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l3fp-expo.dtx
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l3fp-expo.dtx
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% \iffalse meta-comment
%
%% File: l3fp-expo.dtx Copyright (C) 2011-2017 The LaTeX3 Project
%
% It may be distributed and/or modified under the conditions of the
% LaTeX Project Public License (LPPL), either version 1.3c of this
% 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{The \textsf{l3fp-expo} package\\
% Floating point exponential-related functions}
% \author{^^A
% The \LaTeX3 Project\thanks
% {^^A
% E-mail:
% \href{mailto:latex-team@latex-project.org}
% {latex-team@latex-project.org}^^A
% }^^A
% }
% \date{Released 2017/11/14}
%
% \maketitle
%
% \begin{documentation}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{l3fp-expo} implementation}
%
% \begin{macrocode}
%<*initex|package>
% \end{macrocode}
%
% \begin{macrocode}
%<@@=fp>
% \end{macrocode}
%
% \begin{macro}[EXP]
% {
% \@@_parse_word_exp:N ,
% \@@_parse_word_ln: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 ? }
% \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 < 1 \exp_stop_f:
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 e\TeX{} 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 = 1 \exp_stop_f:
\@@_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 > 0 \exp_stop_f:
\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 < 0 \exp_stop_f:
\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 = 1 \exp_stop_f:
\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{macro}[rEXP]
% {
% \@@_exp_pos_large:NnnNwn ,
% \@@_exp_large_after:wwn ,
% \@@_exp_large:w ,
% \@@_exp_large_v:wN,
% \@@_exp_large_iv:wN,
% \@@_exp_large_iii:wN,
% \@@_exp_large_ii:wN,
% \@@_exp_large_i:wN,
% \@@_exp_large_:wN,
% }
% 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 having the auxiliary for one
% exponent call the auxiliary for the next exponent. The current
% total is expressed by leaving the exponent behind in the input
% stream (we are currently within an \cs{__int_eval:w}), and keeping
% track of a fixed point number, |#1| for the numbered auxiliaries.
% Our usage of \cs{if_case:w} is somewhat dirty for optimization:
% \TeX{} jumps to the appropriate case, but we then close the
% \cs{if_case:w} \enquote{by hand}, using \cs{or:} and \cs{fi:} as
% delimiters.
% \begin{macrocode}
\cs_new:Npn \@@_exp_pos_large:NnnNwn #1#2#3 #4#5; #6
{
\exp_after:wN \exp_after:wN
\cs:w @@_exp_large_ \__int_to_roman:w #6 :wN \exp_after:wN \cs_end:
\exp_after:wN \c_@@_one_fixed_tl
\__int_value:w #3 #4 \exp_stop_f:
#5 00000 ;
}
\cs_new:Npn \@@_exp_large:w #1 \or: #2 \fi:
{ \fi: \@@_fixed_mul:wwn #1; }
\cs_new:Npn \@@_exp_large_v:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 4343 \@@_exp_large:w {8806}{8182}{2566}{2921}{5872}{6150} \or:
+ 8686 \@@_exp_large:w {7756}{0047}{2598}{6861}{0458}{3204} \or:
+ 13029 \@@_exp_large:w {6830}{5723}{7791}{4884}{1932}{7351} \or:
+ 17372 \@@_exp_large:w {6015}{5609}{3095}{3052}{3494}{7574} \or:
+ 21715 \@@_exp_large:w {5297}{7951}{6443}{0315}{3251}{3576} \or:
+ 26058 \@@_exp_large:w {4665}{6719}{0099}{3379}{5527}{2929} \or:
+ 30401 \@@_exp_large:w {4108}{9724}{3326}{3186}{5271}{5665} \or:
+ 34744 \@@_exp_large:w {3618}{6973}{3140}{0875}{3856}{4102} \or:
+ 39087 \@@_exp_large:w {3186}{9209}{6113}{3900}{6705}{9685} \or:
\fi:
#1;
\@@_exp_large_iv:wN
}
\cs_new:Npn \@@_exp_large_iv:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 435 \@@_exp_large:w {1970}{0711}{1401}{7046}{9938}{8888} \or:
+ 869 \@@_exp_large:w {3881}{1801}{9428}{4368}{5764}{8232} \or:
+ 1303 \@@_exp_large:w {7646}{2009}{8905}{4704}{8893}{1073} \or:
+ 1738 \@@_exp_large:w {1506}{3559}{7005}{0524}{9009}{7592} \or:
+ 2172 \@@_exp_large:w {2967}{6283}{8402}{3667}{0689}{6630} \or:
+ 2606 \@@_exp_large:w {5846}{4389}{5650}{2114}{7278}{5046} \or:
+ 3041 \@@_exp_large:w {1151}{7900}{5080}{6878}{2914}{4154} \or:
+ 3475 \@@_exp_large:w {2269}{1083}{0850}{6857}{8724}{4002} \or:
+ 3909 \@@_exp_large:w {4470}{3047}{3316}{5442}{6408}{6591} \or:
\fi:
#1;
\@@_exp_large_iii:wN
}
\cs_new:Npn \@@_exp_large_iii:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 44 \@@_exp_large:w {2688}{1171}{4181}{6135}{4484}{1263} \or:
+ 87 \@@_exp_large:w {7225}{9737}{6812}{5749}{2581}{7748} \or:
+ 131 \@@_exp_large:w {1942}{4263}{9524}{1255}{9365}{8421} \or:
+ 174 \@@_exp_large:w {5221}{4696}{8976}{4143}{9505}{8876} \or:
+ 218 \@@_exp_large:w {1403}{5922}{1785}{2837}{4107}{3977} \or:
+ 261 \@@_exp_large:w {3773}{0203}{0092}{9939}{8234}{0143} \or:
+ 305 \@@_exp_large:w {1014}{2320}{5473}{5004}{5094}{5533} \or:
+ 348 \@@_exp_large:w {2726}{3745}{7211}{2566}{5673}{6478} \or:
+ 391 \@@_exp_large:w {7328}{8142}{2230}{7421}{7051}{8866} \or:
\fi:
#1;
\@@_exp_large_ii:wN
}
\cs_new:Npn \@@_exp_large_ii:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 5 \@@_exp_large:w {2202}{6465}{7948}{0671}{6516}{9579} \or:
+ 9 \@@_exp_large:w {4851}{6519}{5409}{7902}{7796}{9107} \or:
+ 14 \@@_exp_large:w {1068}{6474}{5815}{2446}{2146}{9905} \or:
+ 18 \@@_exp_large:w {2353}{8526}{6837}{0199}{8540}{7900} \or:
+ 22 \@@_exp_large:w {5184}{7055}{2858}{7072}{4640}{8745} \or:
+ 27 \@@_exp_large:w {1142}{0073}{8981}{5684}{2836}{6296} \or:
+ 31 \@@_exp_large:w {2515}{4386}{7091}{9167}{0062}{6578} \or:
+ 35 \@@_exp_large:w {5540}{6223}{8439}{3510}{0525}{7117} \or:
+ 40 \@@_exp_large:w {1220}{4032}{9431}{7840}{8020}{0271} \or:
\fi:
#1;
\@@_exp_large_i:wN
}
\cs_new:Npn \@@_exp_large_i:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 1 \@@_exp_large:w {2718}{2818}{2845}{9045}{2353}{6029} \or:
+ 1 \@@_exp_large:w {7389}{0560}{9893}{0650}{2272}{3043} \or:
+ 2 \@@_exp_large:w {2008}{5536}{9231}{8766}{7740}{9285} \or:
+ 2 \@@_exp_large:w {5459}{8150}{0331}{4423}{9078}{1103} \or:
+ 3 \@@_exp_large:w {1484}{1315}{9102}{5766}{0342}{1116} \or:
+ 3 \@@_exp_large:w {4034}{2879}{3492}{7351}{2260}{8387} \or:
+ 4 \@@_exp_large:w {1096}{6331}{5842}{8458}{5992}{6372} \or:
+ 4 \@@_exp_large:w {2980}{9579}{8704}{1728}{2747}{4359} \or:
+ 4 \@@_exp_large:w {8103}{0839}{2757}{5384}{0077}{1000} \or:
\fi:
#1;
\@@_exp_large_:wN
}
\cs_new:Npn \@@_exp_large_:wN #1; #2
{
\if_case:w #2 ~ \exp_after:wN \@@_fixed_continue:wn \or:
+ 1 \@@_exp_large:w {1105}{1709}{1807}{5647}{6248}{1171} \or:
+ 1 \@@_exp_large:w {1221}{4027}{5816}{0169}{8339}{2107} \or:
+ 1 \@@_exp_large:w {1349}{8588}{0757}{6003}{1039}{8374} \or:
+ 1 \@@_exp_large:w {1491}{8246}{9764}{1270}{3178}{2485} \or:
+ 1 \@@_exp_large:w {1648}{7212}{7070}{0128}{1468}{4865} \or:
+ 1 \@@_exp_large:w {1822}{1188}{0039}{0508}{9748}{7537} \or:
+ 1 \@@_exp_large:w {2013}{7527}{0747}{0476}{5216}{2455} \or:
+ 1 \@@_exp_large:w {2225}{5409}{2849}{2467}{6045}{7954} \or:
+ 1 \@@_exp_large:w {2459}{6031}{1115}{6949}{6638}{0013} \or:
\fi:
#1;
\@@_exp_large_after:wwn
}
\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_int_compare:w \__str_if_eq_x:nn { #2 #3 }
{ 1 {1000} {0000} {0000} {0000} } = 0 \exp_stop_f:
\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: