c \XINT_div_cleanR #1#3.{#2}% + \if0#1% + \expandafter\XINT_div_finish_bRzero + \else + \expandafter\XINT_div_finish_bRpos + \fi + #1% }% -\def\XINT_div_III_c #1#2% +\def\XINT_div_finish_bRzero 0.#1#2{{#1}{0}}% +\def\XINT_div_finish_bRpos #1.#2#3% {% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #2\W\W\W\W\W\W\W\Z}{#1}% + \expandafter\xint_exchangetwo_keepbraces\XINT_div_cleanR #1#3.{#2}% }% -\def\XINT_div_start_b #1#2#3#4#5#6% +\def\XINT_div_cleanR #100000000.{{#1}}% +\def\XINT_div_start_c #1% {% - \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}% + \ifnum #1>\xint_c_vi + \expandafter\XINT_div_start_ca + \else + \expandafter\XINT_div_start_cb + \fi {#1}% }% -\def\XINT_div_start_c #1#2.#3#4#5#6% +\def\XINT_div_start_ca #1#2.#3!#4!#5!#6!#7!#8!#9!% {% - \ifnum #1=\xint_c_iv\XINT_div_start_ca\fi \expandafter\XINT_div_start_c\expandafter - {\the\numexpr #1-\xint_c_iv}#2#3#4#5#6.% -}% -\def\XINT_div_start_ca\fi\expandafter\XINT_div_start_c\expandafter - #1#2#3#4#5{\fi\XINT_div_start_d {#2#3#4#5}#2#3#4#5}% -\def\XINT_div_start_d #1#2.#3.#4#5#6% -{% - \XINT_div_I_a {#1}{#4}{#2}{#6}{0000}#5{#3}{#6}{}% + {\the\numexpr #1-\xint_c_vii}#2#3!#4!#5!#6!#7!#8!#9!.% +}% +\def\XINT_div_start_cb #1% + {\csname XINT_div_start_c_\romannumeral\numexpr#1\endcsname}% +\def\XINT_div_start_c_i #1.#2!% + {\XINT_div_start_c_ #1#2!.}% +\def\XINT_div_start_c_ii #1.#2!#3!% + {\XINT_div_start_c_ #1#2!#3!.}% +\def\XINT_div_start_c_iii #1.#2!#3!#4!% + {\XINT_div_start_c_ #1#2!#3!#4!.}% +\def\XINT_div_start_c_iv #1.#2!#3!#4!#5!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!.}% +\def\XINT_div_start_c_v #1.#2!#3!#4!#5!#6!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!#6!.}% +\def\XINT_div_start_c_vi #1.#2!#3!#4!#5!#6!#7!% + {\XINT_div_start_c_ #1#2!#3!#4!#5!#6!#7!.}% +\def\XINT_div_start_c_ 1#1!#2.#3.#4#5#6% +{% + \XINT_div_I_a {#1}{#4}{1#1!#2}{#6}{00000000}#5{#3}{#6}% }% \def\XINT_div_I_a #1#2% {% @@ -1905,11 +1636,10 @@ {% \xint_gob_til_zero #1\XINT_div_I_czero 0\XINT_div_I_c #1% }% -\def\XINT_div_I_czero 0% - \XINT_div_I_c 0.#1#2#3#4#5{\XINT_div_I_g {#5}{#3}}% +\def\XINT_div_I_czero 0\XINT_div_I_c 0.#1#2#3#4#5{1#5\XINT_div_I_g {#3}}% \def\XINT_div_I_c #1.#2#3% {% - \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.% + \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.{#2}{#3}% }% \def\XINT_div_I_da #1.% {% @@ -1925,238 +1655,351 @@ }% \def\XINT_div_I_dN #1.% {% - \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.% + \expandafter\XINT_div_I_b\the\numexpr #1-\xint_c_i.% }% -\def\XINT_div_I_db #1.#2#3% #1=q=un chiffre, #2=alpha, #3=B +\def\XINT_div_I_db #1.#2#3#4#5% {% - \expandafter\XINT_div_I_dc\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% - #1{#2}{#3}% + \expandafter\XINT_div_I_dc\expandafter #1% + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#4\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_verysmallmul #1!#51\Z!}% + \Z {#4}{#5}% }% \def\XINT_div_I_dc #1#2% {% - \if-#1% s'arranger pour que si n\'egatif on ait renvoy\'e alpha=-. - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {\expandafter\XINT_div_I_dP\the\numexpr #2-\xint_c_i.}% - {\XINT_div_I_e {#1}#2}% + \if-#2\expandafter\XINT_div_I_dd\else\expandafter\XINT_div_I_de\fi + #1#2% +}% +\def\XINT_div_I_dd #1-\Z +{% + \if #11\expandafter\XINT_div_I_dz\fi + \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.XX% }% -\def\XINT_div_I_e #1#2#3#4#5% +\def\XINT_div_I_dz #1XX#2#3#4% {% - \expandafter\XINT_div_I_f \the\numexpr \xint_c_x^iv+#2+#5{#1}% + 1#4\XINT_div_I_g {#2}% }% -\def\XINT_div_I_dP #1.#2#3#4% +\def\XINT_div_I_de #1#2\Z #3#4#5{1#5+#1\XINT_div_I_g {#2}}% +\def\XINT_div_I_dP #1.#2#3#4#5#6% {% - \expandafter \XINT_div_I_f \the\numexpr \xint_c_x^iv+#1+#4\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% + 1#6+#1\expandafter\XINT_div_I_g\expandafter + {\romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#4\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_verysmallmul #1!#51\Z!}% + }% }% -\def\XINT_div_I_f 1#1#2#3#4{\XINT_div_I_g {#1#2#3#4}}% -\def\XINT_div_I_g #1#2#3#4#5#6#7#8#9% +\def\XINT_div_I_g #1#2#3#4#5#6#7% {% - \ifnum#3=#4 - \expandafter\XINT_div_III_ab + \expandafter !\the\numexpr + \ifnum#2=#3 + \expandafter\XINT_div_exittofinish \else \expandafter\XINT_div_I_h \fi - {#5}#2.#7.{{#5}{#6}{#4}{#3}}{#8}{#9#1}% + {#4}#1.#6.{{#4}{#5}{#3}{#2}}{#7}% }% -\def\XINT_div_III_ab #1#2.#3.#4#5% +\def\XINT_div_exittofinish #1#2.#3.#4#5% {% - \expandafter\XINT_div_III_b - \romannumeral0\XINT_cuz_loop #2#3\W\W\W\W\W\W\W\Z.% + 1\expandafter\expandafter\expandafter!\expandafter\XINT_unsep_delim + \romannumeral0\XINT_div_unsepR #2#31\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W.% }% -\def\XINT_div_I_h #1.#2#3#4#5#6.#7#8% +\def\XINT_div_I_h #1.#2!#3.#4#5% {% - \XINT_div_II_b #1#2#3#4#5.{#8}{#7}{#6}{#8}% + \XINT_div_II_b #1#2!.{#5}{#4}{#3}{#5}% }% -\def\XINT_div_II_b #1#2#3#4#5#6#7#8#9% +\def\XINT_div_II_b #11#2!#3!% {% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_div_II_skipc 0000% - \XINT_div_II_c #1{#2#3#4#5}{#6#7#8#9}% + \xint_gob_til_eightzeroes #2\XINT_div_II_skipc 00000000% + \XINT_div_II_c #1{1#2}{#3}% }% -\def\XINT_div_II_skipc 0000\XINT_div_II_c #1#2#3#4#5.#6#7% +\def\XINT_div_II_skipc 00000000\XINT_div_II_c #1#2#3#4#5.#6#7% {% - \XINT_div_II_k #7{#4#5}{#6}{0000}% + \XINT_div_II_k #7{#4!#5}{#6}{00000000}% }% \def\XINT_div_II_c #1#2#3#4% {% - \expandafter\XINT_div_II_d\the\numexpr (#3#4+#2)/#1+\xint_c_ixixixix\relax - {#1}{#2}#3#4% + \expandafter\XINT_div_II_d\the\numexpr\XINT_div_mini + #1.#2!#3!#4!{#1}{#2}#3!#4!% }% -\def\XINT_div_II_d 1#1#2#3#4#5#6#7.#8% +\def\XINT_div_II_d 1#1#2#3#4#5!#6#7#8.#9% {% \expandafter\XINT_div_II_e - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#7}}% - {\romannumeral0\XINT_mul_Mr {#1#2#3#4}#8\Z\Z\Z\Z }.% - {#5}{#6}{#8}{#1#2#3#4}% + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#8\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_smallmul_a 100000000.#1#2#3#4.#5!#91\Z!}% + .{#6}{#7}{#9}{#1#2#3#4#5}% }% -\def\XINT_div_II_e #1#2#3#4% +\def\XINT_div_II_e 1#1!% {% - \xint_gob_til_zeros_iv #1#2#3#4\XINT_div_II_skipf 0000% - \XINT_div_II_f #1#2#3#4% + \xint_gob_til_eightzeroes #1\XINT_div_II_skipf 00000000% + \XINT_div_II_f 1#1!% }% -\def\XINT_div_II_skipf 0000\XINT_div_II_f 0000#1.#2#3#4#5#6% +\def\XINT_div_II_skipf 00000000\XINT_div_II_f 100000000!#1.#2#3#4#5#6% {% \XINT_div_II_k #6{#1}{#4}{#5}% }% -\def\XINT_div_II_f #1#2#3#4#5#6#7#8#9.% +\def\XINT_div_II_f #1!#2!#3.% {% - \XINT_div_II_fa {#1#2#3#4#5#6#7#8}{#1#2#3#4#5#6#7#8#9}% + \XINT_div_II_fa {#1!#2!}{#1!#2!#3}% }% \def\XINT_div_II_fa #1#2#3#4% {% - \expandafter\XINT_div_II_g\expandafter - {\the\numexpr (#1+#4)/#3-\xint_c_i}{#2}% + \expandafter\XINT_div_II_g \the\numexpr\XINT_div_mini #3.#4!#1{#2}% }% -\def\XINT_div_II_g #1#2#3#4% +\def\XINT_div_II_g 1#1#2#3#4#5!#6#7#8% {% \expandafter \XINT_div_II_h - \the\numexpr #4+#1+\xint_c_x^iv\expandafter\expandafter\expandafter - {\expandafter\xint_gobble_iv - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}{#3}% + \the\numexpr 1#1#2#3#4#5+#8\expandafter\expandafter\expandafter + .\expandafter\expandafter\expandafter + {\expandafter\xint_gob_til_exclam + \romannumeral0\expandafter\XINT_div_sub\expandafter + {\romannumeral0\XINT_rev_nounsep {}#6\R!\R!\R!\R!\R!\R!\R!\R!\W}% + {\the\numexpr\XINT_div_smallmul_a 100000000.#1#2#3#4.#5!#71\Z!}}% + {#7}% }% -\def\XINT_div_II_h 1#1#2#3#4#5#6#7% +\def\XINT_div_II_h 1#1.#2#3#4% {% - \XINT_div_II_k #7{#5}{#6}{#1#2#3#4}% + \XINT_div_II_k #4{#2}{#3}{#1}% }% \def\XINT_div_II_k #1#2#3#4#5% {% - \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_iv.{#3}#1{#2}#5.% + \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_i.{#3}#1{#2}#5.% }% -\def\XINT_div_II_l #1.#2#3#4#5#6#7#8#9% +\def\XINT_div_II_l #1.#2#3#4#51#6!% {% - \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6#7#8#9}#6#7#8#9% + \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6}1#6!% }% \def\XINT_div_II_m #1#2#3#4.#5#6% {% \XINT_div_I_a {#3}{#2}{#4}{#5}{#6}#1% }% -\def\XINT_div_little_b #1#2#3#4#5#6#7% +\def\XINT_div_minimulwc_a 1#1.#2.#3!#4#5#6#7#8.% {% - \XINT_div_little_c #3.{{#4}{#6}}{#1}% + \expandafter\XINT_div_minimulwc_b + \the\numexpr \xint_c_x^ix+#1+#3*#8.#3*#4#5#6#7+#2*#8.#2*#4#5#6#7.% }% -\def\XINT_div_little_c #1#2#3#4#5.#6#7% +\def\XINT_div_minimulwc_b 1#1#2#3#4#5#6.#7.% {% - \XINT_div_littleI_a {#1#2#3#4}#6{#7}{#5}% + \expandafter\XINT_div_minimulwc_c + \the\numexpr \xint_c_x^ix+#1#2#3#4#5+#7.#6.% }% -\def\XINT_div_littleI_a #1#2#3% +\def\XINT_div_minimulwc_c 1#1#2#3#4#5#6.#7.#8.% {% - \expandafter\XINT_div_littleI_b - \the\numexpr (#1+#2)/#3-\xint_c_i{#1}{#2}{#3}% + 1#6#7\expandafter!% + \the\numexpr\expandafter\XINT_div_smallmul_a + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#8.% }% -\def\XINT_div_littleI_b #1% +\def\XINT_div_smallmul_a #1.#2.#3!1#4!% {% - \xint_gob_til_zero #1\XINT_div_littleI_skip 0\XINT_div_littleI_c #1% + \xint_gob_til_Z #4\XINT_div_smallmul_e\Z + \XINT_div_minimulwc_a #1.#2.#3!#4.#2.#3!% }% -\def\XINT_div_littleI_skip 0\XINT_div_littleI_c 0#1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{{#2}{#3}}{0000}}% -\def\XINT_div_littleI_c #1#2#3#4% +\def\XINT_div_smallmul_e\Z\XINT_div_minimulwc_a 1#1.#2\Z #3!{1\relax #1!}% +\def\XINT_div_verysmallmul #1% + {\xint_gob_til_one #1\XINT_div_verysmallisone 1\XINT_div_verysmallmul_a 0.#1}% +\def\XINT_div_verysmallisone 1\XINT_div_verysmallmul_a 0.1!1#11\Z!% + {1\relax #1100000000!}% +\def\XINT_div_verysmallmul_a #1.#2!1#3!% {% - \expandafter\expandafter\expandafter\XINT_div_littleI_e - \expandafter\expandafter\expandafter - {\expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4}#1{{#3}{#4}}% + \xint_gob_til_Z #3\XINT_div_verysmallmul_e\Z + \expandafter\XINT_div_verysmallmul_b + \the\numexpr \xint_c_x^ix+#2*#3+#1.#2!% }% -\def\XINT_div_littleI_e #1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{#3}{000#2}}% -\def\XINT_div_littleII_a #1% +\def\XINT_div_verysmallmul_b 1#1#2.% + {1#2\expandafter!\the\numexpr\XINT_div_verysmallmul_a #1.}% +\def\XINT_div_verysmallmul_e\Z #1\Z +#2#3!{1\relax 0000000#2!}% +\def\XINT_div_sub #1#2% {% - \ifnum#1=\xint_c_iv - \expandafter\XINT_div_littleIII_ab - \else - \expandafter\XINT_div_littleII_b - \fi {#1}% + \expandafter\XINT_div_sub_clean + \the\numexpr\expandafter\XINT_div_sub_a\expandafter + 1#2\Z!\Z!\Z!\Z!\Z!\W #1\Z!\Z!\Z!\Z!\Z!\W }% -\def\XINT_div_littleIII_ab #1#2#3.#4% +\def\XINT_div_sub_clean #1-#2#3\W {% - \expandafter\XINT_div_III_b\the\numexpr #2#3.% + \if1#2\expandafter\XINT_rev_nounsep\else\expandafter\XINT_div_sub_neg\fi + {}#1\R!\R!\R!\R!\R!\R!\R!\R!\W }% -\def\XINT_div_littleII_b #1% +\def\XINT_div_sub_neg #1\W { -}% +\def\XINT_div_sub_a #1!#2!#3!#4!#5\W #6!#7!#8!#9!% {% - \expandafter\XINT_div_littleII_c\expandafter {\the\numexpr #1-\xint_c_iv}% + \XINT_div_sub_b #1!#6!#2!#7!#3!#8!#4!#9!#5\W }% -\def\XINT_div_littleII_c #1#2#3#4#5#6#7.#8% +\def\XINT_div_sub_b #1#2#3!#4!% {% - \XINT_div_littleII_d {#2#3#4#5#6}#8{#1}{#7}% + \xint_gob_til_Z #4\XINT_div_sub_bi \Z + \expandafter\XINT_div_sub_c\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_littleII_d #1#2#3% +\def\XINT_div_sub_c 1#1#2.% {% - \expandafter\XINT_div_littleII_e\the\numexpr (#1+#2)/#3+\xint_c_ixixixix.% - {#1}{#2}{#3}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_d #1% }% -\def\XINT_div_littleII_e 1#1.#2#3#4% +\def\XINT_div_sub_d #1#2#3!#4!% {% - \expandafter\expandafter\expandafter\XINT_div_littleII_f - \expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4.% - {#1}{{#3}{#4}}% + \xint_gob_til_Z #4\XINT_div_sub_di \Z + \expandafter\XINT_div_sub_e\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_littleII_f #1.#2#3#4#5#6% +\def\XINT_div_sub_e 1#1#2.% {% - \XINT_div_littleII_a {#4}{#1}#5.{#3}{#6#2}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_f #1% }% -\def\XINT_div_sub_xpxp #1#2% #1=alpha d\'ej\`a renvers\'e, #2 se d\'eveloppe en qB +\def\XINT_div_sub_f #1#2#3!#4!% {% - \expandafter\XINT_div_sub_xpxp_b #2\W\X\Y\Z #1\W\X\Y\Z + \xint_gob_til_Z #4\XINT_div_sub_fi \Z + \expandafter\XINT_div_sub_g\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_sub_xpxp_b +\def\XINT_div_sub_g 1#1#2.% {% - \XINT_div_sub_A 1{}% + 1#2\expandafter!\the\numexpr\XINT_div_sub_h #1% }% -\def\XINT_div_sub_A #1#2#3#4#5#6% +\def\XINT_div_sub_h #1#2#3!#4!% {% - \xint_gob_til_W #3\xint_div_sub_az\W - \XINT_div_sub_B #1{#3#4#5#6}{#2}% + \xint_gob_til_Z #4\XINT_div_sub_hi \Z + \expandafter\XINT_div_sub_i\the\numexpr#1-#3+1#4-\xint_c_i.% }% -\def\XINT_div_sub_B #1#2#3#4\W\X\Y\Z #5#6#7#8% +\def\XINT_div_sub_i 1#1#2.% {% - \xint_gob_til_W #5\xint_div_sub_bz\W - \XINT_div_sub_onestep #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z + 1#2\expandafter!\the\numexpr\XINT_div_sub_a #1% }% -\def\XINT_div_sub_onestep #1#2#3#4#5#6% +\def\XINT_div_sub_bi\Z + \expandafter\XINT_div_sub_c\the\numexpr#1-#2+#3.#4!#5!#6!#7!#8!#9!\Z !\W {% - \expandafter\XINT_div_sub_backtoA - \the\numexpr 11#6-#5#4#3#2+#1-\xint_c_i.% + \XINT_div_sub_l #1#2!#5!#7!#9!% }% -\def\XINT_div_sub_backtoA #1#2#3.#4% +\def\XINT_div_sub_di\Z + \expandafter\XINT_div_sub_e\the\numexpr#1-#2+#3.#4!#5!#6!#7!#8\W {% - \XINT_div_sub_A #2{#3#4}% + \XINT_div_sub_l #1#2!#5!#7!% }% -\def\xint_div_sub_bz\W\XINT_div_sub_onestep #1\Z { -}% -\def\xint_div_sub_az\W\XINT_div_sub_B #1#2{\XINT_div_sub_C #1}% -\def\XINT_div_sub_C #1#2#3#4#5#6% +\def\XINT_div_sub_fi\Z + \expandafter\XINT_div_sub_g\the\numexpr#1-#2+#3.#4!#5!#6\W {% - \xint_gob_til_W #3\xint_div_sub_cz\W - \XINT_div_sub_C_onestep #1{#6#5#4#3}{#2}% + \XINT_div_sub_l #1#2!#5!% }% -\def\XINT_div_sub_C_onestep #1#2% +\def\XINT_div_sub_hi\Z + \expandafter\XINT_div_sub_i\the\numexpr#1-#2+#3.#4\W {% - \expandafter\XINT_div_sub_backtoC \the\numexpr 11#2+#1-\xint_c_i.% + \XINT_div_sub_l #1#2!% }% -\def\XINT_div_sub_backtoC #1#2#3.#4% +\def\XINT_div_sub_l #1% {% - \XINT_div_sub_C #2{#3#4}% + \xint_UDzerofork + #1{-2\relax}% + 0\XINT_div_sub_r + \krof }% -\def\xint_div_sub_cz\W\XINT_div_sub_C_onestep #1#2% +\def\XINT_div_sub_r #1!% {% - \if#10% retenue - \expandafter\xint_div_sub_neg - \else\expandafter\xint_div_sub_ok - \fi + -\ifnum 0#1=\xint_c_ 1\else2\fi\relax +}% +%%%%%%%%%%%% +\def\XINT_sdiv_out #1\Z #2\W% + {\expandafter + {\romannumeral0\XINT_unsep_cuzsmall#11\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W}% + {#2}}% +\def\XINT_smalldivx_a #1.1#2!1#3!% +{% + \expandafter\XINT_smalldivx_b + \the\numexpr (#3+#1)/#2-\xint_c_i!#1.#2!#3!% +}% +\def\XINT_smalldivx_b #1!% +{% + \if0#1\else + \xint_c_x^viii+#1\xint_afterfi{\expandafter!\the\numexpr}\fi + \XINT_smalldiv_c #1!% +}% +\def\XINT_smalldiv_c #1!#2.#3!#4!% +{% + \expandafter\XINT_smalldiv_d\the\numexpr #4-#1*#3!#2.#3!% +}% +\def\XINT_smalldiv_d #1!#2!#3#4!% +{% + \xint_gob_til_Z #4\XINT_smalldiv_end \Z + \XINT_smalldiv_e #1!#2!#3#4!% +}% +\def\XINT_smalldiv_end\Z\XINT_smalldiv_e #1!#2!1\Z!{1!\Z #1\W }% +\def\XINT_smalldiv_e #1!#2.#3!% +{% + \expandafter\XINT_smalldiv_f\the\numexpr + \xint_c_xi_e_viii_mone+#1*\xint_c_x^viii/#3!#2.#3!#1!% +}% +\def\XINT_smalldiv_f 1#1#2#3#4#5#6!#7.#8!% +{% + \xint_gob_til_zero #1\XINT_smalldiv_fz 0% + \expandafter\XINT_smalldiv_g + \the\numexpr\XINT_minimul_a #2#3#4#5.#6!#8!#2#3#4#5#6!#7.#8!% +}% +\def\XINT_smalldiv_fz 0% + \expandafter\XINT_smalldiv_g\the\numexpr\XINT_minimul_a + 9999.9999!#1!99999999!#2!0!1#3!% +{% + \XINT_smalldiv_i .#3!\xint_c_!#2!% +}% +\def\XINT_smalldiv_g 1#1!1#2!#3!#4!#5!#6!% +{% + \expandafter\XINT_smalldiv_h + \the\numexpr 1#6-#1.#2!#5!#3!#4!% +}% +\def\XINT_smalldiv_h 1#1#2.#3!#4!% +{% + \expandafter\XINT_smalldiv_i + \the\numexpr #4-#3+#1-\xint_c_i.#2!% +}% +\def\XINT_smalldiv_i #1.#2!#3!#4.#5!% +{% + \expandafter\XINT_smalldiv_j + \the\numexpr (#1#2+#4)/#5-\xint_c_i!#3!#1#2!#4.#5!% +}% +\def\XINT_smalldiv_j #1!#2!% +{% + \xint_c_x^viii+#1+#2\expandafter!\the\numexpr\XINT_smalldiv_k + #1!% +}% +\def\XINT_smalldiv_k #1!#2!#3.#4!% +{% + \expandafter\XINT_smalldiv_d\the\numexpr #2-#1*#4!#3.#4!% +}% +%%%%%%%%%%%% +\def\XINT_div_mini #1.#2!1#3!% +{% + \expandafter\XINT_div_mini_a\the\numexpr + \xint_c_xi_e_viii_mone+#3*\xint_c_x^viii/#1!#1.#2!#3!% +}% +\def\XINT_div_mini_a 1#1#2#3#4#5#6!#7.#8!% +{% + \xint_gob_til_zero #1\XINT_div_mini_w 0% + \expandafter\XINT_div_mini_b + \the\numexpr\XINT_minimul_a #2#3#4#5.#6!#7!#2#3#4#5#6!#7.#8!% +}% +\def\XINT_div_mini_w 0% + \expandafter\XINT_div_mini_b\the\numexpr\XINT_minimul_a + 9999.9999!#1!99999999!#2.#3!00000000!#4!% +{% + \xint_c_x^viii_mone+(#4+#3)/#2!% +}% +\def\XINT_div_mini_b 1#1!1#2!#3!#4!#5!#6!% +{% + \expandafter\XINT_div_mini_c + \the\numexpr 1#6-#1.#2!#5!#3!#4!% +}% +\def\XINT_div_mini_c 1#1#2.#3!#4!% +{% + \expandafter\XINT_div_mini_d + \the\numexpr #4-#3+#1-\xint_c_i.#2!% +}% +\def\XINT_div_mini_d #1.#2!#3!#4.#5!% +{% + \xint_c_x^viii_mone+#3+(#1#2+#5)/#4!% }% -\def\xint_div_sub_neg #1{ -}% -\def\xint_div_sub_ok #1{ #1}% \def\xintiDivRound {\romannumeral0\xintidivround }% -\def\xintidivround #1{\expandafter\XINT_iidivround\romannumeral0\xintnum{#1}\Z }% +\def\xintidivround #1% + {\expandafter\XINT_idivround\romannumeral0\xintnum{#1}\Z }% \def\xintiiDivRound {\romannumeral0\xintiidivround }% \def\xintiidivround #1{\expandafter\XINT_iidivround \romannumeral-`0#1\Z }% -\def\XINT_iidivround #1#2\Z #3{\expandafter\XINT_iidivround_a\expandafter #1% - \romannumeral-`0#3\Z #2\Z }% +\def\XINT_idivround #1#2\Z #3% + {\expandafter\XINT_iidivround_a\expandafter #1% + \romannumeral0\xintnum{#3}\Z #2\Z }% +\def\XINT_iidivround #1#2\Z #3% + {\expandafter\XINT_iidivround_a\expandafter #1\romannumeral-`0#3\Z #2\Z }% \def\XINT_iidivround_a #1#2% #1 de A, #2 de B. {% \if0#2\xint_dothis\XINT_iidivround_divbyzero\fi @@ -2180,19 +2023,50 @@ -{\xintiiopp\XINT_iidivround_pos #1}% \krof }% -\def\XINT_iidivround_pos #1#2\Z #3\Z{\expandafter\XINT_iidivround_pos_a - \romannumeral0\XINT_div_prepare {#2}{#1#30}}% -\def\XINT_iidivround_pos_a #1#2{\xintReverseOrder {#1\XINT_iidivround_pos_b}\Z }% -\def\XINT_iidivround_pos_b #1#2{\xint_gob_til_Z #2\XINT_iidivround_pos_small\Z - \XINT_iidivround_pos_c #1#2}% -\def\XINT_iidivround_pos_c #1#2\Z {\ifnum #1>\xint_c_iv - \expandafter\XINT_iidivround_pos_up - \else \expandafter\xintreverseorder - \fi {#2}}% -\def\XINT_iidivround_pos_up #1{\xintinc {\xintReverseOrder{#1}}}% -\def\XINT_iidivround_pos_small\Z\XINT_iidivround_pos_c #1#2% - {\ifnum #1>\xint_c_iv\expandafter\xint_secondoftwo\else\expandafter - \xint_firstoftwo\fi { 0}{ 1}}% +\def\XINT_iidivround_pos #1#2\Z #3\Z +{% + \expandafter\XINT_iidivround_pos_a + \romannumeral0\XINT_div_prepare {#2}{#1#30}% +}% +\def\XINT_iidivround_pos_a #1#2% +{% + \expandafter\XINT_iidivround_pos_b + \romannumeral0\expandafter\XINT_sepandrev + \romannumeral0\XINT_zeroes_forviii #1\R\R\R\R\R\R\R\R{10}0000001\W + #1\XINT_rsepbyviii_end_A 2345678\XINT_rsepbyviii_end_B 2345678\relax XX% + \R.\R.\R.\R.\R.\R.\R.\R.\W + \Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_iidivround_pos_b 1#1#2#3#4#5#6#7#8!#9% +{% + \xint_gob_til_Z #9\XINT_iidivround_small\Z + \ifnum #8>\xint_c_iv + \expandafter\XINT_iidivround_pos_up + \else \expandafter\XINT_iidivround_pos_finish + \fi + 1#1#2#3#4#5#6#70!#9% +}% +\def\XINT_iidivround_pos_up +{% + \expandafter\XINT_iidivround_pos_finish + \the\numexpr\XINT_add_a\xint_c_ii 100000010!\Z!\Z!\Z!\Z!\Z!\W +}% +\def\XINT_iidivround_pos_finish #10!#2\Z #3\W +{% + \expandafter\XINT_cuz_small\romannumeral0\XINT_unrevbyviii {}% + #1!#21\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W +}% +\def\XINT_iidivround_small\Z\ifnum #1>#2\fi 1#30!#4\W +{% + \ifnum #1>\xint_c_iv + \expandafter\XINT_iidivround_small_up + \else \expandafter\XINT_iidivround_small_trunc + \fi {#3}% +}% +\edef\XINT_iidivround_small_up #1% + {\noexpand\expandafter\space\noexpand\the\numexpr #1+\xint_c_i\relax }% +\edef\XINT_iidivround_small_trunc #1% + {\noexpand\expandafter\space\noexpand\the\numexpr #1\relax }% \def\xintiDivTrunc {\romannumeral0\xintidivtrunc }% \def\xintidivtrunc #1{\expandafter\XINT_iidivtrunc\romannumeral0\xintnum{#1}\Z }% \def\xintiiDivTrunc {\romannumeral0\xintiidivtrunc }% @@ -2221,7 +2095,8 @@ \krof }% \def\XINT_iidivtrunc_pos #1#2\Z #3\Z% - {\expandafter\xint_firstoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}% + {\expandafter\xint_firstoftwo_thenstop + \romannumeral0\XINT_div_prepare {#2}{#1#3}}% \def\xintiMod {\romannumeral0\xintimod }% \def\xintimod #1{\expandafter\XINT_iimod\romannumeral0\xintnum{#1}\Z }% \def\xintiiMod {\romannumeral0\xintiimod }% @@ -2250,7 +2125,16 @@ \krof }% \def\XINT_iimod_pos #1#2\Z #3\Z% - {\expandafter\xint_secondoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}% + {\expandafter\xint_secondoftwo_thenstop\romannumeral0\XINT_div_prepare + {#2}{#1#3}}% +\catcode`! 11 +\def\xintAbs {\Did_you_mean_iiAbs?or_load_xintfrac!}% +\def\xintOpp {\Did_you_mean_iiOpp?or_load_xintfrac!}% +\def\xintAdd {\Did_you_mean_iiAdd?or_load_xintfrac!}% +\def\xintSub {\Did_you_mean_iiSub?or_load_xintfrac!}% +\def\xintMul {\Did_you_mean_iiMul?or_load_xintfrac!}% +\def\xintPow {\Did_you_mean_iiPow?or_load_xintfrac!}% +\def\xintSqr {\Did_you_mean_iiSqr?or_load_xintfrac!}% \XINT_restorecatcodes_endinput% \endinput %% diff --git a/Master/texmf-dist/tex/generic/xint/xintexpr.sty b/Master/texmf-dist/tex/generic/xint/xintexpr.sty index a7bc350cadd..72e5194b49e 100644 --- a/Master/texmf-dist/tex/generic/xint/xintexpr.sty +++ b/Master/texmf-dist/tex/generic/xint/xintexpr.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintexpr: Expandable expression parser %% --------------------------------------------------------------- @@ -81,13 +81,24 @@ \XINTsetupcatcodes% \XINT_providespackage \ProvidesPackage{xintexpr}% - [2015/09/12 v1.1c Expandable expression parser (jfB)]% -\def\xint_gob_til_! #1!{}% this ! has catcode 11 -\edef\XINT_expr_lockscan#1!{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% -\edef\XINT_expr_lockit #1{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% -\def\XINT_expr_inintpart #1!{\XINT_num{#1}}% -\def\XINT_expr_infracpart #1e#2!{#1![\the\numexpr#2-\xintLength{#1}]!}% -\def\XINT_expr_inexppart e#1!{![\the\numexpr #1]!}% + [2015/10/10 v1.2 Expandable expression parser (jfB)]% +\catcode`! 11 +\def\xint_gob_til_! #1!{}% catcode 11 ! default in xintexpr.sty code. +\edef\XINT_expr_lockscan#1!% not used for decimal numbers in xintexpr 1.2 + {\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% +\edef\XINT_expr_lockit + #1{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }% +\def\XINT_expr_unlock_hex_in #1% expanded inside \csname..\endcsname + {\expandafter\XINT_expr_inhex\romannumeral-`0\XINT_expr_unlock#1;}% +\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname +{% + \if#2>\xintHexToDec{#1}% + \else + \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}% + [\the\numexpr-4*\xintLength{#3}]% + \fi +}% +%%%%%%%%%%%% \def\XINT_expr_unlock {\expandafter\XINT_expr_unlock_a\string }% \def\XINT_expr_unlock_a #1.={}% \def\XINT_expr_unexpectedtoken {\xintError:ignored }% @@ -253,14 +264,18 @@ \def\XINT_expr_subexpr !#1\fi !{\expandafter\XINT_expr_getop\xint_gobble_iii }% \def\XINT_expr_countetc #1% {% - \ifx\count#1\else\ifx#1\dimen\else\ifx#1\numexpr\else\ifx#1\dimexpr\else - \ifx\skip#1\else\ifx\glueexpr#1\else\ifx\fontdimen#1\else + \ifx\count#1\else\ifx\dimen#1\else\ifx\numexpr#1\else\ifx\dimexpr#1\else + \ifx\skip#1\else\ifx\glueexpr#1\else\ifx\fontdimen#1\else\ifx\ht#1\else + \ifx\dp#1\else\ifx\wd#1\else\ifx\fontcharht#1\else\ifx\fontcharwd#1\else + \ifx\fontchardp#1\else\ifx\fontcharic#1\else \XINT_expr_unpackvar - \fi\fi\fi\fi\fi\fi\fi + \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi \expandafter\XINT_expr_getnext\number #1% }% -\def\XINT_expr_unpackvar\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getnext\number #1% - {\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getop\csname .=\number#1\endcsname }% +\def\XINT_expr_unpackvar\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi + \expandafter\XINT_expr_getnext\number #1% + {\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi + \expandafter\XINT_expr_getop\csname .=\number#1\endcsname }% \begingroup \lccode`*=`# \lowercase{\endgroup @@ -268,7 +283,7 @@ \if#1*\xint_dothis {\XINT_expr_scan_macropar *}\fi \if#1[\xint_dothis {\xint_c_xviii ({}}\fi \if#1+\xint_dothis \XINT_expr_getnext \fi - \if#1.\xint_dothis {\XINT_expr_scandec_II\XINT_expr_infracpart}\fi + \if#1.\xint_dothis {\XINT_expr_startdec}\fi \if#1-\xint_dothis -\fi \if#1(\xint_dothis {\xint_c_xviii ({}}\fi \xint_orthat {\XINT_expr_scan_nbr_or_func #1}% @@ -279,135 +294,161 @@ {% \if "#1\xint_dothis \XINT_expr_scanhex_I\fi \if `#1\xint_dothis {\XINT_expr_onlitteral_`}\fi - \ifnum \xint_c_ix<1#1 \xint_dothis \XINT_expr_scandec_I\fi + \ifnum \xint_c_ix<1#1 \xint_dothis \XINT_expr_startint\fi \xint_orthat \XINT_expr_scanfunc #1% }% \catcode96 12 % ` -\def\XINT_expr_scandec_I +\def\XINT_expr_startint #1% {% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart - \romannumeral-`0\XINT_expr_scanintpart_b + \if #10\expandafter\XINT_expr_gobz_a\else\XINT_expr_scanint_a\fi #1% }% -\def\XINT_expr_scandec_II +\def\XINT_expr_scanint_a #1#2% + {\expandafter\XINT_expr_getop\csname.=#1% + \expandafter\XINT_expr_scanint_b\romannumeral-`0#2}% +\def\XINT_expr_gobz_a #1% + {\expandafter\XINT_expr_getop\csname.=% + \expandafter\XINT_expr_gobz_scanint_b\romannumeral-`0#1}% +\def\XINT_expr_startdec #1% + {\expandafter\XINT_expr_getop\csname.=% + \expandafter\XINT_expr_scandec_a\romannumeral-`0#1}% +\def\XINT_expr_scanint_b #1% {% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart - \romannumeral-`0\XINT_expr_scanfracpart_b + \ifcat \relax #1\expandafter\XINT_expr_scanint_endbycs\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanint_c\fi + \string#1\XINT_expr_scanint_d }% -\def\XINT_expr_scanintpart_a #1% -{% careful that ! has catcode letter here - \ifcat \relax #1\xint_dothis{!!#1}\fi % stops the scan - \if e#1\xint_dothis{\expandafter\XINT_expr_inexppart - \romannumeral-`0\XINT_expr_scanexppart_a e}\fi - \if E#1\xint_dothis{\expandafter\XINT_expr_inexppart - \romannumeral-`0\XINT_expr_scanexppart_a e}\fi - \ifcat a#1\xint_dothis{!!*#1}\fi % includes subexpressions (#1=! letter) - \xint_orthat {\expandafter\XINT_expr_scanintpart_aa\string #1}% +\def\XINT_expr_scanint_d #1% +{% + \expandafter\XINT_expr_scanint_b\romannumeral-`0#1% }% -\def\XINT_expr_scanintpart_aa #1% +\def\XINT_expr_scanint_endbycs#1#2\XINT_expr_scanint_d{\endcsname #1}% +\def\XINT_expr_scanint_c\string #1\XINT_expr_scanint_d {% - \if .#1\xint_dothis\XINT_expr_scandec_transition\fi - \ifnum \xint_c_ix<1#1 \xint_dothis\XINT_expr_scanintpart_b\fi - \xint_orthat {!!}#1% + \if e#1\xint_dothis{[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{\endcsname*#1}\fi + \if .#1\xint_dothis{\XINT_expr_startdec_a .}\fi + \xint_orthat {\expandafter\endcsname \string#1}% }% -\def\XINT_expr_scanintpart_b #1#2% +\def\XINT_expr_startdec_a .#1% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanintpart_a\romannumeral-`0#2% + \expandafter\XINT_expr_scandec_a\romannumeral-`0#1% }% -\def\XINT_expr_scandec_transition .#1% +\def\XINT_expr_scandec_a #1% {% - \expandafter\XINT_expr_scandec_trans_a\romannumeral-`0#1% + \if .#1\xint_dothis{\endcsname..}\fi + \xint_orthat {\XINT_expr_scandec_b 0.#1}% }% -\def\XINT_expr_scandec_trans_a #1% +\def\XINT_expr_gobz_scanint_b #1% {% - \if .#1\xint_dothis{!!..}\fi - \xint_orthat {\expandafter\XINT_expr_infracpart - \romannumeral-`0\XINT_expr_scanfracpart_a #1}% + \ifcat \relax #1\expandafter\XINT_expr_gobz_scanint_endbycs\expandafter #1\fi + \ifnum\xint_c_x<1\string#1 \else\expandafter\XINT_expr_gobz_scanint_c\fi + \string#1\XINT_expr_scanint_d }% -\def\XINT_expr_scanfracpart_a #1% +\def\XINT_expr_gobz_scanint_endbycs#1#2\XINT_expr_scanint_d{0\endcsname #1}% +\def\XINT_expr_gobz_scanint_c\string #1\XINT_expr_scanint_d {% - \ifcat \relax #1\xint_dothis{e!#1}\fi % stops the scan - \if e#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi - \if E#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi - \ifcat a#1\xint_dothis{e!*#1}\fi % and also the case of subexpressions (!) - \xint_orthat {\expandafter\XINT_expr_scanfracpart_aa\string #1}% + \if e#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{0\endcsname*#1}\fi + \if .#1\xint_dothis{\XINT_expr_gobz_startdec_a .}\fi + \if 0#1\xint_dothis\XINT_expr_gobz_scanint_d\fi + \xint_orthat {0\expandafter\endcsname \string#1}% }% -\def\XINT_expr_scanfracpart_aa #1% +\def\XINT_expr_gobz_scanint_d #1% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanfracpart_b - \else - \xint_afterfi {e!}% - \fi - #1% + \expandafter\XINT_expr_gobz_scanint_b\romannumeral-`0#1% }% -\def\XINT_expr_scanfracpart_b #1#2% +\def\XINT_expr_gobz_startdec_a .#1% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanfracpart_a\romannumeral-`0#2% + \expandafter\XINT_expr_gobz_scandec_a\romannumeral-`0#1% }% -\def\XINT_expr_scanexppart_a #1#2% +\def\XINT_expr_gobz_scandec_a #1% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanexppart_b\romannumeral-`0#2% + \if .#1\xint_dothis{0\endcsname..}\fi + \xint_orthat {\XINT_expr_gobz_scandec_b 0.#1}% }% -\def\XINT_expr_scanexppart_b #1% +\def\XINT_expr_scandec_b #1.#2% {% - \ifcat \relax #1\xint_dothis{0!#1}\fi % stops the scan (incorrect syntax) - \ifcat a#1\xint_dothis{0!*#1}\fi % idem - \if +#1\xint_dothis {\XINT_expr_scanexppart_a +}\fi - \if -#1\xint_dothis {\XINT_expr_scanexppart_a -}\fi - \xint_orthat {\expandafter\XINT_expr_scanexppart_c\string #1}% + \ifcat \relax #2\expandafter\XINT_expr_scandec_endbycs\expandafter#2\fi + \ifnum\xint_c_ix<1\string#2 \else\expandafter\XINT_expr_scandec_c\fi + \string#2\expandafter\XINT_expr_scandec_d\the\numexpr #1-\xint_c_i.% }% -\def\XINT_expr_scanexppart_c #1% +\def\XINT_expr_scandec_endbycs #1#2\XINT_expr_scandec_d + \the\numexpr#3-\xint_c_i.{[#3]\endcsname #1}% +\def\XINT_expr_scandec_d #1.#2% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanexppart_d - \else - \expandafter !% - \fi - #1% + \expandafter\XINT_expr_scandec_b + \the\numexpr #1\expandafter.\romannumeral-`0#2% }% -\def\XINT_expr_scanexppart_d #1#2% +\def\XINT_expr_scandec_c\string #1#2\the\numexpr#3-\xint_c_i.% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanexppart_e\romannumeral-`0#2% + \if e#1\xint_dothis{[\the\numexpr#3\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{[\the\numexpr#3\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{[#3]\endcsname *#1}\fi + \xint_orthat {[#3]\expandafter\endcsname \string#1}% }% -\def\XINT_expr_scanexppart_e #1% +\def\XINT_expr_gobz_scandec_b 0.#1% {% - \ifcat \relax #1\xint_dothis{!#1}\fi % stops the scan - \ifcat a#1\xint_dothis{!*#1}\fi % idem - \xint_orthat {\expandafter\XINT_expr_scanexppart_f\string #1}% + \ifcat \relax #1\expandafter\XINT_expr_gobz_scandec_endbycs\expandafter#1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_gobz_scandec_c\fi + \string#1\expandafter\XINT_expr_scandec_d\the\numexpr\xint_c_mone.% }% -\def\XINT_expr_scanexppart_f #1% +\def\XINT_expr_gobz_scandec_endbycs #1#2\xint_c_mone.{0[0]\endcsname #1}% +\def\XINT_expr_gobz_scandec_c\string #1#2\xint_c_mone.% {% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanexppart_d - \else - \expandafter !% - \fi - #1% + \if e#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \if E#1\xint_dothis{0[\the\numexpr0\XINT_expr_scanexp_a +}\fi + \ifcat a#1\xint_dothis{0[0]\endcsname *#1}\fi + \xint_orthat {0[0]\expandafter\endcsname \string#1}% }% -\def\XINT_expr_scanhex_I #1% +\def\XINT_expr_scanexp_a #1#2% {% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lockscan\expandafter\XINT_expr_inhex - \romannumeral-`0\XINT_expr_scanhexI_a + #1\expandafter\XINT_expr_scanexp_b\romannumeral-`0#2% }% -\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname +\def\XINT_expr_scanexp_b #1% {% - \if#2I\xintHexToDec{#1}% - \else - \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}% - [\the\numexpr-4*\xintLength{#3}]% - \fi + \ifcat \relax #1\expandafter\XINT_expr_scanexp_endbycs\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanexp_c\fi + \string#1\XINT_expr_scanexp_d +}% +\def\XINT_expr_scanexpr_endbycs#1#2\XINT_expr_scanexp_d {]\endcsname #1}% +\def\XINT_expr_scanexp_d #1% +{% + \expandafter\XINT_expr_scanexp_bb\romannumeral-`0#1% +}% +\def\XINT_expr_scanexp_c\string #1\XINT_expr_scanexp_d +{% + \ifcat a#1\xint_dothis {]\endcsname *#1}\fi + \if +#1\xint_dothis {\XINT_expr_scanexp_a +}\fi + \if -#1\xint_dothis {\XINT_expr_scanexp_a -}\fi + \xint_orthat {]\expandafter\endcsname\string #1}% +}% +\def\XINT_expr_scanexp_bb #1% +{% + \ifcat \relax #1\expandafter\XINT_expr_scanexp_endbycs_b\expandafter #1\fi + \ifnum\xint_c_ix<1\string#1 \else\expandafter\XINT_expr_scanexp_cb\fi + \string#1\XINT_expr_scanexp_db +}% +\def\XINT_expr_scanexp_endbycs_b#1#2\XINT_expr_scanexp_db {]\endcsname #1}% +\def\XINT_expr_scanexp_db #1% +{% + \expandafter\XINT_expr_scanexp_bb\romannumeral-`0#1% +}% +\def\XINT_expr_scanexp_cb\string #1\XINT_expr_scanexp_db +{% + \ifcat a#1\xint_dothis {]\endcsname *#1}\fi + \xint_orthat {]\expandafter\endcsname\string #1}% +}% +\def\XINT_expr_scanhex_I #1% #1=" +{% + \expandafter\XINT_expr_getop\csname.=\expandafter + \XINT_expr_unlock_hex_in\csname.=\XINT_expr_scanhexI_a }% \def\XINT_expr_scanhexI_a #1% {% - \ifcat #1\relax\xint_dothis{.I;!#1}\fi - \ifx !#1\xint_dothis{.I;!*!}\fi % tacit multiplication + \ifcat #1\relax\xint_dothis{.>\endcsname\endcsname #1}\fi + \ifx !#1\xint_dothis{.>\endcsname\endcsname*!}\fi % tacit multiplication \xint_orthat {\expandafter\XINT_expr_scanhexI_aa\string #1}% }% \def\XINT_expr_scanhexI_aa #1% @@ -425,24 +466,23 @@ \expandafter\xint_secondoftwo \fi {\expandafter\XINT_expr_scanhex_transition}% - {\xint_afterfi {.I;!}}% + {\xint_afterfi {.>\endcsname\endcsname}}% \fi #1% }% \def\XINT_expr_scanhexI_b #1#2% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexI_a\romannumeral-`0#2% + #1\expandafter\XINT_expr_scanhexI_a\romannumeral-`0#2% }% \def\XINT_expr_scanhex_transition .#1% {% - \expandafter.\expandafter.\romannumeral-`0\expandafter + \expandafter.\expandafter.\expandafter \XINT_expr_scanhexII_a\romannumeral-`0#1% }% \def\XINT_expr_scanhexII_a #1% {% - \ifcat #1\relax\xint_dothis{;!#1}\fi - \ifx !#1\xint_dothis{;!*!}\fi % tacit multiplication + \ifcat #1\relax\xint_dothis{\endcsname\endcsname#1}\fi + \ifx !#1\xint_dothis{\endcsname\endcsname*!}\fi % tacit multiplication \xint_orthat {\expandafter\XINT_expr_scanhexII_aa\string #1}% }% \def\XINT_expr_scanhexII_aa #1% @@ -454,14 +494,13 @@ 0\else1\fi\else0\fi\else1\fi\else0\fi 1% \expandafter\XINT_expr_scanhexII_b \else - \xint_afterfi {;!}% + \xint_afterfi {\endcsname\endcsname}% \fi #1% }% \def\XINT_expr_scanhexII_b #1#2% {% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexII_a\romannumeral-`0#2% + #1\expandafter\XINT_expr_scanhexII_a\romannumeral-`0#2% }% \def\XINT_expr_scanfunc {% @@ -1221,9 +1260,10 @@ \let\XINT_expr_precedence_! \xint_c_x \def\XINT_expr_op_! #1{\expandafter\XINT_expr_getop \csname .=\xintFac{\XINT_expr_unlock #1}\endcsname }% -\let\XINT_flexpr_op_!\XINT_expr_op_! +\def\XINT_flexpr_op_! #1{\expandafter\XINT_expr_getop + \csname .=\XINTinFloatFac{\XINT_expr_unlock #1}\endcsname }% \def\XINT_iiexpr_op_! #1{\expandafter\XINT_expr_getop - \csname .=\xintiFac{\XINT_expr_unlock #1}\endcsname }% + \csname .=\xintiiFac{\XINT_expr_unlock #1}\endcsname }% \catcode`[ 11 \catcode`* 11 \let\XINT_expr_precedence_[ \xint_c_vii @@ -1346,11 +1386,18 @@ {\expandafter\XINT_expr_getop\csname .=\xintToggle{#1}\endcsname }% \def\XINT_expr_onlitteral_protect #1)% {\expandafter\XINT_expr_getop\csname .=\detokenize{#1}\endcsname }% -\def\XINT_expr_func_unknown #1#2#3{\expandafter #1\expandafter #2\csname .=0\endcsname }% +\def\XINT_expr_func_unknown #1#2#3% + {\expandafter #1\expandafter #2\csname .=0\endcsname }% \def\XINT_expr_func_break #1#2#3% -{\expandafter #1\expandafter #2\csname.=?\romannumeral-`0\XINT_expr_unlock #3\endcsname }% + {\expandafter #1\expandafter #2\csname.=?\romannumeral-`0\XINT_expr_unlock #3\endcsname }% \let\XINT_flexpr_func_break \XINT_expr_func_break \let\XINT_iiexpr_func_break \XINT_expr_func_break +\def\XINT_expr_onlitteral_qint #1)% + {\expandafter\XINT_expr_getop\csname .=\xintiNum{#1}\endcsname }% +\def\XINT_expr_onlitteral_qfrac #1)% + {\expandafter\XINT_expr_getop\csname .=\xintRaw{#1}\endcsname }% +\def\XINT_expr_onlitteral_qfloat #1)% + {\expandafter\XINT_expr_getop\csname .=\XINTinFloatdigits{#1}\endcsname }% \def\XINT_expr_onlitteral_seq {\expandafter\XINT_expr_onlitteral_seq_f\romannumeral-`0\XINT_expr_onlitteral_seq_a {}}% \def\XINT_expr_onlitteral_seq_f #1#2{\xint_c_xviii `{seqx}#2)\relax #1}% @@ -1379,7 +1426,7 @@ \def\XINT_isbalanced_a #1({\XINT_isbalanced_b #1)\xint_bye }% \def\XINT_isbalanced_b #1)#2% {\xint_bye #2\XINT_isbalanced_c\xint_bye\XINT_isbalanced_error }% -\def\XINT_isbalanced_error #1)\xint_bye {\m@ne}% +\def\XINT_isbalanced_error #1)\xint_bye {\xint_c_mone}% \def\XINT_isbalanced_c\xint_bye\XINT_isbalanced_error #1% {\xint_bye #1\XINT_isbalanced_yes\xint_bye\XINT_isbalanced_d #1}% \def\XINT_isbalanced_yes\xint_bye\XINT_isbalanced_d\xint_bye )\xint_bye {\xint_c_ }% @@ -2167,10 +2214,11 @@ \romannumeral-`0\expandafter\XINT_xptwo_getab_b \romannumeral-`0####2!{####1}{~xint#1}{xint#1}}% }% -}% +}% cela aurait-il un sens d'ajouter Raw et iNum (à cause de qint, qfrac, + % qfloat?). Pas le temps d'y réfléchir. Je ne fais rien. \xintFor #1 in {Num,Irr,Abs,iiAbs,Sgn,iiSgn,TFrac,Floor,iFloor,Ceil,iCeil,% Sqr,iiSqr,iiSqrt,iiSqrtR,iiIsZero,iiIsNotZero,iiifNotZero,iiifSgn,% - Odd,Even,iiOdd,iiEven,Opp,iiOpp,iiifZero,Fac,iFac,Bool,Toggle}\do + Odd,Even,iiOdd,iiEven,Opp,iiOpp,iiifZero,Fac,iiFac,Bool,Toggle}\do {\toks0 \expandafter{\the\toks0% \expandafter\let\csname xint#1NE\expandafter\endcsname\csname xint#1\expandafter @@ -2178,6 +2226,13 @@ \expandafter\XINT_NEfork_one\romannumeral-`0####1!{~xint#1}{xint#1}{}{}}% }% }% +\toks0 + \expandafter{\the\toks0 + \let\XINTinFloatFacNE\XINTinFloatFac + \def\XINTinFloatFac ##1{% + \expandafter\XINT_NEfork_one + \romannumeral-`0##1!{~XINTinFloatFac}{XINTinFloatFac}{}{}}% + }% \xintFor #1 in {Add,Sub,Mul,Div,Power,E,Mod,SeqA::csv}\do {\toks0 \expandafter{\the\toks0% diff --git a/Master/texmf-dist/tex/generic/xint/xintfrac.sty b/Master/texmf-dist/tex/generic/xint/xintfrac.sty index f0e50614e0d..59030c1e9be 100644 --- a/Master/texmf-dist/tex/generic/xint/xintfrac.sty +++ b/Master/texmf-dist/tex/generic/xint/xintfrac.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintfrac: Expandable operations on fractions %% --------------------------------------------------------------- @@ -70,8 +70,7 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintfrac}% - [2015/09/12 v1.1c Expandable operations on fractions (jfB)]% -\chardef\xint_c_xviii 18 + [2015/10/10 v1.2 Expandable operations on fractions (jfB)]% \def\XINT_cntSgnFork #1% {% \ifcase #1\expandafter\xint_secondofthree @@ -150,126 +149,132 @@ \def\XINT_inFrac {\romannumeral0\XINT_infrac }% \def\XINT_infrac #1% {% - \expandafter\XINT_infrac_ \romannumeral-`0#1[\W]\Z\T + \expandafter\XINT_infrac_fork\romannumeral-`0#1/\XINT_W[\XINT_W\XINT_T }% -\def\XINT_infrac_ #1[#2#3]#4\Z +\def\XINT_infrac_fork #1[#2% {% - \xint_UDwfork - #2\XINT_infrac_A - \W\XINT_infrac_B + \xint_UDXINTWfork + #2\XINT_frac_gen + \XINT_W\XINT_infrac_res_a % strict A[N] or A/B[N] input \krof - #1[#2#3]#4% + #1[#2% }% -\def\XINT_infrac_A #1[\W]\T +\def\XINT_infrac_res_a #1% {% - \XINT_frac #1/\W\Z + \xint_gob_til_zero #1\XINT_infrac_res_zero 0\XINT_infrac_res_b #1% }% -\def\XINT_infrac_B #1% +\def\XINT_infrac_res_zero 0\XINT_infrac_res_b #1\XINT_T {{0}{0}{1}}% +\def\XINT_infrac_res_b #1/#2% {% - \xint_gob_til_zero #1\XINT_infrac_Zero0\XINT_infrac_BB #1% -}% -\def\XINT_infrac_BB #1[\W]\T {\XINT_infrac_BC #1/\W\Z }% -\def\XINT_infrac_BC #1/#2#3\Z -{% - \xint_UDwfork - #2\XINT_infrac_BCa - \W{\expandafter\XINT_infrac_BCb \romannumeral-`0#2}% + \xint_UDXINTWfork + #2\XINT_infrac_res_ca + \XINT_W\XINT_infrac_res_cb \krof - #3\Z #1\Z -}% -\def\XINT_infrac_BCa \Z #1[#2]#3\Z { {#2}{#1}{1}}% -\def\XINT_infrac_BCb #1[#2]/\W\Z #3\Z { {#2}{#3}{#1}}% -\def\XINT_infrac_Zero #1\T { {0}{0}{1}}% -\def\XINT_frac #1/#2#3\Z -{% - \xint_UDwfork - #2\XINT_frac_A - \W{\expandafter\XINT_frac_U \romannumeral-`0#2}% + #1/#2% +}% +\def\XINT_infrac_res_ca #1[#2]/\XINT_W[\XINT_W\XINT_T + {\expandafter{\the\numexpr 0#2}{#1}{1}}% +\def\XINT_infrac_res_cb #1/#2[% + {\expandafter\XINT_infrac_res_cc\romannumeral-`0#2~#1[}% +\def\XINT_infrac_res_cc #1~#2[#3]/\XINT_W[\XINT_W\XINT_T + {\expandafter{\the\numexpr 0#3}{#2}{#1}}% +\def\XINT_frac_gen #1/#2% +{% + \xint_UDXINTWfork + #2\XINT_frac_gen_A + \XINT_W\XINT_frac_gen_B \krof - #3e\W\Z #1e\W\Z + #1/#2% }% -\def\XINT_frac_U #1e#2#3\Z +\def\XINT_frac_gen_A #1/\XINT_W [\XINT_W {\XINT_frac_gen_C 0~1!#1ee.\XINT_W }% +\def\XINT_frac_gen_B #1/#2/\XINT_W[%\XINT_W {% - \xint_UDwfork - #2\XINT_frac_Ua - \W{\XINT_frac_Ub #2}% - \krof - #3\Z #1\Z + \expandafter\XINT_frac_gen_Ba + \romannumeral-`0#2ee.\XINT_W\XINT_Z #1ee.%\XINT_W }% -\def\XINT_frac_Ua \Z #1/\W\Z {\XINT_frac_B #1.\W\Z {0}}% -\def\XINT_frac_Ub #1/\W e\W\Z #2\Z {\XINT_frac_B #2.\W\Z {#1}}% -\def\XINT_frac_B #1.#2#3\Z +\def\XINT_frac_gen_Ba #1.#2% {% - \xint_UDwfork - #2\XINT_frac_Ba - \W{\XINT_frac_Bb #2}% + \xint_UDXINTWfork + #2\XINT_frac_gen_Bb + \XINT_W\XINT_frac_gen_Bc \krof - #3\Z #1\Z + #1.#2% }% -\def\XINT_frac_Ba \Z #1\Z {\XINT_frac_T {0}{#1}}% -\def\XINT_frac_Bb #1.\W\Z #2\Z +\def\XINT_frac_gen_Bb #1e#2e#3\XINT_Z + {\expandafter\XINT_frac_gen_C\the\numexpr 0#2~#1!}% +\def\XINT_frac_gen_Bc #1.#2e% {% - \expandafter \XINT_frac_T \expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% + \expandafter\XINT_frac_gen_Bd\romannumeral-`0#2.#1e% }% -\def\XINT_frac_A e\W\Z {\XINT_frac_T {0}{1}{0}}% -\def\XINT_frac_T #1#2#3#4e#5#6\Z +\def\XINT_frac_gen_Bd #1.#2e#3e#4\XINT_Z {% - \xint_UDwfork - #5\XINT_frac_Ta - \W{\XINT_frac_Tb #5}% - \krof - #6\Z #4\Z {#1}{#2}{#3}% + \expandafter\XINT_frac_gen_C\the\numexpr 0#3-\romannumeral0\expandafter + \XINT_length_loop + 0.#1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye~#2#1!% }% -\def\XINT_frac_Ta \Z #1\Z {\XINT_frac_C #1.\W\Z {0}}% -\def\XINT_frac_Tb #1e\W\Z #2\Z {\XINT_frac_C #2.\W\Z {#1}}% -\def\XINT_frac_C #1.#2#3\Z +\def\XINT_frac_gen_C #1!#2.#3% {% - \xint_UDwfork - #2\XINT_frac_Ca - \W{\XINT_frac_Cb #2}% + \xint_UDXINTWfork + #3\XINT_frac_gen_Ca + \XINT_W\XINT_frac_gen_Cb \krof - #3\Z #1\Z + #1!#2.#3% }% -\def\XINT_frac_Ca \Z #1\Z {\XINT_frac_D {0}{#1}}% -\def\XINT_frac_Cb #1.\W\Z #2\Z +\def\XINT_frac_gen_Ca #1~#2!#3e#4e#5\XINT_T {% - \expandafter\XINT_frac_D\expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% + \expandafter\XINT_frac_gen_F\the\numexpr #4-#1\expandafter + ~\romannumeral0\XINT_num_loop + #2\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#3~% }% -\def\XINT_frac_D #1#2#3#4#5#6% +\def\XINT_frac_gen_Cb #1.#2e% {% - \expandafter \XINT_frac_E \expandafter - {\the\numexpr -#1+#3+#4-#6\expandafter}\expandafter - {\romannumeral0\XINT_num_loop #2% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% - {\romannumeral0\XINT_num_loop #5% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% + \expandafter\XINT_frac_gen_Cc\romannumeral-`0#2.#1e% }% -\def\XINT_frac_E #1#2#3% +\def\XINT_frac_gen_Cc #1.#2~#3!#4e#5e#6\XINT_T {% - \expandafter \XINT_frac_F #3\Z {#2}{#1}% + \expandafter\XINT_frac_gen_F\the\numexpr #5-#2-% + \romannumeral0\XINT_length_loop + 0.#1\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye\expandafter + ~\romannumeral0\XINT_num_loop + #3\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z + ~#4#1~% }% -\def\XINT_frac_F #1% +\def\XINT_frac_gen_F #1~#2% {% \xint_UDzerominusfork - #1-\XINT_frac_Gdivisionbyzero - 0#1\XINT_frac_Gneg - 0-{\XINT_frac_Gpos #1}% - \krof + #2-\XINT_frac_gen_Gdivbyzero + 0#2{\XINT_frac_gen_G -{}}% + 0-{\XINT_frac_gen_G {}#2}% + \krof #1~% }% -\edef\XINT_frac_Gdivisionbyzero #1\Z #2#3% +\def\XINT_frac_gen_Gdivbyzero #1~~#2~% {% - \noexpand\xintError:DivisionByZero\space {0}{#2}{0}% + \expandafter\XINT_frac_gen_Gdivbyzero_a + \romannumeral0\XINT_num_loop + #2\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#1~% }% -\def\XINT_frac_Gneg #1\Z #2#3% +\def\XINT_frac_gen_Gdivbyzero_a #1~#2~% {% - \expandafter\XINT_frac_H \expandafter{\romannumeral0\XINT_opp #2}{#3}{#1}% + \xintError:DivisionByZero {#2}{#1}{0}% }% -\def\XINT_frac_H #1#2{ {#2}{#1}}% -\def\XINT_frac_Gpos #1\Z #2#3{ {#3}{#2}{#1}}% +\def\XINT_frac_gen_G #1#2#3~#4~#5~% +{% + \expandafter\XINT_frac_gen_Ga + \romannumeral0\XINT_num_loop + #1#5\xint_relax\xint_relax\xint_relax\xint_relax + \xint_relax\xint_relax\xint_relax\xint_relax\Z~#3~{#2#4}% +}% +\def\XINT_frac_gen_Ga #1#2~#3~% +{% + \xint_gob_til_zero #1\XINT_frac_gen_zero 0% + {#3}{#1#2}% +}% +\def\XINT_frac_gen_zero 0#1#2#3{{0}{0}{1}}% \def\XINT_factortens #1% {% \expandafter\XINT_cuz_cnt_loop\expandafter @@ -363,6 +368,96 @@ \xint_bye\xint_bye\xint_bye\xint_bye \xint_relax }{#1}% }% +\def\XINT_addm_A #1#2#3#4#5#6% +{% + \xint_gob_til_W #3\xint_addm_az\W + \XINT_addm_AB #1{#3#4#5#6}{#2}% +}% +\def\xint_addm_az\W\XINT_addm_AB #1#2% +{% + \XINT_addm_AC_checkcarry #1% +}% +\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z +}% +\def\XINT_addm_ABE #1#2#3#4#5#6% +{% + \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.% +}% +\def\XINT_addm_ABEA #1#2#3.#4% +{% + \XINT_addm_A #2{#3#4}% +}% +\def\XINT_addm_AC_checkcarry #1% +{% + \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C +}% +\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z +{% + \expandafter + \xint_cleanupzeros_andstop + \romannumeral0% + \XINT_rord_main {}#2% + \xint_relax + \xint_bye\xint_bye\xint_bye\xint_bye + \xint_bye\xint_bye\xint_bye\xint_bye + \xint_relax + #1% +}% +\def\XINT_addm_C #1#2#3#4#5% +{% + \xint_gob_til_W + #5\xint_addm_cw + #4\xint_addm_cx + #3\xint_addm_cy + #2\xint_addm_cz + \W\XINT_addm_CD {#5#4#3#2}{#1}% +}% +\def\XINT_addm_CD #1% +{% + \expandafter\XINT_addm_CC\the\numexpr 1+10#1.% +}% +\def\XINT_addm_CC #1#2#3.#4% +{% + \XINT_addm_AC_checkcarry #2{#3#4}% +}% +\def\xint_addm_cw + #1\xint_addm_cx + #2\xint_addm_cy + #3\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.% +}% +\def\XINT_addm_CDw #1.#2#3\X\Y\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cx + #1\xint_addm_cy + #2\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.% +}% +\def\XINT_addm_CDx #1.#2#3\Y\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cy + #1\xint_addm_cz + \W\XINT_addm_CD +{% + \expandafter\XINT_addm_CDy\the\numexpr 1+#1.% +}% +\def\XINT_addm_CDy #1.#2#3\Z +{% + \XINT_addm_end #1#3% +}% +\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}% +\edef\XINT_addm_end #1#2#3#4#5% + {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}% \def\xintRaw {\romannumeral0\xintraw }% \def\xintraw {% @@ -736,8 +831,8 @@ \def\XINT_jrr_loop_b #1#2#3#4#5#6#7% {% \expandafter \XINT_jrr_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#4}{#1}}{#6}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#1}}{#7}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #4\Z #1\Z}{#6}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #5\Z #1\Z}{#7}}% {#2}{#3}{#4}{#5}% }% \def\XINT_jrr_loop_c #1#2% @@ -1058,33 +1153,7 @@ \expandafter\XINT_xtrunc_negNC\expandafter {\the\numexpr\xintLength {#1}-#2}{#1}% }% -\def\XINT_xtrunc_Q #1% -{% - \expandafter\XINT_xtrunc_prepare_I - \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z -}% -\def\XINT_xtrunc_prepare_I #1.#2#3% -{% - \expandafter\XINT_xtrunc_prepareB_aa\expandafter - {\romannumeral0\xintlength {#2}}{#2}{#1}% -}% -\def\XINT_xtrunc_prepareB_aa #1% -{% - \ifnum #1=\xint_c_i - \expandafter\XINT_xtrunc_prepareB_onedigit - \else - \expandafter\XINT_xtrunc_prepareB_PaBa - \fi - {#1}% -}% -\def\XINT_xtrunc_prepareB_onedigit #1#2% -{% - \ifcase#2 - \or\expandafter\XINT_xtrunc_BisOne - \or\expandafter\XINT_xtrunc_BisTwo - \else\expandafter\XINT_xtrunc_prepareB_PaBe - \fi {000}{0}{4}{#2}% -}% +%%%%%%%%%%%% \def\XINT_xtrunc_BisOne #1#2#3#4#5#6#7% {% #5.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter @@ -1104,48 +1173,76 @@ 0000000000000000000000000000000000000000000000000000000000000000% \repeat }% -\def\XINT_xtrunc_prepareB_PaBa #1#2% +%%%%%%%%%%%% +\def\XINT_xtrunc_Q #1% +{% + \expandafter\XINT_xtrunc_prepare + \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z +}% +\def\XINT_xtrunc_prepare #1.#2#3% {% \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_a {#1}{#2}}% + {\romannumeral0% + \XINT_xtrunc_prepare_a #2\R\R\R\R\R\R\R\R {10}0000001\W !{#2}}{#1}% }% -\def\XINT_xtrunc_prepareB_a #1% +%%%%%%%%%%%% +\def\XINT_xtrunc_prepare_a #1#2#3#4#5#6#7#8#9% {% - \expandafter\XINT_xtrunc_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% + \xint_gob_til_R #9\XINT_xtrunc_prepare_small\R + \XINT_xtrunc_prepare_b #9% }% -\def\XINT_xtrunc_prepareB_c #1#2% +\def\XINT_xtrunc_prepare_small\R #1!#2% {% - \csname XINT_xtrunc_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% + \ifcase #2 + \or\xint_afterfi{ \XINT_div_BisOne}% + \or\xint_afterfi{ \XINT_div_BisTwo}% + \else\expandafter\XINT_xtrunc_small_aa + \fi {#2}% }% -\def\XINT_xtrunc_prepareB_d {\XINT_xtrunc_prepareB_e {}{0000}}% -\def\XINT_xtrunc_prepareB_di {\XINT_xtrunc_prepareB_e {0}{000}}% -\def\XINT_xtrunc_prepareB_dii {\XINT_xtrunc_prepareB_e {00}{00}}% -\def\XINT_xtrunc_prepareB_diii {\XINT_xtrunc_prepareB_e {000}{0}}% -\def\XINT_xtrunc_prepareB_PaBe #1#2#3#4% +\def\XINT_xtrunc_small_aa #1% {% - \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_e {#1}{#2}{#3}{#4}}% + \expandafter\space\expandafter\XINT_xtrunc_small_a + \the\numexpr #1/\xint_c_ii\expandafter + .\the\numexpr \xint_c_x^viii+#1!% }% -\def\XINT_xtrunc_prepareB_e #1#2#3#4% +%%%%%%%%%%%% +\def\XINT_xtrunc_small_a #1.#2!#3% {% - \ifnum#3=\xint_c_iv\expandafter\XINT_xtrunc_prepareLittleB_f - \else\expandafter\XINT_xtrunc_prepareB_f - \fi - #4#1{#3}{#2}{#1}% + \expandafter\XINT_div_small_b\the\numexpr #1\expandafter + .\the\numexpr #2\expandafter!% + \romannumeral0\XINT_div_small_ba #3\R\R\R\R\R\R\R\R{10}0000001\W + #3\XINT_sepbyviii_Z_end 2345678\relax }% -\def\XINT_xtrunc_prepareB_f #1#2#3#4#5#{% - \expandafter\space - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% +%%%%%%%%%%%% +\def\XINT_xtrunc_prepare_b + {\expandafter\XINT_xtrunc_prepare_c\romannumeral0\XINT_zeroes_forviii }% +\def\XINT_xtrunc_prepare_c #1!% +{% + \XINT_xtrunc_prepare_d #1.00000000!{#1}% }% -\def\XINT_xtrunc_prepareLittleB_f #1#{% - \expandafter\space\expandafter - \XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% +\def\XINT_xtrunc_prepare_d #1#2#3#4#5#6#7#8#9% +{% + \expandafter\XINT_xtrunc_prepare_e\xint_gob_til_dot #1#2#3#4#5#6#7#8#9!% }% +\def\XINT_xtrunc_prepare_e #1!#2!#3#4% +{% + \XINT_xtrunc_prepare_f #4#3\X {#1}{#3}% +}% +\def\XINT_xtrunc_prepare_f #1#2#3#4#5#6#7#8#9\X +{% + \expandafter\space\expandafter\XINT_div_prepare_g + \the\numexpr #1#2#3#4#5#6#7#8+\xint_c_i\expandafter + .\the\numexpr (#1#2#3#4#5#6#7#8+\xint_c_i)/\xint_c_ii\expandafter + .\the\numexpr #1#2#3#4#5#6#7#8\expandafter + .\romannumeral0\XINT_sepandrev_andcount + #1#2#3#4#5#6#7#8#9\XINT_rsepbyviii_end_A 2345678% + \XINT_rsepbyviii_end_B 2345678% + \relax\xint_c_ii\xint_c_iii + \R.\xint_c_vi\R.\xint_c_v\R.\xint_c_iv\R.\xint_c_iii + \R.\xint_c_ii\R.\xint_c_i\R.\xint_c_\W + \X +}% +%%%%%%%%%%%% \def\XINT_xtrunc_Pa #1#2% {% \expandafter\XINT_xtrunc_Pb\romannumeral0#1{#2}{#1}% @@ -1540,7 +1637,7 @@ }% \def\XINT_fadd_C #1#2#3% {% - \ifcase\romannumeral0\XINT_cmp_pre {#2}{#3} %<- intentional space here. + \ifcase\romannumeral0\xintiicmp {#2}{#3} %<- intentional space here. \expandafter\XINT_fadd_eq \or\expandafter\XINT_fadd_D \else\expandafter\XINT_fadd_Da @@ -2000,10 +2097,10 @@ \expandafter\XINT_fcmp_Fe\expandafter {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}% }% -\def\XINT_fcmp_Fe #1#2{\XINT_cmp_pre {#2}{#1}}% +\def\XINT_fcmp_Fe #1#2{\xintiicmp {#2}{#1}}% \def\XINT_fcmp_Fn #1\Z #2#3% {% - \expandafter\XINT_cmp_pre\expandafter + \expandafter\xintiicmp\expandafter {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}% }% \def\xintAbs {\romannumeral0\xintabs }% @@ -2391,6 +2488,209 @@ \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax #4{#3}{#5}% }% +\def\xintFloatFac {\romannumeral0\xintfloatfac}% +\def\xintfloatfac #1{\XINT_flfac_chkopt \xintfloat #1\xint_relax }% +\def\XINTinFloatFac {\romannumeral0\XINTinfloatfac }% +\def\XINTinfloatfac #1{\XINT_flfac_chkopt \XINTinfloat #1\xint_relax }% +\def\XINT_flfac_chkopt #1#2% +{% + \ifx [#2\expandafter\XINT_flfac_opt + \else\expandafter\XINT_flfac_noopt + \fi + #1#2% +}% +\def\XINT_flfac_noopt #1#2\xint_relax +{% + \expandafter\XINT_FL_fac_start\expandafter + {\the\numexpr #2}{\XINTdigits}{#1[\XINTdigits]}% +}% +\def\XINT_flfac_opt #1[\xint_relax #2]#3% +{% + \expandafter\XINT_FL_fac_start\expandafter + {\the\numexpr #3\expandafter}\expandafter{\the\numexpr#2}{#1[#2]}% +}% +\def\XINT_FL_fac_start #1% +{% + \ifcase\XINT_cntSgn #1\Z + \expandafter\XINT_FL_fac_iszero + \or + \expandafter\XINT_FL_fac_increaseP + \else + \expandafter\XINT_FL_fac_isneg + \fi {#1}% +}% +\def\XINT_FL_fac_iszero #1#2#3{#3{1/1[0]}}% +\def\XINT_FL_fac_isneg #1#2#3% + {\expandafter\xintError:FactorialOfNegativeNumber #3{1/1[0]}}% +\def\XINT_FL_fac_increaseP #1#2% +{% + \expandafter\XINT_FL_fac_fork + \the\numexpr \xint_c_viii*% + ((\xint_c_v+#2+\XINT_FL_fac_extradigits #187654321\Z)/\xint_c_viii).% + #1.% +}% +\def\XINT_FL_fac_extradigits #1#2#3#4#5#6#7#8{\XINT_FL_fac_extra_a }% +\def\XINT_FL_fac_extra_a #1#2\Z {#1}% +\def\XINT_FL_fac_fork #1.#2.#3% +{% + \ifnum #2>99999999 \xint_dothis{\XINT_FL_fac_toobig }\fi + \ifnum #2>9999 \xint_dothis{\XINT_FL_fac_vbigloop_a }\fi + \ifnum #2>465 \xint_dothis{\XINT_FL_fac_bigloop_a }\fi + \ifnum #2>101 \xint_dothis{\XINT_FL_fac_medloop_a }\fi + \xint_orthat{\XINT_FL_fac_smallloop_a }% + #2.#1.{\XINT_FL_fac_out}{#3}% +}% +\def\XINT_FL_fac_toobig #1.#2.#3#4% + {\expandafter\xintError:FactorialOfTooBigNumber #4{1/1[0]}}% +\def\XINT_FL_fac_out #1\Z![#2]#3{#3{\romannumeral0\XINT_mul_out + #1\Z!1\R!1\R!1\R!1\R!1\R!1\R!1\R!1\R!\W [#2]}}% +\def\XINT_FL_fac_vbigloop_a #1.#2.% +{% + \XINT_FL_fac_bigloop_a 9999.#2.% + {\expandafter\XINT_FL_fac_vbigloop_loop\the\numexpr 100010000\expandafter.% + \the\numexpr \xint_c_x^viii+#1.}% +}% +\def\XINT_FL_fac_vbigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_vbigloop_loop + \the\numexpr #1+\xint_c_i\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_mul #1!% +}% +\def\XINT_FL_fac_bigloop_a #1.% +{% + \expandafter\XINT_FL_fac_bigloop_b \the\numexpr + #1+\xint_c_i-\xint_c_ii*((#1-464)/\xint_c_ii).#1.% +}% +\def\XINT_FL_fac_bigloop_b #1.#2.#3.% +{% + \expandafter\XINT_FL_fac_medloop_a + \the\numexpr #1-\xint_c_i.#3.{\XINT_FL_fac_bigloop_loop #1.#2.}% +}% +\def\XINT_FL_fac_bigloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_bigloop_loop + \the\numexpr #1+\xint_c_ii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_bigloop_mul #1!% +}% +\def\XINT_FL_fac_bigloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr \xint_c_x^viii+#1*(#1+\xint_c_i)!% +}% +\def\XINT_FL_fac_medloop_a #1.% +{% + \expandafter\XINT_FL_fac_medloop_b + \the\numexpr #1+\xint_c_i-\xint_c_iii*((#1-100)/\xint_c_iii).#1.% +}% +\def\XINT_FL_fac_medloop_b #1.#2.#3.% +{% + \expandafter\XINT_FL_fac_smallloop_a + \the\numexpr #1-\xint_c_i.#3.{\XINT_FL_fac_medloop_loop #1.#2.}% +}% +\def\XINT_FL_fac_medloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_medloop_loop + \the\numexpr #1+\xint_c_iii\expandafter.% + \the\numexpr #2\expandafter.\the\numexpr\XINT_FL_fac_medloop_mul #1!% +}% +\def\XINT_FL_fac_medloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)!% +}% +\def\XINT_FL_fac_smallloop_a #1.% +{% + \csname + XINT_FL_fac_smallloop_\the\numexpr #1-\xint_c_iv*(#1/\xint_c_iv)\relax + \endcsname #1.% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_1\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000001!.{2.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_-2\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000002!.{3.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_-1\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000006!.{4.#1.}{#2}% +}% +\expandafter\def\csname XINT_FL_fac_smallloop_0\endcsname #1.#2.% +{% + \XINT_FL_fac_addzeros #2.100000024!.{5.#1.}{#2}% +}% +\def\XINT_FL_fac_addzeros #1.% +{% + \ifnum #1=\xint_c_viii \expandafter\XINT_FL_fac_addzeros_exit\fi + \expandafter\XINT_FL_fac_addzeros\the\numexpr #1-\xint_c_viii.100000000!% +}% +\def\XINT_FL_fac_addzeros_exit #1.#2.#3#4% + {\XINT_FL_fac_smallloop_loop #3#21\Z![-#4]}% +\def\XINT_FL_fac_smallloop_loop #1.#2.% +{% + \ifnum #1>#2 \expandafter\XINT_FL_fac_loop_exit\fi + \expandafter\XINT_FL_fac_smallloop_loop + \the\numexpr #1+\xint_c_iv\expandafter.% + \the\numexpr #2\expandafter.\romannumeral0\XINT_FL_fac_smallloop_mul #1!% +}% +\def\XINT_FL_fac_smallloop_mul #1!% +{% + \expandafter\XINT_FL_fac_mul + \the\numexpr + \xint_c_x^viii+#1*(#1+\xint_c_i)*(#1+\xint_c_ii)*(#1+\xint_c_iii)!% +}%[[ +\def\XINT_FL_fac_loop_exit #1!#2]#3{#3#2]}% +\def\XINT_FL_fac_mul 1#1!% + {\expandafter\XINT_FL_fac_mul_a\the\numexpr\XINT_FL_fac_smallmul 10!{#1}}% +\def\XINT_FL_fac_mul_a #1-#2% +{% + \if#21\xint_afterfi{\expandafter\space\xint_gob_til_exclam}\else + \expandafter\space\fi #11\Z!% +}% +\def\XINT_FL_fac_minimulwc_a #1#2#3#4#5!#6#7#8#9% +{% + \XINT_FL_fac_minimulwc_b {#1#2#3#4}{#5}{#6#7#8#9}% +}% +\def\XINT_FL_fac_minimulwc_b #1#2#3#4!#5% +{% + \expandafter\XINT_FL_fac_minimulwc_c + \the\numexpr \xint_c_x^ix+#5+#2*#4.{{#1}{#2}{#3}{#4}}% +}% +\def\XINT_FL_fac_minimulwc_c 1#1#2#3#4#5#6.#7% +{% + \expandafter\XINT_FL_fac_minimulwc_d {#1#2#3#4#5}#7{#6}% +}% +\def\XINT_FL_fac_minimulwc_d #1#2#3#4#5% +{% + \expandafter\XINT_FL_fac_minimulwc_e + \the\numexpr \xint_c_x^ix+#1+#2*#5+#3*#4.{#2}{#4}% +}% +\def\XINT_FL_fac_minimulwc_e 1#1#2#3#4#5#6.#7#8#9% +{% + 1#6#9\expandafter!% + \the\numexpr\expandafter\XINT_FL_fac_smallmul + \the\numexpr \xint_c_x^viii+#1#2#3#4#5+#7*#8!% +}% +\def\XINT_FL_fac_smallmul 1#1!#21#3!% +{% + \xint_gob_til_Z #3\XINT_FL_fac_smallmul_end\Z + \XINT_FL_fac_minimulwc_a #2!#3!{#1}{#2}% +}% +\def\XINT_FL_fac_smallmul_end\Z\XINT_FL_fac_minimulwc_a #1!\Z!#2#3[#4]% +{% + \ifnum #2=\xint_c_ + \expandafter\xint_firstoftwo\else + \expandafter\xint_secondoftwo + \fi + {-2\relax[#4]}% + {1#2\expandafter!\expandafter-\expandafter1\expandafter + [\the\numexpr #4+\xint_c_viii]}% +}% \def\xintFloatSqrt {\romannumeral0\xintfloatsqrt }% \def\xintfloatsqrt #1{\XINT_flsqrt_chkopt \xintfloat #1\xint_relax }% \def\XINTinFloatSqrt {\romannumeral0\XINTinfloatsqrt }% @@ -2539,7 +2839,7 @@ {% \expandafter\XINT_flsqrt_big_j \romannumeral0\xintiidivision - {#1}{\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% + {#1}{\romannumeral0\XINT_dbl_pos #2\Z}{#2}% }% \def\XINT_flsqrt_big_j #1% {% @@ -2551,8 +2851,8 @@ \def\XINT_flsqrt_big_k #1#2#3% {% \expandafter\XINT_flsqrt_big_l\expandafter - {\romannumeral0\XINT_sub_pre {#3}{#1}}% - {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr {#1}}}% + {\romannumeral0\xintiisub {#3}{#1}}% + {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr #1\Z}}% }% \def\XINT_flsqrt_big_l #1#2% {% diff --git a/Master/texmf-dist/tex/generic/xint/xintgcd.sty b/Master/texmf-dist/tex/generic/xint/xintgcd.sty index b5905103847..187a37133df 100644 --- a/Master/texmf-dist/tex/generic/xint/xintgcd.sty +++ b/Master/texmf-dist/tex/generic/xint/xintgcd.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintgcd: Euclidean algorithm with xint package %% --------------------------------------------------------------- @@ -70,7 +70,7 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintgcd}% - [2015/09/12 v1.1c Euclide algorithm with xint package (jfB)]% + [2015/10/10 v1.2 Euclide algorithm with xint package (jfB)]% \def\xintGCD {\romannumeral0\xintgcd }% \def\xintgcd #1% {% @@ -250,8 +250,8 @@ \def\XINT_bezout_loop_b #1#2#3#4#5#6#7#8% {% \expandafter \XINT_bezout_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#2}}{#7}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#6}{#2}}{#8}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #5\Z #2\Z}{#7}}% + {\romannumeral0\xintiiadd{\XINT_mul_fork #6\Z #2\Z}{#8}}% {#1}{#3}{#4}{#5}{#6}% }% \def\XINT_bezout_loop_c #1#2% diff --git a/Master/texmf-dist/tex/generic/xint/xintkernel.sty b/Master/texmf-dist/tex/generic/xint/xintkernel.sty index 8904861217e..2b16addfa2b 100644 --- a/Master/texmf-dist/tex/generic/xint/xintkernel.sty +++ b/Master/texmf-dist/tex/generic/xint/xintkernel.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintkernel: Paraphernalia for the xint packages %% --------------------------------------------------------------- @@ -42,19 +42,21 @@ \else \def\y#1#2{\PackageInfo{#1}{#2}}% \fi + \let\z\relax \expandafter \ifx\csname numexpr\endcsname\relax \y{xintkernel}{\numexpr not available, aborting input}% - \aftergroup\endinput + \def\z{\endgroup\endinput}% \else \expandafter \ifx\csname XINTsetupcatcodes\endcsname\relax \else \y{xintkernel}{I was already loaded, aborting input}% - \aftergroup\endinput + \def\z{\endgroup\endinput}% \fi \fi - \def\SetCatcodesIfInputNotAborted + \ifx\z\relax\else\expandafter\z\fi% + \def\PrepareCatcodes {% \endgroup \def\XINT_restorecatcodes @@ -123,19 +125,19 @@ \catcode36=3 % $ \catcode91=12 % [ \catcode93=12 % ] - \catcode33=11 % ! LETTER + \catcode33=12 % ! \catcode64=11 % @ LETTER \catcode38=12 % & \catcode124=12 % | \catcode63=11 % ? LETTER \catcode34=12 % " \catcode39=12 % ' - \catcode126=3 % ~ + \catcode126=3 % ~ MATH \catcode59=12 % ; }% \XINT_setcatcodes }% -\SetCatcodesIfInputNotAborted +\PrepareCatcodes \def\XINTsetupcatcodes {% for use by other modules \edef\XINT_restorecatcodes_endinput {% @@ -152,7 +154,47 @@ \fi \XINT_providespackage \ProvidesPackage {xintkernel}% - [2015/09/12 v1.1c Paraphernalia for the xint packages (jfB)]% + [2015/10/10 v1.2 Paraphernalia for the xint packages (jfB)]% +\chardef\xint_c_ 0 +\chardef\xint_c_i 1 +\chardef\xint_c_ii 2 +\chardef\xint_c_iii 3 +\chardef\xint_c_iv 4 +\chardef\xint_c_v 5 +\chardef\xint_c_vi 6 +\chardef\xint_c_vii 7 +\chardef\xint_c_viii 8 +\chardef\xint_c_ix 9 +\chardef\xint_c_x 10 +\chardef\xint_c_xiv 14 +\chardef\xint_c_xvi 16 +\chardef\xint_c_xviii 18 +\chardef\xint_c_xxii 22 +\chardef\xint_c_ii^v 32 +\chardef\xint_c_ii^vi 64 +\chardef\xint_c_ii^vii 128 +\mathchardef\xint_c_ii^viii 256 +\mathchardef\xint_c_ii^xii 4096 +\mathchardef\xint_c_x^iv 10000 +\def\XINT_tmpa { }% +\ifx\XINT_tmpa\space\else + \immediate\write-1{Package xintkernel Warning: ATTENTION!}% + \immediate\write-1{\string\space\XINT_tmpa macro does not have its normal + meaning.}% + \immediate\write-1{\XINT_tmpa\XINT_tmpa\XINT_tmpa\XINT_tmpa + All kinds of catastrophes will ensue!!!!}% +\fi +\def\XINT_tmpb {}% +\ifx\XINT_tmpb\empty\else + \immediate\write-1{Package xintkernel Warning: ATTENTION!}% + \immediate\write-1{\string\empty\XINT_tmpa macro does not have its normal + meaning.}% + \immediate\write-1{\XINT_tmpa\XINT_tmpa\XINT_tmpa\XINT_tmpa + All kinds of catastrophes will ensue!!!!}% +\fi +\let\XINT_tmpa\relax \let\XINT_tmpb\relax +\ifdefined\space\else\def\space { }\fi +\ifdefined\empty\else\def\empty {}\fi \long\def\xint_gobble_ {}% \long\def\xint_gobble_i #1{}% \long\def\xint_gobble_ii #1#2{}% @@ -168,11 +210,29 @@ \long\def\xint_firstofone_thenstop #1{ #1}% \long\def\xint_firstoftwo_thenstop #1#2{ #1}% \long\def\xint_secondoftwo_thenstop #1#2{ #2}% -\def\xint_gob_til_zero #10{}% -\def\xint_UDzerominusfork #10-#2#3\krof {#2}% +\def\xint_minus_thenstop { -}% +\def\xint_exchangetwo_keepbraces #1#2{{#2}{#1}}% \long\def\xint_gob_til_R #1\R {}% \long\def\xint_gob_til_W #1\W {}% \long\def\xint_gob_til_Z #1\Z {}% +\def\xint_gob_til_zero #10{}% +\def\xint_gob_til_one #11{}% +\def\xint_gob_til_zeros_iii #1000{}% +\def\xint_gob_til_zeros_iv #10000{}% +\def\xint_gob_til_eightzeroes #100000000{}% +\def\xint_gob_til_exclam #1!{}% catcode 12 exclam +\def\xint_gob_til_dot #1.{}% +\def\xint_gob_til_G #1G{}% +\def\xint_gob_til_minus #1-{}% +\def\xint_gob_til_relax #1\relax {}% +\def\xint_UDzerominusfork #10-#2#3\krof {#2}% +\def\xint_UDzerofork #10#2#3\krof {#2}% +\def\xint_UDsignfork #1-#2#3\krof {#2}% +\def\xint_UDwfork #1\W#2#3\krof {#2}% +\def\xint_UDXINTWfork #1\XINT_W#2#3\krof {#2}% +\def\xint_UDzerosfork #100#2#3\krof {#2}% +\def\xint_UDonezerofork #110#2#3\krof {#2}% +\def\xint_UDsignsfork #1--#2#3\krof {#2}% \let\xint_relax\relax \def\xint_brelax {\xint_relax }% \long\def\xint_gob_til_xint_relax #1\xint_relax {}% @@ -180,16 +240,9 @@ \long\def\xint_bye #1\xint_bye {}% \long\def\xint_dothis #1#2\xint_orthat #3{\fi #1}% v1.1 \let\xint_orthat \xint_firstofone +\long\def\xintdothis #1#2\xintorthat #3{\fi #1}% +\let\xintorthat \xint_firstofone \def\xint_zapspaces #1 #2{#1#2\xint_zapspaces }% v1.1 -\chardef\xint_c_ 0 -\chardef\xint_c_i 1 -\chardef\xint_c_ii 2 -\chardef\xint_c_iii 3 -\chardef\xint_c_iv 4 -\chardef\xint_c_v 5 -\chardef\xint_c_vi 6 -\chardef\xint_c_vii 7 -\chardef\xint_c_viii 8 \def\xintodef #1{\expandafter\def\expandafter#1\expandafter }% \def\xintoodef #1{\expandafter\expandafter\expandafter\def \expandafter\expandafter\expandafter#1% diff --git a/Master/texmf-dist/tex/generic/xint/xintseries.sty b/Master/texmf-dist/tex/generic/xint/xintseries.sty index 1bec170ddc4..40cfb759761 100644 --- a/Master/texmf-dist/tex/generic/xint/xintseries.sty +++ b/Master/texmf-dist/tex/generic/xint/xintseries.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xintseries: Expandable partial sums with xint package %% --------------------------------------------------------------- @@ -70,7 +70,7 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xintseries}% - [2015/09/12 v1.1c Expandable partial sums with xint package (jfB)]% + [2015/10/10 v1.2 Expandable partial sums with xint package (jfB)]% \def\xintSeries {\romannumeral0\xintseries }% \def\xintseries #1#2% {% diff --git a/Master/texmf-dist/tex/generic/xint/xinttools.sty b/Master/texmf-dist/tex/generic/xint/xinttools.sty index 345777e41d8..94b8839bea3 100644 --- a/Master/texmf-dist/tex/generic/xint/xinttools.sty +++ b/Master/texmf-dist/tex/generic/xint/xinttools.sty @@ -21,7 +21,7 @@ %% same distribution. (The sources need not necessarily be %% in the same archive or directory.) %% --------------------------------------------------------------- -%% The xint bundle v1.1c 2015/09/12 +%% The xint bundle v1.2 2015/10/10 %% Copyright (C) 2013-2015 by Jean-Francois Burnol %% xinttools: Expandable and non-expandable utilities %% --------------------------------------------------------------- @@ -70,7 +70,7 @@ \XINTsetupcatcodes% defined in xintkernel.sty \XINT_providespackage \ProvidesPackage{xinttools}% - [2015/09/12 v1.1c Expandable and non-expandable utilities (jfB)]% + [2015/10/10 v1.2 Expandable and non-expandable utilities (jfB)]% \newtoks\XINT_toks \xint_firstofone{\let\XINT_sptoken= } %<- space here! \def\xintgodef {\global\xintodef }% |