summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx')
-rw-r--r--Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx581
1 files changed, 581 insertions, 0 deletions
diff --git a/Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx b/Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx
new file mode 100644
index 00000000000..16a3fb465f6
--- /dev/null
+++ b/Master/texmf-dist/source/latex/l3kernel/l3fp-trig.dtx
@@ -0,0 +1,581 @@
+% \iffalse meta-comment
+%
+%% File: l3fp-trig.dtx Copyright (C) 2011-2012 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
+%%
+%% http://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 released version of this bundle is available from CTAN.
+%%
+%% -----------------------------------------------------------------------
+%%
+%% The development version of the bundle can be found at
+%%
+%% http://www.latex-project.org/svnroot/experimental/trunk/
+%%
+%% for those people who are interested.
+%%
+%%%%%%%%%%%
+%% NOTE: %%
+%%%%%%%%%%%
+%%
+%% Snapshots taken from the repository represent work in progress and may
+%% not work or may contain conflicting material! We therefore ask
+%% people _not_ to put them into distributions, archives, etc. without
+%% prior consultation with the LaTeX Project Team.
+%%
+%% -----------------------------------------------------------------------
+%%
+%
+%<*driver>
+\RequirePackage{l3names}
+\GetIdInfo$Id: l3fp-trig.dtx 3514 2012-03-08 06:14:48Z bruno $
+ {L3 Floating-point trigonometric functions}
+\documentclass[full]{l3doc}
+\begin{document}
+ \DocInput{\jobname.dtx}
+\end{document}
+%</driver>
+% \fi
+%
+% \title{The \textsf{l3fp-trig} package\thanks{This file
+% has version number \ExplFileVersion, last
+% revised \ExplFileDate.}\\
+% Floating point trigonometric 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 \ExplFileDate}
+%
+% \maketitle
+%
+% \begin{documentation}
+%
+% \end{documentation}
+%
+% \begin{implementation}
+%
+% \section{Implementation}
+%
+% \begin{macrocode}
+%<*initex|package>
+% \end{macrocode}
+%
+% \begin{macrocode}
+%<@@=fp>
+% \end{macrocode}
+%
+%^^A todo: check EXP/rEXP everywhere.
+%
+% \subsection{Inverting a floating point number}
+%
+% \begin{macro}[int, EXP]{\@@_one_over:w}
+% Expects a floating point of the form \cs{s_@@} \ldots{} |;| and
+% computes its multiplicative inverse. This is used to compute the
+% cotangent function very near $0$.
+% \begin{macrocode}
+\cs_new_nopar:Npx \@@_one_over:w
+ {
+ \exp_not:N \exp_after:wN
+ \exp_not:c { @@_/_o:ww }
+ \exp_not:N \c_one_fp
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsection{Direct trigonometric functions}
+%
+% The approach for all trigonometric functions (sine, cosine, tangent,
+% and cotangent) is the same.
+% \begin{itemize}
+% \item Filter out special cases ($\pm 0$, $\pm\inf$ and \nan{}).
+% \item Keep the sign for later, and work with the absolute value $|x|$
+% of the argument.
+% \item For numbers less than $1$, shift the mantissa to convert them to
+% fixed point numbers. Very small numbers take a slightly different
+% route.
+% \item For numbers $\geq 1$, subtract a multiple of $\pi/2$ to bring
+% them to the range to $[0, \pi/2]$.
+% \item Reduce further to $[0, \pi/4]$ using $\sin x = \cos (\pi/2-x)$.
+% \item Use the appropriate power series depending on the octant
+% $\lfloor\frac{|x|}{\pi/4}\rfloor \mod 8$, the sign, and the function
+% to compute.
+% \end{itemize}
+%
+% \subsubsection{Sign and special numbers}
+%
+% \begin{macro}[int, EXP]{\@@_sin:w}
+% The sine of $\pm 0$ or \nan{} is the same floating point number.
+% The sine of $\pm\infty$ raises an invalid operation exception.
+% Otherwise, check the exponent, preparing to use
+% \cs{@@_sin_series:NNwww} for the calculation, with a sign |#2|, and
+% an initial octant of $0$. The question mark is an argument which is
+% not used in this case.
+% \begin{macrocode}
+\cs_new:Npn \@@_sin:w \s_@@ \@@_chk:w #1#2
+ {
+ \if_case:w #1 \exp_stop_f:
+ \@@_case_return_same_o:w
+ \or:
+ \exp_after:wN \@@_trig_exponent:NNNNwn
+ \exp_after:wN \@@_sin_series:NNwww
+ \exp_after:wN ?
+ \exp_after:wN #2
+ \exp_after:wN \c_zero
+ \or:
+ \@@_case_use:nw
+ { \@@_invalid_operation:Nnw \c_nan_fp { sin } }
+ \else: \@@_case_return_same_o:w
+ \fi:
+ \s_@@ \@@_chk:w #1#2
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[int, EXP]{\@@_cos:w}
+% The cosine of $\pm 0$ is $1$. The cosine of $\pm\infty$ raises an
+% invalid operation exception. The cosine of \nan{} is itself.
+% Otherwise, check the exponent, preparing to use
+% \cs{@@_sin_series:NNwww} for the calculation, with a positive sign
+% ($0$), and an initial octant of $2$, because $\cos x = \sin ( \pi/2
+% + |x|)$. The question mark is an argument which is not used in this
+% case.
+% \begin{macrocode}
+\cs_new:Npn \@@_cos:w \s_@@ \@@_chk:w #1#2
+ {
+ \if_case:w #1 \exp_stop_f:
+ \@@_case_return_o:Nw \c_one_fp
+ \or:
+ \@@_case_use:nw %^^A todo: is that faster than the exp_after route?
+ {
+ \@@_trig_exponent:NNNNwn
+ \@@_sin_series:NNwww
+ ?
+ 0
+ \c_two
+ }
+ \or:
+ \@@_case_use:nw
+ { \@@_invalid_operation:Nnw \c_nan_fp { cos } }
+ \else: \@@_case_return_same_o:w
+ \fi:
+ \s_@@ \@@_chk:w #1#2
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[int, EXP]{\@@_tan:w}
+% The tangent of $\pm 0$ or \nan{} is the same floating point number.
+% The tangent of $\pm\infty$ raises an invalid operation exception.
+% Otherwise, check the exponent, preparing to use
+% \cs{@@_tan_series:NNwww} for the calculation, with a positive sign
+% ($0$), and an initial octant of $1$, chosen to be distinct from the
+% octants for sine and cosine. See \cs{@@_cot:w} for an
+% explanation of the $0$ argument.
+% \begin{macrocode}
+\cs_new:Npn \@@_tan:w \s_@@ \@@_chk:w #1#2
+ {
+ \if_case:w #1 \exp_stop_f:
+ \@@_case_return_same_o:w
+ \or:
+ \exp_after:wN \@@_trig_exponent:NNNNwn
+ \exp_after:wN \@@_tan_series:NNwww
+ \exp_after:wN 0
+ \exp_after:wN #2
+ \exp_after:wN \c_one
+ \or:
+ \@@_case_use:nw
+ { \@@_invalid_operation:Nnw \c_nan_fp { tan } }
+ \else: \@@_case_return_same_o:w
+ \fi:
+ \s_@@ \@@_chk:w #1#2
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[int, EXP]{\@@_cot:w}
+% The cotangent of $\pm 0$ is $\pm \infty$ with the same sign,
+% produced by \cs{@@_one_over:w}. The cotangent of $\pm\infty$ raises
+% an invalid operation exception. The cotangent of \nan{} is itself.
+% We use $\cot x = - \tan (\pi/2 + x)$, and the initial octant for the
+% tangent was chosen to be $1$, so the octant here starts at $3$. The
+% change in sign is obtained by feeding \cs{@@_tan_series:NNwww} two
+% signs rather than just the sign of the argument: the first of those
+% indicates whether we compute tangent or cotangent. Those signs are
+% eventually combined.
+% \begin{macrocode}
+\cs_new:Npn \@@_cot:w \s_@@ \@@_chk:w #1#2
+ {
+ \if_case:w #1 \exp_stop_f:
+ \exp_after:wN \@@_one_over:w
+ \or:
+ \exp_after:wN \@@_trig_exponent:NNNNwn
+ \exp_after:wN \@@_tan_series:NNwww
+ \exp_after:wN 2
+ \exp_after:wN #2
+ \exp_after:wN \c_three
+ \or:
+ \@@_case_use:nw
+ { \@@_invalid_operation:Nnw \c_nan_fp { cot } }
+ \else: \@@_case_return_same_o:w
+ \fi:
+ \s_@@ \@@_chk:w #1#2
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsubsection{Small and tiny arguments}
+%
+% \begin{macro}[aux, EXP]{\@@_trig_exponent:NNNNwn}
+% The first four arguments control what trigonometric function we
+% compute, then follows a normal floating point number. If the
+% floating point is smaller than $10^{-8}$, then call the appropriate
+% \texttt{_epsilon} auxiliary. Otherwise, call the function |#1|,
+% with arguments |#2|, |#3|, the octant, computed in an integer
+% expression starting with |#4|, and a fixed point number obtained
+% from the floating point number by argument reduction. Numbers less
+% than $1$ are converted using \cs{@@_trig_small:w} which simply
+% shifts the mantissa, while large numbers need argument reduction.
+% \begin{macrocode}
+\cs_new:Npn \@@_trig_exponent:NNNNwn #1#2#3#4 \s_@@ \@@_chk:w 1#5#6
+ {
+ \if_int_compare:w #6 > - \c_eight
+ \exp_after:wN #1
+ \exp_after:wN #2
+ \exp_after:wN #3
+ \int_use:N \__int_eval:w #4
+ \if_int_compare:w #6 > \c_zero
+ \exp_after:wN \@@_trig_large:w \__int_value:w
+ \else:
+ \exp_after:wN \@@_trig_small:w \__int_value:w
+ \fi:
+ \else:
+ \if_case:w #4
+ \@@_sin_epsilon:w
+ \or: \@@_sin_epsilon:w
+ \or: \@@_cos_epsilon:w
+ \else: \@@_cot_epsilon:w
+ \fi:
+ #5
+ \fi:
+ #6 ;
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[aux, EXP]
+% {\@@_sin_epsilon:w, \@@_cos_epsilon:w, \@@_cot_epsilon:w}
+% Sine and tangent of tiny numbers give the number itself: the
+% relative error is less than $5 \cdot 10^{-17}$, which is
+% appropriate. Cosine simply gives $1$. Cotangent computes the
+% inverse. This is actually slightly wrong because further terms in
+% the power series could affect the rounding for cotangent.
+% \begin{macrocode}
+\cs_new:Npn \@@_sin_epsilon:w #1 \fi: #2 \fi: #3 ;
+ { \fi: \fi: \@@_exp_after_o:w \s_@@ \@@_chk:w 1 #2 {#3} }
+\cs_new:Npn \@@_cos_epsilon:w #1 \fi: #2 \fi: #3 ; #4 ;
+ { \fi: \fi: \exp_after:wN \c_one_fp }
+\cs_new:Npn \@@_cot_epsilon:w \fi: #1 \fi: #2 ;
+ { \fi: \fi: \@@_one_over:w \s_@@ \@@_chk:w 1 #1 {#2} }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[aux, EXP]{\@@_trig_small:w, \@@_trig_small_aux:wwNN}
+% Floating point numbers less than $1$ are converted to fixed point
+% numbers by shifting the mantissa. Since we have already filtered
+% out numbers less than $10^{-8}$, no digit is lost in converting to
+% a fixed point number.
+% \begin{macrocode}
+\cs_new:Npn \@@_trig_small:w #1;
+ {
+ \exp_after:wN \exp_after:wN \exp_after:wN \@@_trig_small_aux:wwNN
+ \prg_replicate:nn { - #1 } { 0 } ;
+ }
+\cs_new:Npn \@@_trig_small_aux:wwNN #1; #2#3#4#5;
+ {
+ \@@_pack_twice_four:wNNNNNNNN
+ \@@_pack_twice_four:wNNNNNNNN
+ \@@_pack_twice_four:wNNNNNNNN
+ .
+ ;
+ #1#2#3#4#5 0000 0000;
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsubsection{Reduction of large arguments}
+%
+% In the case of a floating point argument greater or equal to $1$, we
+% need to perform argument reduction.
+%
+% \begin{macro}[aux, rEXP]
+% {
+% \@@_trig_large:w, \@@_trig_large_i:www,
+% \@@_trig_large_ii:wnnnnnn, \@@_trig_large_break:w
+% }
+% We shift the mantissa by one digit at a time, subtracting a multiple
+% of $2\pi$ at each step. We use a value of $2\pi$ rounded up,
+% consistent with the choice of \cs{c_pi_fp}. This is not quite
+% correct from an accuracy perspective, but has the nice property that
+% $\sin(180\mathrm{deg}) = 0$ exactly. The arguments of
+% \cs{@@_trig_large_i:www} are a leading block of up to $5$ digits,
+% three brace groups of $4$ digits each, and the exponent, decremented
+% at each step. The multiple of $2\pi$ to subtract is estimated as
+% $\lfloor |#1| / 6283\rfloor$ (the formula chosen always gives a
+% non-negative integer). The subtraction has a form similar to our
+% usual multiplications (see \pkg{l3fp-basics} or
+% \pkg{l3fp-extended}). Once the exponent reaches $0$, we are done
+% subtracting $2\pi$, and we call \cs{@@_trig_octant_loop:nw} to do
+% the reduction by $\pi/2$.
+% \begin{macrocode}
+\cs_new:Npn \@@_trig_large:w #1; #2#3;
+ { \@@_trig_large_i:www #2; #3 ; #1; }
+\cs_new:Npn \@@_trig_large_i:www #1; #2; #3;
+ {
+ \if_meaning:w 0 #3 \@@_trig_large_break:w \fi:
+ \exp_after:wN \@@_trig_large_ii:wnnnnnn
+ \int_use:N \__int_eval:w ( #1 - 3141 ) / 6283 ;
+ {#1} #2;
+ \int_use:N \__int_eval:w \c_minus_one + #3;
+ }
+\cs_new:Npn \@@_trig_large_ii:wnnnnnn #1; #2#3#4#5;
+ {
+ \exp_after:wN \@@_trig_large_i:www
+ \int_use:N \__int_eval:w -5 0000 + #20 - #1*62831
+ \exp_after:wN \@@_fixed_mul_pack:NNNNNw
+ \int_use:N \__int_eval:w 4 9995 0000 + #30 - #1*8530
+ \exp_after:wN \@@_fixed_mul_pack:NNNNNw
+ \int_use:N \__int_eval:w 4 9995 0000 + #40 - #1*7179
+ \exp_after:wN \@@_fixed_mul_pack:NNNNNw
+ \int_use:N \__int_eval:w 5 0000 0000 + #50 - #1*5880
+ \exp_after:wN ;
+ \exp_after:wN ;
+ }
+\cs_new:Npn \@@_trig_large_break:w \fi: #1; #2;
+ { \fi: \@@_trig_octant_loop:nw #2 {0000} {0000} ; }
+% \end{macrocode}
+% \end{macro}
+%
+%^^A todo: optimize: we don't need 6x4 digits here, only 4x4.
+%
+% \begin{macro}[aux, rEXP]
+% {
+% \@@_trig_octant_loop:nw, \@@_trig_octant_break:w,
+% \@@_trig_octant_neg:w
+% }
+% We receive a fixed point number as argument. As long as it is
+% greater than $1.5707$ (a slight underestimate of $\pi/2$), subtract
+% $\pi/2$, and leave |+ \c_two| in the integer expression for the
+% octant. Once it becomes smaller, if it is greater than $0.7854$
+% (overestimate of $\pi/4$), then compute $\pi/2 - x$ and increment
+% the octant. If it is negative, correct this by changing the sign
+% and decrementing the octant (by adding $7$). The result is in all
+% cases in the range $[0, 0.7854]$, appropriate for a series
+% expansion.
+% \begin{macrocode}
+\cs_new:Npn \@@_trig_octant_loop:nw #1#2;
+ {
+ \if_int_compare:w #1 < 15707 \exp_stop_f:
+ \@@_trig_octant_break:w
+ \fi:
+ + \c_two
+ \@@_fixed_sub_back:wwN
+ {15707} {9632} {6794} {8970} {0000} {0000} ;
+ {#1} #2;
+ \@@_trig_octant_loop:nw
+ }
+\cs_new:Npn \@@_trig_octant_break:w #1 \fi: + #2#3 #4; #5#6; #7;
+ {
+ \fi:
+ \if_int_compare:w #5 < 7854 \exp_stop_f:
+ \if_int_compare:w #5 < \c_zero
+ \exp_after:wN \@@_trig_octant_neg:w
+ \fi:
+ \exp_after:wN \@@_use_i_until_s:nw
+ \exp_after:wN .
+ \fi:
+ + \c_one
+ \@@_fixed_sub:wwN
+ {15707} {9632} {6794} {8970} {0000} {0000} ;
+ {#5} #6 ; . ;
+ }
+\cs_new:Npn \@@_trig_octant_neg:w #1\fi: #2; #3#4#5#6#7#8; #9
+ {
+ \fi:
+ + \c_seven
+ \exp_after:wN \@@_fixed_add_after:NNNNNwN
+ \int_use:N \__int_eval:w 1 9999 9998 - #30000 - #4
+ \exp_after:wN \@@_fixed_add_pack:NNNNNwN
+ \int_use:N \__int_eval:w 1 9999 9998 - #5#6
+ \exp_after:wN \@@_fixed_add_pack:NNNNNwN
+ \int_use:N \__int_eval:w 2 0000 0000 - #7#8 ; {#9} ;
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsection{Computing the power series}
+%
+% \begin{macro}[aux, EXP]{\@@_sin_series:NNwww, \@@_sin_series_aux:Nnww}
+% Here we receive an unused |?|, a \meta{sign} ($0$ or $2$), a
+% (non-negative) \meta{octant} delimited by a dot, a \meta{fixed
+% point} number, and junk delimited by a semicolon. The auxiliary
+% receives:
+% \begin{itemize}
+% \item The final sign, which depends on the octant |#3| and the
+% original sign |#2|,
+% \item The octant |#3|, which will control the series we use.
+% \item The square |#4 * #4| of the argument, computed with
+% \cs{@@_fixed_mul:wwn}.
+% \item The number itself.
+% \end{itemize}
+% If the octant is in $\{1,2,5,6,\ldots{}\}$, we are near an extremum
+% of the function and we use the series
+% \[
+% \cos(x) = 1 - x^2 \bigg( \frac{1}{2!} - x^2 \bigg( \frac{1}{4!}
+% - x^2 \bigg( \cdots \bigg) \bigg) \bigg) .
+% \]
+% Otherwise, the series
+% \[
+% \sin(x) = x \bigg( 1 - x^2 \bigg( \frac{1}{3!} - x^2 \bigg(
+% \frac{1}{5!} - x^2 \bigg( \cdots \bigg) \bigg) \bigg) \bigg)
+% \]
+% is used. Finally, the fixed point number is converted to a floating
+% point number with the given sign, and we check for overflow or
+% underflow. %^^A todo: can over/underflow really happen??
+% \begin{macrocode}
+\cs_new:Npn \@@_sin_series:NNwww #1#2#3 . #4; #5;
+ {
+ \@@_fixed_mul:wwn #4; #4;
+ {
+ \exp_after:wN \@@_sin_series_aux:Nnww
+ \__int_value:w
+ \if_int_odd:w \__int_eval:w ( #3 + \c_two ) / \c_four \__int_eval_end:
+ #2
+ \else:
+ \if_meaning:w #2 0 2 \else: 0 \fi:
+ \fi:
+ {#3}
+ }
+ #4 ;
+ }
+\cs_new:Npn \@@_sin_series_aux:Nnww #1#2 #3; #4;
+ {
+ \if_int_odd:w \__int_eval:w #2 / \c_two \__int_eval_end:
+ \exp_after:wN \use_i:nn
+ \else:
+ \exp_after:wN \use_ii:nn
+ \fi:
+ {
+ \@@_fixed_continue:wn {0000}{0000}{0000}{0001}{5619}{2070}; % 1/18!
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0000}{0000}{0477}{9477}{3324};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0000}{0011}{4707}{4559}{7730};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0000}{2087}{6756}{9878}{6810};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0027}{5573}{1922}{3985}{8907};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{2480}{1587}{3015}{8730}{1587};
+ \@@_fixed_mul_sub_back:wwwn #3; {0013}{8888}{8888}{8888}{8888}{8889};
+ \@@_fixed_mul_sub_back:wwwn #3; {0416}{6666}{6666}{6666}{6666}{6667};
+ \@@_fixed_mul_sub_back:wwwn #3; {5000}{0000}{0000}{0000}{0000}{0000};
+ \@@_fixed_mul_sub_back:wwwn #3;{10000}{0000}{0000}{0000}{0000}{0000};
+ }
+ {
+ \@@_fixed_continue:wn {0000}{0000}{0000}{0028}{1145}{7254}; % 1/17!
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0000}{0000}{7647}{1637}{3182};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0000}{0160}{5904}{3836}{8216};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0002}{5052}{1083}{8544}{1719};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0275}{5731}{9223}{9858}{9065};
+ \@@_fixed_mul_sub_back:wwwn #3; {0001}{9841}{2698}{4126}{9841}{2698};
+ \@@_fixed_mul_sub_back:wwwn #3; {0083}{3333}{3333}{3333}{3333}{3333};
+ \@@_fixed_mul_sub_back:wwwn #3; {1666}{6666}{6666}{6666}{6666}{6667};
+ \@@_fixed_mul_sub_back:wwwn #3;{10000}{0000}{0000}{0000}{0000}{0000};
+ \@@_fixed_mul:wwn #4;
+ }
+ {
+ \exp_after:wN \@@_sanitize:Nw
+ \exp_after:wN #1
+ \int_use:N \__int_eval:w \@@_fixed_to_float:wN
+ }
+ #1
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[aux, EXP]{\@@_tan_series:NNwww, \@@_tan_series_aux:Nnww}
+% Similar to \cs{@@_sin_series:NNwww}, but with slightly different
+% rules to find the sign. The result is expressed as a ratio of
+% polynomials, of the form
+% \[
+% \tan(x) \simeq
+% \frac{x (1 - x^2 (a_1 - x^2 (a_2 - x^2 (a_3 - x^2 (a_4 - x^2 a_5)))))}
+% {1 - x^2 (b_1 - x^2 (b_2 - x^2 (b_3 - x^2 (b_4 - x^2 b_5))))} .
+% \]
+% The ratio of the two fixed point numbers is converted to a floating
+% point number directly to avoid rounding issues. The two fixed
+% points may be exchanged before computing the ratio, depending on the
+% quadrant.
+% \begin{macrocode}
+\cs_new:Npn \@@_tan_series:NNwww #1#2#3. #4; #5;
+ {
+ \@@_fixed_mul:wwn #4; #4;
+ {
+ \exp_after:wN \@@_tan_series_aux:Nnww
+ \__int_value:w
+ \if_int_odd:w \__int_eval:w #3 / \c_two \__int_eval_end:
+ \exp_after:wN \reverse_if:N
+ \fi:
+ \if_meaning:w #1#2 2 \else: 0 \fi:
+ {#3}
+ }
+ #4 ;
+ }
+\cs_new:Npn \@@_tan_series_aux:Nnww #1 #2 #3; #4;
+ {
+ \@@_fixed_continue:wn {0000}{0000}{1527}{3493}{0856}{7059};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{0159}{6080}{0274}{5257}{6472};
+ \@@_fixed_mul_sub_back:wwwn #3; {0002}{4571}{2320}{0157}{2558}{8481};
+ \@@_fixed_mul_sub_back:wwwn #3; {0115}{5830}{7533}{5397}{3168}{2147};
+ \@@_fixed_mul_sub_back:wwwn #3; {1929}{8245}{6140}{3508}{7719}{2982};
+ \@@_fixed_mul_sub_back:wwwn #3;{10000}{0000}{0000}{0000}{0000}{0000};
+ \@@_fixed_mul:wwn #4;
+ {
+ \@@_fixed_continue:wn {0000}{0007}{0258}{0681}{9408}{4706};
+ \@@_fixed_mul_sub_back:wwwn #3; {0000}{2343}{7175}{1399}{6151}{7670};
+ \@@_fixed_mul_sub_back:wwwn #3; {0019}{2638}{4588}{9232}{8861}{3691};
+ \@@_fixed_mul_sub_back:wwwn #3; {0536}{6357}{0691}{4344}{6852}{4252};
+ \@@_fixed_mul_sub_back:wwwn #3; {5263}{1578}{9473}{6842}{1052}{6315};
+ \@@_fixed_mul_sub_back:wwwn #3;{10000}{0000}{0000}{0000}{0000}{0000};
+ {
+ \exp_after:wN \@@_sanitize:Nw
+ \exp_after:wN #1
+ \int_use:N \__int_eval:w
+ \reverse_if:N \if_int_odd:w
+ \__int_eval:w (#2 - \c_one) / \c_two \__int_eval_end:
+ \exp_after:wN \@@_reverse_args:Nww
+ \fi:
+ \@@_fixed_div_to_float:ww
+ }
+ }
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macrocode}
+%</initex|package>
+% \end{macrocode}
+%
+% \end{implementation}
+%
+% \PrintChanges
+%
+% \PrintIndex \ No newline at end of file