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Diffstat (limited to 'Master/texmf-dist/source/latex/l3kernel/l3fp-extended.dtx')
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1 files changed, 656 insertions, 216 deletions
diff --git a/Master/texmf-dist/source/latex/l3kernel/l3fp-extended.dtx b/Master/texmf-dist/source/latex/l3kernel/l3fp-extended.dtx index 79e96d7715c..9bb7145d635 100644 --- a/Master/texmf-dist/source/latex/l3kernel/l3fp-extended.dtx +++ b/Master/texmf-dist/source/latex/l3kernel/l3fp-extended.dtx @@ -1,6 +1,6 @@ % \iffalse meta-comment % -%% File: l3fp-extended.dtx Copyright (C) 2011-2012 The LaTeX3 Project +%% File: l3fp-extended.dtx Copyright (C) 2011-2013 The LaTeX3 Project %% %% It may be distributed and/or modified under the conditions of the %% LaTeX Project Public License (LPPL), either version 1.3c of this @@ -36,8 +36,8 @@ % %<*driver> \RequirePackage{l3bootstrap} -\GetIdInfo$Id: l3fp-extended.dtx 4482 2013-04-24 21:05:12Z joseph $ - {L3 Floating-point extended precision fixed-points} +\GetIdInfo$Id: l3fp-extended.dtx 4601 2013-11-18 23:13:28Z bruno $ + {L3 Floating-points with extended precision} \documentclass[full]{l3doc} \begin{document} \DocInput{\jobname.dtx} @@ -48,7 +48,7 @@ % \title{The \textsf{l3fp-extended} package\thanks{This file % has version number \ExplFileVersion, last % revised \ExplFileDate.}\\ -% Fixed points with extended precision for internal use} +% Manipulating numbers with extended precision, for internal use} % \author{^^A % The \LaTeX3 Project\thanks % {^^A @@ -77,10 +77,12 @@ %<@@=fp> % \end{macrocode} % -% \subsection{Description of extended fixed points} +% \subsection{Description of fixed point numbers} % -% In this module, we work on (almost) fixed-point numbers with -% extended ($24$ digits) precision. This is used in the computation of +% This module provides a few functions to manipulate positive floating +% point numbers with extended precision ($24$ digits), but mostly +% provides functions for fixed-point numbers with this precision ($24$ +% digits). Those are used in the computation of % Taylor series for the logarithm, exponential, and trigonometric % functions. Since we eventually only care about the $16$ first digits % of the final result, some of the calculations are not performed with @@ -93,7 +95,7 @@ % \end{quote} % where each \meta{a_i} is exactly $4$ digits (ranging from |0000| to % |9999|), except \meta{a_1}, which may be any \enquote{not-too-large} -% non-negative integer, with or without trailing zeros. Here, +% non-negative integer, with or without leading zeros. Here, % \enquote{not-too-large} depends on the specific function (see the % corresponding comments for details). Checking for overflow is the % responsibility of the code calling those functions. The fixed point @@ -118,15 +120,16 @@ % appropriate for computing continued fractions and Taylor series. % % At the end of the calculation, the result is turned back to a floating -% point number using \cs{@@_fixed_to_float:Nw}. This function has to +% point number using \cs{@@_fixed_to_float:wN}. This function has to % change the exponent of the floating point number: it must be used % after starting an integer expression for the overall exponent of the % result. % -% \subsection{Helpers for extended fixed points} +% \subsection{Helpers for numbers with extended precision} % +% ^^A todo: put trailing semicolon here? % \begin{variable}[int]{\c_@@_one_fixed_tl} -% The extended fixed-point number~$1$, used in \pkg{l3fp-expo}. +% The fixed-point number~$1$, used in \pkg{l3fp-expo}. % \begin{macrocode} \tl_const:Nn \c_@@_one_fixed_tl { {10000} {0000} {0000} {0000} {0000} {0000} } @@ -157,6 +160,28 @@ % \end{macrocode} % \end{macro} % +% \begin{macro}[int, EXP]{\@@_fixed_div_myriad:wn} +% Divide a fixed point number by $10000$. This is a little bit more +% subtle than just removing the last group and adding a leading group +% of zeros: the first group~|#1| may have any number of digits, and we +% must split~|#1| into the new first group and a second group of +% exactly $4$~digits. The choice of shifts allows~|#1| to be in the +% range $[0, 5\cdot 10^{8}-1]$. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_div_myriad:wn #1#2#3#4#5#6; #7 + { + \exp_after:wN \@@_fixed_mul_after:wn + \int_use:N \__int_eval:w \c_@@_leading_shift_int + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_trailing_shift_int + + #1 ; {#7} {#2}{#3}{#4}{#5}; + } +% \end{macrocode} +% \end{macro} +% +% ^^A todo:\cs_new:Npn \@@_fixed_mul_after:wn #1; #2; #3 { #3 {#1} #2; } +% ^^A and do not bring the continuation up while packing. +% ^^A possibly delete use_braced_s function afterwards. % \begin{macro}[aux, EXP]{\@@_fixed_mul_after:wn} % The fixed point operations which involve multiplication end by % calling this auxiliary. It braces the last block of digits, and @@ -168,6 +193,51 @@ % \end{macrocode} % \end{macro} % +% \subsection{Multiplying a fixed point number by a short one} +% +% \begin{macro}[int, EXP]{\@@_fixed_mul_short:wwn} +% \begin{syntax} +% \cs{@@_fixed_mul_short:wwn} +% \ \ \Arg{a_1} \Arg{a_2} \Arg{a_3} \Arg{a_4} \Arg{a_5} \Arg{a_6} |;| +% \ \ \Arg{b_0} \Arg{b_1} \Arg{b_2} |;| \Arg{continuation} +% \end{syntax} +% Computes the product $c=ab$ of $a=\sum_i \meta{a_i} 10^{-4i}$ and +% $b=\sum_i \meta{b_i} 10^{-4i}$, rounds it to the closest multiple of +% $10^{-24}$, and leaves \meta{continuation} \Arg{c_1} \ldots{} +% \Arg{c_6} |;| in the input stream, where each of the \meta{c_i} are +% blocks of $4$~digits, except \meta{c_1}, which is any \TeX{} +% integer. Note that indices for \meta{b} start at~$0$: a second +% operand of |{0001}{0000}{0000}| will leave the first operand +% unchanged (rather than dividing it by $10^{4}$, as +% \cs{@@_fixed_mul:wwn} would). +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_short:wwn #1#2#3#4#5#6; #7#8#9; + { + \exp_after:wN \@@_fixed_mul_after:wn + \int_use:N \__int_eval:w \c_@@_leading_shift_int + + #1*#7 + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_middle_shift_int + + #1*#8 + #2*#7 + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_middle_shift_int + + #1*#9 + #2*#8 + #3*#7 + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_middle_shift_int + + #2*#9 + #3*#8 + #4*#7 + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_middle_shift_int + + #3*#9 + #4*#8 + #5*#7 + \exp_after:wN \@@_pack:NNNNNwn + \int_use:N \__int_eval:w \c_@@_trailing_shift_int + + #4*#9 + #5*#8 + #6*#7 + + ( #5*#9 + #6*#8 + #6*#9 / \c_ten_thousand ) + / \c_ten_thousand + \exp_after:wN ; \@@_use_braced_s:n + } +% \end{macrocode} +% \end{macro} +% % \subsection{Dividing a fixed point number by a small integer} % % \begin{macro}[int, EXP]{\@@_fixed_div_int:wwN} @@ -270,12 +340,13 @@ % \cs{@@_fixed_add:wwn} \meta{a} |;| \meta{b} |;| \Arg{continuation} % \end{syntax} % Computes $a+b$ (resp.\ $a-b$) and feeds the result to the -% \meta{continuation}. This function requires $0\leq -% a_{1},b_{1}<50000$, and requires the result to be positive (this -% happens automatically for addition). The two functions only differ +% \meta{continuation}. This function requires $0\leq a_{1},b_{1}\leq +% 114748$, its result must be positive (this happens automatically for +% addition) and its first group must have at most~$5$ digits: $(a\pm +% b)_{1}<100000$. The two functions only differ by % a sign, hence use a common auxiliary. It would be nice to grab the % $12$ brace groups in one go; only $9$ parameters are allowed. Start -% by grabbing the two signs, $a_{1}, \ldots, a_{4}$, the rest of $a$, +% by grabbing the sign, $a_{1}, \ldots, a_{4}$, the rest of $a$, % and $b_{1}$ and $b_{2}$. The second auxiliary receives the rest of % $a$, the sign multiplying $b$, the rest of $b$, and the % \meta{continuation} as arguments. After going down through the @@ -309,6 +380,7 @@ % % \subsection{Multiplying fixed points} % +% ^^A todo: may a_1 or b_1 be = 10000? Used in ediv_epsi later. % \begin{macro}[int, EXP]{\@@_fixed_mul:wwn} % \begin{macro}[aux, EXP]{\@@_fixed_mul:nnnnnnnwn} % \begin{syntax} @@ -564,8 +636,576 @@ % \end{macrocode} % \end{macro} % +% \subsection{Extended-precision floating point numbers} +% +% In this section we manipulate floating point numbers with roughly $24$ +% significant figures (``extended-precision'' numbers, in short, +% ``ep''), which take the form of an integer exponent, followed by a +% comma, then six groups of digits, ending with a semicolon. The first +% group of digit may be any non-negative integer, while other groups of +% digits have $4$~digits. In other words, an extended-precision number +% is an exponent ending in a comma, then a fixed point number. +% +% \begin{macro}[int, EXP]{\@@_ep_to_fixed:wwn} +% \begin{macro}[aux, EXP] +% {\@@_ep_to_fixed_auxi:www, \@@_ep_to_fixed_auxii:nnnnnnnwn} +% Converts an extended-precision number with an exponent at most~$4$ +% to a fixed point number whose first block will have $12$~digits, +% most often starting with many zeros. +% \begin{macrocode} +\cs_new:Npn \@@_ep_to_fixed:wwn #1,#2 + { + \exp_after:wN \@@_ep_to_fixed_auxi:www + \int_use:N \__int_eval:w 1 0000 0000 + #2 \exp_after:wN ; + \tex_romannumeral:D -`0 + \prg_replicate:nn { \c_four - \int_max:nn {#1} { -32 } } { 0 } ; + } +\cs_new:Npn \@@_ep_to_fixed_auxi:www 1#1; #2; #3#4#5#6#7; + { + \@@_pack_eight:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_ep_to_fixed_auxii:nnnnnnnwn ; + #2 #1#3#4#5#6#7 0000 ! + } +\cs_new:Npn \@@_ep_to_fixed_auxii:nnnnnnnwn #1#2#3#4#5#6#7; #8! #9 + { #9 {#1#2}{#3}{#4}{#5}{#6}{#7}; } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% ^^A todo: make it work when the arg is zero. +% ^^A todo: remove the unused(?) 'n' arg. +% \begin{macro}[aux, EXP]{\@@_ep_to_ep:wwN} +% \begin{macro}[aux, rEXP]{\@@_ep_to_ep_loop:N, \@@_ep_to_ep_end:www} +% \begin{macro}[aux, EXP]{\@@_ep_to_ep_zero:ww} +% Normalize an extended-precision number. More precisely, leading +% zeros are removed from the mantissa of the argument, decreasing its +% exponent as appropriate. Then the digits are packed into $6$~groups +% of~$4$ (discarding any remaining digit, not rounding). Finally, the +% continuation~|#8| is placed before the resulting exponent--mantissa +% pair. The input exponent may in fact be given as an integer +% expression. The \texttt{loop} auxiliary grabs a digit: if it +% is~$0$, decrement the exponent and continue looping, and otherwise +% call the \texttt{end} auxiliary, which places all digits in the +% right order (the digit that was not~$0$, and any remaining digits), +% followed by some~$0$, then packs them up neatly in $3\times2=6$ +% blocks of four. At the end of the day, remove with \cs{@@_use_i:ww} +% any digit that did not make it in the final mantissa (typically only +% zeros, unless the original first block has more than~$4$ digits). +% \begin{macrocode} +\cs_new:Npn \@@_ep_to_ep:wwN #1,#2#3#4#5#6#7; #8 + { + \exp_after:wN #8 + \int_use:N \__int_eval:w #1 + \c_four + \exp_after:wN \use_i:nn + \exp_after:wN \@@_ep_to_ep_loop:N + \int_use:N \__int_eval:w 1 0000 0000 + #2 \__int_eval_end: + #3#4#5#6#7 ; ; ! + } +\cs_new:Npn \@@_ep_to_ep_loop:N #1 + { + \if_meaning:w 0 #1 + - \c_one + \else: + \@@_ep_to_ep_end:www #1 + \fi: + \@@_ep_to_ep_loop:N + } +\cs_new:Npn \@@_ep_to_ep_end:www + #1 \fi: \@@_ep_to_ep_loop:N #2; #3! + { + \fi: + \if_meaning:w ; #1 + - \c_two * \c_@@_max_exponent_int + \@@_ep_to_ep_zero:ww + \fi: + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_use_i:ww , ; + #1 #2 0000 0000 0000 0000 0000 0000 ; + } +\cs_new:Npn \@@_ep_to_ep_zero:ww \fi: #1; #2; #3; + { \fi: , {1000}{0000}{0000}{0000}{0000}{0000} ; } +% \end{macrocode} +% \end{macro} +% \end{macro} +% \end{macro} +% +% \begin{macro}[int, EXP]{\@@_ep_compare:wwww} +% \begin{macro}[aux, EXP]{\@@_ep_compare_aux:wwww} +% In \pkg{l3fp-trig} we need to compare two extended-precision +% numbers. This is based on the same function for positive floating +% point numbers, with an extra test if comparing only $16$ decimals is +% not enough to distinguish the numbers. Note that this function only +% works if the numbers are normalized so that their first block is +% in~$[1000,9999]$. +% \begin{macrocode} +\cs_new:Npn \@@_ep_compare:wwww #1,#2#3#4#5#6#7; + { \@@_ep_compare_aux:wwww {#1}{#2}{#3}{#4}{#5}; #6#7; } +\cs_new:Npn \@@_ep_compare_aux:wwww #1;#2;#3,#4#5#6#7#8#9; + { + \if_case:w + \@@_compare_npos:nwnw #1; {#3}{#4}{#5}{#6}{#7}; \exp_stop_f: + \if_int_compare:w #2 = #8#9 \exp_stop_f: + 0 + \else: + \if_int_compare:w #2 < #8#9 - \fi: 1 + \fi: + \or: 1 + \else: -1 + \fi: + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% ^^A todo: doc that neither operand may be zero (or fix ep_to_ep above) +% \begin{macro}[int, EXP]{\@@_ep_mul:wwwwn, \@@_ep_mul_raw:wwwwN} +% Multiply two extended-precision numbers: first normalize them to +% avoid losing too much precision, then multiply the mantissas |#2| +% and~|#4| as fixed point numbers, and sum the exponents |#1| +% and~|#3|. The result's first block is in $[100,9999]$. +% \begin{macrocode} +\cs_new:Npn \@@_ep_mul:wwwwn #1,#2; #3,#4; + { + \@@_ep_to_ep:wwN #3,#4; + \@@_fixed_continue:wn + { + \@@_ep_to_ep:wwN #1,#2; + \@@_ep_mul_raw:wwwwN + } + \@@_fixed_continue:wn + } +\cs_new:Npn \@@_ep_mul_raw:wwwwN #1,#2; #3,#4; #5 + { + \@@_fixed_mul:wwn #2; #4; + { \exp_after:wN #5 \int_use:N \__int_eval:w #1 + #3 , } + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Dividing extended-precision numbers} +% +% \newcommand{\eTeXfrac}[2]{\left[\frac{#1}{#2}\right]} +% +% Divisions of extended-precision numbers are difficult to perform with +% exact rounding: the technique used in \pkg{l3fp-basics} for $16$-digit +% floating point numbers does not generalize easily to $24$-digit +% numbers. Thankfully, there is no need for exact rounding. +% +% Let us call \meta{n} the numerator and \meta{d} the denominator. +% After a simple normalization step, we can assume that +% $\meta{n}\in[0.1,1)$ and $\meta{d}\in[0.1,1)$, and compute +% $\meta{n}/(10\meta{d})\in(0.01,1)$. In terms of the $6$~blocks of +% digits $\meta{n_1}\cdots\meta{n_6}$ and the $6$~blocks +% $\meta{d_1}\cdots\meta{d_6}$, the condition translates to +% $\meta{n_1},\meta{d_1}\in[1000,9999]$. +% +% We will first find an integer estimate $a \simeq 10^{8} / \meta{d}$ by +% computing +% \begin{align*} +% \alpha &= \eTeXfrac{10^{9}}{\meta{d_1}+1} \\ +% \beta &= \eTeXfrac{10^{9}}{\meta{d_1}} \\ +% a &= 10^{3} \alpha + (\beta-\alpha) \cdot +% \left(10^{3}-\eTeXfrac{\meta{d_2}}{10}\right) - 1250, +% \end{align*} +% where $\eTeXfrac{\bullet}{\bullet}$ denotes \eTeX{}'s rounding +% division, which rounds ties away from zero. The idea is to +% interpolate between $10^{3}\alpha$ and $10^{3}\beta$ with a parameter +% $\meta{d_2}/10^{4}$, so that when $\meta{d_2}=0$ one gets $a = +% 10^{3}\beta-1250 \simeq 10^{12} / \meta{d_1} \simeq 10^{8} / +% \meta{d}$, while when $\meta{d_2}=9999$ one gets $a = +% 10^{3}\alpha-1250 \simeq 10^{12} / (\meta{d_1} + 1) \simeq 10^{8} / +% \meta{d}$. The shift by $1250$ helps to ensure that $a$ is an +% underestimate of the correct value. We will prove that +% \[ +% 1 - 1.755\cdot 10^{-5} < \frac{\meta{d}a}{10^{8}} < 1 . +% \] +% We can then compute the inverse of $\meta{d}a/10^{8} = 1 - \epsilon$ +% using the relation $1/(1-\epsilon) \simeq (1+\epsilon)(1+\epsilon^{2}) +% + \epsilon^{4}$, which is correct up to a relative error of +% $\epsilon^5 < 1.6\cdot 10^{-24}$. This allows us to find the desired +% ratio as +% \[ +% \frac{\meta{n}}{\meta{d}} +% = \frac{\meta{n}a}{10^{8}} +% \bigl( (1+\epsilon)(1+\epsilon^{2}) + \epsilon^{4}\bigr) . +% \] +% +% Let us prove the upper bound first (multiplied by $10^{15}$). Note +% that $10^{7} \meta{d} < 10^{3} \meta{d_1} + 10^{-1} (\meta{d_2} + 1)$, +% and that \eTeX{}'s division $\eTeXfrac{\meta{d_2}}{10}$ will at most +% underestimate $10^{-1}(\meta{d_2} + 1)$ by $0.5$, as can be checked +% for each possible last digit of \meta{d_2}. Then, +% \begin{align} +% 10^{7} \meta{d}a +% & < +% \left(10^{3}\meta{d_1} +% + \eTeXfrac{\meta{d_2}}{10} + \frac{1}{2}\right) +% \left(\left(10^{3}-\eTeXfrac{\meta{d_2}}{10}\right) \beta +% + \eTeXfrac{\meta{d_2}}{10} \alpha - 1250\right) +% \\ +% & < +% \left(10^{3}\meta{d_1} +% + \eTeXfrac{\meta{d_2}}{10} + \frac{1}{2}\right) +% \left( +% \left(10^{3}-\eTeXfrac{\meta{d_2}}{10}\right) +% \left(\frac{10^{9}}{\meta{d_1}} + \frac{1}{2} \right) +% + \eTeXfrac{\meta{d_2}}{10} +% \left(\frac{10^{9}}{\meta{d_1}+1} + \frac{1}{2} \right) +% - 1250 +% \right) +% \\ +% & < +% \left(10^{3} \meta{d_1} +% + \eTeXfrac{\meta{d_2}}{10} + \frac{1}{2}\right) +% \left(\frac{10^{12}}{\meta{d_1}} +% - \eTeXfrac{\meta{d_2}}{10} +% \frac{10^{9}}{\meta{d_1}(\meta{d_1}+1)} +% - 750\right) +% \end{align} +% We recognize a quadratic polynomial in $[\meta{d_2}/10]$ with a +% negative leading coefficient: this polynomial is bounded above, +% according to $([\meta{d_2}/10]+a)(b-c[\meta{d_2}/10]) \leq +% (b+ca)^2/(4c)$. Hence, +% \[ +% 10^{7} \meta{d}a +% < \frac{10^{15}}{\meta{d_1}(\meta{d_1}+1)} \left( +% \meta{d_1} + \frac{1}{2} + \frac{1}{4} 10^{-3} +% - \frac{3}{8} \cdot 10^{-9} \meta{d_1}(\meta{d_1}+1) \right)^2 +% \] +% Since \meta{d_1} takes integer values within $[1000,9999]$, it is a +% simple programming exercise to check that the squared expression is +% always less than $\meta{d_1}(\meta{d_1}+1)$, hence $10^{7} \meta{d} a +% < 10^{15}$. The upper bound is proven. We also find that +% $\frac{3}{8}$ can be replaced by slightly smaller numbers, but nothing +% less than $0.374563\ldots$, and going back through the derivation of +% the upper bound, we find that $1250$ is as small a shift as we can +% obtain without breaking the bound. +% +% Now, the lower bound. The same computation as for the upper bound +% implies +% \[ +% 10^{7} \meta{d}a +% > \left(10^{3} \meta{d_1} + \eTeXfrac{\meta{d_2}}{10} +% - \frac{1}{2}\right) +% \left(\frac{10^{12}}{\meta{d_1}} +% - \eTeXfrac{\meta{d_2}}{10} \frac{10^{9}}{\meta{d_1}(\meta{d_1}+1)} +% - 1750\right) +% \] +% This time, we want to find the minimum of this quadratic polynomial. +% Since the leading coefficient is still negative, the minimum is +% reached for one of the extreme values $[y/10]=0$ or $[y/10]=100$, and +% we easily check the bound for those values. +% +% We have proven that the algorithm will give us a precise enough +% answer. Incidentally, the upper bound that we derived tells us that +% $a < 10^{8}/\meta{d} \leq 10^{9}$, hence we can compute $a$ safely as +% a \TeX{} integer, and even add $10^{9}$ to it to ease grabbing of all +% the digits. The lower bound implies $10^{8} - 1755 < a$, which we do +% not care about. +% +% ^^A todo: provide ep_inv, not ep_div? +% ^^A todo: make extra sure that the result's first block cannot be 99 +% ^^A todo: doc that neither operand may be zero (or fix ep_to_ep) +% \begin{macro}[int, EXP]{\@@_ep_div:wwwwn} +% Compute the ratio of two extended-precision numbers. The result is +% an extended-precision number whose first block lies in the range +% $[100,9999]$, and is placed after the \meta{continuation} once we +% are done. First normalize the inputs so that both first block lie +% in $[1000,9999]$, then call \cs{@@_ep_div_esti:wwwwn} +% \meta{denominator} \meta{numerator}, responsible for estimating the +% inverse of the denominator. +% \begin{macrocode} +\cs_new:Npn \@@_ep_div:wwwwn #1,#2; #3,#4; + { + \@@_ep_to_ep:wwN #1,#2; + \@@_fixed_continue:wn + { + \@@_ep_to_ep:wwN #3,#4; + \@@_ep_div_esti:wwwwn + } + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[aux, EXP] +% { +% \@@_ep_div_esti:wwwwn, +% \@@_ep_div_estii:wwnnwwn, +% \@@_ep_div_estiii:NNNNNwwwn +% } +% The \texttt{esti} function evaluates $\alpha=10^{9} / (\meta{d_1} + +% 1)$, which is used twice in the expression for $a$, and combines the +% exponents |#1| and~|#4| (with a shift by~$1$ because we will compute +% $\meta{n}/(10\meta{d})$. Then the \texttt{estii} function evaluates +% $10^{9} + a$, and puts the exponent~|#2| after the +% continuation~|#7|: from there on we can forget exponents and focus +% on the mantissa. The \texttt{estiii} function multiplies the +% denominator~|#7| by $10^{-8}a$ (obtained as $a$ split into the +% single digit~|#1| and two blocks of $4$~digits, |#2#3#4#5| +% and~|#6|). The result $10^{-8}a\meta{d}=(1-\epsilon)$, and a +% partially packed $10^{-9}a$ (as a block of four digits, and five +% individual digits, not packed by lack of available macro parameters +% here) are passed to \cs{@@_ep_div_epsi:wnNNNNn}, which computes +% $10^{-9}a/(1-\epsilon)$, that is, $1/(10\meta{d})$ and we finally +% multiply this by the numerator~|#8|. +% \begin{macrocode} +\cs_new:Npn \@@_ep_div_esti:wwwwn #1,#2#3; #4, + { + \exp_after:wN \@@_ep_div_estii:wwnnwwn + \int_use:N \__int_eval:w 10 0000 0000 / ( #2 + \c_one ) + \exp_after:wN ; + \int_use:N \__int_eval:w #4 - #1 + \c_one , + {#2} #3; + } +\cs_new:Npn \@@_ep_div_estii:wwnnwwn #1; #2,#3#4#5; #6; #7 + { + \exp_after:wN \@@_ep_div_estiii:NNNNNwwwn + \int_use:N \__int_eval:w 10 0000 0000 - 1750 + + #1 000 + (10 0000 0000 / #3 - #1) * (1000 - #4 / 10) ; + {#3}{#4}#5; #6; { #7 #2, } + } +\cs_new:Npn \@@_ep_div_estiii:NNNNNwwwn 1#1#2#3#4#5#6; #7; + { + \@@_fixed_mul_short:wwn #7; {#1}{#2#3#4#5}{#6}; + \@@_ep_div_epsi:wnNNNNNn {#1#2#3#4}#5#6 + \@@_fixed_mul:wwn + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[aux, EXP] +% { +% \@@_ep_div_epsi:wnNNNNNn, +% \@@_ep_div_eps_pack:NNNNNw, +% \@@_ep_div_epsii:wwnNNNNNn, +% } +% The bounds shown above imply that the \texttt{epsi} function's first +% operand is $(1-\epsilon)$ with $\epsilon\in[0,1.755\cdot 10^{-5}]$. +% The \texttt{epsi} function computes $\epsilon$ as $1-(1-\epsilon)$. +% Since $\epsilon<10^{-4}$, its first block vanishes and there is no +% need to explicitly use~|#1| (which is $9999$). Then \texttt{epsii} +% evaluates $10^{-9}a/(1-\epsilon)$ as +% $(1+\epsilon^2)(1+\epsilon)(10^{-9}a \epsilon) + 10^{-9}a$. +% Importantly, we compute $10^{-9}a \epsilon$ before multiplying it +% with the rest, rather than multiplying by $\epsilon$ and then +% $10^{-9}a$, as this second option loses more precision. Also, the +% combination of \texttt{short_mul} and \texttt{div_myriad} is both +% faster and more precise than a simple \texttt{mul}. +% \begin{macrocode} +\cs_new:Npn \@@_ep_div_epsi:wnNNNNNn #1#2#3#4#5#6; + { + \exp_after:wN \@@_ep_div_epsii:wwnNNNNNn + \int_use:N \__int_eval:w 1 9998 - #2 + \exp_after:wN \@@_ep_div_eps_pack:NNNNNw + \int_use:N \__int_eval:w 1 9999 9998 - #3#4 + \exp_after:wN \@@_ep_div_eps_pack:NNNNNw + \int_use:N \__int_eval:w 2 0000 0000 - #5#6 ; ; + } +\cs_new:Npn \@@_ep_div_eps_pack:NNNNNw #1#2#3#4#5#6; + { + #1 ; {#2#3#4#5} {#6} } +\cs_new:Npn \@@_ep_div_epsii:wwnNNNNNn 1#1; #2; #3#4#5#6#7#8 + { + \@@_fixed_mul:wwn {0000}{#1}#2; {0000}{#1}#2; + \@@_fixed_add_one:wN + \@@_fixed_mul:wwn {10000} {#1} #2 ; + { + \@@_fixed_mul_short:wwn {0000}{#1}#2; {#3}{#4#5#6#7}{#8000}; + \@@_fixed_div_myriad:wn + \@@_fixed_mul:wwn + } + \@@_fixed_add:wwn {#3}{#4#5#6#7}{#8000}{0000}{0000}{0000}; + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Inverse square root of extended precision numbers} +% +% The idea here is similar to division. Normalize the input, +% multiplying by powers of $100$ until we have $x\in[0.01,1)$. Then +% find an integer approximation $r \in [101, 1003]$ of +% $10^{2}/\sqrt{x}$, as the fixed point of iterations of the Newton +% method: essentially $r \mapsto (r + 10^{8} / (x_{1} r)) / 2$, starting +% from a guess that optimizes the number of steps before convergence. +% In fact, just as there is a slight shift when computing divisions to +% ensure that some inequalities hold, we will replace $10^{8}$ by a +% slightly larger number which will ensure that $r^2 x \geq 10^{4}$. +% This also causes $r \in [101, 1003]$. Another correction to the above +% is that the input is actually normalized to $[0.1,1)$, and we use +% either $10^{8}$ or $10^{9}$ in the Newton method, depending on the +% parity of the exponent. Skipping those technical hurdles, once we +% have the approximation~$r$, we set $y = 10^{-4} r^{2} x$ (or rather, +% the correct power of~$10$ to get $y\simeq 1$) and compute $y^{-1/2}$ +% through another application of Newton's method. This time, the +% starting value is $z=1$, each step maps $z \mapsto z(1.5-0.5yz^2)$, +% and we perform a fixed number of steps. Our final result combines~$r$ +% with $y^{-1/2}$ as $x^{-1/2} = 10^{-2} r y^{-1/2}$. +% +% ^^A todo: doc that the operand may not be zero (or fix ep_to_ep above) +% \begin{macro}[int, EXP]{\@@_ep_isqrt:wwn} +% \begin{macro}[aux, EXP] +% {\@@_ep_isqrt_aux:wwn, \@@_ep_isqrt_auxii:wwnnnwn} +% First normalize the input, then check the parity of the +% exponent~|#1|. If it is even, the result's exponent will be +% $-|#1|/2$, otherwise it will be $(|#1|-1)/2$ (except in the case +% where the input was an exact power of $100$). The \texttt{auxii} +% function receives as~|#1| the result's exponent just computed, as +% |#2| the starting value for the iteration giving~$r$ (the +% values~$168$ and~$535$ lead to the least number of iterations before +% convergence, on average), as |#3| and~|#4| one empty argument and +% one~|0|, depending on the parity of the original exponent, as |#5| +% and~|#6| the normalized mantissa ($|#5|\in[1000,9999]$), and as |#7| +% the continuation. It sets up the iteration giving~$r$: the +% \texttt{esti} function thus receives the initial two guesses |#2| +% and~$0$, an approximation~|#5| of~$10^{4}x$ (its first block of +% digits), and the empty/zero arguments |#3| and~|#4|, followed by the +% mantissa and an altered continuation where we have stored the +% result's exponent. +% \begin{macrocode} +\cs_new:Npn \@@_ep_isqrt:wwn #1,#2; + { + \@@_ep_to_ep:wwN #1,#2; + \@@_ep_isqrt_auxi:wwn + } +\cs_new:Npn \@@_ep_isqrt_auxi:wwn #1, + { + \exp_after:wN \@@_ep_isqrt_auxii:wwnnnwn + \int_use:N \__int_eval:w + \int_if_odd:nTF {#1} + { (\c_one - #1) / \c_two , 535 , { 0 } { } } + { \c_one - #1 / \c_two , 168 , { } { 0 } } + } +\cs_new:Npn \@@_ep_isqrt_auxii:wwnnnwn #1, #2, #3#4 #5#6; #7 + { + \@@_ep_isqrt_esti:wwwnnwn #2, 0, #5, {#3} {#4} + {#5} #6 ; { #7 #1 , } + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{macro}[aux, EXP] +% { +% \@@_ep_isqrt_esti:wwwnnwn, +% \@@_ep_isqrt_estii:wwwnnwn, +% \@@_ep_isqrt_estiii:NNNNNwwwn +% } +% If the last two approximations gave the same result, we are done: +% call the \texttt{estii} function to clean up. Otherwise, evaluate +% $(\meta{prev} + 1.005 \cdot 10^{\text{$8$ or $9$}} / (\meta{prev} +% \cdot x)) / 2$, as the next approximation: omitting the $1.005$ +% factor, this would be Newton's method. We can check by brute force +% that if |#4| is empty (the original exponent was even), the process +% computes an integer slightly larger than $100 / \sqrt{x}$, while if +% |#4| is~$0$ (the original exponent was odd), the result is an +% integer slightly larger than $100 / \sqrt{x/10}$. Once we are done, +% we evaluate $100 r^2 / 2$ or $10 r^2 / 2$ (when the exponent is even +% or odd, respectively) and feed that to \texttt{estiii}. This third +% auxiliary finds $y_{\text{even}} / 2 = 10^{-4} r^2 x / 2$ or +% $y_{\text{odd}} / 2 = 10^{-5} r^2 x / 2$ (again, depending on +% earlier parity). A simple program shows that $y\in [1, 1.0201]$. +% The number $y/2$ is fed to \cs{@@_ep_isqrt_epsi:wN}, which computes +% $1/\sqrt{y}$, and we finally multiply the result by~$r$. +% \begin{macrocode} +\cs_new:Npn \@@_ep_isqrt_esti:wwwnnwn #1, #2, #3, #4 + { + \if_int_compare:w #1 = #2 \exp_stop_f: + \exp_after:wN \@@_ep_isqrt_estii:wwwnnwn + \fi: + \exp_after:wN \@@_ep_isqrt_esti:wwwnnwn + \int_use:N \__int_eval:w + (#1 + 1 0050 0000 #4 / (#1 * #3)) / \c_two , + #1, #3, {#4} + } +\cs_new:Npn \@@_ep_isqrt_estii:wwwnnwn #1, #2, #3, #4#5 + { + \exp_after:wN \@@_ep_isqrt_estiii:NNNNNwwwn + \int_use:N \__int_eval:w 1000 0000 + #2 * #2 #5 * \c_five + \exp_after:wN , \int_use:N \__int_eval:w 10000 + #2 ; + } +\cs_new:Npn \@@_ep_isqrt_estiii:NNNNNwwwn 1#1#2#3#4#5#6, 1#7#8; #9; + { + \@@_fixed_mul_short:wwn #9; {#1} {#2#3#4#5} {#600} ; + \@@_ep_isqrt_epsi:wN + \@@_fixed_mul_short:wwn {#7} {#80} {0000} ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[aux, EXP]{\@@_ep_isqrt_epsi:wN, \@@_ep_isqrt_epsii:wwN} +% Here, we receive a fixed point number $y/2$ with $y\in[1,1.0201]$. +% Starting from $z = 1$ we iterate $z \mapsto z(3/2 - z^2 y/2)$. In +% fact, we start from the first iteration $z=3/2-y/2$ to avoid useless +% multiplications. The \texttt{epsii} auxiliary receives $z$ as~|#1| +% and $y$ as~|#2|. +% \begin{macrocode} +\cs_new:Npn \@@_ep_isqrt_epsi:wN #1; + { + \@@_fixed_sub:wwn {15000}{0000}{0000}{0000}{0000}{0000}; #1; + \@@_ep_isqrt_epsii:wwN #1; + \@@_ep_isqrt_epsii:wwN #1; + \@@_ep_isqrt_epsii:wwN #1; + } +\cs_new:Npn \@@_ep_isqrt_epsii:wwN #1; #2; + { + \@@_fixed_mul:wwn #1; #1; + \@@_fixed_mul_sub_back:wwwn #2; + {15000}{0000}{0000}{0000}{0000}{0000}; + \@@_fixed_mul:wwn #1; + } +% \end{macrocode} +% \end{macro} +% % \subsection{Converting from fixed point to floating point} +% ^^A todo: doc and turn ..._to_float:... -> ..._to_float_o:... +% +% After computing Taylor series, we wish to convert the result from +% extended precision (with or without an exponent) to the public +% floating point format. The functions here should be called within an +% integer expression for the overall exponent of the floating point. % +% \begin{macro}[int, rEXP]{\@@_ep_to_float:wwN, \@@_ep_inv_to_float:wwN} +% An extended-precision number is simply a comma-delimited exponent +% followed by a fixed point number. Leave the exponent in the current +% integer expression then convert the fixed point number. +% \begin{macrocode} +\cs_new:Npn \@@_ep_to_float:wwN #1, + { + \__int_eval:w #1 \@@_fixed_to_float:wN } +\cs_new:Npn \@@_ep_inv_to_float:wwN #1,#2; + { + \@@_ep_div:wwwwn 1,{1000}{0000}{0000}{0000}{0000}{0000}; #1,#2; + \@@_ep_to_float:wwN + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[rEXP, int]{\@@_fixed_inv_to_float:wN} +% Another function which reduces to converting an extended precision +% number to a float. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_inv_to_float:wN + { \@@_ep_inv_to_float:wwN 0, } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[rEXP, int]{\@@_fixed_to_float_rad:wN} +% Converts the fixed point number~|#1| from degrees to radians then to +% a floating point number. This could perhaps remain in +% \pkg{l3fp-trig}. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_to_float_rad:wN #1; + { + \@@_fixed_mul:wwn #1; {5729}{5779}{5130}{8232}{0876}{7981}; + { \@@_ep_to_float:wwN 2, } + } +% \end{macrocode} +% \end{macro} +% +% ^^A todo: make exponents end in ',' consistently throughout l3fp % \begin{macro}[int, rEXP] % {\@@_fixed_to_float:wN, \@@_fixed_to_float:Nw} % \begin{syntax} @@ -586,7 +1226,7 @@ \cs_new:Npn \@@_fixed_to_float:Nw #1#2; { \@@_fixed_to_float:wN #2; #1 } \cs_new:Npn \@@_fixed_to_float:wN #1#2#3#4#5#6; #7 { - + \c_four % for the 8-digit-at-the-start thing. + + \__int_eval:w \c_four % for the 8-digit-at-the-start thing. \exp_after:wN \exp_after:wN \exp_after:wN \@@_fixed_to_loop:N \exp_after:wN \use_none:n @@ -642,206 +1282,6 @@ % \end{macrocode} % \end{macro} % -% \begin{macro}[rEXP, int]{\@@_fixed_inv_to_float:wN, \@@_fixed_div_to_float:ww} -% Starting from \texttt{fixed_dtf} $A$ |;| $B$ |;| we want to compute -% $A/B$, and express it as a floating point number. Normalize both -% numbers by removing leading brace groups of zeros and leaving the -% appropriate exponent shift in the input stream. -% \begin{macrocode} -\cs_new:Npn \@@_fixed_inv_to_float:wN #1#2; #3 - { - + \__int_eval:w % ^^A todo: remove the +? - \if_int_compare:w #1 < \c_one_thousand - \@@_fixed_dtf_zeros:wNnnnnnn - \fi: - \@@_fixed_dtf_no_zero:Nwn + {#1} #2 \s_@@ - \@@_fixed_dtf_approx:n - {10000} {0000} {0000} {0000} {0000} {0000} ; - } -\cs_new:Npn \@@_fixed_div_to_float:ww #1#2; #3#4; - { - \if_int_compare:w #1 < \c_one_thousand - \@@_fixed_dtf_zeros:wNnnnnnn - \fi: - \@@_fixed_dtf_no_zero:Nwn - {#1} #2 \s_@@ - { - \if_int_compare:w #3 < \c_one_thousand - \@@_fixed_dtf_zeros:wNnnnnnn - \fi: - \@@_fixed_dtf_no_zero:Nwn + {#3} #4 \s_@@ - \@@_fixed_dtf_approx:n - } - } -\cs_new:Npn \@@_fixed_dtf_no_zero:Nwn #1#2 \s_@@ #3 { #3 #2; } -\cs_new:Npn \@@_fixed_dtf_zeros:wNnnnnnn - \fi: \@@_fixed_dtf_no_zero:Nwn #1#2#3#4#5#6#7 - { - \fi: - #1 \c_minus_one - \exp_after:wN \use_i_ii:nnn - \exp_after:wN \@@_fixed_dtf_zeros:NN - \exp_after:wN #1 - \int_use:N \__int_eval:w 10 0000 + #2 \__int_eval_end: #3#4#5#6#7 - ; 1 ; - } -\cs_new:Npn \@@_fixed_dtf_zeros:NN #1#2 - { - \if_meaning:w 0 #2 - #1 \c_one - \else: - \@@_fixed_dtf_zeros_end:wNww #2 - \fi: - \@@_fixed_dtf_zeros:NN #1 - } -\cs_new:Npn \@@_fixed_dtf_zeros_end:wNww - #1 \fi: \@@_fixed_dtf_zeros:NN #2 #3; #4 \s_@@ - { - \fi: - \if_meaning:w ; #1 - #2 \c_two * \c_@@_max_exponent_int - \use_i_ii:nnn - \fi: - \@@_fixed_dtf_zeros_auxi:ww - #1#3 0000 0000 0000 0000 0000 0000 ; - } -\cs_new:Npn \@@_fixed_dtf_zeros_auxi:ww - { - \@@_pack_twice_four:wNNNNNNNN - \@@_pack_twice_four:wNNNNNNNN - \@@_pack_twice_four:wNNNNNNNN - \@@_fixed_dtf_zeros_auxii:ww - ; - } -\cs_new:Npn \@@_fixed_dtf_zeros_auxii:ww #1; #2; #3 { #3 #1; } -% \end{macrocode} -% \newcommand{\eTeXfrac}[2]{\left[\frac{#1}{#2}\right]} -% We get -% \begin{quote} -% \cs{@@_fixed_dtf_approx:n} \meta{B'} |;| \meta{A'} |;| -% \end{quote} -% where \meta{B'} and \meta{A'} are each $6$ brace groups, -% representing fixed point numbers in the range $[0.1,1)$. Denote by -% $x\in[1000,9999]$ and $y\in[0,9999]$ the first two groups of -% \meta{B'}. We first find an estimate $a$ for the inverse of $B'$ by -% computing -% \begin{align*} -% \alpha &= \eTeXfrac{10^{9}}{x+1} \\ -% \beta &= \eTeXfrac{10^{9}}{x} \\ -% a &= 10^{3} \alpha + (\beta-\alpha) \cdot -% \left(10^{3}-\eTeXfrac{y}{10}\right) - 1750, -% \end{align*} -% where $\eTeXfrac{\bullet}{\bullet}$ denotes \eTeX{}'s rounding -% division. The idea is to interpolate between $\alpha$ and $\beta$ -% with a parameter $y/10^{4}$. The shift by $1750$ helps to ensure -% that $a$ is an underestimate of the correct value. We will prove -% that -% \[ -% 1 - 2.255\cdot 10^{-5} < \frac{B'a}{10^{8}} < 1 . -% \] -% We can then compute the inverse $B'a/10^{8}$ using $1/(1-\epsilon) -% \simeq (1+\epsilon)(1+\epsilon^{2})$, which is correct up to a -% relative error of $\epsilon^4 < 2.6\cdot 10^{-19}$. Since we target -% a $16$-digit value, this is small enough. -% -% Let us prove the upper bound first. -% \begin{align}\label{l3fp-fixed-eTeXfrac} -% 10^{7} B'a -% & < \left(10^{3} x + \eTeXfrac{y}{10} + \frac{3}{2}\right) -% \left(\left(10^{3}-\eTeXfrac{y}{10}\right) \beta -% + \eTeXfrac{y}{10} \alpha - 1750\right) -% \\& < \left(10^{3} x + \eTeXfrac{y}{10} + \frac{3}{2}\right) -% \left(\left(10^{3}-\eTeXfrac{y}{10}\right) -% \left(\frac{10^{9}}{x} + \frac{1}{2} \right) -% + \eTeXfrac{y}{10} \left(\frac{10^{9}}{x+1} + \frac{1}{2} \right) -% - 1750\right) -% \\& < \left(10^{3} x + \eTeXfrac{y}{10} + \frac{3}{2}\right) -% \left(\frac{10^{12}}{x} -% - \eTeXfrac{y}{10} \frac{10^{9}}{x(x+1)} -% - 1250\right) -% \end{align} -% We recognize a quadratic polynomial in $[y/10]$ with a negative -% leading coefficient, $([y/10]+a)(b-c[y/10]) \leq (b+ca)^2/(4c)$. -% Hence, -% \[ -% 10^{7} B'a -% < \frac{10^{15}}{x(x+1)} \left( -% x + \frac{1}{2} + \frac{3}{4} 10^{-3} -% - 6.25\cdot 10^{-10} x(x+1) \right)^2 -% \] -% We want to prove that the squared expression is less than $x(x+1)$, -% which we do by simplifying the difference, and checking its sign, -% \[ -% x(x+1) - \left(x + \frac{1}{2} + \frac{3}{4} 10^{-3} -% - 6.25\cdot 10^{-10} x(x+1) \right)^2 -% > - \frac{1}{4} (1+1.5\cdot 10^{-3})^2 - 10^{-3} x -% + 1.25\cdot 10^{-9} x(x+1)(x+0.5) -% > 0. -% \] -% -% Now, the lower bound. The same computation as -% \eqref{l3fp-fixed-eTeXfrac} imply -% \[ -% 10^{7} B'a -% > \left(10^{3} x + \eTeXfrac{y}{10} - \frac{1}{2}\right) -% \left(\frac{10^{12}}{x} - \eTeXfrac{y}{10} \frac{10^{9}}{x(x+1)} -% - 2250\right) -% \] -% This time, we want to find the minimum of this quadratic polynomial. -% Since the leading coefficient is still negative, the minimum is -% reached for one of the extreme values $y=0$ or $y=9999$, and we -% easily check the bound for those values. -% -% We have proven that the algorithm will give us a precise enough -% answer. Incidentally, the upper bound that we derived tells us that -% $a < 10^{8}/B \leq 10^{9}$, hence we can compute $a$ safely as a -% \TeX{} integer, and even add $10^{9}$ to it to ease grabbing of all -% the digits. -% \begin{macrocode} -\cs_new:Npn \@@_fixed_dtf_approx:n #1 - { - \exp_after:wN \@@_fixed_dtf_approx:wnn - \int_use:N \__int_eval:w 10 0000 0000 / ( #1 + \c_one ) ; - {#1} - } -\cs_new:Npn \@@_fixed_dtf_approx:wnn #1; #2#3 - { -%<assert> \assert:n { \tl_count:n {#1} = 6 } - \exp_after:wN \@@_fixed_dtf_approx:NNNNNw - \int_use:N \__int_eval:w 10 0000 0000 - 1750 - + #1000 + (10 0000 0000/#2-#1) * (1000-#3/10) ; - {#2}{#3} - } -\cs_new:Npn \@@_fixed_dtf_approx:NNNNNw 1#1#2#3#4#5#6; #7; #8; - { - + \c_four % because of the line below "dtf_epsilon" here. - \@@_fixed_mul:wwn {000#1}{#2#3#4#5}{#6}{0000}{0000}{0000} ; #7; - \@@_fixed_dtf_epsilon:wN - \@@_fixed_mul:wwn {000#1}{#2#3#4#5}{#6}{0000}{0000}{0000} ; - \@@_fixed_mul:wwn #8; - \@@_fixed_to_float:wN ? - } -\cs_new:Npn \@@_fixed_dtf_epsilon:wN #1#2#3#4#5#6; - { -%<assert> \assert:n { #1 = 0000 } -%<assert> \assert:n { #2 = 9999 } - \exp_after:wN \@@_fixed_dtf_epsilon:NNNNNww - \int_use:N \__int_eval:w 1 9999 9998 - #3#4 + - \exp_after:wN \@@_fixed_dtf_epsilon_pack:NNNNNw - \int_use:N \__int_eval:w 2 0000 0000 - #5#6 ; {0000} ; - } -\cs_new:Npn \@@_fixed_dtf_epsilon_pack:NNNNNw #1#2#3#4#5#6; - { #1 ; {#2#3#4#5} {#6} } -\cs_new:Npn \@@_fixed_dtf_epsilon:NNNNNww #1#2#3#4#5#6; #7; - { - \@@_fixed_mul:wwn %^^A todo: optimize to use \@@_mul_significand. - {0000} {#2#3#4#5} {#6} #7 ; - {0000} {#2#3#4#5} {#6} #7 ; - \@@_fixed_add_one:wN - \@@_fixed_mul:wwn {10000} {#2#3#4#5} {#6} #7 ; - } -% \end{macrocode} -% \end{macro} -% % \begin{macrocode} %</initex|package> % \end{macrocode} |