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diff --git a/macros/latex/required/l3kernel/l3fp-extended.dtx b/macros/latex/required/l3kernel/l3fp-extended.dtx new file mode 100644 index 0000000000..356c32b232 --- /dev/null +++ b/macros/latex/required/l3kernel/l3fp-extended.dtx @@ -0,0 +1,1275 @@ +% \iffalse meta-comment +% +%% File: l3fp-extended.dtx +% +% Copyright (C) 2011-2024 The LaTeX Project +% +% It may be distributed and/or modified under the conditions of the +% LaTeX Project Public License (LPPL), either version 1.3c of this +% license or (at your option) any later version. The latest version +% of this license is in the file +% +% https://www.latex-project.org/lppl.txt +% +% This file is part of the "l3kernel bundle" (The Work in LPPL) +% and all files in that bundle must be distributed together. +% +% ----------------------------------------------------------------------- +% +% The development version of the bundle can be found at +% +% https://github.com/latex3/latex3 +% +% for those people who are interested. +% +%<*driver> +\documentclass[full,kernel]{l3doc} +\begin{document} + \DocInput{\jobname.dtx} +\end{document} +%</driver> +% \fi +% +% \title{^^A +% The \pkg{l3fp-extended} module\\ +% Manipulating numbers with extended precision, for internal use^^A +% } +% \author{^^A +% The \LaTeX{} Project\thanks +% {^^A +% E-mail: +% \href{mailto:latex-team@latex-project.org} +% {latex-team@latex-project.org}^^A +% }^^A +% } +% \date{Released 2024-04-11} +% +% \maketitle +% +% \begin{documentation} +% +% \end{documentation} +% +% \begin{implementation} +% +% \section{\pkg{l3fp-extended} implementation} +% +% \begin{macrocode} +%<*package> +% \end{macrocode} +% +% \begin{macrocode} +%<@@=fp> +% \end{macrocode} +% +% \subsection{Description of fixed point numbers} +% +% 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 +% the full $24$-digit precision. In other words, the last two blocks of +% each fixed point number may be wrong as long as the error is small +% enough to be rounded away when converting back to a floating point +% number. The fixed point numbers are expressed as +% \begin{quote} +% \Arg{a_1} \Arg{a_2} \Arg{a_3} \Arg{a_4} \Arg{a_5} \Arg{a_6} |;| +% \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 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 +% number $a$ corresponding to the representation above is $a = +% \sum_{i=1}^{6} \meta{a_i} \cdot 10^{-4i}$. +% +% Most functions we define here have the form +% \begin{syntax} +% \cs{@@_fixed_\meta{calculation}:wwn} \meta{operand_1} |;| \meta{operand_2} |;| \Arg{continuation} +% \end{syntax} +% They perform the \meta{calculation} on the two \meta{operands}, then +% feed the result ($6$ brace groups followed by a semicolon) to the +% \meta{continuation}, responsible for the next step of the calculation. +% Some functions only accept an \texttt{N}-type \meta{continuation}. +% This allows constructions such as +% \begin{quote} +% \cs{@@_fixed_add:wwn} \meta{X_1} |;| \meta{X_2} |;| \\ +% \cs{@@_fixed_mul:wwn} \meta{X_3} |;| \\ +% \cs{@@_fixed_add:wwn} \meta{X_4} |;| \\ +% \end{quote} +% to compute $(X_1+X_2)\cdot X_3 + X_4$. This turns out to be very +% 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_o: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 numbers with extended precision} +% +% \begin{variable}{\c_@@_one_fixed_tl} +% 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} ; } +% \end{macrocode} +% \end{variable} +% +% \begin{macro}[EXP]{\@@_fixed_continue:wn} +% This function simply calls the next function. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_continue:wn #1; #2 { #2 #1; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fixed_add_one:wN} +% \begin{syntax} +% \cs{@@_fixed_add_one:wN} \meta{a} |;| \meta{continuation} +% \end{syntax} +% This function adds $1$ to the fixed point \meta{a}, by changing +% $a_1$ to $10000+a_1$, then calls the \meta{continuation}. This +% requires $a_1 + 10000 < 2^{31}$. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_add_one:wN #1#2; #3 + { + \exp_after:wN #3 \exp_after:wN + { \int_value:w \@@_int_eval:w \c_@@_myriad_int + #1 } #2 ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[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; + { + \exp_after:wN \@@_fixed_mul_after:wwn + \int_value:w \@@_int_eval:w \c_@@_leading_shift_int + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_trailing_shift_int + + #1 ; {#2}{#3}{#4}{#5}; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fixed_mul_after:wwn} +% The fixed point operations which involve multiplication end by +% calling this auxiliary. It braces the last block of digits, and +% places the \meta{continuation} |#3| in front. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_after:wwn #1; #2; #3 { #3 {#1} #2; } +% \end{macrocode} +% \end{macro} +% +% \subsection{Multiplying a fixed point number by a short one} +% +% \begin{macro}[EXP]{\@@_fixed_mul_short:wwn} +% \begin{syntax}\parskip=0pt\obeylines +% \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~$0$: for instance +% a second operand of |{0001}{0000}{0000}| leaves 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:wwn + \int_value:w \@@_int_eval:w \c_@@_leading_shift_int + + #1*#7 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#8 + #2*#7 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#9 + #2*#8 + #3*#7 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #2*#9 + #3*#8 + #4*#7 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #3*#9 + #4*#8 + #5*#7 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_trailing_shift_int + + #4*#9 + #5*#8 + #6*#7 + + ( #5*#9 + #6*#8 + #6*#9 / \c_@@_myriad_int ) + / \c_@@_myriad_int ; ; + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Dividing a fixed point number by a small integer} +% +% \begin{macro}[EXP]{\@@_fixed_div_int:wwN} +% \begin{macro}[EXP] +% { +% \@@_fixed_div_int:wnN, \@@_fixed_div_int_auxi:wnn, +% \@@_fixed_div_int_auxii:wnn, \@@_fixed_div_int_pack:Nw, +% \@@_fixed_div_int_after:Nw +% } +% \begin{syntax} +% \cs{@@_fixed_div_int:wwN} \meta{a} |;| \meta{n} |;| \meta{continuation} +% \end{syntax} +% Divides the fixed point number \meta{a} by the (small) integer +% $0<\meta{n}<10^4$ and feeds the result to the \meta{continuation}. +% There is no bound on $a_1$. +% +% The arguments of the \texttt{i} auxiliary are 1: one of the $a_{i}$, +% 2: $n$, 3: the \texttt{ii} or the \texttt{iii} auxiliary. It +% computes a (somewhat tight) lower bound $Q_{i}$ for the ratio +% $a_{i}/n$. +% +% The \texttt{ii} auxiliary receives $Q_{i}$, $n$, and $a_{i}$ as +% arguments. It adds $Q_{i}$ to a surrounding integer expression, and +% starts a new one with the initial value $9999$, which ensures that +% the result of this expression has $5$ digits. The auxiliary +% also computes $a_{i}-n\cdot Q_{i}$, placing the result in front of +% the $4$ digits of $a_{i+1}$. The resulting $a'_{i+1} = 10^{4} +% (a_{i} - n \cdot Q_{i}) + a_{i+1}$ serves as the first argument for +% a new call to the \texttt{i} auxiliary. +% +% When the \texttt{iii} auxiliary is called, the situation looks like +% this: +% \begin{quote} +% \cs{@@_fixed_div_int_after:Nw} \meta{continuation} \\ +% $-1 + Q_{1}$ \\ +% \cs{@@_fixed_div_int_pack:Nw} $9999 + Q_{2}$ \\ +% \cs{@@_fixed_div_int_pack:Nw} $9999 + Q_{3}$ \\ +% \cs{@@_fixed_div_int_pack:Nw} $9999 + Q_{4}$ \\ +% \cs{@@_fixed_div_int_pack:Nw} $9999 + Q_{5}$ \\ +% \cs{@@_fixed_div_int_pack:Nw} $9999$ \\ +% \cs{@@_fixed_div_int_auxii:wnn} $Q_{6}$ |;| \Arg{n} \Arg{a_{6}} +% \end{quote} +% where expansion is happening from the last line up. The +% \texttt{iii} auxiliary adds $Q_{6} + 2 \simeq a_{6}/n + 1$ to the +% last $9999$, giving the integer closest to $10000 + a_{6}/n$. +% +% Each \texttt{pack} auxiliary receives $5$ digits followed by a +% semicolon. The first digit is added as a carry to the integer +% expression above, and the $4$ other digits are braced. Each call to +% the \texttt{pack} auxiliary thus produces one brace group. The last +% brace group is produced by the \texttt{after} auxiliary, which +% places the \meta{continuation} as appropriate. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_div_int:wwN #1#2#3#4#5#6 ; #7 ; #8 + { + \exp_after:wN \@@_fixed_div_int_after:Nw + \exp_after:wN #8 + \int_value:w \@@_int_eval:w - 1 + \@@_fixed_div_int:wnN + #1; {#7} \@@_fixed_div_int_auxi:wnn + #2; {#7} \@@_fixed_div_int_auxi:wnn + #3; {#7} \@@_fixed_div_int_auxi:wnn + #4; {#7} \@@_fixed_div_int_auxi:wnn + #5; {#7} \@@_fixed_div_int_auxi:wnn + #6; {#7} \@@_fixed_div_int_auxii:wnn ; + } +\cs_new:Npn \@@_fixed_div_int:wnN #1; #2 #3 + { + \exp_after:wN #3 + \int_value:w \@@_int_eval:w #1 / #2 - 1 ; + {#2} + {#1} + } +\cs_new:Npn \@@_fixed_div_int_auxi:wnn #1; #2 #3 + { + + #1 + \exp_after:wN \@@_fixed_div_int_pack:Nw + \int_value:w \@@_int_eval:w 9999 + \exp_after:wN \@@_fixed_div_int:wnN + \int_value:w \@@_int_eval:w #3 - #1*#2 \@@_int_eval_end: + } +\cs_new:Npn \@@_fixed_div_int_auxii:wnn #1; #2 #3 { + #1 + 2 ; } +\cs_new:Npn \@@_fixed_div_int_pack:Nw #1 #2; { + #1; {#2} } +\cs_new:Npn \@@_fixed_div_int_after:Nw #1 #2; { #1 {#2} } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Adding and subtracting fixed points} +% +% \begin{macro}[EXP]{\@@_fixed_add:wwn, \@@_fixed_sub:wwn} +% \begin{macro}[EXP] +% { +% \@@_fixed_add:Nnnnnwnn, +% \@@_fixed_add:nnNnnnwn, +% \@@_fixed_add_pack:NNNNNwn, +% \@@_fixed_add_after:NNNNNwn +% } +% \begin{syntax} +% \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 $0\leq a_{1},b_{1}\leq +% 114748$, 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 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 +% various level, we go back up, packing digits and bringing the +% \meta{continuation} (|#8|, then |#7|) from the end of the argument +% list to its start. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_add:wwn { \@@_fixed_add:Nnnnnwnn + } +\cs_new:Npn \@@_fixed_sub:wwn { \@@_fixed_add:Nnnnnwnn - } +\cs_new:Npn \@@_fixed_add:Nnnnnwnn #1 #2#3#4#5 #6; #7#8 + { + \exp_after:wN \@@_fixed_add_after:NNNNNwn + \int_value:w \@@_int_eval:w 9 9999 9998 + #2#3 #1 #7#8 + \exp_after:wN \@@_fixed_add_pack:NNNNNwn + \int_value:w \@@_int_eval:w 1 9999 9998 + #4#5 + \@@_fixed_add:nnNnnnwn #6 #1 + } +\cs_new:Npn \@@_fixed_add:nnNnnnwn #1#2 #3 #4#5 #6#7 ; #8 + { + #3 #4#5 + \exp_after:wN \@@_fixed_add_pack:NNNNNwn + \int_value:w \@@_int_eval:w 2 0000 0000 #3 #6#7 + #1#2 ; {#8} ; + } +\cs_new:Npn \@@_fixed_add_pack:NNNNNwn #1 #2#3#4#5 #6; #7 + { + #1 ; {#7} {#2#3#4#5} {#6} } +\cs_new:Npn \@@_fixed_add_after:NNNNNwn 1 #1 #2#3#4#5 #6; #7 + { #7 {#1#2#3#4#5} {#6} } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Multiplying fixed points} +% +% ^^A todo: may a_1 or b_1 be = 10000? Used in ediv_epsi later. +% \begin{macro}[EXP]{\@@_fixed_mul:wwn} +% \begin{macro}[EXP]{\@@_fixed_mul:nnnnnnnw} +% \begin{syntax} +% \cs{@@_fixed_mul:wwn} \meta{a} |;| \meta{b} |;| \Arg{continuation} +% \end{syntax} +% Computes $a\times b$ and feeds the result to \meta{continuation}. +% This function requires $0\leq a_{1}, b_{1} < 10000$. Once more, we +% need to play around the limit of $9$ arguments for \TeX{} macros. +% Note that we don't need to obtain an exact rounding, contrarily to +% the |*| operator, so things could be harder. We wish to perform +% carries in +% \begin{align*} +% a \times b = +% & a_{1} \cdot b_{1} \cdot 10^{-8} \\ +% & + (a_{1} \cdot b_{2} + a_{2} \cdot b_{1}) \cdot 10^{-12} \\ +% & + (a_{1} \cdot b_{3} + a_{2} \cdot b_{2} +% + a_{3} \cdot b_{1}) \cdot 10^{-16} \\ +% & + (a_{1} \cdot b_{4} + a_{2} \cdot b_{3} +% + a_{3} \cdot b_{2} + a_{4} \cdot b_{1}) \cdot 10^{-20} \\ +% & + \Bigl(a_{2} \cdot b_{4} + a_{3} \cdot b_{3} + a_{4} \cdot b_{2} +% \\ & \qquad +% + \frac{a_{3} \cdot b_{4} + a_{4} \cdot b_{3} +% + a_{1} \cdot b_{6} + a_{2} \cdot b_{5} +% + a_{5} \cdot b_{2} + a_{6} \cdot b_{1}}{10^{4}} +% \\ & \qquad +% + a_{1} \cdot b_{5} + a_{5} \cdot b_{1}\Bigr) \cdot 10^{-24} +% + O(10^{-24}), +% \end{align*} +% where the $O(10^{-24})$ stands for terms which are at most $5\cdot +% 10^{-24}$; ignoring those leads to an error of at most +% $5$~\texttt{ulp}. Note how the first $15$~terms only depend on +% $a_{1},\ldots{},a_{4}$ and $b_{1},\ldots,b_{4}$, while the last +% $6$~terms only depend on $a_{1},a_{2},a_{5},a_{6}$, and the +% corresponding parts of~$b$. Hence, the first function grabs +% $a_{1},\ldots,a_{4}$, the rest of $a$, and $b_{1},\ldots,b_{4}$, and +% writes the $15$ first terms of the expression, including a left +% parenthesis for the fraction. The \texttt{i} auxiliary receives +% $a_{5}$, $a_{6}$, $b_{1}$, $b_{2}$, $a_{1}$, $a_{2}$, $b_{5}$, +% $b_{6}$ and finally the \meta{continuation} as arguments. It writes +% the end of the expression, including the right parenthesis and the +% denominator of the fraction. The \meta{continuation} +% is finally placed in front of the $6$ brace groups by +% \cs{@@_fixed_mul_after:wwn}. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul:wwn #1#2#3#4 #5; #6#7#8#9 + { + \exp_after:wN \@@_fixed_mul_after:wwn + \int_value:w \@@_int_eval:w \c_@@_leading_shift_int + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#6 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#7 + #2*#6 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#8 + #2*#7 + #3*#6 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + + #1*#9 + #2*#8 + #3*#7 + #4*#6 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_trailing_shift_int + + #2*#9 + #3*#8 + #4*#7 + + ( #3*#9 + #4*#8 + + \@@_fixed_mul:nnnnnnnw #5 {#6}{#7} {#1}{#2} + } +\cs_new:Npn \@@_fixed_mul:nnnnnnnw #1#2 #3#4 #5#6 #7#8 ; + { + #1*#4 + #2*#3 + #5*#8 + #6*#7 ) / \c_@@_myriad_int + + #1*#3 + #5*#7 ; ; + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Combining product and sum of fixed points} +% +% \begin{macro}[EXP] +% { +% \@@_fixed_mul_add:wwwn, +% \@@_fixed_mul_sub_back:wwwn, +% \@@_fixed_mul_one_minus_mul:wwn, +% } +% \begin{syntax} +% \cs{@@_fixed_mul_add:wwwn} \meta{a} |;| \meta{b} |;| \meta{c} |;| \Arg{continuation} +% \cs{@@_fixed_mul_sub_back:wwwn} \meta{a} |;| \meta{b} |;| \meta{c} |;| \Arg{continuation} +% \cs{@@_fixed_one_minus_mul:wwn} \meta{a} |;| \meta{b} |;| \Arg{continuation} +% \end{syntax} +% Sometimes called |FMA| (fused multiply-add), these functions +% compute $a\times b + c$, $c - a\times b$, and $1 - a\times b$ and +% feed the result to the \meta{continuation}. Those functions require +% $0\leq a_{1}, b_{1}, c_{1} \leq 10000$. Since those functions are +% at the heart of the computation of Taylor expansions, we +% over-optimize them a bit, and in particular we do not factor out the +% common parts of the three functions. +% +% For definiteness, consider the task of computing $a\times b + c$. +% We perform carries in +% \begin{align*} +% a \times b + c = +% & (a_{1} \cdot b_{1} + c_{1} c_{2})\cdot 10^{-8} \\ +% & + (a_{1} \cdot b_{2} + a_{2} \cdot b_{1}) \cdot 10^{-12} \\ +% & + (a_{1} \cdot b_{3} + a_{2} \cdot b_{2} + a_{3} \cdot b_{1} +% + c_{3} c_{4}) \cdot 10^{-16} \\ +% & + (a_{1} \cdot b_{4} + a_{2} \cdot b_{3} + a_{3} \cdot b_{2} +% + a_{4} \cdot b_{1}) \cdot 10^{-20} \\ +% & + \Big(a_{2} \cdot b_{4} + a_{3} \cdot b_{3} + a_{4} \cdot b_{2} +% \\ & \qquad +% + \frac{a_{3} \cdot b_{4} + a_{4} \cdot b_{3} +% + a_{1} \cdot b_{6} + a_{2} \cdot b_{5} +% + a_{5} \cdot b_{2} + a_{6} \cdot b_{1}}{10^{4}} +% \\ & \qquad +% + a_{1} \cdot b_{5} + a_{5} \cdot b_{1} +% + c_{5} c_{6} \Big) \cdot 10^{-24} +% + O(10^{-24}), +% \end{align*} +% where $c_{1} c_{2}$, $c_{3} c_{4}$, $c_{5} c_{6}$ denote the +% $8$-digit number obtained by juxtaposing the two blocks of digits of +% $c$, and $\cdot$ denotes multiplication. The task is obviously +% tough because we have $18$ brace groups in front of us. +% +% Each of the three function starts the first two levels (the first, +% corresponding to $10^{-4}$, is empty), with $c_{1} c_{2}$ in the +% first level, calls the \texttt{i} auxiliary with arguments described +% later, and adds a trailing ${} + c_{5}c_{6}$ |;| +% \Arg{continuation}~|;|. The ${} + c_{5}c_{6}$ piece, which is +% omitted for \cs{@@_fixed_one_minus_mul:wwn}, is taken in the +% integer expression for the $10^{-24}$ level. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_add:wwwn #1; #2; #3#4#5#6#7#8; + { + \exp_after:wN \@@_fixed_mul_after:wwn + \int_value:w \@@_int_eval:w \c_@@_big_leading_shift_int + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + #3 #4 + \@@_fixed_mul_add:Nwnnnwnnn + + + #5 #6 ; #2 ; #1 ; #2 ; + + + #7 #8 ; ; + } +\cs_new:Npn \@@_fixed_mul_sub_back:wwwn #1; #2; #3#4#5#6#7#8; + { + \exp_after:wN \@@_fixed_mul_after:wwn + \int_value:w \@@_int_eval:w \c_@@_big_leading_shift_int + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + #3 #4 + \@@_fixed_mul_add:Nwnnnwnnn - + + #5 #6 ; #2 ; #1 ; #2 ; - + + #7 #8 ; ; + } +\cs_new:Npn \@@_fixed_one_minus_mul:wwn #1; #2; + { + \exp_after:wN \@@_fixed_mul_after:wwn + \int_value:w \@@_int_eval:w \c_@@_big_leading_shift_int + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + + 1 0000 0000 + \@@_fixed_mul_add:Nwnnnwnnn - + ; #2 ; #1 ; #2 ; - + ; ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fixed_mul_add:Nwnnnwnnn} +% \begin{syntax} +% \cs{@@_fixed_mul_add:Nwnnnwnnn} \meta{op} |+| \meta{c_3} \meta{c_4} |;| +% ~~\meta{b} |;| \meta{a} |;| \meta{b} |;| \meta{op} +% ~~|+| \meta{c_5} \meta{c_6} |;| +% \end{syntax} +% Here, \meta{op} is either |+| or |-|. Arguments |#3|, |#4|, |#5| +% are \meta{b_1}, \meta{b_2}, \meta{b_3}; arguments |#7|, |#8|, |#9| +% are \meta{a_1}, \meta{a_2}, \meta{a_3}. We can build three levels: +% $a_{1} \cdot b_{1}$ for $10^{-8}$, $(a_{1} \cdot b_{2} + a_{2} \cdot +% b_{1})$ for $10^{-12}$, and $(a_{1} \cdot b_{3} + a_{2} \cdot b_{2} +% + a_{3} \cdot b_{1} + c_{3} c_{4})$ for $10^{-16}$. The $a$--$b$ +% products use the sign |#1|. Note that |#2| is empty for +% \cs{@@_fixed_one_minus_mul:wwn}. We call the \texttt{ii} auxiliary +% for levels $10^{-20}$ and $10^{-24}$, keeping the pieces of \meta{a} +% we've read, but not \meta{b}, since there is another copy later in +% the input stream. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_add:Nwnnnwnnn #1 #2; #3#4#5#6; #7#8#9 + { + #1 #7*#3 + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + #1 #7*#4 #1 #8*#3 + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + #1 #7*#5 #1 #8*#4 #1 #9*#3 #2 + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int + #1 \@@_fixed_mul_add:nnnnwnnnn {#7}{#8}{#9} + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fixed_mul_add:nnnnwnnnn} +% \begin{syntax} +% \cs{@@_fixed_mul_add:nnnnwnnnn} \meta{a} |;| \meta{b} |;| \meta{op} +% ~~|+| \meta{c_5} \meta{c_6} |;| +% \end{syntax} +% Level $10^{-20}$ is $(a_{1} \cdot b_{4} + a_{2} \cdot b_{3} + a_{3} +% \cdot b_{2} + a_{4} \cdot b_{1})$, multiplied by the sign, which was +% inserted by the \texttt{i} auxiliary. Then we prepare level +% $10^{-24}$. We don't have access to all parts of \meta{a} and +% \meta{b} needed to make all products. Instead, we prepare the +% partial expressions +% \begin{align*} +% & b_{1} + a_{4} \cdot b_{2} + a_{3} \cdot b_{3} + a_{2} \cdot b_{4} + a_{1} \\ +% & b_{2} + a_{4} \cdot b_{3} + a_{3} \cdot b_{4} + a_{2} . +% \end{align*} +% Obviously, those expressions make no mathematical sense: we +% complete them with $a_{5} \cdot {}$ and ${} \cdot b_{5}$, and with +% $a_{6} \cdot b_{1} + a_{5} \cdot {}$ and ${} \cdot b_{5} + a_{1} +% \cdot b_{6}$, and of course with the trailing ${} + c_{5} c_{6}$. +% To do all this, we keep $a_{1}$, $a_{5}$, $a_{6}$, and the +% corresponding pieces of \meta{b}. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_add:nnnnwnnnn #1#2#3#4#5; #6#7#8#9 + { + ( #1*#9 + #2*#8 + #3*#7 + #4*#6 ) + \exp_after:wN \@@_pack_big:NNNNNNw + \int_value:w \@@_int_eval:w \c_@@_big_trailing_shift_int + \@@_fixed_mul_add:nnnnwnnwN + { #6 + #4*#7 + #3*#8 + #2*#9 + #1 } + { #7 + #4*#8 + #3*#9 + #2 } + {#1} #5; + {#6} + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fixed_mul_add:nnnnwnnwN} +% \begin{syntax} +% \cs{@@_fixed_mul_add:nnnnwnnwN} \Arg{partial_1} \Arg{partial_2} +% ~~\Arg{a_1} \Arg{a_5} \Arg{a_6} |;| \Arg{b_1} \Arg{b_5} \Arg{b_6} |;| +% ~~\meta{op} |+| \meta{c_5} \meta{c_6} |;| +% \end{syntax} +% Complete the \meta{partial_1} and \meta{partial_2} expressions as +% explained for the \texttt{ii} auxiliary. The second one is divided +% by $10000$: this is the carry from level $10^{-28}$. The trailing +% ${} + c_{5} c_{6}$ is taken into the expression for level +% $10^{-24}$. Note that the total of level $10^{-24}$ is in the +% interval $[-5\cdot 10^{8}, 6\cdot 10^{8}$ (give or take a couple of +% $10000$), hence adding it to the shift gives a $10$-digit number, as +% expected by the packing auxiliaries. See \pkg{l3fp-aux} for the +% definition of the shifts and packing auxiliaries. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_mul_add:nnnnwnnwN #1#2 #3#4#5; #6#7#8; #9 + { + #9 (#4* #1 *#7) + #9 (#5*#6+#4* #2 *#7+#3*#8) / \c_@@_myriad_int + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Extended-precision floating point numbers} +% +% In this section we manipulate floating point numbers with roughly $24$ +% significant figures (\enquote{extended-precision} numbers, in short, +% \enquote{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. The +% corresponding value is $0.\meta{digits}\cdot 10^{\meta{exponent}}$. +% This convention differs from floating points. +% +% \begin{macro}[EXP]{\@@_ep_to_fixed:wwn} +% \begin{macro}[EXP] +% {\@@_ep_to_fixed_auxi:www, \@@_ep_to_fixed_auxii:nnnnnnnwn} +% Converts an extended-precision number with an exponent at most~$4$ +% and a first block less than $10^{8}$ to a fixed point number whose +% first block has $12$~digits, hopefully starting with many zeros. +% \begin{macrocode} +\cs_new:Npn \@@_ep_to_fixed:wwn #1,#2 + { + \exp_after:wN \@@_ep_to_fixed_auxi:www + \int_value:w \@@_int_eval:w 1 0000 0000 + #2 \exp_after:wN ; + \exp:w \exp_end_continue_f:w + \prg_replicate:nn { 4 - \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. +% \begin{macro}[EXP]{\@@_ep_to_ep:wwN} +% \begin{macro}[rEXP]{\@@_ep_to_ep_loop:N, \@@_ep_to_ep_end:www} +% \begin{macro}[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_value:w \@@_int_eval:w #1 + 4 + \exp_after:wN \use_i:nn + \exp_after:wN \@@_ep_to_ep_loop:N + \int_value:w \@@_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 + - 1 + \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 + - 2 * \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}[EXP]{\@@_ep_compare:wwww} +% \begin{macro}[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}[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_value:w \@@_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 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 shall 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 $\eTeXfrac{\meta{d_2}}{10}$ underestimates +% $10^{-1}(\meta{d_2} + 1)$ by $0.5$ at most, 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) +% \\ & \qquad +% \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 gives 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}[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}[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 later 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_value:w \@@_int_eval:w 10 0000 0000 / ( #2 + 1 ) + \exp_after:wN ; + \int_value:w \@@_int_eval:w #4 - #1 + 1 , + {#2} #3; + } +\cs_new:Npn \@@_ep_div_estii:wwnnwwn #1; #2,#3#4#5; #6; #7 + { + \exp_after:wN \@@_ep_div_estiii:NNNNNwwwn + \int_value:w \@@_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}[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_value:w \@@_int_eval:w 1 9998 - #2 + \exp_after:wN \@@_ep_div_eps_pack:NNNNNw + \int_value:w \@@_int_eval:w 1 9999 9998 - #3#4 + \exp_after:wN \@@_ep_div_eps_pack:NNNNNw + \int_value:w \@@_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 replace $10^{8}$ by a +% slightly larger number which ensures 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}[EXP]{\@@_ep_isqrt:wwn} +% \begin{macro}[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_value:w \@@_int_eval:w + \int_if_odd:nTF {#1} + { (1 - #1) / 2 , 535 , { 0 } { } } + { 1 - #1 / 2 , 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}[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_value:w \@@_int_eval:w + (#1 + 1 0050 0000 #4 / (#1 * #3)) / 2 , + #1, #3, {#4} + } +\cs_new:Npn \@@_ep_isqrt_estii:wwwnnwn #1, #2, #3, #4#5 + { + \exp_after:wN \@@_ep_isqrt_estiii:NNNNNwwwn + \int_value:w \@@_int_eval:w 1000 0000 + #2 * #2 #5 * 5 + \exp_after:wN , \int_value:w \@@_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}[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 +% +% 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}[rEXP]{\@@_ep_to_float_o:wwN, \@@_ep_inv_to_float_o: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_o:wwN #1, + { + \@@_int_eval:w #1 \@@_fixed_to_float_o:wN } +\cs_new:Npn \@@_ep_inv_to_float_o:wwN #1,#2; + { + \@@_ep_div:wwwwn 1,{1000}{0000}{0000}{0000}{0000}{0000}; #1,#2; + \@@_ep_to_float_o:wwN + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[rEXP]{\@@_fixed_inv_to_float_o:wN} +% Another function which reduces to converting an extended precision +% number to a float. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_inv_to_float_o:wN + { \@@_ep_inv_to_float_o:wwN 0, } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[rEXP]{\@@_fixed_to_float_rad_o: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_o:wN #1; + { + \@@_fixed_mul:wwn #1; {5729}{5779}{5130}{8232}{0876}{7981}; + { \@@_ep_to_float_o:wwN 2, } + } +% \end{macrocode} +% \end{macro} +% +% ^^A todo: make exponents end in ',' consistently throughout l3fp +% \begin{macro}[rEXP] +% {\@@_fixed_to_float_o:wN, \@@_fixed_to_float_o:Nw} +% \begin{syntax} +% \ldots{} \cs{@@_int_eval:w} \meta{exponent} \cs{@@_fixed_to_float_o:wN} \Arg{a_1} \Arg{a_2} \Arg{a_3} \Arg{a_4} \Arg{a_5} \Arg{a_6} |;| \meta{sign} +% \end{syntax} +% yields +% \begin{quote} +% \meta{exponent'} |;| \Arg{a'_1} \Arg{a'_2} \Arg{a'_3} \Arg{a'_4} |;| +% \end{quote} +% And the \texttt{to_fixed} version gives six brace groups instead of +% $4$, ensuring that $1000\leq\meta{a'_1}\leq 9999$. At this stage, we +% know that \meta{a_1} is positive (otherwise, it is sign of an error +% before), and we assume that it is less than $10^8$.\footnote{Bruno: +% I must double check this assumption.} +% +%^^A todo: round properly when rounding to infinity: I need the sign. +% \begin{macrocode} +\cs_new:Npn \@@_fixed_to_float_o:Nw #1#2; + { \@@_fixed_to_float_o:wN #2; #1 } +\cs_new:Npn \@@_fixed_to_float_o:wN #1#2#3#4#5#6; #7 + { % for the 8-digit-at-the-start thing + + \@@_int_eval:w \c_@@_block_int + \exp_after:wN \exp_after:wN + \exp_after:wN \@@_fixed_to_loop:N + \exp_after:wN \use_none:n + \int_value:w \@@_int_eval:w + 1 0000 0000 + #1 \exp_after:wN \@@_use_none_stop_f:n + \int_value:w 1#2 \exp_after:wN \@@_use_none_stop_f:n + \int_value:w 1#3#4 \exp_after:wN \@@_use_none_stop_f:n + \int_value:w 1#5#6 + \exp_after:wN ; + \exp_after:wN ; + } +\cs_new:Npn \@@_fixed_to_loop:N #1 + { + \if_meaning:w 0 #1 + - 1 + \exp_after:wN \@@_fixed_to_loop:N + \else: + \exp_after:wN \@@_fixed_to_loop_end:w + \exp_after:wN #1 + \fi: + } +\cs_new:Npn \@@_fixed_to_loop_end:w #1 #2 ; + { + \if_meaning:w ; #1 + \exp_after:wN \@@_fixed_to_float_zero:w + \else: + \exp_after:wN \@@_pack_twice_four:wNNNNNNNN + \exp_after:wN \@@_pack_twice_four:wNNNNNNNN + \exp_after:wN \@@_fixed_to_float_pack:ww + \exp_after:wN ; + \fi: + #1 #2 0000 0000 0000 0000 ; + } +\cs_new:Npn \@@_fixed_to_float_zero:w ; 0000 0000 0000 0000 ; + { + - 2 * \c_@@_max_exponent_int ; + {0000} {0000} {0000} {0000} ; + } +\cs_new:Npn \@@_fixed_to_float_pack:ww #1 ; #2#3 ; ; + { + \if_int_compare:w #2 > 4 \exp_stop_f: + \exp_after:wN \@@_fixed_to_float_round_up:wnnnnw + \fi: + ; #1 ; + } +\cs_new:Npn \@@_fixed_to_float_round_up:wnnnnw ; #1#2#3#4 ; + { + \exp_after:wN \@@_basics_pack_high:NNNNNw + \int_value:w \@@_int_eval:w 1 #1#2 + \exp_after:wN \@@_basics_pack_low:NNNNNw + \int_value:w \@@_int_eval:w 1 #3#4 + 1 ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macrocode} +%</package> +% \end{macrocode} +% +% \end{implementation} +% +% \PrintChanges +% +% \PrintIndex |