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diff --git a/macros/latex/required/l3kernel/l3fp-expo.dtx b/macros/latex/required/l3kernel/l3fp-expo.dtx new file mode 100644 index 0000000000..6fbc9de52c --- /dev/null +++ b/macros/latex/required/l3kernel/l3fp-expo.dtx @@ -0,0 +1,1380 @@ +% \iffalse meta-comment +% +%% File: l3fp-expo.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-expo} module\\ +% Floating point exponential-related functions^^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-expo} implementation} +% +% \begin{macrocode} +%<*package> +% \end{macrocode} +% +% \begin{macrocode} +%<@@=fp> +% \end{macrocode} +% +% \begin{macro}[EXP] +% { +% \@@_parse_word_exp:N , +% \@@_parse_word_ln:N , +% \@@_parse_word_fact:N, +% } +% Unary functions. +% \begin{macrocode} +\cs_new:Npn \@@_parse_word_exp:N + { \@@_parse_unary_function:NNN \@@_exp_o:w ? } +\cs_new:Npn \@@_parse_word_ln:N + { \@@_parse_unary_function:NNN \@@_ln_o:w ? } +\cs_new:Npn \@@_parse_word_fact:N + { \@@_parse_unary_function:NNN \@@_fact_o:w ? } +% \end{macrocode} +% \end{macro} +% +% \subsection{Logarithm} +% +% \subsubsection{Work plan} +% +% As for many other functions, we filter out special cases in +% \cs{@@_ln_o:w}. Then \cs{@@_ln_npos_o:w} receives a positive normal +% number, which we write in the form $a\cdot 10^{b}$ with $a\in[0.1,1)$. +% +% \emph{The rest of this section is actually not in sync with the code. +% Or is the code not in sync with the section? In the current code, +% $c\in [1,10]$ is such that $0.7\leq ac < 1.4$.} +% +% We are given a positive normal number, of the form $a\cdot 10^{b}$ +% with $a\in[0.1,1)$. To compute its logarithm, we find a small integer +% $5\leq c < 50$ such that $0.91 \leq a c / 5 < 1.1$, and use the +% relation +% \begin{equation*} +% \ln (a \cdot 10^b) = b \cdot \ln (10) - \ln (c/5) + \ln (ac/5). +% \end{equation*} +% The logarithms $\ln(10)$ and $\ln(c/5)$ are looked up in a table. The +% last term is computed using the following Taylor series of $\ln$ near +% $1$: +% \begin{equation*} +% \ln\left(\frac{ac}{5}\right) +% = \ln\left(\frac{1+t}{1-t}\right) +% = 2t\left(1 + t^2 \left(\frac{1}{3} + t^2 \left(\frac{1}{5} +% + t^2 \left(\frac{1}{7} + t^2 \left( \frac{1}{9} + \cdots +% \right)\right)\right)\right)\right) +% \end{equation*} +% where $t=1-10/(ac+5)$. We can now see one reason for the choice of +% $ac\sim 5$: then $ac+5=10(1-\epsilon)$ with $-0.05<\epsilon\leq +% 0.045$, hence +% \begin{equation*} +% t = \frac{\epsilon}{1-\epsilon} +% = \epsilon (1+\epsilon)(1+\epsilon^2)(1+\epsilon^4)\ldots, +% \end{equation*} +% is not too difficult to compute. +% +% \subsubsection{Some constants} +% +% \begin{variable} +% { +% \c_@@_ln_i_fixed_tl , +% \c_@@_ln_ii_fixed_tl , +% \c_@@_ln_iii_fixed_tl , +% \c_@@_ln_iv_fixed_tl , +% \c_@@_ln_vi_fixed_tl , +% \c_@@_ln_vii_fixed_tl , +% \c_@@_ln_viii_fixed_tl , +% \c_@@_ln_ix_fixed_tl , +% \c_@@_ln_x_fixed_tl, +% } +% A few values of the logarithm as extended fixed point numbers. +% Those are needed in the implementation. It turns out that we don't +% need the value of $\ln(5)$. +% \begin{macrocode} +\tl_const:Nn \c_@@_ln_i_fixed_tl { {0000}{0000}{0000}{0000}{0000}{0000};} +\tl_const:Nn \c_@@_ln_ii_fixed_tl { {6931}{4718}{0559}{9453}{0941}{7232};} +\tl_const:Nn \c_@@_ln_iii_fixed_tl {{10986}{1228}{8668}{1096}{9139}{5245};} +\tl_const:Nn \c_@@_ln_iv_fixed_tl {{13862}{9436}{1119}{8906}{1883}{4464};} +\tl_const:Nn \c_@@_ln_vi_fixed_tl {{17917}{5946}{9228}{0550}{0081}{2477};} +\tl_const:Nn \c_@@_ln_vii_fixed_tl {{19459}{1014}{9055}{3133}{0510}{5353};} +\tl_const:Nn \c_@@_ln_viii_fixed_tl{{20794}{4154}{1679}{8359}{2825}{1696};} +\tl_const:Nn \c_@@_ln_ix_fixed_tl {{21972}{2457}{7336}{2193}{8279}{0490};} +\tl_const:Nn \c_@@_ln_x_fixed_tl {{23025}{8509}{2994}{0456}{8401}{7991};} +% \end{macrocode} +% \end{variable} +% +% \subsubsection{Sign, exponent, and special numbers} +% +% \begin{macro}[EXP]{\@@_ln_o:w} +% The logarithm of negative numbers (including $-\infty$ and $-0$) +% raises the \enquote{invalid} exception. The logarithm of $+0$ is +% $-\infty$, raising a division by zero exception. The logarithm of +% $+\infty$ or a \texttt{nan} is itself. Positive normal numbers call +% \cs{@@_ln_npos_o:w}. +% \begin{macrocode} +\cs_new:Npn \@@_ln_o:w #1 \s_@@ \@@_chk:w #2#3#4; @ + { + \if_meaning:w 2 #3 + \@@_case_use:nw { \@@_invalid_operation_o:nw { ln } } + \fi: + \if_case:w #2 \exp_stop_f: + \@@_case_use:nw + { \@@_division_by_zero_o:Nnw \c_minus_inf_fp { ln } } + \or: + \else: + \@@_case_return_same_o:w + \fi: + \@@_ln_npos_o:w \s_@@ \@@_chk:w #2#3#4; + } +% \end{macrocode} +% \end{macro} +% +% \subsubsection{Absolute ln} +% +% \begin{macro}[EXP]{\@@_ln_npos_o:w} +% We catch the case of a significand very close to $0.1$ or to $1$. +% In all other cases, the final result is at least $10^{-4}$, and +% then an error of $0.5\cdot 10^{-20}$ is acceptable. +% \begin{macrocode} +\cs_new:Npn \@@_ln_npos_o:w \s_@@ \@@_chk:w 10#1#2#3; + { %^^A todo: ln(1) should be "exact zero", not "underflow" + \exp_after:wN \@@_sanitize:Nw + \int_value:w % for the overall sign + \if_int_compare:w #1 < \c_one_int + 2 + \else: + 0 + \fi: + \exp_after:wN \exp_stop_f: + \int_value:w \@@_int_eval:w % for the exponent + \@@_ln_significand:NNNNnnnN #2#3 + \@@_ln_exponent:wn {#1} + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_ln_significand:NNNNnnnN} +% \begin{syntax} +% \cs{@@_ln_significand:NNNNnnnN} \meta{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \meta{continuation} +% \end{syntax} +% This function expands to +% \begin{syntax} +% \meta{continuation} \Arg{Y_1} \Arg{Y_2} \Arg{Y_3} \Arg{Y_4} \Arg{Y_5} \Arg{Y_6} |;| +% \end{syntax} +% where $Y = - \ln(X)$ as an extended fixed point. +% \begin{macrocode} +\cs_new:Npn \@@_ln_significand:NNNNnnnN #1#2#3#4 + { + \exp_after:wN \@@_ln_x_ii:wnnnn + \int_value:w + \if_case:w #1 \exp_stop_f: + \or: + \if_int_compare:w #2 < 4 \exp_stop_f: + \@@_int_eval:w 10 - #2 + \else: + 6 + \fi: + \or: 4 + \or: 3 + \or: 2 + \or: 2 + \or: 2 + \else: 1 + \fi: + ; { #1 #2 #3 #4 } + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_ln_x_ii:wnnnn} +% We have thus found $c \in [1,10]$ such that $0.7\leq ac < 1.4$ +% in all cases. Compute $ 1 + x = 1 + ac \in [1.7,2.4)$. +% \begin{macrocode} +\cs_new:Npn \@@_ln_x_ii:wnnnn #1; #2#3#4#5 + { + \exp_after:wN \@@_ln_div_after:Nw + \cs:w c_@@_ln_ \@@_int_to_roman:w #1 _fixed_tl \exp_after:wN \cs_end: + \int_value:w + \exp_after:wN \@@_ln_x_iv:wnnnnnnnn + \int_value:w \@@_int_eval:w + \exp_after:wN \@@_ln_x_iii_var:NNNNNw + \int_value:w \@@_int_eval:w 9999 9990 + #1*#2#3 + + \exp_after:wN \@@_ln_x_iii:NNNNNNw + \int_value:w \@@_int_eval:w 10 0000 0000 + #1*#4#5 ; + {20000} {0000} {0000} {0000} + } %^^A todo: reoptimize (a generalization attempt failed). +\cs_new:Npn \@@_ln_x_iii:NNNNNNw #1#2 #3#4#5#6 #7; + { #1#2; {#3#4#5#6} {#7} } +\cs_new:Npn \@@_ln_x_iii_var:NNNNNw #1 #2#3#4#5 #6; + { + #1#2#3#4#5 + 1 ; + {#1#2#3#4#5} {#6} + } +% \end{macrocode} +% The Taylor series to be used is expressed in terms of +% $t = (x-1)/(x+1) = 1 - 2/(x+1)$. We now compute the +% quotient with extended precision, reusing some code +% from \cs{@@_/_o:ww}. Note that $1+x$ is known exactly. +% +% To reuse notations from \pkg{l3fp-basics}, we want to +% compute $ A / Z $ with $ A = 2 $ and $ Z = x + 1 $. +% In \pkg{l3fp-basics}, we considered the case where +% both $A$ and $Z$ are arbitrary, in the range $[0.1,1)$, +% and we had to monitor the growth of the sequence of +% remainders $A$, $B$, $C$, etc. to ensure that no overflow +% occurred during the computation of the next quotient. +% The main source of risk was our choice to define the +% quotient as roughly $10^9 \cdot A / 10^5 \cdot Z$: then +% $A$ was bound to be below $2.147\cdots$, and this limit +% was never far. +% +% In our case, we can simply work with $10^8 \cdot A$ and +% $10^4 \cdot Z$, because our reason to work with higher +% powers has gone: we needed the integer $y \simeq 10^5 \cdot Z$ +% to be at least $10^4$, and now, the definition +% $y \simeq 10^4 \cdot Z$ suffices. +% +% Let us thus define $y = \left\lfloor 10^4 \cdot Z \right\rfloor + 1 +% \in ( 1.7 \cdot 10^4, 2.4 \cdot 10^4 ] $, and +% \[ +% Q_{1} +% = +% \left\lfloor +% \frac {\left\lfloor 10^8 \cdot A\right\rfloor} {y} - \frac{1}{2} +% \right\rfloor. +% \] +% (The $1/2$ comes from how \eTeX{} rounds.) As for division, it is +% easy to see that $Q_{1} \leq 10^4 A / Z$, \emph{i.e.}, $Q_{1}$ +% is an underestimate. +% +% Exactly as we did for division, we set $B = 10^4 A - Q_{1}Z$. Then +% \begin{align*} +% 10^4 B +% & \leq +% A_{1}A_{2}.A_{3}A_{4} +% - \left( \frac{A_{1}A_{2}}{y} - \frac{3}{2} \right) 10^4 Z +% \\ +% & \leq +% A_{1}A_{2} \left( 1 - \frac{10^4 Z}{y} \right) + 1 + \frac{3}{2} y +% \\ +% & \leq +% 10^8 \frac{A}{y} + 1 + \frac{3}{2} y +% \end{align*} +% In the same way, and using $1.7\cdot 10^4\leq y\leq 2.4\cdot 10^4$, +% and convexity, we get +% \begin{align*} +% 10^4 A &= 2\cdot 10^4 \\ +% 10^4 B &\leq 10^8 \frac{A}{y} + 1.6 y \leq 4.7\cdot 10^4\\ +% 10^4 C &\leq 10^8 \frac{B}{y} + 1.6 y \leq 5.8\cdot 10^4\\ +% 10^4 D &\leq 10^8 \frac{C}{y} + 1.6 y \leq 6.3\cdot 10^4\\ +% 10^4 E &\leq 10^8 \frac{D}{y} + 1.6 y \leq 6.5\cdot 10^4\\ +% 10^4 F &\leq 10^8 \frac{E}{y} + 1.6 y \leq 6.6\cdot 10^4\\ +% \end{align*} +% Note that we compute more steps than for division: since $t$ is +% not the end result, we need to know it with more accuracy +% (on the other hand, the ending is much simpler, as we don't +% need an exact rounding for transcendental functions, but just +% a faithful rounding). +% ^^A todo: doc +% +% \begin{syntax} +% \cs{@@_ln_x_iv:wnnnnnnnn} \meta{1 or 2} \meta{8d} |;| \Arg{4d} \Arg{4d} \meta{fixed-tl} +% \end{syntax} +% The number is $x$. Compute $y$ by adding 1 to the five first digits. +% \begin{macrocode} +\cs_new:Npn \@@_ln_x_iv:wnnnnnnnn #1; #2#3#4#5 #6#7#8#9 + { + \exp_after:wN \@@_div_significand_pack:NNN + \int_value:w \@@_int_eval:w + \@@_ln_div_i:w #1 ; + #6 #7 ; {#8} {#9} + {#2} {#3} {#4} {#5} + { \exp_after:wN \@@_ln_div_ii:wwn \int_value:w #1 } + { \exp_after:wN \@@_ln_div_ii:wwn \int_value:w #1 } + { \exp_after:wN \@@_ln_div_ii:wwn \int_value:w #1 } + { \exp_after:wN \@@_ln_div_ii:wwn \int_value:w #1 } + { \exp_after:wN \@@_ln_div_vi:wwn \int_value:w #1 } + } +\cs_new:Npn \@@_ln_div_i:w #1; + { + \exp_after:wN \@@_div_significand_calc:wwnnnnnnn + \int_value:w \@@_int_eval:w 999999 + 2 0000 0000 / #1 ; % Q1 + } +\cs_new:Npn \@@_ln_div_ii:wwn #1; #2;#3 % y; B1;B2 <- for k=1 + { + \exp_after:wN \@@_div_significand_pack:NNN + \int_value:w \@@_int_eval:w + \exp_after:wN \@@_div_significand_calc:wwnnnnnnn + \int_value:w \@@_int_eval:w 999999 + #2 #3 / #1 ; % Q2 + #2 #3 ; + } +\cs_new:Npn \@@_ln_div_vi:wwn #1; #2;#3#4#5 #6#7#8#9 %y;F1;F2F3F4x1x2x3x4 + { + \exp_after:wN \@@_div_significand_pack:NNN + \int_value:w \@@_int_eval:w 1000000 + #2 #3 / #1 ; % Q6 + } +% \end{macrocode} +% We now have essentially +% ^^A todo: determine error on $Q_{6}$ (probably $6.7$), +% ^^A todo: conclude the final result is off by $<10^{-23}$ +% \begin{syntax} +% \cs{@@_ln_div_after:Nw} \meta{fixed tl} +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{1}$ +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{2}$ +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{3}$ +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{4}$ +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{5}$ +% \cs{@@_div_significand_pack:NNN} $10^6 + Q_{6}$ |;| +% \meta{exponent} |;| \meta{continuation} +% \end{syntax} +% where \meta{fixed tl} holds the logarithm of a number +% in $[1,10]$, and \meta{exponent} is +% the exponent. Also, the expansion is done backwards. Then +% \cs{@@_div_significand_pack:NNN} puts things in the +% correct order to add the $Q_{i}$ together and put semicolons +% between each piece. Once those have been expanded, we get +% \begin{syntax} +% \cs{@@_ln_div_after:Nw} \meta{fixed-tl} \meta{1d} |;| \meta{4d} |;| \meta{4d} |;| +% ~~\meta{4d} |;| \meta{4d} |;| \meta{4d} |;| \meta{4d} |;| \meta{exponent} |;| +% \end{syntax} +% ^^A todo: redoc. +% Just as with division, we know that the first two digits +% are |1| and |0| because of bounds on the final result of +% the division $2/(x+1)$, which is between roughly $0.8$ and $1.2$. +% We then compute $1-2/(x+1)$, after testing whether $2/(x+1)$ is +% greater than or smaller than $1$. +% \begin{macrocode} +\cs_new:Npn \@@_ln_div_after:Nw #1#2; + { + \if_meaning:w 0 #2 + \exp_after:wN \@@_ln_t_small:Nw + \else: + \exp_after:wN \@@_ln_t_large:NNw + \exp_after:wN - + \fi: + #1 + } +\cs_new:Npn \@@_ln_t_small:Nw #1 #2; #3; #4; #5; #6; #7; + { + \exp_after:wN \@@_ln_t_large:NNw + \exp_after:wN + % <sign> + \exp_after:wN #1 + \int_value:w \@@_int_eval:w 9999 - #2 \exp_after:wN ; + \int_value:w \@@_int_eval:w 9999 - #3 \exp_after:wN ; + \int_value:w \@@_int_eval:w 9999 - #4 \exp_after:wN ; + \int_value:w \@@_int_eval:w 9999 - #5 \exp_after:wN ; + \int_value:w \@@_int_eval:w 9999 - #6 \exp_after:wN ; + \int_value:w \@@_int_eval:w 1 0000 - #7 ; + } +% \end{macrocode} +% +% \begin{syntax} +% \cs{@@_ln_t_large:NNw} \meta{sign} \meta{fixed tl} +% ~~\meta{t_1}|;| \meta{t_2} |;| \meta{t_3}|;| \meta{t_4}|;| \meta{t_5} |;| \meta{t_6}|;| +% ~~\meta{exponent} |;| \meta{continuation} +% \end{syntax} +% Compute the square $|t|^2$, and keep $|t|$ at the end with its +% sign. We know that $|t|<0.1765$, so every piece has at most $4$ +% digits. However, since we were not careful in \cs{@@_ln_t_small:w}, +% they can have less than $4$ digits. +% \begin{macrocode} +\cs_new:Npn \@@_ln_t_large:NNw #1 #2 #3; #4; #5; #6; #7; #8; + { + \exp_after:wN \@@_ln_square_t_after:w + \int_value:w \@@_int_eval:w 9999 0000 + #3*#3 + \exp_after:wN \@@_ln_square_t_pack:NNNNNw + \int_value:w \@@_int_eval:w 9999 0000 + 2*#3*#4 + \exp_after:wN \@@_ln_square_t_pack:NNNNNw + \int_value:w \@@_int_eval:w 9999 0000 + 2*#3*#5 + #4*#4 + \exp_after:wN \@@_ln_square_t_pack:NNNNNw + \int_value:w \@@_int_eval:w 9999 0000 + 2*#3*#6 + 2*#4*#5 + \exp_after:wN \@@_ln_square_t_pack:NNNNNw + \int_value:w \@@_int_eval:w + 1 0000 0000 + 2*#3*#7 + 2*#4*#6 + #5*#5 + + (2*#3*#8 + 2*#4*#7 + 2*#5*#6) / 1 0000 + % ; ; ; + \exp_after:wN \@@_ln_twice_t_after:w + \int_value:w \@@_int_eval:w -1 + 2*#3 + \exp_after:wN \@@_ln_twice_t_pack:Nw + \int_value:w \@@_int_eval:w 9999 + 2*#4 + \exp_after:wN \@@_ln_twice_t_pack:Nw + \int_value:w \@@_int_eval:w 9999 + 2*#5 + \exp_after:wN \@@_ln_twice_t_pack:Nw + \int_value:w \@@_int_eval:w 9999 + 2*#6 + \exp_after:wN \@@_ln_twice_t_pack:Nw + \int_value:w \@@_int_eval:w 9999 + 2*#7 + \exp_after:wN \@@_ln_twice_t_pack:Nw + \int_value:w \@@_int_eval:w 10000 + 2*#8 ; ; + { \@@_ln_c:NwNw #1 } + #2 + } +\cs_new:Npn \@@_ln_twice_t_pack:Nw #1 #2; { + #1 ; {#2} } +\cs_new:Npn \@@_ln_twice_t_after:w #1; { ;;; {#1} } +\cs_new:Npn \@@_ln_square_t_pack:NNNNNw #1 #2#3#4#5 #6; + { + #1#2#3#4#5 ; {#6} } +\cs_new:Npn \@@_ln_square_t_after:w 1 0 #1#2#3 #4; + { \@@_ln_Taylor:wwNw {0#1#2#3} {#4} } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\@@_ln_Taylor:wwNw} +% Denoting $T=t^2$, we get +% \begin{syntax} +% \cs{@@_ln_Taylor:wwNw} +% ~~\Arg{T_1} \Arg{T_2} \Arg{T_3} \Arg{T_4} \Arg{T_5} \Arg{T_6} |;| |;| +% ~~\Arg{(2t)_1} \Arg{(2t)_2} \Arg{(2t)_3} \Arg{(2t)_4} \Arg{(2t)_5} \Arg{(2t)_6} |;| +% ~~|{| \cs{@@_ln_c:NwNw} \meta{sign} |}| +% ~~\meta{fixed tl} \meta{exponent} |;| \meta{continuation} +% \end{syntax} +% And we want to compute +% \[ +% \ln\left(\frac{1+t}{1-t}\right) +% = 2t\left(1 + T \left(\frac{1}{3} + T \left(\frac{1}{5} +% + T \left(\frac{1}{7} + T \left( \frac{1}{9} + \cdots +% \right)\right)\right)\right)\right) +% \] +% The process looks as follows +% \begin{verbatim} +% \loop 5; A; +% \div_int 5; 1.0; \add A; \mul T; {\loop \eval 5-2;} +% \add 0.2; A; \mul T; {\loop \eval 5-2;} +% \mul B; T; {\loop 3;} +% \loop 3; C; +% \end{verbatim} +% ^^A todo: doc +% +% This uses the routine for dividing a number by a small integer +% (${}<10^4$). +% \begin{macrocode} +\cs_new:Npn \@@_ln_Taylor:wwNw + { \@@_ln_Taylor_loop:www 21 ; {0000}{0000}{0000}{0000}{0000}{0000} ; } +\cs_new:Npn \@@_ln_Taylor_loop:www #1; #2; #3; + { + \if_int_compare:w #1 = \c_one_int + \@@_ln_Taylor_break:w + \fi: + \exp_after:wN \@@_fixed_div_int:wwN \c_@@_one_fixed_tl #1; + \@@_fixed_add:wwn #2; + \@@_fixed_mul:wwn #3; + { + \exp_after:wN \@@_ln_Taylor_loop:www + \int_value:w \@@_int_eval:w #1 - 2 ; + } + #3; + } +\cs_new:Npn \@@_ln_Taylor_break:w \fi: #1 \@@_fixed_add:wwn #2#3; #4 ;; + { + \fi: + \exp_after:wN \@@_fixed_mul:wwn + \exp_after:wN { \int_value:w \@@_int_eval:w 10000 + #2 } #3; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\@@_ln_c:NwNw} +% \begin{syntax} +% \cs{@@_ln_c:NwNw} \meta{sign} +% ~~\Arg{r_1} \Arg{r_2} \Arg{r_3} \Arg{r_4} \Arg{r_5} \Arg{r_6} |;| +% ~~\meta{fixed tl} \meta{exponent} |;| \meta{continuation} +% \end{syntax} +% We are now reduced to finding $\ln(c)$ and $\meta{exponent}\ln(10)$ +% in a table, and adding it to the mixture. The first step is to +% get $\ln(c) - \ln(x) = - \ln(a)$, then we get $|b|\ln(10)$ and add +% or subtract. +% +% For now, $\ln(x)$ is given as $\cdot 10^0$. Unless both the exponent +% is $1$ and $c=1$, we shift to working in units of $\cdot 10^4$, +% since the final result is at least $\ln(10/7) \simeq 0.35$. +% \begin{macrocode} +\cs_new:Npn \@@_ln_c:NwNw #1 #2; #3 + { + \if_meaning:w + #1 + \exp_after:wN \exp_after:wN \exp_after:wN \@@_fixed_sub:wwn + \else: + \exp_after:wN \exp_after:wN \exp_after:wN \@@_fixed_add:wwn + \fi: + #3 #2 ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\@@_ln_exponent:wn} +% \begin{syntax} +% \cs{@@_ln_exponent:wn} +% ~~\Arg{s_1} \Arg{s_2} \Arg{s_3} \Arg{s_4} \Arg{s_5} \Arg{s_6} |;| +% ~~\Arg{exponent} +% \end{syntax} +% Compute \meta{exponent} times $\ln(10)$. Apart from the cases where +% \meta{exponent} is $0$ or $1$, the result is necessarily at +% least $\ln(10) \simeq 2.3$ in magnitude. We can thus drop the least +% significant $4$ digits. In the case of a very large (positive or +% negative) exponent, we can (and we need to) drop $4$ additional +% digits, since the result is of order $10^4$. Naively, one would +% think that in both cases we can drop $4$ more digits than we do, +% but that would be slightly too tight for rounding to happen correctly. +% Besides, we already have addition and subtraction for $24$ digits +% fixed point numbers. +% \begin{macrocode} +\cs_new:Npn \@@_ln_exponent:wn #1; #2 + { + \if_case:w #2 \exp_stop_f: + 0 \@@_case_return:nw { \@@_fixed_to_float_o:Nw 2 } + \or: + \exp_after:wN \@@_ln_exponent_one:ww \int_value:w + \else: + \if_int_compare:w #2 > \c_zero_int + \exp_after:wN \@@_ln_exponent_small:NNww + \exp_after:wN 0 + \exp_after:wN \@@_fixed_sub:wwn \int_value:w + \else: + \exp_after:wN \@@_ln_exponent_small:NNww + \exp_after:wN 2 + \exp_after:wN \@@_fixed_add:wwn \int_value:w - + \fi: + \fi: + #2; #1; + } +% \end{macrocode} +% Now we painfully write all the cases.\footnote{Bruno: do rounding.} +% No overflow nor underflow can happen, except when computing \texttt{ln(1)}. +% \begin{macrocode} +\cs_new:Npn \@@_ln_exponent_one:ww 1; #1; + { + 0 + \exp_after:wN \@@_fixed_sub:wwn \c_@@_ln_x_fixed_tl #1; + \@@_fixed_to_float_o:wN 0 + } +% \end{macrocode} +% For small exponents, we just drop one block of digits, and set the +% exponent of the log to $4$ (minus any shift coming from leading zeros +% in the conversion from fixed point to floating point). Note that here +% the exponent has been made positive. +% \begin{macrocode} +\cs_new:Npn \@@_ln_exponent_small:NNww #1#2#3; #4#5#6#7#8#9; + { + 4 + \exp_after:wN \@@_fixed_mul:wwn + \c_@@_ln_x_fixed_tl + {#3}{0000}{0000}{0000}{0000}{0000} ; + #2 + {0000}{#4}{#5}{#6}{#7}{#8}; + \@@_fixed_to_float_o:wN #1 + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Exponential} +% +% \subsubsection{Sign, exponent, and special numbers} +% +% \begin{macro}[EXP]{\@@_exp_o:w} +% \begin{macrocode} +\cs_new:Npn \@@_exp_o:w #1 \s_@@ \@@_chk:w #2#3#4; @ + { + \if_case:w #2 \exp_stop_f: + \@@_case_return_o:Nw \c_one_fp + \or: + \exp_after:wN \@@_exp_normal_o:w + \or: + \if_meaning:w 0 #3 + \exp_after:wN \@@_case_return_o:Nw + \exp_after:wN \c_inf_fp + \else: + \exp_after:wN \@@_case_return_o:Nw + \exp_after:wN \c_zero_fp + \fi: + \or: + \@@_case_return_same_o:w + \fi: + \s_@@ \@@_chk:w #2#3#4; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_normal_o:w, \@@_exp_pos_o:Nnwnw, \@@_exp_overflow:NN} +% \begin{macrocode} +\cs_new:Npn \@@_exp_normal_o:w \s_@@ \@@_chk:w 1#1 + { + \if_meaning:w 0 #1 + \@@_exp_pos_o:NNwnw + \@@_fixed_to_float_o:wN + \else: + \@@_exp_pos_o:NNwnw - \@@_fixed_inv_to_float_o:wN + \fi: + } +\cs_new:Npn \@@_exp_pos_o:NNwnw #1#2#3 \fi: #4#5; + { + \fi: + \if_int_compare:w #4 > \c_@@_max_exp_exponent_int + \token_if_eq_charcode:NNTF + #1 + { \@@_exp_overflow:NN \@@_overflow:w \c_inf_fp } + { \@@_exp_overflow:NN \@@_underflow:w \c_zero_fp } + \exp:w + \else: + \exp_after:wN \@@_sanitize:Nw + \exp_after:wN 0 + \int_value:w #1 \@@_int_eval:w + \if_int_compare:w #4 < \c_zero_int + \exp_after:wN \use_i:nn + \else: + \exp_after:wN \use_ii:nn + \fi: + { + 0 + \@@_decimate:nNnnnn { - #4 } + \@@_exp_Taylor:Nnnwn + } + { + \@@_decimate:nNnnnn { \c_@@_prec_int - #4 } + \@@_exp_pos_large:NnnNwn + } + #5 + {#4} + #1 #2 0 + \exp:w + \fi: + \exp_after:wN \exp_end: + } +\cs_new:Npn \@@_exp_overflow:NN #1#2 + { + \exp_after:wN \exp_after:wN + \exp_after:wN #1 + \exp_after:wN #2 + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_Taylor:Nnnwn} +% \begin{macro}[EXP]{\@@_exp_Taylor_loop:www, \@@_exp_Taylor_break:Nww} +% This function is called for numbers in the range $[10^{-9}, +% 10^{-1})$. We compute $10$ terms of the Taylor series. The +% first argument is irrelevant (rounding digit used by some other +% functions). The next three arguments, at least $16$ digits, +% delimited by a semicolon, form a fixed point number, so we pack it +% in blocks of $4$ digits. +% \begin{macrocode} +\cs_new:Npn \@@_exp_Taylor:Nnnwn #1#2#3 #4; #5 #6 + { + #6 + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_pack_twice_four:wNNNNNNNN + \@@_exp_Taylor_ii:ww + ; #2#3#4 0000 0000 ; + } +\cs_new:Npn \@@_exp_Taylor_ii:ww #1; #2; + { \@@_exp_Taylor_loop:www 10 ; #1 ; #1 ; \s_@@_stop } +\cs_new:Npn \@@_exp_Taylor_loop:www #1; #2; #3; + { + \if_int_compare:w #1 = \c_one_int + \exp_after:wN \@@_exp_Taylor_break:Nww + \fi: + \@@_fixed_div_int:wwN #3 ; #1 ; + \@@_fixed_add_one:wN + \@@_fixed_mul:wwn #2 ; + { + \exp_after:wN \@@_exp_Taylor_loop:www + \int_value:w \@@_int_eval:w #1 - 1 ; + #2 ; + } + } +\cs_new:Npn \@@_exp_Taylor_break:Nww #1 #2; #3 \s_@@_stop + { \@@_fixed_add_one:wN #2 ; } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{variable}{\c_@@_exp_intarray} +% The integer array has $6\times 9\times 4=216$ items encoding the +% values of $\exp(j\times 10^i)$ for $j=1,\dots,9$ and $i=-1,\dots,4$. +% Each value is expressed as $\simeq 10^p \times 0.m_1m_2m_3$ with +% three $8$-digit blocks $m_1$, $m_2$, $m_3$ and an integer +% exponent~$p$ (one more than the scientific exponent), and these are +% stored in the integer array as four items: $p$, $10^8+m_1$, +% $10^8+m_2$, $10^8+m_3$. The various exponentials are stored in +% increasing order of $j\times 10^i$. +% +% Storing this data in an integer array makes it slightly harder to +% access (slower, too), but uses $16$ bytes of memory per exponential +% stored, while storing as tokens used around $40$ tokens; tokens have +% an especially large footprint in Unicode-aware engines. +% \begin{macrocode} +\intarray_const_from_clist:Nn \c_@@_exp_intarray + { + 1 , 1 1105 1709 , 1 1807 5647 , 1 6248 1171 , + 1 , 1 1221 4027 , 1 5816 0169 , 1 8339 2107 , + 1 , 1 1349 8588 , 1 0757 6003 , 1 1039 8374 , + 1 , 1 1491 8246 , 1 9764 1270 , 1 3178 2485 , + 1 , 1 1648 7212 , 1 7070 0128 , 1 1468 4865 , + 1 , 1 1822 1188 , 1 0039 0508 , 1 9748 7537 , + 1 , 1 2013 7527 , 1 0747 0476 , 1 5216 2455 , + 1 , 1 2225 5409 , 1 2849 2467 , 1 6045 7954 , + 1 , 1 2459 6031 , 1 1115 6949 , 1 6638 0013 , + 1 , 1 2718 2818 , 1 2845 9045 , 1 2353 6029 , + 1 , 1 7389 0560 , 1 9893 0650 , 1 2272 3043 , + 2 , 1 2008 5536 , 1 9231 8766 , 1 7740 9285 , + 2 , 1 5459 8150 , 1 0331 4423 , 1 9078 1103 , + 3 , 1 1484 1315 , 1 9102 5766 , 1 0342 1116 , + 3 , 1 4034 2879 , 1 3492 7351 , 1 2260 8387 , + 4 , 1 1096 6331 , 1 5842 8458 , 1 5992 6372 , + 4 , 1 2980 9579 , 1 8704 1728 , 1 2747 4359 , + 4 , 1 8103 0839 , 1 2757 5384 , 1 0077 1000 , + 5 , 1 2202 6465 , 1 7948 0671 , 1 6516 9579 , + 9 , 1 4851 6519 , 1 5409 7902 , 1 7796 9107 , + 14 , 1 1068 6474 , 1 5815 2446 , 1 2146 9905 , + 18 , 1 2353 8526 , 1 6837 0199 , 1 8540 7900 , + 22 , 1 5184 7055 , 1 2858 7072 , 1 4640 8745 , + 27 , 1 1142 0073 , 1 8981 5684 , 1 2836 6296 , + 31 , 1 2515 4386 , 1 7091 9167 , 1 0062 6578 , + 35 , 1 5540 6223 , 1 8439 3510 , 1 0525 7117 , + 40 , 1 1220 4032 , 1 9431 7840 , 1 8020 0271 , + 44 , 1 2688 1171 , 1 4181 6135 , 1 4484 1263 , + 87 , 1 7225 9737 , 1 6812 5749 , 1 2581 7748 , + 131 , 1 1942 4263 , 1 9524 1255 , 1 9365 8421 , + 174 , 1 5221 4696 , 1 8976 4143 , 1 9505 8876 , + 218 , 1 1403 5922 , 1 1785 2837 , 1 4107 3977 , + 261 , 1 3773 0203 , 1 0092 9939 , 1 8234 0143 , + 305 , 1 1014 2320 , 1 5473 5004 , 1 5094 5533 , + 348 , 1 2726 3745 , 1 7211 2566 , 1 5673 6478 , + 391 , 1 7328 8142 , 1 2230 7421 , 1 7051 8866 , + 435 , 1 1970 0711 , 1 1401 7046 , 1 9938 8888 , + 869 , 1 3881 1801 , 1 9428 4368 , 1 5764 8232 , + 1303 , 1 7646 2009 , 1 8905 4704 , 1 8893 1073 , + 1738 , 1 1506 3559 , 1 7005 0524 , 1 9009 7592 , + 2172 , 1 2967 6283 , 1 8402 3667 , 1 0689 6630 , + 2606 , 1 5846 4389 , 1 5650 2114 , 1 7278 5046 , + 3041 , 1 1151 7900 , 1 5080 6878 , 1 2914 4154 , + 3475 , 1 2269 1083 , 1 0850 6857 , 1 8724 4002 , + 3909 , 1 4470 3047 , 1 3316 5442 , 1 6408 6591 , + 4343 , 1 8806 8182 , 1 2566 2921 , 1 5872 6150 , + 8686 , 1 7756 0047 , 1 2598 6861 , 1 0458 3204 , + 13029 , 1 6830 5723 , 1 7791 4884 , 1 1932 7351 , + 17372 , 1 6015 5609 , 1 3095 3052 , 1 3494 7574 , + 21715 , 1 5297 7951 , 1 6443 0315 , 1 3251 3576 , + 26058 , 1 4665 6719 , 1 0099 3379 , 1 5527 2929 , + 30401 , 1 4108 9724 , 1 3326 3186 , 1 5271 5665 , + 34744 , 1 3618 6973 , 1 3140 0875 , 1 3856 4102 , + 39087 , 1 3186 9209 , 1 6113 3900 , 1 6705 9685 , + } +% \end{macrocode} +% \end{variable} +% +% \begin{macro}[rEXP] +% { +% \@@_exp_pos_large:NnnNwn , +% \@@_exp_large_after:wwn , +% \@@_exp_large:NwN , +% \@@_exp_intarray:w , +% \@@_exp_intarray_aux:w , +% } +% The first two arguments are irrelevant (a rounding digit, and a +% brace group with $8$ zeros). The third argument is the integer part +% of our number, then we have the decimal part delimited by a +% semicolon, and finally the exponent, in the range $[0,5]$. Remove +% leading zeros from the integer part: putting |#4| in there too +% ensures that an integer part of $0$ is also removed. Then read +% digits one by one, looking up $\exp(\meta{digit}\cdot +% 10^{\meta{exponent}})$ in a table, and multiplying that to the +% current total. The loop is done by \cs{@@_exp_large:NwN}, whose +% |#1| is the \meta{exponent}, |#2| is the current mantissa, and |#3| +% is the \meta{digit}. At the end, \cs{@@_exp_large_after:wwn} moves +% on to the Taylor series, eventually multiplied with the mantissa +% that we have just computed. +% \begin{macrocode} +\cs_new:Npn \@@_exp_pos_large:NnnNwn #1#2#3 #4#5; #6 + { + \exp_after:wN \exp_after:wN \exp_after:wN \@@_exp_large:NwN + \exp_after:wN \exp_after:wN \exp_after:wN #6 + \exp_after:wN \c_@@_one_fixed_tl + \int_value:w #3 #4 \exp_stop_f: + #5 00000 ; + } +\cs_new:Npn \@@_exp_large:NwN #1#2; #3 + { + \if_case:w #3 ~ + \exp_after:wN \@@_fixed_continue:wn + \else: + \exp_after:wN \@@_exp_intarray:w + \int_value:w \@@_int_eval:w 36 * #1 + 4 * #3 \exp_after:wN ; + \fi: + #2; + { + \if_meaning:w 0 #1 + \exp_after:wN \@@_exp_large_after:wwn + \else: + \exp_after:wN \@@_exp_large:NwN + \int_value:w \@@_int_eval:w #1 - 1 \exp_after:wN \scan_stop: + \fi: + } + } +\cs_new:Npn \@@_exp_intarray:w #1 ; + { + + + \__kernel_intarray_item:Nn \c_@@_exp_intarray + { \@@_int_eval:w #1 - 3 \scan_stop: } + \exp_after:wN \use_i:nnn + \exp_after:wN \@@_fixed_mul:wwn + \int_value:w 0 + \exp_after:wN \@@_exp_intarray_aux:w + \int_value:w \__kernel_intarray_item:Nn + \c_@@_exp_intarray { \@@_int_eval:w #1 - 2 } + \exp_after:wN \@@_exp_intarray_aux:w + \int_value:w \__kernel_intarray_item:Nn + \c_@@_exp_intarray { \@@_int_eval:w #1 - 1 } + \exp_after:wN \@@_exp_intarray_aux:w + \int_value:w \__kernel_intarray_item:Nn \c_@@_exp_intarray {#1} ; ; + } +\cs_new:Npn \@@_exp_intarray_aux:w 1 #1#2#3#4#5 ; { ; {#1#2#3#4} {#5} } +\cs_new:Npn \@@_exp_large_after:wwn #1; #2; #3 + { + \@@_exp_Taylor:Nnnwn ? { } { } 0 #2; {} #3 + \@@_fixed_mul:wwn #1; + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Power} +% +% Raising a number $a$ to a power $b$ leads to many distinct situations. +% \begin{center}\def\abs#1{\lvert #1\rvert} +% \begin{tabular}{>{$}c<{$}|*8{>{$}l<{$}}} +% a^b &-\infty &(-\infty,-0) &-\text{integer} &\pm 0 &+\text{integer} &(0,\infty) &+\infty &\nan \\ \hline +% +\infty &+0 &\multicolumn{2}{c}{$+0$} &+1 &\multicolumn{2}{c}{$+\infty$} &+\infty &\nan \\ +% (1,\infty) &+0 &\multicolumn{2}{c}{$+\abs{a}^{b}$} &+1 &\multicolumn{2}{c}{$+\abs{a}^{b}$} &+\infty &\nan \\ +% +1 &+1 &\multicolumn{2}{c}{$+1$} &+1 &\multicolumn{2}{c}{$+1$} &+1 &+1 \\ +% (0,1) &+\infty &\multicolumn{2}{c}{$+\abs{a}^{b}$} &+1 &\multicolumn{2}{c}{$+\abs{a}^{b}$} &+0 &\nan \\ +% +0 &+\infty &\multicolumn{2}{c}{$+\infty$} &+1 &\multicolumn{2}{c}{$+0$} &+0 &\nan \\ +% -0 &+\infty &\nan &(-1)^b\infty &+1 &(-1)^b 0 &+0 &+0 &\nan \\ +% (-1,0) &+\infty &\nan &(-1)^b\abs{a}^{b} &+1 &(-1)^b\abs{a}^{b} &\nan &+0 &\nan \\ +% -1 &+1 &\nan &(-1)^b &+1 &(-1)^b &\nan &+1 &\nan \\ +% (-\infty,-1) &+0 &\nan &(-1)^b\abs{a}^{b} &+1 &(-1)^b\abs{a}^{b} &\nan &+\infty &\nan \\ +% -\infty &+0 &+0 &(-1)^b 0 &+1 &(-1)^b\infty &\nan &+\infty &\nan \\ +% \nan &\nan &\nan &\nan &+1 &\nan &\nan &\nan &\nan \\ +% \end{tabular} +% \end{center} +% We distinguished in this table the cases of finite (positive or +% negative) integer exponents, as $(-1)^b$ is defined in that case. +% One peculiarity of this operation is that $\nan^0 = 1^\nan = 1$, +% because this relation is obeyed for any number, even $\pm\infty$. +% +% \begin{macro}[EXP]+\@@_^_o:ww+ +% We cram most of the tests into a single function to save csnames. +% First treat the case $b=0$: $a^0=1$ for any $a$, even \texttt{nan}. +% Then test the sign of $a$. +% \begin{itemize} +% \item If it is positive, and $a$ is a normal number, call +% \cs{@@_pow_normal_o:ww} followed by the two \texttt{fp} $a$ and $b$. +% For $a=+0$ or $+\inf$, call \cs{@@_pow_zero_or_inf:ww} instead, to +% return either $+0$ or $+\infty$ as appropriate. +% \item If $a$ is a \texttt{nan}, then skip to the next semicolon +% (which happens to be conveniently the end of $b$) and return +% \texttt{nan}. +% \item Finally, if $a$ is negative, compute $|a|^b$ +% (\cs{@@_pow_normal_o:ww} which ignores the sign of its first +% operand), and keep an extra copy of $a$ and $b$ (the second brace +% group, containing \{~$b$~$a$~\}, is inserted between $a$ and $b$). +% Then do some tests to find the final sign of the result if it +% exists. +% \end{itemize} +% \begin{macrocode} +\cs_new:cpn { @@_ \iow_char:N \^ _o:ww } + \s_@@ \@@_chk:w #1#2#3; \s_@@ \@@_chk:w #4#5#6; + { + \if_meaning:w 0 #4 + \@@_case_return_o:Nw \c_one_fp + \fi: + \if_case:w #2 \exp_stop_f: + \exp_after:wN \use_i:nn + \or: + \@@_case_return_o:Nw \c_nan_fp + \else: + \exp_after:wN \@@_pow_neg:www + \exp:w \exp_end_continue_f:w \exp_after:wN \use:nn + \fi: + { + \if_meaning:w 1 #1 + \exp_after:wN \@@_pow_normal_o:ww + \else: + \exp_after:wN \@@_pow_zero_or_inf:ww + \fi: + \s_@@ \@@_chk:w #1#2#3; + } + { \s_@@ \@@_chk:w #4#5#6; \s_@@ \@@_chk:w #1#2#3; } + \s_@@ \@@_chk:w #4#5#6; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_pow_zero_or_inf:ww} +% Raising $-0$ or $-\infty$ to \texttt{nan} yields \texttt{nan}. For +% other powers, the result is $+0$ if $0$ is raised to a positive +% power or $\infty$ to a negative power, and $+\infty$ otherwise. +% Thus, if the type of $a$ and the sign of $b$ coincide, the result +% is~$0$, since those conveniently take the same possible values, $0$ +% and~$2$. Otherwise, either $a=\pm\infty$ and $b>0$ and the result +% is $+\infty$, or $a=\pm 0$ with $b<0$ and we have a division by zero +% unless $b=-\infty$. +% \begin{macrocode} +\cs_new:Npn \@@_pow_zero_or_inf:ww + \s_@@ \@@_chk:w #1#2; \s_@@ \@@_chk:w #3#4 + { + \if_meaning:w 1 #4 + \@@_case_return_same_o:w + \fi: + \if_meaning:w #1 #4 + \@@_case_return_o:Nw \c_zero_fp + \fi: + \if_meaning:w 2 #1 + \@@_case_return_o:Nw \c_inf_fp + \fi: + \if_meaning:w 2 #3 + \@@_case_return_o:Nw \c_inf_fp + \else: + \@@_case_use:nw + { + \@@_division_by_zero_o:NNww \c_inf_fp ^ + \s_@@ \@@_chk:w #1 #2 ; + } + \fi: + \s_@@ \@@_chk:w #3#4 + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_pow_normal_o:ww} +% We have in front of us $a$, and $b\neq 0$, we know that $a$ is a +% normal number, and we wish to compute $\lvert a\rvert^{b}$. If +% $\lvert a\rvert=1$, we return $1$, unless $a=-1$ and $b$ is +% \texttt{nan}. Indeed, returning $1$ at this point would wrongly +% raise \enquote{invalid} when the sign is considered. If $\lvert +% a\rvert\neq 1$, test the type of $b$: +% \begin{itemize} +% \item[0] Impossible, we already filtered $b=\pm 0$. +% \item[1] Call \cs{@@_pow_npos_o:Nww}. +% \item[2] Return $+\infty$ or $+0$ depending on the sign of $b$ and +% whether the exponent of $a$ is positive or not. +% \item[3] Return $b$. +% \end{itemize} +% \begin{macrocode} +\cs_new:Npn \@@_pow_normal_o:ww + \s_@@ \@@_chk:w 1 #1#2#3; \s_@@ \@@_chk:w #4#5 + { + \if:w 0 \@@_str_if_eq:nn { #2 #3 } { 1 {1000} {0000} {0000} {0000} } + \if_int_compare:w #4 #1 = 32 \exp_stop_f: + \exp_after:wN \@@_case_return_ii_o:ww + \fi: + \@@_case_return_o:Nww \c_one_fp + \fi: + \if_case:w #4 \exp_stop_f: + \or: + \exp_after:wN \@@_pow_npos_o:Nww + \exp_after:wN #5 + \or: + \if_meaning:w 2 #5 \exp_after:wN \reverse_if:N \fi: + \if_int_compare:w #2 > \c_zero_int + \exp_after:wN \@@_case_return_o:Nww + \exp_after:wN \c_inf_fp + \else: + \exp_after:wN \@@_case_return_o:Nww + \exp_after:wN \c_zero_fp + \fi: + \or: + \@@_case_return_ii_o:ww + \fi: + \s_@@ \@@_chk:w 1 #1 {#2} #3 ; + \s_@@ \@@_chk:w #4 #5 + } +% \end{macrocode} +% \end{macro} +% +% ^^A todo: check that we compute ln to 21 digits! +% \begin{macro}[EXP]{\@@_pow_npos_o:Nww} +% We now know that $a\neq\pm 1$ is a normal number, and $b$ is a +% normal number too. We want to compute $\lvert a\rvert^{b} = (\lvert +% x\rvert\cdot 10^{n})^{y\cdot 10^{p}} = \exp((\ln\lvert x\rvert + n +% \ln(10))\cdot y \cdot 10^{p}) = \exp(z)$. To compute the +% exponential accurately, we need to know the digits of $z$ up to the +% $16$-th position. Since the exponential of $10^{5}$ is infinite, we +% only need at most $21$ digits, hence the fixed point result of +% \cs{@@_ln_o:w} is precise enough for our needs. Start an integer +% expression for the decimal exponent of $e^{\lvert z\rvert}$. If $z$ +% is negative, negate that decimal exponent, and prepare to take the +% inverse when converting from the fixed point to the floating point result. +% \begin{macrocode} +\cs_new:Npn \@@_pow_npos_o:Nww #1 \s_@@ \@@_chk:w 1#2#3 + { + \exp_after:wN \@@_sanitize:Nw + \exp_after:wN 0 + \int_value:w + \if:w #1 \if_int_compare:w #3 > \c_zero_int 0 \else: 2 \fi: + \exp_after:wN \@@_pow_npos_aux:NNnww + \exp_after:wN + + \exp_after:wN \@@_fixed_to_float_o:wN + \else: + \exp_after:wN \@@_pow_npos_aux:NNnww + \exp_after:wN - + \exp_after:wN \@@_fixed_inv_to_float_o:wN + \fi: + {#3} + } +% \end{macrocode} +% \end{macro} +% +%^^A begin[todo] +% \begin{macro}[EXP]{\@@_pow_npos_aux:NNnww} +% The first argument is the conversion function from fixed point to +% float. Then comes an exponent and the $4$ brace groups of $x$, +% followed by $b$. Compute $-\ln(x)$. +% \begin{macrocode} +\cs_new:Npn \@@_pow_npos_aux:NNnww #1#2#3#4#5; \s_@@ \@@_chk:w 1#6#7#8; + { + #1 + \@@_int_eval:w + \@@_ln_significand:NNNNnnnN #4#5 + \@@_pow_exponent:wnN {#3} + \@@_fixed_mul:wwn #8 {0000}{0000} ; + \@@_pow_B:wwN #7; + #1 #2 0 % fixed_to_float_o:wN + } +\cs_new:Npn \@@_pow_exponent:wnN #1; #2 + { + \if_int_compare:w #2 > \c_zero_int + \exp_after:wN \@@_pow_exponent:Nwnnnnnw % n\ln(10) - (-\ln(x)) + \exp_after:wN + + \else: + \exp_after:wN \@@_pow_exponent:Nwnnnnnw % -(|n|\ln(10) + (-\ln(x))) + \exp_after:wN - + \fi: + #2; #1; + } +\cs_new:Npn \@@_pow_exponent:Nwnnnnnw #1#2; #3#4#5#6#7#8; + { %^^A todo: use that in ln. + \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#2*23025 - #1 #3 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + #1 #2*8509 - #1 #4 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + #1 #2*2994 - #1 #5 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_middle_shift_int + #1 #2*0456 - #1 #6 + \exp_after:wN \@@_pack:NNNNNw + \int_value:w \@@_int_eval:w \c_@@_trailing_shift_int + #1 #2*8401 - #1 #7 + #1 ( #2*7991 - #8 ) / 1 0000 ; ; + } +\cs_new:Npn \@@_pow_B:wwN #1#2#3#4#5#6; #7; + { + \if_int_compare:w #7 < \c_zero_int + \exp_after:wN \@@_pow_C_neg:w \int_value:w - + \else: + \if_int_compare:w #7 < 22 \exp_stop_f: + \exp_after:wN \@@_pow_C_pos:w \int_value:w + \else: + \exp_after:wN \@@_pow_C_overflow:w \int_value:w + \fi: + \fi: + #7 \exp_after:wN ; + \int_value:w \@@_int_eval:w 10 0000 + #1 \@@_int_eval_end: + #2#3#4#5#6 0000 0000 0000 0000 0000 0000 ; %^^A todo: how many 0? + } +\cs_new:Npn \@@_pow_C_overflow:w #1; #2; #3 + { + + 2 * \c_@@_max_exponent_int + \exp_after:wN \@@_fixed_continue:wn \c_@@_one_fixed_tl + } +\cs_new:Npn \@@_pow_C_neg:w #1 ; 1 + { + \exp_after:wN \exp_after:wN \exp_after:wN \@@_pow_C_pack:w + \prg_replicate:nn {#1} {0} + } +\cs_new:Npn \@@_pow_C_pos:w #1; 1 + { \@@_pow_C_pos_loop:wN #1; } +\cs_new:Npn \@@_pow_C_pos_loop:wN #1; #2 + { + \if_meaning:w 0 #1 + \exp_after:wN \@@_pow_C_pack:w + \exp_after:wN #2 + \else: + \if_meaning:w 0 #2 + \exp_after:wN \@@_pow_C_pos_loop:wN \int_value:w + \else: + \exp_after:wN \@@_pow_C_overflow:w \int_value:w + \fi: + \@@_int_eval:w #1 - 1 \exp_after:wN ; + \fi: + } +\cs_new:Npn \@@_pow_C_pack:w + { + \exp_after:wN \@@_exp_large:NwN + \exp_after:wN 5 + \c_@@_one_fixed_tl + } +% \end{macrocode} +% \end{macro} +%^^A end[todo] +% +% \begin{macro}[EXP]{\@@_pow_neg:www, \@@_pow_neg_aux:wNN} +% This function is followed by three floating point numbers: $|a|^b$, +% $a\in[-\infty,-0]$, and $b$. If $b$ is an even integer (case $-1$), +% $a^b=|a|^b$. If $b$ is an odd integer (case $0$), $a^b=-|a|^b$, +% obtained by a call to \cs{@@_pow_neg_aux:wNN}. Otherwise, the sign is +% undefined. This is invalid, unless $|a|^b$ turns out to be $+0$ or +% \texttt{nan}, in which case we return that as $a^b$. In particular, +% since the underflow detection occurs before \cs{@@_pow_neg:www} is +% called, |(-0.1)**(12345.67)| gives $+0$ rather than complaining +% that the sign is not defined. +% \begin{macrocode} +\cs_new:Npn \@@_pow_neg:www \s_@@ \@@_chk:w #1#2; #3; #4; + { + \if_case:w \@@_pow_neg_case:w #4 ; + \exp_after:wN \@@_pow_neg_aux:wNN + \or: + \if_int_compare:w \@@_int_eval:w #1 / 2 = \c_one_int + \@@_invalid_operation_o:Nww ^ #3; #4; + \exp:w \exp_end_continue_f:w + \exp_after:wN \exp_after:wN + \exp_after:wN \@@_use_none_until_s:w + \fi: + \fi: + \@@_exp_after_o:w + \s_@@ \@@_chk:w #1#2; + } +\cs_new:Npn \@@_pow_neg_aux:wNN #1 \s_@@ \@@_chk:w #2#3 + { + \exp_after:wN \@@_exp_after_o:w + \exp_after:wN \s_@@ + \exp_after:wN \@@_chk:w + \exp_after:wN #2 + \int_value:w \@@_int_eval:w 2 - #3 \@@_int_eval_end: + } +% \end{macrocode} +% ^^A todo: is this \@@_exp_after_o:w necessary? Appropriate? +% \end{macro} +% +% \begin{macro}[rEXP] +% { +% \@@_pow_neg_case:w, \@@_pow_neg_case_aux:nnnnn, +% \@@_pow_neg_case_aux:Nnnw +% } +% This function expects a floating point number, and determines its +% \enquote{parity}. It should be used after \cs{if_case:w} or in an +% integer expression. It gives $-1$ if the number is an even integer, +% $0$~if the number is an odd integer, and $1$~otherwise. Zeros and +% $\pm\infty$ are even (because very large finite floating points are +% even), while \texttt{nan} is a non-integer. The sign of normal +% numbers is irrelevant to parity. After \cs{@@_decimate:nNnnnn} the +% argument |#1| of \cs{@@_pow_neg_case_aux:Nnnw} is a rounding digit, +% |0|~if and only if the number was an integer, and |#3| is the $8$ +% least significant digits of that integer. +% \begin{macrocode} +\cs_new:Npn \@@_pow_neg_case:w \s_@@ \@@_chk:w #1#2#3; + { + \if_case:w #1 \exp_stop_f: + -1 + \or: \@@_pow_neg_case_aux:nnnnn #3 + \or: -1 + \else: 1 + \fi: + \exp_stop_f: + } +\cs_new:Npn \@@_pow_neg_case_aux:nnnnn #1#2#3#4#5 + { + \if_int_compare:w #1 > \c_@@_prec_int + -1 + \else: + \@@_decimate:nNnnnn { \c_@@_prec_int - #1 } + \@@_pow_neg_case_aux:Nnnw + {#2} {#3} {#4} {#5} + \fi: + } +\cs_new:Npn \@@_pow_neg_case_aux:Nnnw #1#2#3#4 ; + { + \if_meaning:w 0 #1 + \if_int_odd:w #3 \exp_stop_f: + 0 + \else: + -1 + \fi: + \else: + 1 + \fi: + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Factorial} +% +% \begin{variable}{\c_@@_fact_max_arg_int} +% The maximum integer whose factorial fits in the exponent range is +% $3248$, as $3249!\sim 10^{10000.8}$ +% \begin{macrocode} +\int_const:Nn \c_@@_fact_max_arg_int { 3248 } +% \end{macrocode} +% \end{variable} +% +% \begin{macro}[EXP]{\@@_fact_o:w} +% First detect $\pm 0$ and $+\infty$ and \texttt{nan}. Then note that +% factorial of anything with a negative sign (except $-0$) is +% undefined. Then call \cs{@@_small_int:wTF} to get an integer as the +% argument, and start a loop. This is not the most efficient way of +% computing the factorial, but it works all right. Of course we work +% with $24$ digits instead of~$16$. It is easy to check that +% computing factorials with this precision is enough. +% \begin{macrocode} +\cs_new:Npn \@@_fact_o:w #1 \s_@@ \@@_chk:w #2#3#4; @ + { + \if_case:w #2 \exp_stop_f: + \@@_case_return_o:Nw \c_one_fp + \or: + \or: + \if_meaning:w 0 #3 + \exp_after:wN \@@_case_return_same_o:w + \fi: + \or: + \@@_case_return_same_o:w + \fi: + \if_meaning:w 2 #3 + \@@_case_use:nw { \@@_invalid_operation_o:fw { fact } } + \fi: + \@@_fact_pos_o:w + \s_@@ \@@_chk:w #2 #3 #4 ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fact_pos_o:w, \@@_fact_int_o:w} +% Then check the input is an integer, and call +% \cs{@@_facorial_int_o:n} with that \texttt{int} as an argument. If +% it's too big the factorial overflows. Otherwise call +% \cs{@@_sanitize:Nw} with a positive sign marker~|0| and an integer +% expression that will mop up any exponent in the calculation. +% \begin{macrocode} +\cs_new:Npn \@@_fact_pos_o:w #1; + { + \@@_small_int:wTF #1; + { \@@_fact_int_o:n } + { \@@_invalid_operation_o:fw { fact } #1; } + } +\cs_new:Npn \@@_fact_int_o:n #1 + { + \if_int_compare:w #1 > \c_@@_fact_max_arg_int + \@@_case_return:nw + { + \exp_after:wN \exp_after:wN \exp_after:wN \@@_overflow:w + \exp_after:wN \c_inf_fp + } + \fi: + \exp_after:wN \@@_sanitize:Nw + \exp_after:wN 0 + \int_value:w \@@_int_eval:w + \@@_fact_loop_o:w #1 . 4 , { 1 } { } { } { } { } { } ; + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_fact_loop_o:w} +% The loop receives an integer |#1| whose factorial we want to +% compute, which we progressively decrement, and the result so far as +% an extended-precision number |#2| in the form +% \meta{exponent}|,|\meta{mantissa}|;|. The loop goes in steps of two +% because we compute |#1*#1-1| as an integer expression (it must fit +% since |#1| is at most $3248$), then multiply with the result so far. +% We don't need to fill in most of the mantissa with zeros because +% \cs{@@_ep_mul:wwwwn} first normalizes the extended precision number +% to avoid loss of precision. When reaching a small enough number +% simply use a table of factorials less than $10^8$. This limit is +% chosen because the normalization step cannot deal with larger +% integers. +% \begin{macrocode} +\cs_new:Npn \@@_fact_loop_o:w #1 . #2 ; + { + \if_int_compare:w #1 < 12 \exp_stop_f: + \@@_fact_small_o:w #1 + \fi: + \exp_after:wN \@@_ep_mul:wwwwn + \exp_after:wN 4 \exp_after:wN , + \exp_after:wN { \int_value:w \@@_int_eval:w #1 * (#1 - 1) } + { } { } { } { } { } ; + #2 ; + { + \exp_after:wN \@@_fact_loop_o:w + \int_value:w \@@_int_eval:w #1 - 2 . + } + } +\cs_new:Npn \@@_fact_small_o:w #1 \fi: #2 ; #3 ; #4 + { + \fi: + \exp_after:wN \@@_ep_mul:wwwwn + \exp_after:wN 4 \exp_after:wN , + \exp_after:wN + { + \int_value:w + \if_case:w #1 \exp_stop_f: + 1 \or: 1 \or: 2 \or: 6 \or: 24 \or: 120 \or: 720 \or: 5040 + \or: 40320 \or: 362880 \or: 3628800 \or: 39916800 + \fi: + } { } { } { } { } { } ; + #3 ; + \@@_ep_to_float_o:wwN 0 + } +% \end{macrocode} +% \end{macro} +% +% \begin{macrocode} +%</package> +% \end{macrocode} +% +% \end{implementation} +% +% \PrintChanges +% +% \PrintIndex |