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@@ -1,1380 +0,0 @@
-% \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 2024-03-14}
-%
-% \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