summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/l3kernel/l3fp.dtx
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/source/latex/l3kernel/l3fp.dtx')
-rw-r--r--Master/texmf-dist/source/latex/l3kernel/l3fp.dtx5676
1 files changed, 5676 insertions, 0 deletions
diff --git a/Master/texmf-dist/source/latex/l3kernel/l3fp.dtx b/Master/texmf-dist/source/latex/l3kernel/l3fp.dtx
new file mode 100644
index 00000000000..36ac323debd
--- /dev/null
+++ b/Master/texmf-dist/source/latex/l3kernel/l3fp.dtx
@@ -0,0 +1,5676 @@
+% \iffalse meta-comment
+%
+%% File: l3fp.dtx Copyright (C) 2010,2011 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 "expl3 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 LaTeX3 Project.
+%%
+%% -----------------------------------------------------------------------
+%
+%<*driver|package>
+\RequirePackage{l3names}
+\GetIdInfo$Id: l3fp.dtx 2478 2011-06-19 21:34:23Z joseph $
+ {L3 Experimental floating-point operations}
+%</driver|package>
+%<*driver>
+\documentclass[full]{l3doc}
+\begin{document}
+ \DocInput{\jobname.dtx}
+\end{document}
+%</driver>
+% \fi
+%
+% \title{^^A
+% The \pkg{l3fp} package\\ Floating-point operations^^A
+% \thanks{This file describes v\ExplFileVersion,
+% last revised \ExplFileDate.}^^A
+% }
+%
+% \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}
+%
+% A floating point number is one which is stored as a mantissa and
+% a separate exponent. This module implements arithmetic using radix
+% $10$ floating point numbers. This means that the mantissa should
+% be a real number in the range $1 \le \expandafter\mathopen\string|
+% x \expandafter\mathclose\string| < 10$, with the
+% exponent given as an integer between $-99$ and $99$. In the
+% input, the exponent part is represented starting with an \texttt{e}.
+% As this is a low-level module, error-checking is minimal. Numbers
+% which are too large for the floating point unit to handle will result
+% in errors, either from \TeX{} or from \LaTeX{}. The \LaTeX{} code does not
+% check that the input will not overflow, hence the possibility of a
+% \TeX{} error. On the other hand, numbers which are too small will be
+% dropped, which will mean that extra decimal digits will simply be
+% lost.
+%
+% When parsing numbers, any missing parts will be interpreted as
+% zero. So for example
+%\begin{verbatim}
+% \fp_set:Nn \l_my_fp { }
+% \fp_set:Nn \l_my_fp { . }
+% \fp_set:Nn \l_my_fp { - }
+% \end{verbatim}
+% will all be interpreted as zero values without raising an error.
+%
+% Operations which give an undefined result (such as division by
+% $0$) will not lead to errors. Instead special marker values are
+% returned, which can be tested for using fr example
+% \cs{fp_if_undefined:N(TF)}. In this way it is possible to work with
+% asymptotic functions without first checking the input. If these
+% special values are carried forward in calculations they will be
+% treated as $0$.
+%
+% Floating point numbers are stored in the \texttt{fp} floating point
+% variable type. This has a standard range of functions for
+% variable management.
+%
+% \section{Floating-point variables}
+%
+% \begin{function}{\fp_new:N, \fp_new:c}
+% \begin{syntax}
+% \cs{fp_new:N} \meta{floating point variable}
+% \end{syntax}
+% Creates a new \meta{floating point variable} or raises an error if
+% the name is already taken. The declaration global. The
+% \meta{floating point} will initially be set to |+0.000000000e0|
+% (the zero floating point).
+% \end{function}
+%
+% \begin{function}{\fp_const:Nn, \fp_const:cn}
+% \begin{syntax}
+% \cs{fp_const:Nn} \meta{floating point variable} \Arg{value}
+% \end{syntax}
+% Creates a new constant \meta{floating point variable} or raises an
+% error if the name is already taken. The value of the
+% \meta{floating point variable} will be set globally to the
+% \meta{value}.
+% \end{function}
+%
+% \begin{function}{\fp_set_eq:NN, \fp_set_eq:cN, \fp_set_eq:Nc, \fp_set_eq:cc}
+% \begin{syntax}
+% \cs{fp_set_eq:NN} \meta{fp var1} \meta{fp var2}
+% \end{syntax}
+% Sets the value of \meta{floating point variable1} equal to that of
+% \meta{floating point variable2}. This assignment is restricted to the
+% current \TeX{} group level.
+% \end{function}
+%
+% \begin{function}
+% {\fp_gset_eq:NN, \fp_gset_eq:cN, \fp_gset_eq:Nc, \fp_gset_eq:cc}
+% \begin{syntax}
+% \cs{fp_gset_eq:NN} \meta{fp var1} \meta{fp var2}
+% \end{syntax}
+% Sets the value of \meta{floating point variable1} equal to that of
+% \meta{floating point variable2}. This assignment is global and so is
+% not limited by the current \TeX{} group level.
+% \end{function}
+%
+% \begin{function}{\fp_zero:N, \fp_zero:c}
+% \begin{syntax}
+% \cs{fp_zero:N} \meta{floating point variable}
+% \end{syntax}
+% Sets the \meta{floating point variable} to |+0.000000000e0| within
+% the current scope.
+% \end{function}
+%
+% \begin{function}{\fp_gzero:N, \fp_gzero:c}
+% \begin{syntax}
+% \cs{fp_gzero:N} \meta{floating point variable}
+% \end{syntax}
+% Sets the \meta{floating point variable} to |+0.000000000e0| globally.
+% \end{function}
+%
+% \begin{function}{\fp_set:Nn, \fp_set:cn}
+% \begin{syntax}
+% \cs{fp_set:Nn} \meta{floating point variable} \Arg{value}
+% \end{syntax}
+% Sets the \meta{floating point variable} variable to \meta{value}
+% within the scope of the current \TeX{} group.
+% \end{function}
+%
+% \begin{function}{\fp_gset:Nn, \fp_gset:cn}
+% \begin{syntax}
+% \cs{fp_gset:Nn} \meta{floating point variable} \Arg{value}
+% \end{syntax}
+% Sets the \meta{floating point variable} variable to \meta{value}
+% globally.
+% \end{function}
+%
+% \begin{function}{\fp_set_from_dim:Nn, \fp_set_from_dim:cn}
+% \begin{syntax}
+% \cs{fp_set_from_dim:Nn} \meta{floating point variable} \Arg{dimexpr}
+% \end{syntax}
+% Sets the \meta{floating point variable} to the distance represented
+% by the \meta{dimension expression} in the units points. This means
+% that distances given in other units are first converted to points
+% before being assigned to the \meta{floating point variable}. The
+% assignment is local.
+% \end{function}
+%
+% \begin{function}{\fp_gset_from_dim:Nn, \fp_gset_from_dim:cn}
+% \begin{syntax}
+% \cs{fp_gset_from_dim:Nn} \meta{floating point variable} \Arg{dimexpr}
+% \end{syntax}
+% Sets the \meta{floating point variable} to the distance represented
+% by the \meta{dimension expression} in the units points. This means
+% that distances given in other units are first converted to points
+% before being assigned to the \meta{floating point variable}. The
+% assignment is global.
+% \end{function}
+%
+% \begin{function}[EXP]{\fp_use:N, \fp_use:c}
+% \begin{syntax}
+% \cs{fp_use:N} \meta{floating point variable}
+% \end{syntax}
+% Inserts the value of the \meta{floating point variable} into the
+% input stream. The value will be given as a real number without any
+% exponent part, and will always include a decimal point. For example,
+% \begin{verbatim}
+% \fp_new:Nn \test
+% \fp_set:Nn \test { 1.234 e 5 }
+% \fp_use:N \test
+% \end{verbatim}
+% will insert |12345.00000| into the input stream.
+% As illustrated, a floating point will always be inserted with ten
+% significant digits given. Very large and very small values will
+% include additional zeros for place value.
+% \end{function}
+%
+% \begin{function}{\fp_show:N, \fp_show:c}
+% \begin{syntax}
+% \cs{fp_show:N} \meta{floating point variable}
+% \end{syntax}
+% Displays the content of the \meta{floating point variable} on the
+% terminal.
+% \end{function}
+%
+% \section{Conversion of floating point values to other formats}
+%
+% It is useful to be able to convert floating point variables to
+% other forms. These functions are expandable, so that the material
+% can be used in a variety of contexts. The \cs{fp_use:N} function
+% should also be consulted in this context, as it will insert the
+% value of the floating point variable as a real number.
+%
+% \begin{function}[EXP]{\fp_to_dim:N, \fp_to_dim:c}
+% \begin{syntax}
+% \cs{fp_to_dim:N} \meta{floating point variable}
+% \end{syntax}
+% Inserts the value of the \meta{floating point variable}
+% into the input stream converted into a dimension in points.
+% \end{function}
+%
+% \begin{function}[EXP]{\fp_to_int:N, \fp_to_int:c}
+% \begin{syntax}
+% \cs{fp_to_int:N} \meta{floating point variable}
+% \end{syntax}
+% Inserts the integer value of the \meta{floating point variable}
+% into the input stream. The decimal part of the number will not be
+% included, but will be used to round the integer.
+% \end{function}
+%
+% \begin{function}[EXP]{\fp_to_tl:N, \fp_to_tl:c}
+% \begin{syntax}
+% \cs{fp_to_tl:N} \meta{floating point variable}
+% \end{syntax}
+% Inserts a representation of the \meta{floating point variable} into
+% the input stream as a token list. The representation follows the
+% conventions of a pocket calculator:
+% \begin{center}
+% \ttfamily
+% \begin{tabular}{r@{.}lr@{.}l}
+% \toprule
+% \multicolumn{2}{l}{\rmfamily{Floating point value}} &
+% \multicolumn{2}{l}{\rmfamily{Representation}} \\
+% \midrule
+% 1 & 234000000000e0 & 1 & 234 \\
+% -1 & 234000000000e0 & -1 & 234 \\
+% 1 & 234000000000e3 & \multicolumn{2}{l}{1234} \\
+% 1 & 234000000000e13 & \multicolumn{2}{l}{1234e13} \\
+% 1 & 234000000000e-1 & 0 & 1234 \\
+% 1 & 234000000000e-2 & 0 & 01234 \\
+% 1 & 234000000000e-3 & 1 & 234e-3 \\
+% \bottomrule
+% \end{tabular}
+% \end{center}
+% Notice that trailing zeros are removed in this process, and that
+% numbers which do not require a decimal part do \emph{not} include
+% a decimal marker.
+% \end{function}
+%
+% \section{Rounding floating point values}
+%
+% The module can round floating point values to either decimal places
+% or significant figures using the usual method in which exact halves
+% are rounded up.
+%
+% \begin{function}{\fp_round_figures:Nn, \fp_round_figures:cn}
+% \begin{syntax}
+% \cs{fp_round_figures:Nn} \meta{floating point variable} \Arg{target}
+% \end{syntax}
+% Rounds the \meta{floating point variable} to the \meta{target} number
+% of significant figures (an integer expression). The rounding is
+% carried out locally.
+% \end{function}
+%
+% \begin{function}{\fp_ground_figures:Nn, \fp_ground_figures:cn}
+% \begin{syntax}
+% \cs{fp_ground_figures:Nn} \meta{floating point variable} \Arg{target}
+% \end{syntax}
+% Rounds the \meta{floating point variable} to the \meta{target} number
+% of significant figures (an integer expression). The rounding is
+% carried out globally.
+% \end{function}
+%
+% \begin{function}{\fp_round_places:Nn, \fp_round_places:cn}
+% \begin{syntax}
+% \cs{fp_round_places:Nn} \meta{floating point variable} \Arg{target}
+% \end{syntax}
+% Rounds the \meta{floating point variable} to the \meta{target} number
+% of decimal places (an integer expression). The rounding is
+% carried out locally.
+% \end{function}
+%
+% \begin{function}{\fp_ground_places:Nn, \fp_ground_places:cn}
+% \begin{syntax}
+% \cs{fp_ground_places:Nn} \meta{floating point variable} \Arg{target}
+% \end{syntax}
+% Rounds the \meta{floating point variable} to the \meta{target} number
+% of decimal places (an integer expression). The rounding is
+% carried out globally.
+% \end{function}
+%
+% \section{Floating-point conditionals}
+%
+% \begin{function}[EXP,pTF]{\fp_if_undefined:N}
+% \begin{syntax}
+% \cs{fp_if_undefined_p:N} \meta{fixed-point}
+% \cs{fp_if_undefined:NTF} \meta{fixed-point}
+% ~~\Arg{true code} \Arg{false code}
+% \end{syntax}
+% Tests if \meta{floating point} is undefined (\emph{i.e.}~equal to the
+% special \cs{c_undefined_fp} variable). The branching versions then
+% leave either \meta{true code} or \meta{false code} in the input
+% stream, as appropriate to the truth of the test and the variant of
+% the function chosen. The logical truth of the test is left in the
+% input stream by the predicate version.
+% \end{function}
+%
+% \begin{function}[EXP]{\fp_if_zero:N}
+% \begin{syntax}
+% \cs{fp_if_zero_p:N} \meta{fixed-point}
+% \cs{fp_if_zero:NTF} \meta{fixed-point} \Arg{true code} \Arg{false code}
+% \end{syntax}
+% Tests if \meta{floating point} is equal to zero (\emph{i.e.}~equal to
+% the special \cs{c_zero_fp} variable). The branching versions then
+% leave either \meta{true code} or \meta{false code} in the input
+% stream, as appropriate to the truth of the test and the variant of
+% the function chosen. The logical truth of the test is left in the
+% input stream by the predicate version.
+% \end{function}
+%
+% \begin{function}[TF]{\fp_compare:nNn}
+% \begin{syntax}
+% \cs{fp_compare:nNnTF}
+% ~~\Arg{floating point1} \meta{relation} \Arg{floating point2}
+% ~~\Arg{true code} \Arg{false code}
+% \end{syntax}
+% This function compared the two \meta{floating point} values, which
+% may be stored as \texttt{fp} variables, using the \meta{relation}:
+% \begin{center}
+% \begin{tabular}{ll}
+% Equal & |=| \\
+% Greater than & |>| \\
+% Less than & |<| \\
+% \end{tabular}
+% \end{center}
+% Either \meta{true code} or \meta{false code} is then left in the
+% input stream, as appropriate to the truth of the test and the variant
+% of the function chosen. The tests treat undefined floating points as
+% zero as the comparison is intended for real numbers only.
+% \end{function}
+%
+% \begin{function}[TF]{\fp_compare:n}
+% \begin{syntax}
+% \cs{fp_compare:nTF}
+% ~~\{ \meta{floating point1} \meta{relation} \meta{floating point2} \}
+% ~~\Arg{true code} \Arg{false code}
+% \end{syntax}
+% This function compared the two \meta{floating point} values, which
+% may be stored as \texttt{fp} variables, using the \meta{relation}:
+% \begin{center}
+% \begin{tabular}{ll}
+% Equal & |=| or |==| \\
+% Greater than & |>| \\
+% Greater than or equal & |>=| \\
+% Less than & |<| \\
+% Less than or equal & |<=| \\
+% Not equal & |!=| \\
+% \end{tabular}
+% \end{center}
+% Either \meta{true code} or \meta{false code} is then left in the
+% input stream, as appropriate to the truth of the test and the variant
+% of the function chosen. The tests treat undefined floating points as
+% zero as the comparison is intended for real numbers only.
+% \end{function}
+%
+% \section{Unary floating-point operations}
+%
+% The unary operations alter the value stored within an \texttt{fp}
+% variable.
+%
+% \begin{function}{\fp_abs:N, \fp_abs:c}
+% \begin{syntax}
+% \cs{fp_abs:N} \meta{floating point variable}
+% \end{syntax}
+% Converts the \meta{floating point variable} to its absolute value,
+% assigning the result within the current \TeX\ group.
+% \end{function}
+%
+% \begin{function}{\fp_gabs:N, \fp_gabs:c}
+% \begin{syntax}
+% \cs{fp_gabs:N} \meta{floating point variable}
+% \end{syntax}
+% Converts the \meta{floating point variable} to its absolute value,
+% assigning the result globally.
+% \end{function}
+%
+% \begin{function}{\fp_neg:N, \fp_neg:c}
+% \begin{syntax}
+% \cs{fp_neg:N} \meta{floating point variable}
+% \end{syntax}
+% Reverse the sign of the \meta{floating point variable}, assigning the
+% result within the current \TeX\ group.
+% \end{function}
+%
+% \begin{function}{\fp_gneg:N, \fp_gneg:c}
+% \begin{syntax}
+% \cs{fp_gneg:N} \meta{floating point variable}
+% \end{syntax}
+% Reverse the sign of the \meta{floating point variable}, assigning the
+% result globally.
+% \end{function}
+%
+% \section{Floating-point arithmetic}
+%
+% Binary arithmetic operations act on the value stored in an
+% \texttt{fp}, so for example
+% \begin{verbatim}
+% \fp_set:Nn \l_my_fp { 1.234 }
+% \fp_sub:Nn \l_my_fp { 5.678 }
+% \end{verbatim}
+% sets \cs{l_my_fp} to the result of $1.234 - 5.678$
+% (\emph{i.e.}~$-4.444$).
+%
+% \begin{function}{\fp_add:Nn, \fp_add:cn}
+% \begin{syntax}
+% \cs{fp_add:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Adds the \meta{value} to the \meta{floating point}, making the
+% assignment within the current \TeX{} group level.
+% \end{function}
+%
+% \begin{function}{\fp_gadd:Nn, \fp_gadd:cn}
+% \begin{syntax}
+% \cs{fp_gadd:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Adds the \meta{value} to the \meta{floating point}, making the
+% assignment globally.
+% \end{function}
+%
+% \begin{function}{\fp_sub:Nn, \fp_sub:cn}
+% \begin{syntax}
+% \cs{fp_sub:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Subtracts the \meta{value} from the \meta{floating point}, making the
+% assignment within the current \TeX{} group level.
+% \end{function}
+%
+% \begin{function}{\fp_gsub:Nn, \fp_gsub:cn}
+% \begin{syntax}
+% \cs{fp_gsub:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Subtracts the \meta{value} from the \meta{floating point}, making the
+% assignment globally.
+% \end{function}
+%
+% \begin{function}{\fp_mul:Nn, \fp_mul:cn}
+% \begin{syntax}
+% \cs{fp_mul:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Multiples the \meta{floating point} by the \meta{value}, making the
+% assignment within the current \TeX{} group level.
+% \end{function}
+%
+% \begin{function}{\fp_gmul:Nn, \fp_gmul:cn}
+% \begin{syntax}
+% \cs{fp_gmul:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Multiples the \meta{floating point} by the \meta{value}, making the
+% assignment globally.
+% \end{function}
+%
+% \begin{function}{\fp_div:Nn, \fp_div:cn}
+% \begin{syntax}
+% \cs{fp_div:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Divides the \meta{floating point} by the \meta{value}, making the
+% assignment within the current \TeX{} group level. If the \meta{value}
+% is zero, the \meta{floating point} will be set to
+% \cs{c_undefined_fp}. The assignment is local.
+% \end{function}
+%
+% \begin{function}{\fp_gdiv:Nn, \fp_gdiv:cn}
+% \begin{syntax}
+% \cs{fp_gdiv:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Divides the \meta{floating point} by the \meta{value}, making the
+% assignment globally. If the \meta{value} is zero, the
+% \meta{floating point} will be set to \cs{c_undefined_fp}.
+% The assignment is global.
+% \end{function}
+%
+% \section{Floating-point power operations}
+%
+% \begin{function}{\fp_pow:Nn, \fp_pow:cn}
+% \begin{syntax}
+% \cs{fp_pow:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Raises the \meta{floating point} to the given \meta{value}. If the
+% \meta{floating point} is negative, then the \meta{value} should be
+% either a positive real number or a negative integer. If the
+% \meta{floating point} is positive, then the \meta{value} may be any
+% real value. Mathematically invalid operations such as $0^{0}$
+% will give set the \meta{floating point} to to \cs{c_undefined_fp}.
+% The assignment is local.
+% \end{function}
+%
+% \begin{function}{\fp_gpow:Nn, \fp_gpow:cn}
+% \begin{syntax}
+% \cs{fp_gpow:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Raises the \meta{floating point} to the given \meta{value}. If the
+% \meta{floating point} is negative, then the \meta{value} should be
+% either a positive real number or a negative integer. If the
+% \meta{floating point} is positive, then the \meta{value} may be any
+% real value. Mathematically invalid operations such as $0^{0}$
+% will give set the \meta{floating point} to to \cs{c_undefined_fp}.
+% The assignment is global.
+% \end{function}
+%
+% \section{Exponential and logarithm functions}
+%
+% \begin{function}{\fp_exp:Nn, \fp_exp:cn}
+% \begin{syntax}
+% \cs{fp_exp:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Calculates the exponential of the \meta{value} and assigns this
+% to the \meta{floating point}. The assignment is local.
+% \end{function}
+%
+% \begin{function}{\fp_gexp:Nn, \fp_gexp:cn}
+% \begin{syntax}
+% \cs{fp_gexp:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Calculates the exponential of the \meta{value} and assigns this
+% to the \meta{floating point}. The assignment is global.
+% \end{function}
+%
+% \begin{function}{\fp_ln:Nn, \fp_ln:cn}
+% \begin{syntax}
+% \cs{fp_ln:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Calculates the natural logarithm of the \meta{value} and assigns
+% this to the \meta{floating point}. The assignment is local.
+% \end{function}
+%
+% \begin{function}{\fp_gln:Nn, \fp_gln:cn}
+% \begin{syntax}
+% \cs{fp_gln:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Calculates the natural logarithm of the \meta{value} and assigns
+% this to the \meta{floating point}. The assignment is global.
+% \end{function}
+%
+% \section{Trigonometric functions}
+%
+% The trigonometric functions all work in radians. They accept a maximum
+% input value of $100\,000\,000$, as there are issues with range
+% reduction and very large input values.
+%
+% \begin{function}{\fp_sin:Nn, \fp_sin:cn}
+% \begin{syntax}
+% \cs{fp_sin:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the sine of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% local.
+% \end{function}
+%
+% \begin{function}{\fp_gsin:Nn, \fp_gsin:cn}
+% \begin{syntax}
+% \cs{fp_gsin:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the sine of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% global.
+% \end{function}
+%
+% \begin{function}{\fp_cos:Nn, \fp_cos:cn}
+% \begin{syntax}
+% \cs{fp_cos:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the cosine of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% local.
+% \end{function}
+%
+% \begin{function}{\fp_gcos:Nn, \fp_gcos:cn}
+% \begin{syntax}
+% \cs{fp_gcos:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the cosine of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% global.
+% \end{function}
+%
+% \begin{function}{\fp_tan:Nn, \fp_tan:cn}
+% \begin{syntax}
+% \cs{fp_tan:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the tangent of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% local.
+% \end{function}
+%
+% \begin{function}{\fp_gtan:Nn, \fp_gtan:cn}
+% \begin{syntax}
+% \cs{fp_gtan:Nn} \meta{floating point} \Arg{value}
+% \end{syntax}
+% Assigns the tangent of the \meta{value} to the \meta{floating point}.
+% The \meta{value} should be given in radians. The assignment is
+% global.
+% \end{function}
+%
+% \section{Constant floating point values}
+%
+% \begin{variable}{\c_e_fp}
+% The value of the base of natural numbers, $\mathrm{e}$.
+% \end{variable}
+%
+% \begin{variable}{\c_one_fp}
+% A floating point variable with permanent value $1$: used for
+% speeding up some comparisons.
+% \end{variable}
+%
+% \begin{variable}{\c_pi_fp}
+% The value of $\pi$.
+% \end{variable}
+%
+% \begin{variable}{\c_undefined_fp}
+% A special marker floating point variable representing the result of
+% an operation which does not give a defined result (such as division
+% by $0$).
+% \end{variable}
+%
+% \begin{variable}{\c_zero_fp}
+% A permanently zero floating point variable.
+% \end{variable}
+%
+% \section{Notes on the floating point unit}
+%
+% As calculation of the elemental transcendental functions is
+% computationally expensive compared to storage of results, after
+% calculating a trigonometric function, exponent, \emph{etc.}~the module
+% stored the result for reuse. Thus the performance of the module for
+% repeated operations, most probably trigonometric functions, should be
+% much higher than if the values were re-calculated every time they
+% were needed.
+%
+% Anyone with experience of programming floating point calculations will
+% know that this is a complex area. The aim of the unit is to be
+% accurate enough for the likely applications in a typesetting context.
+% The arithmetic operations are therefore intended to provide ten digit
+% accuracy with the last digit accurate to $\pm 1$. The elemental
+% transcendental functions may not provide such high accuracy in every
+% case, although the design aim has been to provide $10$ digit
+% accuracy for cases likely to be relevant in typesetting situations.
+% A good overview of the challenges in this area can be found in
+% J.-M.~Muller, \emph{Elementary functions: algorithms and
+% implementation}, 2nd edition, Birkh{\"a}uer Boston, New York, USA,
+% 2006.
+%
+% The internal representation of numbers is tuned to the needs of the
+% underlying \TeX{} system. This means that the format is somewhat
+% different from that used in, for example, computer floating point
+% units. Programming in \TeX{} makes it most convenient to use a
+% radix $10$ system, using \TeX{} \texttt{count} registers for
+% storage and taking advantage where possible of delimited arguments.
+%
+% \end{documentation}
+%
+% \begin{implementation}
+%
+% \section{\pkg{l3fp} Implementation}
+%
+% \TestFiles{m3fp003.lvt}
+%
+% \begin{macrocode}
+%<*initex|package>
+% \end{macrocode}
+%
+% \begin{macrocode}
+%<*package>
+\ProvidesExplPackage
+ {\ExplFileName}{\ExplFileDate}{\ExplFileVersion}{\ExplFileDescription}
+\package_check_loaded_expl:
+%</package>
+% \end{macrocode}
+%
+% \subsection{Constants}
+%
+% \begin{variable}{\c_forty_four}
+% \begin{variable}{\c_one_million}
+% \begin{variable}{\c_one_hundred_million}
+% \begin{variable}{\c_five_hundred_million}
+% \begin{variable}{\c_one_thousand_million}
+% There is some speed to gain by moving numbers into fixed positions.
+% \begin{macrocode}
+\int_const:Nn \c_forty_four { 44 }
+\int_const:Nn \c_one_million { 1 000 000 }
+\int_const:Nn \c_one_hundred_million { 100 000 000 }
+\int_const:Nn \c_five_hundred_million { 500 000 000 }
+\int_const:Nn \c_one_thousand_million { 1 000 000 000 }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\c_fp_pi_by_four_decimal_int}
+% \begin{variable}{\c_fp_pi_by_four_extended_int}
+% \begin{variable}{\c_fp_pi_decimal_int}
+% \begin{variable}{\c_fp_pi_extended_int}
+% \begin{variable}{\c_fp_two_pi_decimal_int}
+% \begin{variable}{\c_fp_two_pi_extended_int}
+% Parts of $\pi$ for trigonometric range reduction, implemented
+% as \texttt{int} variables for speed.
+% \begin{macrocode}
+\int_new:N \c_fp_pi_by_four_decimal_int
+\int_set:Nn \c_fp_pi_by_four_decimal_int { 785 398 158 }
+\int_new:N \c_fp_pi_by_four_extended_int
+\int_set:Nn \c_fp_pi_by_four_extended_int { 897 448 310 }
+\int_new:N \c_fp_pi_decimal_int
+\int_set:Nn \c_fp_pi_decimal_int { 141 592 653 }
+\int_new:N \c_fp_pi_extended_int
+\int_set:Nn \c_fp_pi_extended_int { 589 793 238 }
+\int_new:N \c_fp_two_pi_decimal_int
+\int_set:Nn \c_fp_two_pi_decimal_int { 283 185 307 }
+\int_new:N \c_fp_two_pi_extended_int
+\int_set:Nn \c_fp_two_pi_extended_int { 179 586 477 }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\c_e_fp}
+% The value $\mathrm{e}$ as a \enquote{machine number}.
+% \begin{macrocode}
+\tl_const:Nn \c_e_fp { + 2.718281828 e 0 }
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\c_one_fp}
+% The constant value $1$: used for fast comparisons.
+% \begin{macrocode}
+\tl_const:Nn \c_one_fp { + 1.000000000 e 0 }
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\c_pi_fp}
+% The value $\pi$ as a \enquote{machine number}.
+% \begin{macrocode}
+\tl_const:Nn \c_pi_fp { + 3.141592654 e 0 }
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\c_undefined_fp}
+% A marker for undefined values.
+% \begin{macrocode}
+\tl_const:Nn \c_undefined_fp { X 0.000000000 e 0 }
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\c_zero_fp}
+% The constant zero value.
+% \begin{macrocode}
+\tl_const:Nn \c_zero_fp { + 0.000000000 e 0 }
+% \end{macrocode}
+% \end{variable}
+%
+% \subsection{Variables}
+%
+% \begin{variable}{\l_fp_arg_tl}
+% A token list to store the formalised representation of the input
+% for transcendental functions.
+% \begin{macrocode}
+\tl_new:N \l_fp_arg_tl
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_count_int}
+% A counter for things like the number of divisions possible.
+% \begin{macrocode}
+\int_new:N \l_fp_count_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_div_offset_int}
+% When carrying out division, an offset is used for the results to
+% get the decimal part correct.
+% \begin{macrocode}
+\int_new:N \l_fp_div_offset_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_exp_integer_int}
+% \begin{variable}{\l_fp_exp_decimal_int}
+% \begin{variable}{\l_fp_exp_extended_int}
+% \begin{variable}{\l_fp_exp_exponent_int}
+% Used for the calculation of exponent values.
+% \begin{macrocode}
+\int_new:N \l_fp_exp_integer_int
+\int_new:N \l_fp_exp_decimal_int
+\int_new:N \l_fp_exp_extended_int
+\int_new:N \l_fp_exp_exponent_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_input_a_sign_int}
+% \begin{variable}{\l_fp_input_a_integer_int}
+% \begin{variable}{\l_fp_input_a_decimal_int}
+% \begin{variable}{\l_fp_input_a_exponent_int}
+% \begin{variable}{\l_fp_input_b_sign_int}
+% \begin{variable}{\l_fp_input_b_integer_int}
+% \begin{variable}{\l_fp_input_b_decimal_int}
+% \begin{variable}{\l_fp_input_b_exponent_int}
+% Storage for the input: two storage areas as there are at most two
+% inputs.
+% \begin{macrocode}
+\int_new:N \l_fp_input_a_sign_int
+\int_new:N \l_fp_input_a_integer_int
+\int_new:N \l_fp_input_a_decimal_int
+\int_new:N \l_fp_input_a_exponent_int
+\int_new:N \l_fp_input_b_sign_int
+\int_new:N \l_fp_input_b_integer_int
+\int_new:N \l_fp_input_b_decimal_int
+\int_new:N \l_fp_input_b_exponent_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_input_a_extended_int}
+% \begin{variable}{\l_fp_input_b_extended_int}
+% For internal use, \enquote{extended} floating point numbers are
+% needed.
+% \begin{macrocode}
+\int_new:N \l_fp_input_a_extended_int
+\int_new:N \l_fp_input_b_extended_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_mul_a_i_int}
+% \begin{variable}{\l_fp_mul_a_ii_int}
+% \begin{variable}{\l_fp_mul_a_iii_int}
+% \begin{variable}{\l_fp_mul_a_iv_int}
+% \begin{variable}{\l_fp_mul_a_v_int}
+% \begin{variable}{\l_fp_mul_a_vi_int}
+% \begin{variable}{\l_fp_mul_b_i_int}
+% \begin{variable}{\l_fp_mul_b_ii_int}
+% \begin{variable}{\l_fp_mul_b_iii_int}
+% \begin{variable}{\l_fp_mul_b_iv_int}
+% \begin{variable}{\l_fp_mul_b_v_int}
+% \begin{variable}{\l_fp_mul_b_vi_int}
+% Multiplication requires that the decimal part is split into parts
+% so that there are no overflows.
+% \begin{macrocode}
+\int_new:N \l_fp_mul_a_i_int
+\int_new:N \l_fp_mul_a_ii_int
+\int_new:N \l_fp_mul_a_iii_int
+\int_new:N \l_fp_mul_a_iv_int
+\int_new:N \l_fp_mul_a_v_int
+\int_new:N \l_fp_mul_a_vi_int
+\int_new:N \l_fp_mul_b_i_int
+\int_new:N \l_fp_mul_b_ii_int
+\int_new:N \l_fp_mul_b_iii_int
+\int_new:N \l_fp_mul_b_iv_int
+\int_new:N \l_fp_mul_b_v_int
+\int_new:N \l_fp_mul_b_vi_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_mul_output_int}
+% \begin{variable}{\l_fp_mul_output_tl}
+% Space for multiplication results.
+% \begin{macrocode}
+\int_new:N \l_fp_mul_output_int
+\tl_new:N \l_fp_mul_output_tl
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_output_sign_int}
+% \begin{variable}{\l_fp_output_integer_int}
+% \begin{variable}{\l_fp_output_decimal_int}
+% \begin{variable}{\l_fp_output_exponent_int}
+% Output is stored in the same way as input.
+% \begin{macrocode}
+\int_new:N \l_fp_output_sign_int
+\int_new:N \l_fp_output_integer_int
+\int_new:N \l_fp_output_decimal_int
+\int_new:N \l_fp_output_exponent_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_output_extended_int}
+% Again, for calculations an extended part.
+% \begin{macrocode}
+\int_new:N \l_fp_output_extended_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_round_carry_bool}
+% To indicate that a digit needs to be carried forward.
+% \begin{macrocode}
+\bool_new:N \l_fp_round_carry_bool
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_round_decimal_tl}
+% A temporary store when rounding, to build up the decimal part without
+% needing to do any maths.
+% \begin{macrocode}
+\tl_new:N \l_fp_round_decimal_tl
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_round_position_int}
+% \begin{variable}{\l_fp_round_target_int}
+% Used to check the position for rounding.
+% \begin{macrocode}
+\int_new:N \l_fp_round_position_int
+\int_new:N \l_fp_round_target_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_sign_tl}
+% There are places where the sign needs to be set up \enquote{early},
+% so that the registers can be re-used.
+% \begin{macrocode}
+\tl_new:N \l_fp_sign_tl
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_split_sign_int}
+% When splitting the input it is fastest to use a fixed name for the
+% sign part, and to transfer it after the split is complete.
+% \begin{macrocode}
+\int_new:N \l_fp_split_sign_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_tmp_int}
+% A scratch \texttt{int}: used only where the value is not carried
+% forward.
+% \begin{macrocode}
+\int_new:N \l_fp_tmp_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_tmp_tl}
+% A scratch token list variable for expanding material.
+% \begin{macrocode}
+\tl_new:N \l_fp_tmp_tl
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_trig_octant_int}
+% To track which octant the trigonometric input is in.
+% \begin{macrocode}
+\int_new:N \l_fp_trig_octant_int
+% \end{macrocode}
+% \end{variable}
+%
+% \begin{variable}{\l_fp_trig_sign_int}
+% \begin{variable}{\l_fp_trig_decimal_int}
+% \begin{variable}{\l_fp_trig_extended_int}
+% Used for the calculation of trigonometric values.
+% \begin{macrocode}
+\int_new:N \l_fp_trig_sign_int
+\int_new:N \l_fp_trig_decimal_int
+\int_new:N \l_fp_trig_extended_int
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \subsection{Parsing numbers}
+%
+% \begin{macro}{\fp_read:N}
+% \begin{macro}[aux]{\fp_read_aux:w}
+% Reading a stored value is made easier as the format is designed to
+% match the delimited function. This is always used to read the first
+% value (register |a|).
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_read:N #1
+ { \exp_after:wN \fp_read_aux:w #1 \q_stop }
+\cs_new_protected_nopar:Npn \fp_read_aux:w #1#2 . #3 e #4 \q_stop
+ {
+ \if:w #1 -
+ \l_fp_input_a_sign_int \c_minus_one
+ \else:
+ \l_fp_input_a_sign_int \c_one
+ \fi:
+ \l_fp_input_a_integer_int #2 \scan_stop:
+ \l_fp_input_a_decimal_int #3 \scan_stop:
+ \l_fp_input_a_exponent_int #4 \scan_stop:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_split:Nn}
+% \begin{macro}[aux]{\fp_split_sign:}
+% \begin{macro}[aux]{\fp_split_exponent:}
+% \begin{macro}[aux]{\fp_split_aux_i:w}
+% \begin{macro}[aux]{\fp_split_aux_ii:w}
+% \begin{macro}[aux]{\fp_split_aux_iii:w}
+% \begin{macro}[aux]{\fp_split_decimal:w}
+% \begin{macro}[aux]{\fp_split_decimal_aux:w}
+% The aim here is to use as much of \TeX{}'s mechanism as possible to pick
+% up the numerical input without any mistakes. In particular, negative
+% numbers have to be filtered out first in case the integer part is
+% $0$ (in which case \TeX{} would drop the |-| sign). That process
+% has to be done in a loop for cases where the sign is repeated.
+% Finding an exponent is relatively easy, after which the next phase is
+% to find the integer part, which will terminate with a |.|, and trigger
+% the decimal-finding code. The later will allow the decimal to be too
+% long, truncating the result.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_split:Nn #1#2
+ {
+ \tl_set:Nx \l_fp_tmp_tl {#2}
+ \tl_set_rescan:Nno \l_fp_tmp_tl { \char_set_catcode_ignore:n { 32 } }
+ { \l_fp_tmp_tl }
+ \l_fp_split_sign_int \c_one
+ \fp_split_sign:
+ \use:c { l_fp_input_ #1 _sign_int } \l_fp_split_sign_int
+ \exp_after:wN \fp_split_exponent:w \l_fp_tmp_tl e e \q_stop #1
+ }
+\cs_new_protected_nopar:Npn \fp_split_sign:
+ {
+ \if_int_compare:w \pdftex_strcmp:D
+ { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { - }
+ = \c_zero
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \exp_after:wN
+ \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop
+ }
+ \l_fp_split_sign_int -\l_fp_split_sign_int
+ \exp_after:wN \fp_split_sign:
+ \else:
+ \if_int_compare:w \pdftex_strcmp:D
+ { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { + }
+ = \c_zero
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \exp_after:wN
+ \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop
+ }
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_split_sign:
+ \fi:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_split_exponent:w #1 e #2 e #3 \q_stop #4
+ {
+ \use:c { l_fp_input_ #4 _exponent_int }
+ \int_eval:w 0 #2 \scan_stop:
+ \tex_afterassignment:D \fp_split_aux_i:w
+ \use:c { l_fp_input_ #4 _integer_int }
+ \int_eval:w 0 #1 . . \q_stop #4
+ }
+\cs_new_protected_nopar:Npn \fp_split_aux_i:w #1 . #2 . #3 \q_stop
+ { \fp_split_aux_ii:w #2 000000000 \q_stop }
+\cs_new_protected_nopar:Npn \fp_split_aux_ii:w #1#2#3#4#5#6#7#8#9
+ { \fp_split_aux_iii:w {#1#2#3#4#5#6#7#8#9} }
+\cs_new_protected_nopar:Npn \fp_split_aux_iii:w #1#2 \q_stop
+ {
+ \l_fp_tmp_int 1 #1 \scan_stop:
+ \exp_after:wN \fp_split_decimal:w
+ \int_use:N \l_fp_tmp_int 000000000 \q_stop
+ }
+\cs_new_protected_nopar:Npn \fp_split_decimal:w #1#2#3#4#5#6#7#8#9
+ { \fp_split_decimal_aux:w {#2#3#4#5#6#7#8#9} }
+\cs_new_protected_nopar:Npn \fp_split_decimal_aux:w #1#2#3 \q_stop #4
+ {
+ \use:c { l_fp_input_ #4 _decimal_int } #1#2 \scan_stop:
+ \if_int_compare:w
+ \int_eval:w
+ \use:c { l_fp_input_ #4 _integer_int } +
+ \use:c { l_fp_input_ #4 _decimal_int }
+ \scan_stop:
+ = \c_zero
+ \use:c { l_fp_input_ #4 _sign_int } \c_one
+ \fi:
+ \if_int_compare:w
+ \use:c { l_fp_input_ #4 _integer_int } < \c_one_thousand_million
+ \else:
+ \exp_after:wN \fp_overflow_msg:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_standardise:NNNN}
+% \begin{macro}[aux]{\fp_standardise_aux:NNNN}
+% \begin{macro}[aux]{\fp_standardise_aux:}
+% \begin{macro}[aux]{\fp_standardise_aux:w}
+% The idea here is to shift the input into a known exponent range. This
+% is done using \TeX{} tokens where possible, as this is faster than
+% arithmetic.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_standardise:NNNN #1#2#3#4
+ {
+ \if_int_compare:w
+ \int_eval:w #2 + #3 = \c_zero
+ #1 \c_one
+ #4 \c_zero
+ \exp_after:wN \use_none:nnnn
+ \else:
+ \exp_after:wN \fp_standardise_aux:NNNN
+ \fi:
+ #1#2#3#4
+ }
+\cs_new_protected_nopar:Npn \fp_standardise_aux:NNNN #1#2#3#4
+ {
+ \cs_set_protected_nopar:Npn \fp_standardise_aux:
+ {
+ \if_int_compare:w #2 = \c_zero
+ \tex_advance:D #3 \c_one_thousand_million
+ \exp_after:wN \fp_standardise_aux:w
+ \int_use:N #3 \q_stop
+ \exp_after:wN \fp_standardise_aux:
+ \fi:
+ }
+ \cs_set_protected_nopar:Npn
+ \fp_standardise_aux:w ##1##2##3##4##5##6##7##8##9 \q_stop
+ {
+ #2 ##2 \scan_stop:
+ #3 ##3##4##5##6##7##8##9 0 \scan_stop:
+ \tex_advance:D #4 \c_minus_one
+ }
+ \fp_standardise_aux:
+ \cs_set_protected_nopar:Npn \fp_standardise_aux:
+ {
+ \if_int_compare:w #2 > \c_nine
+ \tex_advance:D #2 \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_standardise_aux:w \int_use:N #2
+ \exp_after:wN \fp_standardise_aux:
+ \fi:
+ }
+ \cs_set_protected_nopar:Npn
+ \fp_standardise_aux:w ##1##2##3##4##5##6##7##8##9
+ {
+ #2 ##1##2##3##4##5##6##7##8 \scan_stop:
+ \tex_advance:D #3 \c_one_thousand_million
+ \tex_divide:D #3 \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ ##9
+ \exp_after:wN \use_none:n \int_use:N #3
+ }
+ #3 \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D #4 \c_one
+ }
+ \fp_standardise_aux:
+ \if_int_compare:w #4 < \c_one_hundred
+ \if_int_compare:w #4 > -\c_one_hundred
+ \else:
+ #1 \c_one
+ #2 \c_zero
+ #3 \c_zero
+ #4 \c_zero
+ \fi:
+ \else:
+ \exp_after:wN \fp_overflow_msg:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_standardise_aux: { }
+\cs_new_protected_nopar:Npn \fp_standardise_aux:w { }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Internal utilities}
+%
+% \begin{macro}{\fp_level_input_exponents:}
+% \begin{macro}[aux]{\fp_level_input_exponents_a:}
+% \begin{macro}[aux]{\fp_level_input_exponents_a:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_level_input_exponents_b:}
+% \begin{macro}[aux]{\fp_level_input_exponents_b:NNNNNNNNN}
+% The routines here are similar to those used to standardise the
+% exponent. However, the aim here is different: the two exponents need
+% to end up the same.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_level_input_exponents:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int
+ \exp_after:wN \fp_level_input_exponents_a:
+ \else:
+ \exp_after:wN \fp_level_input_exponents_b:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_level_input_exponents_a:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int
+ \tex_advance:D \l_fp_input_b_integer_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_level_input_exponents_a:NNNNNNNNN
+ \int_use:N \l_fp_input_b_integer_int
+ \exp_after:wN \fp_level_input_exponents_a:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_level_input_exponents_a:NNNNNNNNN
+ #1#2#3#4#5#6#7#8#9
+ {
+ \l_fp_input_b_integer_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \tex_advance:D \l_fp_input_b_decimal_int \c_one_thousand_million
+ \tex_divide:D \l_fp_input_b_decimal_int \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #9
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_b_decimal_int
+ }
+ \l_fp_input_b_decimal_int \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D \l_fp_input_b_exponent_int \c_one
+ }
+\cs_new_protected_nopar:Npn \fp_level_input_exponents_b:
+ {
+ \if_int_compare:w \l_fp_input_b_exponent_int > \l_fp_input_a_exponent_int
+ \tex_advance:D \l_fp_input_a_integer_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_level_input_exponents_b:NNNNNNNNN
+ \int_use:N \l_fp_input_a_integer_int
+ \exp_after:wN \fp_level_input_exponents_b:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_level_input_exponents_b:NNNNNNNNN
+ #1#2#3#4#5#6#7#8#9
+ {
+ \l_fp_input_a_integer_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \tex_divide:D \l_fp_input_a_decimal_int \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #9
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ }
+ \l_fp_input_a_decimal_int \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D \l_fp_input_a_exponent_int \c_one
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}[aux]{\fp_tmp:w}
+% Used for output of results, cutting down on \cs{exp_after:wN}.
+% This is just a place holder definition.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_tmp:w #1#2 { }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsection{Operations for \texttt{fp} variables}
+%
+% The format of \texttt{fp} variables is tightly defined, so that
+% they can be read quickly by the internal code. The format is a single
+% sign token, a single number, the decimal point, nine decimal numbers,
+% an |e| and finally the exponent. This final part may vary in length.
+% When stored, floating points will always be stored with a value in
+% the integer position unless the number is zero.
+%
+% \begin{macro}{\fp_new:N, \fp_new:c}
+% \UnitTested
+% Fixed-points always have a value, and of course this has to be
+% initialised globally.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_new:N #1
+ {
+ \tl_new:N #1
+ \tl_gset_eq:NN #1 \c_zero_fp
+ }
+\cs_generate_variant:Nn \fp_new:N { c }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_const:Nn, \fp_const:cn}
+% A simple wrapper.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_const:Nn #1#2
+ {
+ \fp_new:N #1
+ \fp_gset:Nn #1 {#2}
+ }
+\cs_generate_variant:Nn \fp_const:Nn { c }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_zero:N, \fp_zero:c}
+% \UnitTested
+% \begin{macro}{\fp_gzero:N, \fp_gzero:c}
+% \UnitTested
+% Zeroing fixed-points is pretty obvious.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_zero:N #1
+ { \tl_set_eq:NN #1 \c_zero_fp }
+\cs_new_protected_nopar:Npn \fp_gzero:N #1
+ { \tl_gset_eq:NN #1 \c_zero_fp }
+\cs_generate_variant:Nn \fp_zero:N { c }
+\cs_generate_variant:Nn \fp_gzero:N { c }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_set:Nn, \fp_set:cn}
+% \UnitTested
+% \begin{macro}{\fp_gset:Nn, \fp_gset:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_set_aux:NNn}
+% To trap any input errors, a very simple version of the parser is run
+% here. This will pick up any invalid characters at this stage, saving
+% issues later. The splitting approach is the same as the more
+% advanced function later.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_set:Nn { \fp_set_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gset:Nn { \fp_set_aux:NNn \tl_gset:Nn }
+\cs_new_protected_nopar:Npn \fp_set_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+\cs_generate_variant:Nn \fp_set:Nn { c }
+\cs_generate_variant:Nn \fp_gset:Nn { c }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+%
+% \begin{macro}{\fp_set_from_dim:Nn, \fp_set_from_dim:cn}
+% \UnitTested
+% \begin{macro}{\fp_gset_from_dim:Nn, \fp_gset_from_dim:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_set_from_dim_aux:NNn}
+% \begin{macro}[aux]{\fp_set_from_dim_aux:w}
+% \begin{variable}{\l_fp_tmp_dim}
+% \begin{variable}{\l_fp_tmp_skip}
+% Here, dimensions are converted to fixed-points \emph{via} a
+% temporary variable. This ensures that they always convert as points.
+% The code is then essentially the same as for \cs{fp_set:Nn}, but with
+% the dimension passed so that it will be striped of the |pt| on the
+% way through. The passage through a skip is used to remove any rubber
+% part.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_set_from_dim:Nn
+ { \fp_set_from_dim_aux:NNn \tl_set:Nx }
+\cs_new_protected_nopar:Npn \fp_gset_from_dim:Nn
+ { \fp_set_from_dim_aux:NNn \tl_gset:Nx }
+\cs_new_protected_nopar:Npn \fp_set_from_dim_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \l_fp_tmp_skip \etex_glueexpr:D #3 \scan_stop:
+ \l_fp_tmp_dim \l_fp_tmp_skip
+ \fp_split:Nn a
+ {
+ \exp_after:wN \fp_set_from_dim_aux:w
+ \dim_use:N \l_fp_tmp_dim
+ }
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+\cs_set_protected_nopar:Npx \fp_set_from_dim_aux:w
+ {
+ \cs_set_nopar:Npn \exp_not:N \fp_set_from_dim_aux:w
+ ##1 \tl_to_str:n { pt } {##1}
+ }
+\fp_set_from_dim_aux:w
+\cs_generate_variant:Nn \fp_set_from_dim:Nn { c }
+\cs_generate_variant:Nn \fp_gset_from_dim:Nn { c }
+\dim_new:N \l_fp_tmp_dim
+\skip_new:N \l_fp_tmp_skip
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_set_eq:NN, \fp_set_eq:cN, \fp_set_eq:Nc, \fp_set_eq:cc}
+% \UnitTested
+% \begin{macro}{\fp_gset_eq:NN, \fp_gset_eq:cN, \fp_gset_eq:Nc, \fp_gset_eq:cc}
+% \UnitTested
+% Pretty simple, really.
+% \begin{macrocode}
+\cs_new_eq:NN \fp_set_eq:NN \tl_set_eq:NN
+\cs_new_eq:NN \fp_set_eq:cN \tl_set_eq:cN
+\cs_new_eq:NN \fp_set_eq:Nc \tl_set_eq:Nc
+\cs_new_eq:NN \fp_set_eq:cc \tl_set_eq:cc
+\cs_new_eq:NN \fp_gset_eq:NN \tl_gset_eq:NN
+\cs_new_eq:NN \fp_gset_eq:cN \tl_gset_eq:cN
+\cs_new_eq:NN \fp_gset_eq:Nc \tl_gset_eq:Nc
+\cs_new_eq:NN \fp_gset_eq:cc \tl_gset_eq:cc
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_show:N, \fp_show:c}
+% \UnitTested
+% Simple showing of the underlying variable.
+% \begin{macrocode}
+\cs_new_eq:NN \fp_show:N \tl_show:N
+\cs_new_eq:NN \fp_show:c \tl_show:c
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_use:N, \fp_use:c}
+% \UnitTested
+% \begin{macro}[aux]{\fp_use_aux:w}
+% \begin{macro}[aux]{\fp_use_none:w}
+% \begin{macro}[aux]{\fp_use_small:w}
+% \begin{macro}[aux]{\fp_use_large:w}
+% \begin{macro}[aux]{\fp_use_large_aux_i:w}
+% \begin{macro}[aux]{\fp_use_large_aux_1:w}
+% \begin{macro}[aux]{\fp_use_large_aux_2:w}
+% \begin{macro}[aux]{\fp_use_large_aux_3:w}
+% \begin{macro}[aux]{\fp_use_large_aux_4:w}
+% \begin{macro}[aux]{\fp_use_large_aux_5:w}
+% \begin{macro}[aux]{\fp_use_large_aux_6:w}
+% \begin{macro}[aux]{\fp_use_large_aux_7:w}
+% \begin{macro}[aux]{\fp_use_large_aux_8:w}
+% \begin{macro}[aux]{\fp_use_large_aux_i:w}
+% \begin{macro}[aux]{\fp_use_large_aux_ii:w}
+% The idea of the \cs{fp_use:N} function to convert the stored
+% value into something suitable for \TeX{} to use as a number in an
+% expandable manner. The first step is to deal with the sign, then
+% work out how big the input is.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_use:N #1
+ { \exp_after:wN \fp_use_aux:w #1 \q_stop }
+\cs_generate_variant:Nn \fp_use:N { c }
+\cs_new_nopar:Npn \fp_use_aux:w #1#2 e #3 \q_stop
+ {
+ \if:w #1 -
+ -
+ \fi:
+ \if_int_compare:w #3 > \c_zero
+ \exp_after:wN \fp_use_large:w
+ \else:
+ \if_int_compare:w #3 < \c_zero
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_use_small:w
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_use_none:w
+ \fi:
+ \fi:
+ #2 e #3 \q_stop
+ }
+% \end{macrocode}
+% When the exponent is zero, the input is simply returned as output.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_use_none:w #1 e #2 \q_stop {#1}
+% \end{macrocode}
+% For small numbers (less than $1$) the correct number of zeros
+% have to be inserted, but the decimal point is easy.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_use_small:w #1 . #2 e #3 \q_stop
+ {
+ 0 .
+ \prg_replicate:nn { -#3 - 1 } { 0 }
+ #1#2
+ }
+% \end{macrocode}
+% Life is more complex for large numbers. The decimal point needs to
+% be shuffled, with potentially some zero-filling for very large values.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_use_large:w #1 . #2 e #3 \q_stop
+ {
+ \if_int_compare:w #3 < \c_ten
+ \exp_after:wN \fp_use_large_aux_i:w
+ \else:
+ \exp_after:wN \fp_use_large_aux_ii:w
+ \fi:
+ #1#2 e #3 \q_stop
+ }
+\cs_new_nopar:Npn \fp_use_large_aux_i:w #1#2 e #3 \q_stop
+ {
+ #1
+ \use:c { fp_use_large_aux_ #3 :w } #2 \q_stop
+ }
+\cs_new_nopar:cpn { fp_use_large_aux_1:w } #1#2 \q_stop { #1 . #2 }
+\cs_new_nopar:cpn { fp_use_large_aux_2:w } #1#2#3 \q_stop
+ { #1#2 . #3 }
+\cs_new_nopar:cpn { fp_use_large_aux_3:w } #1#2#3#4 \q_stop
+ { #1#2#3 . #4 }
+\cs_new_nopar:cpn { fp_use_large_aux_4:w } #1#2#3#4#5 \q_stop
+ { #1#2#3#4 . #5 }
+\cs_new_nopar:cpn { fp_use_large_aux_5:w } #1#2#3#4#5#6 \q_stop
+ { #1#2#3#4#5 . #6 }
+\cs_new_nopar:cpn { fp_use_large_aux_6:w } #1#2#3#4#5#6#7 \q_stop
+ { #1#2#3#4#5#6 . #7 }
+\cs_new_nopar:cpn { fp_use_large_aux_7:w } #1#2#3#4#5#6#7#8 \q_stop
+ { #1#2#3#4#6#7 . #8 }
+\cs_new_nopar:cpn { fp_use_large_aux_8:w } #1#2#3#4#5#6#7#8#9 \q_stop
+ { #1#2#3#4#5#6#7#8 . #9 }
+\cs_new_nopar:cpn { fp_use_large_aux_9:w } #1 \q_stop { #1 . }
+\cs_new_nopar:Npn \fp_use_large_aux_ii:w #1 e #2 \q_stop
+ {
+ #1
+ \prg_replicate:nn { #2 - 9 } { 0 }
+ .
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Transferring to other types}
+%
+% The \cs{fp_use:N} function converts a floating point variable to
+% a form that can be used by \TeX{}. Here, the functions are slightly
+% different, as some information may be discarded.
+%
+% \begin{macro}{\fp_to_dim:N, \fp_to_dim:c}
+% A very simple wrapper.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_dim:N #1 { \fp_use:N #1 pt }
+\cs_generate_variant:Nn \fp_to_dim:N { c }
+% \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\fp_to_int:N, \fp_to_int:c}
+% \UnitTested
+% \begin{macro}[aux]{\fp_to_int_aux:w}
+% \begin{macro}[aux]{\fp_to_int_none:w}
+% \begin{macro}[aux]{\fp_to_int_small:w}
+% \begin{macro}[aux]{\fp_to_int_large:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_i:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_1:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_2:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_3:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_4:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_5:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_6:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_7:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_8:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux_i:w}
+% \begin{macro}[aux]{\fp_to_int_large_aux:nnn}
+% \begin{macro}[aux]{\fp_to_int_large_aux_ii:w}
+% Converting to integers in an expandable manner is very similar to
+% simply using floating point variables, particularly in the lead-off.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_int:N #1
+ { \exp_after:wN \fp_to_int_aux:w #1 \q_stop }
+\cs_generate_variant:Nn \fp_to_int:N { c }
+\cs_new_nopar:Npn \fp_to_int_aux:w #1#2 e #3 \q_stop
+ {
+ \if:w #1 -
+ -
+ \fi:
+ \if_int_compare:w #3 < \c_zero
+ \exp_after:wN \fp_to_int_small:w
+ \else:
+ \exp_after:wN \fp_to_int_large:w
+ \fi:
+ #2 e #3 \q_stop
+ }
+% \end{macrocode}
+% For small numbers, if the decimal part is greater than a half then
+% there is rounding up to do.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_int_small:w #1 . #2 e #3 \q_stop
+ {
+ \if_int_compare:w #3 > \c_one
+ \else:
+ \if_int_compare:w #1 < \c_five
+ 0
+ \else:
+ 1
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% For large numbers, the idea is to split off the part for rounding,
+% do the rounding and fill if needed.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_int_large:w #1 . #2 e #3 \q_stop
+ {
+ \if_int_compare:w #3 < \c_ten
+ \exp_after:wN \fp_to_int_large_aux_i:w
+ \else:
+ \exp_after:wN \fp_to_int_large_aux_ii:w
+ \fi:
+ #1#2 e #3 \q_stop
+ }
+\cs_new_nopar:Npn \fp_to_int_large_aux_i:w #1#2 e #3 \q_stop
+ { \use:c { fp_to_int_large_aux_ #3 :w } #2 \q_stop {#1} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_1:w } #1#2 \q_stop
+ { \fp_to_int_large_aux:nnn { #2 0 } {#1} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_2:w } #1#2#3 \q_stop
+ { \fp_to_int_large_aux:nnn { #3 00 } {#1#2} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_3:w } #1#2#3#4 \q_stop
+ { \fp_to_int_large_aux:nnn { #4 000 } {#1#2#3} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_4:w } #1#2#3#4#5 \q_stop
+ { \fp_to_int_large_aux:nnn { #5 0000 } {#1#2#3#4} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_5:w } #1#2#3#4#5#6 \q_stop
+ { \fp_to_int_large_aux:nnn { #6 00000 } {#1#2#3#4#5} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_6:w } #1#2#3#4#5#6#7 \q_stop
+ { \fp_to_int_large_aux:nnn { #7 000000 } {#1#2#3#4#5#6} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_7:w } #1#2#3#4#5#6#7#8 \q_stop
+ { \fp_to_int_large_aux:nnn { #8 0000000 } {#1#2#3#4#5#6#7} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_8:w } #1#2#3#4#5#6#7#8#9 \q_stop
+ { \fp_to_int_large_aux:nnn { #9 00000000 } {#1#2#3#4#5#6#7#8} }
+\cs_new_nopar:cpn { fp_to_int_large_aux_9:w } #1 \q_stop {#1}
+\cs_new_nopar:Npn \fp_to_int_large_aux:nnn #1#2#3
+ {
+ \if_int_compare:w #1 < \c_five_hundred_million
+ #3#2
+ \else:
+ \int_value:w \int_eval:w #3#2 + 1 \int_eval_end:
+ \fi:
+ }
+\cs_new_nopar:Npn \fp_to_int_large_aux_ii:w #1 e #2 \q_stop
+ {
+ #1
+ \prg_replicate:nn { #2 - 9 } { 0 }
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_to_tl:N, \fp_to_tl:c}
+% \UnitTested
+% \begin{macro}[aux]{\fp_to_tl_aux:w}
+% \begin{macro}[aux]{\fp_to_tl_large:w}
+% \begin{macro}[aux]{\fp_to_tl_large_aux_i:w}
+% \begin{macro}[aux]{\fp_to_tl_large_aux_ii:w}
+% \begin{macro}[aux]{\fp_to_tl_large_0:w}
+% \begin{macro}[aux]{\fp_to_tl_large_1:w}
+% \begin{macro}[aux]{\fp_to_tl_large_2:w}
+% \begin{macro}[aux]{\fp_to_tl_large_3:w}
+% \begin{macro}[aux]{\fp_to_tl_large_4:w}
+% \begin{macro}[aux]{\fp_to_tl_large_5:w}
+% \begin{macro}[aux]{\fp_to_tl_large_6:w}
+% \begin{macro}[aux]{\fp_to_tl_large_7:w}
+% \begin{macro}[aux]{\fp_to_tl_large_8:w}
+% \begin{macro}[aux]{\fp_to_tl_large_8_aux:w}
+% \begin{macro}[aux]{\fp_to_tl_large_9:w}
+% \begin{macro}[aux]{\fp_to_tl_small:w}
+% \begin{macro}[aux]{\fp_to_tl_small_one:w}
+% \begin{macro}[aux]{\fp_to_tl_small_two:w}
+% \begin{macro}[aux]{\fp_to_tl_small_aux:w}
+% \begin{macro}[aux]{\fp_to_tl_large_zeros:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_to_tl_small_zeros:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_use_iix_ix:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_use_ix:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_use_i_to_vii:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_use_i_to_iix:NNNNNNNNN}
+% Converting to integers in an expandable manner is very similar to
+% simply using floating point variables, particularly in the lead-off.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_tl:N #1
+ { \exp_after:wN \fp_to_tl_aux:w #1 \q_stop }
+\cs_generate_variant:Nn \fp_to_tl:N { c }
+\cs_new_nopar:Npn \fp_to_tl_aux:w #1#2 e #3 \q_stop
+ {
+ \if:w #1 -
+ -
+ \fi:
+ \if_int_compare:w #3 < \c_zero
+ \exp_after:wN \fp_to_tl_small:w
+ \else:
+ \exp_after:wN \fp_to_tl_large:w
+ \fi:
+ #2 e #3 \q_stop
+ }
+% \end{macrocode}
+% For \enquote{large} numbers (exponent $\ge 0$) there are two
+% cases. For very large exponents ($ \ge 10 $) life is easy: apart
+% from dropping extra zeros there is no work to do. On the other hand,
+% for intermediate exponent values the decimal needs to be moved, then
+% zeros can be dropped.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_tl_large:w #1 e #2 \q_stop
+ {
+ \if_int_compare:w #2 < \c_ten
+ \exp_after:wN \fp_to_tl_large_aux_i:w
+ \else:
+ \exp_after:wN \fp_to_tl_large_aux_ii:w
+ \fi:
+ #1 e #2 \q_stop
+ }
+\cs_new_nopar:Npn \fp_to_tl_large_aux_i:w #1 e #2 \q_stop
+ { \use:c { fp_to_tl_large_ #2 :w } #1 \q_stop }
+\cs_new_nopar:Npn \fp_to_tl_large_aux_ii:w #1 . #2 e #3 \q_stop
+ {
+ #1
+ \fp_to_tl_large_zeros:NNNNNNNNN #2
+ e #3
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_0:w } #1 . #2 \q_stop
+ {
+ #1
+ \fp_to_tl_large_zeros:NNNNNNNNN #2
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_1:w } #1 . #2#3 \q_stop
+ {
+ #1#2
+ \fp_to_tl_large_zeros:NNNNNNNNN #3 0
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_2:w } #1 . #2#3#4 \q_stop
+ {
+ #1#2#3
+ \fp_to_tl_large_zeros:NNNNNNNNN #4 00
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_3:w } #1 . #2#3#4#5 \q_stop
+ {
+ #1#2#3#4
+ \fp_to_tl_large_zeros:NNNNNNNNN #5 000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_4:w } #1 . #2#3#4#5#6 \q_stop
+ {
+ #1#2#3#4#5
+ \fp_to_tl_large_zeros:NNNNNNNNN #6 0000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_5:w } #1 . #2#3#4#5#6#7 \q_stop
+ {
+ #1#2#3#4#5#6
+ \fp_to_tl_large_zeros:NNNNNNNNN #7 00000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_6:w } #1 . #2#3#4#5#6#7#8 \q_stop
+ {
+ #1#2#3#4#5#6#7
+ \fp_to_tl_large_zeros:NNNNNNNNN #8 000000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_7:w } #1 . #2#3#4#5#6#7#8#9 \q_stop
+ {
+ #1#2#3#4#5#6#7#8
+ \fp_to_tl_large_zeros:NNNNNNNNN #9 0000000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_8:w } #1 .
+ {
+ #1
+ \use:c { fp_to_tl_large_8_aux:w }
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_8_aux:w } #1#2#3#4#5#6#7#8#9 \q_stop
+ {
+ #1#2#3#4#5#6#7#8
+ \fp_to_tl_large_zeros:NNNNNNNNN #9 00000000
+ }
+\cs_new_nopar:cpn { fp_to_tl_large_9:w } #1 . #2 \q_stop {#1#2}
+% \end{macrocode}
+% Dealing with small numbers is a bit more complex as there has to be
+% rounding. This makes life rather awkward, as there need to be a series
+% of tests and calculations, as things cannot be stored in an
+% expandable system.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_tl_small:w #1 e #2 \q_stop
+ {
+ \if_int_compare:w #2 = \c_minus_one
+ \exp_after:wN \fp_to_tl_small_one:w
+ \else:
+ \if_int_compare:w #2 = -\c_two
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_to_tl_small_two:w
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_to_tl_small_aux:w
+ \fi:
+ \fi:
+ #1 e #2 \q_stop
+ }
+\cs_new_nopar:Npn \fp_to_tl_small_one:w #1 . #2 e #3 \q_stop
+ {
+ \if_int_compare:w \fp_use_ix:NNNNNNNNN #2 > \c_four
+ \if_int_compare:w
+ \int_eval:w #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1
+ < \c_one_thousand_million
+ 0.
+ \exp_after:wN \fp_to_tl_small_zeros:NNNNNNNNN
+ \int_value:w \int_eval:w
+ #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1
+ \int_eval_end:
+ \else:
+ 1
+ \fi:
+ \else:
+ 0. #1
+ \fp_to_tl_small_zeros:NNNNNNNNN #2
+ \fi:
+ }
+\cs_new_nopar:Npn \fp_to_tl_small_two:w #1 . #2 e #3 \q_stop
+ {
+ \if_int_compare:w \fp_use_iix_ix:NNNNNNNNN #2 > \c_forty_four
+ \if_int_compare:w
+ \int_eval:w #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten
+ < \c_one_thousand_million
+ 0.0
+ \exp_after:wN \fp_to_tl_small_zeros:NNNNNNNNN
+ \int_value:w \int_eval:w
+ #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten
+ \int_eval_end:
+ \else:
+ 0.1
+ \fi:
+ \else:
+ 0.0
+ #1
+ \fp_to_tl_small_zeros:NNNNNNNNN #2
+ \fi:
+ }
+\cs_new_nopar:Npn \fp_to_tl_small_aux:w #1 . #2 e #3 \q_stop
+ {
+ #1
+ \fp_to_tl_large_zeros:NNNNNNNNN #2
+ e #3
+ }
+% \end{macrocode}
+% Rather than a complex recursion, the tests for finding trailing zeros
+% are written out long-hand. The difference between the two is only the
+% need for a decimal marker.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_to_tl_large_zeros:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \if_int_compare:w #9 = \c_zero
+ \if_int_compare:w #8 = \c_zero
+ \if_int_compare:w #7 = \c_zero
+ \if_int_compare:w #6 = \c_zero
+ \if_int_compare:w #5 = \c_zero
+ \if_int_compare:w #4 = \c_zero
+ \if_int_compare:w #3 = \c_zero
+ \if_int_compare:w #2 = \c_zero
+ \if_int_compare:w #1 = \c_zero
+ \else:
+ . #1
+ \fi:
+ \else:
+ . #1#2
+ \fi:
+ \else:
+ . #1#2#3
+ \fi:
+ \else:
+ . #1#2#3#4
+ \fi:
+ \else:
+ . #1#2#3#4#5
+ \fi:
+ \else:
+ . #1#2#3#4#5#6
+ \fi:
+ \else:
+ . #1#2#3#4#5#6#7
+ \fi:
+ \else:
+ . #1#2#3#4#5#6#7#8
+ \fi:
+ \else:
+ . #1#2#3#4#5#6#7#8#9
+ \fi:
+ }
+\cs_new_nopar:Npn \fp_to_tl_small_zeros:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \if_int_compare:w #9 = \c_zero
+ \if_int_compare:w #8 = \c_zero
+ \if_int_compare:w #7 = \c_zero
+ \if_int_compare:w #6 = \c_zero
+ \if_int_compare:w #5 = \c_zero
+ \if_int_compare:w #4 = \c_zero
+ \if_int_compare:w #3 = \c_zero
+ \if_int_compare:w #2 = \c_zero
+ \if_int_compare:w #1 = \c_zero
+ \else:
+ #1
+ \fi:
+ \else:
+ #1#2
+ \fi:
+ \else:
+ #1#2#3
+ \fi:
+ \else:
+ #1#2#3#4
+ \fi:
+ \else:
+ #1#2#3#4#5
+ \fi:
+ \else:
+ #1#2#3#4#5#6
+ \fi:
+ \else:
+ #1#2#3#4#5#6#7
+ \fi:
+ \else:
+ #1#2#3#4#5#6#7#8
+ \fi:
+ \else:
+ #1#2#3#4#5#6#7#8#9
+ \fi:
+ }
+% \end{macrocode}
+% Some quick \enquote{return a few} functions.
+% \begin{macrocode}
+\cs_new_nopar:Npn \fp_use_iix_ix:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#8#9}
+\cs_new_nopar:Npn \fp_use_ix:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#9}
+\cs_new_nopar:Npn \fp_use_i_to_vii:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {#1#2#3#4#5#6#7}
+\cs_new_nopar:Npn \fp_use_i_to_iix:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {#1#2#3#4#5#6#7#8}
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Rounding numbers}
+%
+% The results may well need to be rounded. A couple of related functions
+% to do this for a stored value.
+%
+% \begin{macro}{\fp_round_figures:Nn, \fp_round_figures:cn}
+% \UnitTested
+% \begin{macro}{\fp_ground_figures:Nn, \fp_ground_figures:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_round_figures_aux:NNn}
+% Rounding to figures needs only an adjustment to the target by one
+% (as the target is in decimal places).
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_round_figures:Nn
+ { \fp_round_figures_aux:NNn \tl_set:Nn }
+\cs_generate_variant:Nn \fp_round_figures:Nn { c }
+\cs_new_protected_nopar:Npn \fp_ground_figures:Nn
+ { \fp_round_figures_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_ground_figures:Nn { c }
+\cs_new_protected_nopar:Npn \fp_round_figures_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \int_set:Nn \l_fp_round_target_int { #3 - 1 }
+ \if_int_compare:w \l_fp_round_target_int < \c_ten
+ \exp_after:wN \fp_round:
+ \fi:
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_round_places:Nn, \fp_round_places:cn}
+% \UnitTested
+% \begin{macro}{\fp_ground_places:Nn, \fp_ground_places:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_round_places_aux:NNn}
+% Rounding to places needs an adjustment for the exponent value, which
+% will mean that everything should be correct.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_round_places:Nn
+ { \fp_round_places_aux:NNn \tl_set:Nn }
+\cs_generate_variant:Nn \fp_round_places:Nn { c }
+\cs_new_protected_nopar:Npn \fp_ground_places:Nn
+ { \fp_round_places_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_ground_places:Nn { c }
+\cs_new_protected_nopar:Npn \fp_round_places_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \int_set:Nn \l_fp_round_target_int
+ { #3 + \l_fp_input_a_exponent_int }
+ \if_int_compare:w \l_fp_round_target_int < \c_ten
+ \exp_after:wN \fp_round:
+ \fi:
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_round:}
+% \begin{macro}[aux]{\fp_round_aux:NNNNNNNNN}
+% \begin{macro}{\fp_round_loop:N}
+% The rounding approach is the same for decimal places and significant
+% figures. There are always nine decimal digits to round, so the code
+% can be written to account for this. The basic logic is simply to
+% find the rounding, track any carry digit and move along. At the end
+% of the loop there is a possible shuffle if the integer part has
+% become $10$.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_round:
+ {
+ \bool_set_false:N \l_fp_round_carry_bool
+ \l_fp_round_position_int \c_eight
+ \tl_clear:N \l_fp_round_decimal_tl
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_round_aux:NNNNNNNNN \int_use:N \l_fp_input_a_decimal_int
+ }
+\cs_new_protected_nopar:Npn \fp_round_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \fp_round_loop:N #9#8#7#6#5#4#3#2#1
+ \bool_if:NT \l_fp_round_carry_bool
+ { \tex_advance:D \l_fp_input_a_integer_int \c_one }
+ \l_fp_input_a_decimal_int \l_fp_round_decimal_tl \scan_stop:
+ \if_int_compare:w \l_fp_input_a_integer_int < \c_ten
+ \else:
+ \l_fp_input_a_integer_int \c_one
+ \tex_divide:D \l_fp_input_a_decimal_int \c_ten
+ \tex_advance:D \l_fp_input_a_exponent_int \c_one
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_round_loop:N #1
+ {
+ \if_int_compare:w \l_fp_round_position_int < \l_fp_round_target_int
+ \bool_if:NTF \l_fp_round_carry_bool
+ { \l_fp_tmp_int \int_eval:w #1 + \c_one \scan_stop: }
+ { \l_fp_tmp_int \int_eval:w #1 \scan_stop: }
+ \if_int_compare:w \l_fp_tmp_int = \c_ten
+ \l_fp_tmp_int \c_zero
+ \else:
+ \bool_set_false:N \l_fp_round_carry_bool
+ \fi:
+ \tl_set:Nx \l_fp_round_decimal_tl
+ { \int_use:N \l_fp_tmp_int \l_fp_round_decimal_tl }
+ \else:
+ \tl_set:Nx \l_fp_round_decimal_tl { 0 \l_fp_round_decimal_tl }
+ \if_int_compare:w \l_fp_round_position_int = \l_fp_round_target_int
+ \if_int_compare:w #1 > \c_four
+ \bool_set_true:N \l_fp_round_carry_bool
+ \fi:
+ \fi:
+ \fi:
+ \tex_advance:D \l_fp_round_position_int \c_minus_one
+ \if_int_compare:w \l_fp_round_position_int > \c_minus_one
+ \exp_after:wN \fp_round_loop:N
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Unary functions}
+%
+% \begin{macro}{\fp_abs:N, \fp_abs:c}
+% \UnitTested
+% \begin{macro}{\fp_gabs:N, \fp_gabs:c}
+% \UnitTested
+% \begin{macro}[aux]{\fp_abs_aux:NN}
+% Setting the absolute value is easy: read the value, ignore the sign,
+% return the result.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_abs:N { \fp_abs_aux:NN \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gabs:N { \fp_abs_aux:NN \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_abs:N { c }
+\cs_generate_variant:Nn \fp_gabs:N { c }
+\cs_new_protected_nopar:Npn \fp_abs_aux:NN #1#2
+ {
+ \group_begin:
+ \fp_read:N #2
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ +
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_neg:N, \fp_neg:c}
+% \UnitTested
+% \begin{macro}{\fp_gneg:N, \fp_gneg:c}
+% \UnitTested
+% \begin{macro}[aux]{\fp_neg:NN}
+% Just a bit more complex: read the input, reverse the sign and
+% output the result.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_neg:N { \fp_neg_aux:NN \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gneg:N { \fp_neg_aux:NN \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_neg:N { c }
+\cs_generate_variant:Nn \fp_gneg:N { c }
+\cs_new_protected_nopar:Npn \fp_neg_aux:NN #1#2
+ {
+ \group_begin:
+ \fp_read:N #2
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_input_a_decimal_int
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \exp_after:wN \group_end: \exp_after:wN
+ #1 \exp_after:wN #2 \exp_after:wN { \l_fp_tmp_tl }
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Basic arithmetic}
+%
+% \begin{macro}{\fp_add:Nn, \fp_add:cn}
+% \UnitTested
+% \begin{macro}{\fp_gadd:Nn,\fp_gadd:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_add_aux:NNn}
+% \begin{macro}[aux]{\fp_add_core:}
+% \begin{macro}[aux]{\fp_add_sum:}
+% \begin{macro}[aux]{\fp_add_difference:}
+% The various addition functions are simply different ways to call the
+% single master function below. This pattern is repeated for the
+% other arithmetic functions.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_add:Nn { \fp_add_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gadd:Nn { \fp_add_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_add:Nn { c }
+\cs_generate_variant:Nn \fp_gadd:Nn { c }
+% \end{macrocode}
+% Addition takes place using one of two paths. If the signs of the
+% two parts are the same, they are simply combined. On the other
+% hand, if the signs are different the calculation finds this
+% difference.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_add_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \fp_split:Nn b {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \fp_add_core:
+ \fp_tmp:w #1#2
+ }
+\cs_new_protected_nopar:Npn \fp_add_core:
+ {
+ \fp_level_input_exponents:
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
+ > \c_zero
+ \exp_after:wN \fp_add_sum:
+ \else:
+ \exp_after:wN \fp_add_difference:
+ \fi:
+ \l_fp_output_exponent_int \l_fp_input_a_exponent_int
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2
+ {
+ \if_int_compare:w \l_fp_output_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+ }
+% \end{macrocode}
+% Finding the sum of two numbers is trivially easy.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_add_sum:
+ {
+ \l_fp_output_sign_int \l_fp_input_a_sign_int
+ \l_fp_output_integer_int
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_b_integer_int
+ \scan_stop:
+ \l_fp_output_decimal_int
+ \int_eval:w
+ \l_fp_input_a_decimal_int + \l_fp_input_b_decimal_int
+ \scan_stop:
+ \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_output_integer_int \c_one
+ \tex_advance:D \l_fp_output_decimal_int -\c_one_thousand_million
+ \fi:
+ }
+% \end{macrocode}
+% When the signs of the two parts of the input are different, the
+% absolute difference is worked out first. There is then a calculation
+% to see which way around everything has worked out, so that the final
+% sign is correct. The difference might also give a zero result with
+% a negative sign, which is reversed as zero is regarded as positive.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_add_difference:
+ {
+ \l_fp_output_integer_int
+ \int_eval:w
+ \l_fp_input_a_integer_int - \l_fp_input_b_integer_int
+ \scan_stop:
+ \l_fp_output_decimal_int
+ \int_eval:w
+ \l_fp_input_a_decimal_int - \l_fp_input_b_decimal_int
+ \scan_stop:
+ \if_int_compare:w \l_fp_output_decimal_int < \c_zero
+ \tex_advance:D \l_fp_output_integer_int \c_minus_one
+ \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_output_integer_int < \c_zero
+ \l_fp_output_sign_int \l_fp_input_b_sign_int
+ \if_int_compare:w \l_fp_output_decimal_int = \c_zero
+ \l_fp_output_integer_int -\l_fp_output_integer_int
+ \else:
+ \l_fp_output_decimal_int
+ \int_eval:w
+ \c_one_thousand_million - \l_fp_output_decimal_int
+ \scan_stop:
+ \l_fp_output_integer_int
+ \int_eval:w
+ - \l_fp_output_integer_int - \c_one
+ \scan_stop:
+ \fi:
+ \else:
+ \l_fp_output_sign_int \l_fp_input_a_sign_int
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_sub:Nn, \fp_sub:cn}
+% \UnitTested
+% \begin{macro}{\fp_gsub:Nn,\fp_gsub:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_sub_aux:NNn}
+% Subtraction is essentially the same as addition, but with the sign
+% of the second component reversed. Thus the core of the two function
+% groups is the same, with just a little set up here.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_sub:Nn { \fp_sub_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gsub:Nn { \fp_sub_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_sub:Nn { c }
+\cs_generate_variant:Nn \fp_gsub:Nn { c }
+\cs_new_protected_nopar:Npn \fp_sub_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \fp_split:Nn b {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \tex_multiply:D \l_fp_input_b_sign_int \c_minus_one
+ \fp_add_core:
+ \fp_tmp:w #1#2
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_mul:Nn, \fp_mul:cn}
+% \UnitTested
+% \begin{macro}{\fp_gmul:Nn,\fp_gmul:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_mul_aux:NNn}
+% \begin{macro}[aux]{\fp_mul_internal:}
+% \begin{macro}[aux]{\fp_mul_split:NNNN}
+% \begin{macro}[aux]{\fp_mul_split:w}
+% \begin{macro}[aux]{\fp_mul_end_level:}
+% \begin{macro}[aux]{\fp_mul_end_level:NNNNNNNNN}
+% The pattern is much the same for multiplication.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul:Nn { \fp_mul_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gmul:Nn { \fp_mul_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_mul:Nn { c }
+\cs_generate_variant:Nn \fp_gmul:Nn { c }
+% \end{macrocode}
+% The approach to multiplication is as follows. First, the two numbers
+% are split into blocks of three digits. These are then multiplied
+% together to find products for each group of three output digits. This
+% is al written out in full for speed reasons. Between each block of
+% three digits in the output, there is a carry step. The very lowest
+% digits are not calculated, while
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \fp_split:Nn b {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \fp_mul_internal:
+ \l_fp_output_exponent_int
+ \int_eval:w
+ \l_fp_input_a_exponent_int + \l_fp_input_b_exponent_int
+ \scan_stop:
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
+ < \c_zero
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_output_integer_int + \l_fp_output_decimal_int
+ = \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% Done separately so that the internal use is a bit easier.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul_internal:
+ {
+ \fp_mul_split:NNNN \l_fp_input_a_decimal_int
+ \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int
+ \fp_mul_split:NNNN \l_fp_input_b_decimal_int
+ \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int
+ \l_fp_mul_output_int \c_zero
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int
+ \tex_divide:D \l_fp_mul_output_int \c_one_thousand
+ \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_input_b_integer_int
+ \fp_mul_end_level:
+ \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_input_b_integer_int
+ \fp_mul_end_level:
+ \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_input_b_integer_int
+ \fp_mul_end_level:
+ \l_fp_output_decimal_int 0 \l_fp_mul_output_tl \scan_stop:
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_input_b_integer_int
+ \fp_mul_end_level:
+ \l_fp_output_integer_int 0 \l_fp_mul_output_tl \scan_stop:
+ }
+% \end{macrocode}
+% The split works by making a $10$ digit number, from which
+% the first digit can then be dropped using a delimited argument. The
+% groups of three digits are then assigned to the various parts of
+% the input: notice that |##9| contains the last two digits of the
+% smallest part of the input.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul_split:NNNN #1#2#3#4
+ {
+ \tex_advance:D #1 \c_one_thousand_million
+ \cs_set_protected_nopar:Npn \fp_mul_split_aux:w
+ ##1##2##3##4##5##6##7##8##9 \q_stop {
+ #2 ##2##3##4 \scan_stop:
+ #3 ##5##6##7 \scan_stop:
+ #4 ##8##9 \scan_stop:
+ }
+ \exp_after:wN \fp_mul_split_aux:w \int_use:N #1 \q_stop
+ \tex_advance:D #1 -\c_one_thousand_million
+ }
+\cs_new_protected_nopar:Npn \fp_mul_product:NN #1#2
+ {
+ \l_fp_mul_output_int
+ \int_eval:w \l_fp_mul_output_int + #1 * #2 \scan_stop:
+ }
+% \end{macrocode}
+% At the end of each output group of three, there is a transfer of
+% information so that there is no danger of an overflow. This is done by
+% expansion to keep the number of calculations down.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul_end_level:
+ {
+ \tex_advance:D \l_fp_mul_output_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_mul_end_level:NNNNNNNNN \int_use:N \l_fp_mul_output_int
+ }
+\cs_new_protected_nopar:Npn \fp_mul_end_level:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \tl_set:Nx \l_fp_mul_output_tl { #7#8#9 \l_fp_mul_output_tl }
+ \l_fp_mul_output_int #1#2#3#4#5#6 \scan_stop:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_div:Nn, \fp_div:cn}
+% \UnitTested
+% \begin{macro}{\fp_gdiv:Nn,\fp_gdiv:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_div_aux:NNn}
+% \begin{macro}{\fp_div_internal:}
+% \begin{macro}[aux]{\fp_div_loop:}
+% \begin{macro}[aux]{\fp_div_divide:}
+% \begin{macro}[aux]{\fp_div_divide_aux:}
+% \begin{macro}[aux]{\fp_div_store:}
+% \begin{macro}[aux]{\fp_div_store_integer:}
+% \begin{macro}[aux]{\fp_div_store_decimal:}
+% The pattern is much the same for multiplication.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div:Nn { \fp_div_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gdiv:Nn { \fp_div_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_div:Nn { c }
+\cs_generate_variant:Nn \fp_gdiv:Nn { c }
+% \end{macrocode}
+% Division proper starts with a couple of tests. If the denominator is
+% zero then a error is issued. On the other hand, if the numerator is
+% zero then the result must be $0.0$ and can be given with no
+% further work.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \fp_split:Nn b {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
+ = \c_zero
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ #1 \exp_not:N #2 { \c_undefined_fp }
+ }
+ \else:
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ = \c_zero
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ #1 \exp_not:N #2 { \c_zero_fp }
+ }
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_internal:
+ \fi:
+ \fi:
+ \fp_tmp:w #1#2
+ }
+% \end{macrocode}
+% The main division algorithm works by finding how many times |b| can
+% be removed from |a|, storing the result and doing the subtraction.
+% Input |a| is then multiplied by $10$, and the process is repeated.
+% The looping ends either when there is nothing left of |a|
+% (\emph{i.e.}~an exact result) or when the code reaches the ninth
+% decimal place. Most of the process takes place in the loop function
+% below.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_internal: {
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \cs_set_eq:NN \fp_div_store: \fp_div_store_integer:
+ \l_fp_div_offset_int \c_one_hundred_million
+ \fp_div_loop:
+ \l_fp_output_exponent_int
+ \int_eval:w
+ \l_fp_input_a_exponent_int - \l_fp_input_b_exponent_int
+ \scan_stop:
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2
+ {
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
+ < \c_zero
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_output_integer_int + \l_fp_output_decimal_int
+ = \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ \int_eval_end:
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+}
+% \end{macrocode}
+% The main loop implements the approach described above. The storing
+% function is done as a function so that the integer and decimal parts
+% can be done separately but rapidly.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_loop:
+ {
+ \l_fp_count_int \c_zero
+ \fp_div_divide:
+ \fp_div_store:
+ \tex_multiply:D \l_fp_input_a_integer_int \c_ten
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \exp_after:wN \fp_div_loop_step:w
+ \int_use:N \l_fp_input_a_decimal_int \q_stop
+ \if_int_compare:w
+ \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ > \c_zero
+ \if_int_compare:w \l_fp_div_offset_int > \c_zero
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_div_loop:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% Checking to see if the numerator can be divides needs quite an
+% involved check. Either the integer part has to be bigger for the
+% numerator or, if it is not smaller then the decimal part of the
+% numerator must not be smaller than that of the denominator. Once
+% the test is right the rest is much as elsewhere.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_divide:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \l_fp_input_b_integer_int
+ \exp_after:wN \fp_div_divide_aux:
+ \else:
+ \if_int_compare:w \l_fp_input_a_integer_int < \l_fp_input_b_integer_int
+ \else:
+ \if_int_compare:w
+ \l_fp_input_a_decimal_int < \l_fp_input_b_decimal_int
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_div_divide_aux:
+ \fi:
+ \fi:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_div_divide_aux:
+ {
+ \tex_advance:D \l_fp_count_int \c_one
+ \tex_advance:D \l_fp_input_a_integer_int -\l_fp_input_b_integer_int
+ \tex_advance:D \l_fp_input_a_decimal_int -\l_fp_input_b_decimal_int
+ \if_int_compare:w \l_fp_input_a_decimal_int < \c_zero
+ \tex_advance:D \l_fp_input_a_integer_int \c_minus_one
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \fi:
+ \fp_div_divide:
+ }
+% \end{macrocode}
+% Storing the number of each division is done differently for the
+% integer and decimal. The integer is easy and a one-off, while the
+% decimal also needs to account for the position of the digit to store.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_store: { }
+\cs_new_protected_nopar:Npn \fp_div_store_integer:
+ {
+ \l_fp_output_integer_int \l_fp_count_int
+ \cs_set_eq:NN \fp_div_store: \fp_div_store_decimal:
+ }
+\cs_new_protected_nopar:Npn \fp_div_store_decimal:
+ {
+ \l_fp_output_decimal_int
+ \int_eval:w
+ \l_fp_output_decimal_int +
+ \l_fp_count_int * \l_fp_div_offset_int
+ \int_eval_end:
+ \tex_divide:D \l_fp_div_offset_int \c_ten
+ }
+\cs_new_protected_nopar:Npn \fp_div_loop_step:w #1#2#3#4#5#6#7#8#9 \q_stop
+ {
+ \l_fp_input_a_integer_int
+ \int_eval:w #2 + \l_fp_input_a_integer_int \int_eval_end:
+ \l_fp_input_a_decimal_int #3#4#5#6#7#8#9 0 \scan_stop:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Arithmetic for internal use}
+%
+% For the more complex functions, it is only possible to deliver
+% reliable $10$ digit accuracy if the internal calculations are
+% carried out to a higher degree of precision. This is done using a
+% second set of functions so that the `user' versions are not
+% slowed down. These versions are also focussed on the needs of internal
+% calculations. No error checking, sign checking or exponent levelling
+% is done. For addition and subtraction, the arguments are:
+% \begin{itemize}
+% \item Integer part of input |a|.
+% \item Decimal part of input |a|.
+% \item Additional decimal part of input |a|.
+% \item Integer part of input |b|.
+% \item Decimal part of input |b|.
+% \item Additional decimal part of input |b|.
+% \item Integer part of output.
+% \item Decimal part of output.
+% \item Additional decimal part of output.
+% \end{itemize}
+% The situation for multiplication and division is a little different as
+% they only deal with the decimal part.
+%
+% \begin{macro}{\fp_add:NNNNNNNNN}
+% The internal sum is always exactly that: it is always a sum and there
+% is no sign check.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_add:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ #7 \int_eval:w #1 + #4 \int_eval_end:
+ #8 \int_eval:w #2 + #5 \int_eval_end:
+ #9 \int_eval:w #3 + #6 \int_eval_end:
+ \if_int_compare:w #9 < \c_one_thousand_million
+ \else:
+ \tex_advance:D #8 \c_one
+ \tex_advance:D #9 -\c_one_thousand_million
+ \fi:
+ \if_int_compare:w #8 < \c_one_thousand_million
+ \else:
+ \tex_advance:D #7 \c_one
+ \tex_advance:D #8 -\c_one_thousand_million
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_sub:NNNNNNNNN}
+% Internal subtraction is needed only when the first number is bigger
+% than the second, so there is no need to worry about the sign. This is
+% a good job as there are no arguments left. The flipping flag is
+% used in the rare case where a sign change is possible.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_sub:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ #7 \int_eval:w #1 - #4 \int_eval_end:
+ #8 \int_eval:w #2 - #5 \int_eval_end:
+ #9 \int_eval:w #3 - #6 \int_eval_end:
+ \if_int_compare:w #9 < \c_zero
+ \tex_advance:D #8 \c_minus_one
+ \tex_advance:D #9 \c_one_thousand_million
+ \fi:
+ \if_int_compare:w #8 < \c_zero
+ \tex_advance:D #7 \c_minus_one
+ \tex_advance:D #8 \c_one_thousand_million
+ \fi:
+ \if_int_compare:w #7 < \c_zero
+ \if_int_compare:w \int_eval:w #8 + #9 = \c_zero
+ #7 -#7
+ \else:
+ \tex_advance:D #7 \c_one
+ #8 \int_eval:w \c_one_thousand_million - #8 \int_eval_end:
+ #9 \int_eval:w \c_one_thousand_million - #9 \int_eval_end:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_mul:NNNNNN}
+% Decimal-part only multiplication but with higher accuracy than the
+% user version.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul:NNNNNN #1#2#3#4#5#6
+ {
+ \fp_mul_split:NNNN #1
+ \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int
+ \fp_mul_split:NNNN #2
+ \l_fp_mul_a_iv_int \l_fp_mul_a_v_int \l_fp_mul_a_vi_int
+ \fp_mul_split:NNNN #3
+ \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int
+ \fp_mul_split:NNNN #4
+ \l_fp_mul_b_iv_int \l_fp_mul_b_v_int \l_fp_mul_b_vi_int
+ \l_fp_mul_output_int \c_zero
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_vi_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_v_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_vi_int \l_fp_mul_b_i_int
+ \tex_divide:D \l_fp_mul_output_int \c_one_thousand
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_v_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_i_int
+ \fp_mul_end_level:
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_i_int
+ \fp_mul_end_level:
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int
+ \fp_mul_end_level:
+ #6 0 \l_fp_mul_output_tl \scan_stop:
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int
+ \fp_mul_end_level:
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int
+ \fp_mul_end_level:
+ \fp_mul_end_level:
+ #5 0 \l_fp_mul_output_tl \scan_stop:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_mul:NNNNNNNNN}
+% For internal multiplication where the integer does need to be
+% retained. This means of course that this code is quite slow, and so
+% is only used when necessary.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_mul:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \fp_mul_split:NNNN #2
+ \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int
+ \fp_mul_split:NNNN #3
+ \l_fp_mul_a_iv_int \l_fp_mul_a_v_int \l_fp_mul_a_vi_int
+ \fp_mul_split:NNNN #5
+ \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int
+ \fp_mul_split:NNNN #6
+ \l_fp_mul_b_iv_int \l_fp_mul_b_v_int \l_fp_mul_b_vi_int
+ \l_fp_mul_output_int \c_zero
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_vi_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_v_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_vi_int \l_fp_mul_b_i_int
+ \tex_divide:D \l_fp_mul_output_int \c_one_thousand
+ \fp_mul_product:NN #1 \l_fp_mul_b_vi_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_v_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_vi_int #4
+ \fp_mul_end_level:
+ \fp_mul_product:NN #1 \l_fp_mul_b_v_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_v_int #4
+ \fp_mul_end_level:
+ \fp_mul_product:NN #1 \l_fp_mul_b_iv_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_iv_int #4
+ \fp_mul_end_level:
+ #9 0 \l_fp_mul_output_tl \scan_stop:
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN #1 \l_fp_mul_b_iii_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_iii_int #4
+ \fp_mul_end_level:
+ \fp_mul_product:NN #1 \l_fp_mul_b_ii_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_ii_int #4
+ \fp_mul_end_level:
+ \fp_mul_product:NN #1 \l_fp_mul_b_i_int
+ \fp_mul_product:NN \l_fp_mul_a_i_int #4
+ \fp_mul_end_level:
+ #8 0 \l_fp_mul_output_tl \scan_stop:
+ \tl_clear:N \l_fp_mul_output_tl
+ \fp_mul_product:NN #1 #4
+ \fp_mul_end_level:
+ #7 0 \l_fp_mul_output_tl \scan_stop:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_div_integer:NNNNN}
+% Here, division is always by an integer, and so it is possible to
+% use \TeX{}'s native calculations rather than doing it in macros.
+% The idea here is to divide the decimal part, find any remainder,
+% then do the real division of the two parts before adding in what
+% is needed for the remainder.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_div_integer:NNNNN #1#2#3#4#5
+ {
+ \l_fp_tmp_int #1
+ \tex_divide:D \l_fp_tmp_int #3
+ \l_fp_tmp_int \int_eval:w #1 - \l_fp_tmp_int * #3 \int_eval_end:
+ #4 #1
+ \tex_divide:D #4 #3
+ #5 #2
+ \tex_divide:D #5 #3
+ \tex_multiply:D \l_fp_tmp_int \c_one_thousand
+ \tex_divide:D \l_fp_tmp_int #3
+ #5 \int_eval:w #5 + \l_fp_tmp_int * \c_one_million \int_eval_end:
+ \if_int_compare:w #5 > \c_one_thousand_million
+ \tex_advance:D #4 \c_one
+ \tex_advance:D #5 -\c_one_thousand_million
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_extended_normalise:}
+% \begin{macro}[aux]{\fp_extended_normalise_aux_i:}
+% \begin{macro}[aux]{\fp_extended_normalise_aux_i:w}
+% \begin{macro}[aux]{\fp_extended_normalise_aux_ii:w}
+% \begin{macro}[aux]{\fp_extended_normalise_aux_ii:}
+% \begin{macro}[aux]{\fp_extended_normalise_aux:NNNNNNNNN}
+% The \enquote{extended} integers for internal use are mainly used in
+% fixed-point mode. This comes up in a few places, so a generalised
+% utility is made available to carry out the change. This function
+% simply calls the two loops to shift the input to the point of
+% having a zero exponent.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_extended_normalise:
+ {
+ \fp_extended_normalise_aux_i:
+ \fp_extended_normalise_aux_ii:
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_aux_i:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int > \c_zero
+ \tex_multiply:D \l_fp_input_a_integer_int \c_ten
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \exp_after:wN \fp_extended_normalise_aux_i:w
+ \int_use:N \l_fp_input_a_decimal_int \q_stop
+ \exp_after:wN \fp_extended_normalise_aux_i:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_aux_i:w
+ #1#2#3#4#5#6#7#8#9 \q_stop
+ {
+ \l_fp_input_a_integer_int
+ \int_eval:w \l_fp_input_a_integer_int + #2 \scan_stop:
+ \l_fp_input_a_decimal_int #3#4#5#6#7#8#9 0 \scan_stop:
+ \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million
+ \exp_after:wN \fp_extended_normalise_aux_ii:w
+ \int_use:N \l_fp_input_a_extended_int \q_stop
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_aux_ii:w
+ #1#2#3#4#5#6#7#8#9 \q_stop
+ {
+ \l_fp_input_a_decimal_int
+ \int_eval:w \l_fp_input_a_decimal_int + #2 \scan_stop:
+ \l_fp_input_a_extended_int #3#4#5#6#7#8#9 0 \scan_stop:
+ \tex_advance:D \l_fp_input_a_exponent_int \c_minus_one
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_aux_ii:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_zero
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_extended_normalise_ii_aux:NNNNNNNNN
+ \int_use:N \l_fp_input_a_decimal_int
+ \exp_after:wN \fp_extended_normalise_aux_ii:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_ii_aux:NNNNNNNNN
+ #1#2#3#4#5#6#7#8#9
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int = \c_zero
+ \l_fp_input_a_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \else:
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \int_use:N \l_fp_input_a_integer_int
+ #1#2#3#4#5#6#7#8
+ }
+ \l_fp_input_a_integer_int \c_zero
+ \l_fp_input_a_decimal_int \l_fp_tmp_tl \scan_stop:
+ \fi:
+ \tex_divide:D \l_fp_input_a_extended_int \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #9
+ \int_use:N \l_fp_input_a_extended_int
+ }
+ \l_fp_input_a_extended_int \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D \l_fp_input_a_exponent_int \c_one
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_extended_normalise_output:}
+% \begin{macro}[aux]{\fp_extended_normalise_output_aux_i:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_extended_normalise_output_aux_ii:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_extended_normalise_output_aux:N}
+% At some stages in working out extended output, it is possible for the
+% value to need shifting to keep the integer part in range. This only
+% ever happens such that the integer needs to be made smaller.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_extended_normalise_output:
+ {
+ \if_int_compare:w \l_fp_output_integer_int > \c_nine
+ \tex_advance:D \l_fp_output_integer_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_extended_normalise_output_aux_i:NNNNNNNNN
+ \int_use:N \l_fp_output_integer_int
+ \exp_after:wN \fp_extended_normalise_output:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux_i:NNNNNNNNN
+ #1#2#3#4#5#6#7#8#9
+ {
+ \l_fp_output_integer_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #9
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_output_decimal_int
+ }
+ \exp_after:wN \fp_extended_normalise_output_aux_ii:NNNNNNNNN
+ \l_fp_tmp_tl
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux_ii:NNNNNNNNN
+ #1#2#3#4#5#6#7#8#9
+ {
+ \l_fp_output_decimal_int #1#2#3#4#5#6#7#8#9 \scan_stop:
+ \fp_extended_normalise_output_aux:N
+ }
+\cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux:N #1
+ {
+ \tex_advance:D \l_fp_output_extended_int \c_one_thousand_million
+ \tex_divide:D \l_fp_output_extended_int \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #1
+ \exp_after:wN \use_none:n
+ \int_use:N \l_fp_output_extended_int
+ }
+ \l_fp_output_extended_int \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D \l_fp_output_exponent_int \c_one
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Trigonometric functions}
+%
+% \begin{macro}{\fp_trig_normalise:}
+% \begin{macro}[aux]{\fp_trig_normalise_aux:}
+% \begin{macro}[aux]{\fp_trig_sub:NNN}
+% For normalisation, the code essentially switches to fixed-point
+% arithmetic. There is a shift of the exponent, then repeated
+% subtractions. The end result is a number in the range
+% $ -\pi < x \le \pi $.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_trig_normalise:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten
+ \l_fp_input_a_extended_int \c_zero
+ \fp_extended_normalise:
+ \fp_trig_normalise_aux:
+ \if_int_compare:w \l_fp_input_a_integer_int < \c_zero
+ \l_fp_input_a_sign_int -\l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int -\l_fp_input_a_integer_int
+ \fi:
+ \exp_after:wN \fp_trig_octant:
+ \else:
+ \l_fp_input_a_sign_int \c_one
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \l_fp_output_exponent_int \c_zero
+ \exp_after:wN \fp_trig_overflow_msg:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_trig_normalise_aux:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_three
+ \fp_trig_sub:NNN
+ \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int
+ \exp_after:wN \fp_trig_normalise_aux:
+ \else:
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_two
+ \if_int_compare:w \l_fp_input_a_decimal_int > \c_fp_pi_decimal_int
+ \fp_trig_sub:NNN
+ \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_trig_normalise_aux:
+ \fi:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% Here, there may be a sign change but there will never be any
+% variation in the input. So a dedicated function can be used.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_trig_sub:NNN #1#2#3
+ {
+ \l_fp_input_a_integer_int
+ \int_eval:w \l_fp_input_a_integer_int - #1 \int_eval_end:
+ \l_fp_input_a_decimal_int
+ \int_eval:w \l_fp_input_a_decimal_int - #2 \int_eval_end:
+ \l_fp_input_a_extended_int
+ \int_eval:w \l_fp_input_a_extended_int - #3 \int_eval_end:
+ \if_int_compare:w \l_fp_input_a_extended_int < \c_zero
+ \tex_advance:D \l_fp_input_a_decimal_int \c_minus_one
+ \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_input_a_decimal_int < \c_zero
+ \tex_advance:D \l_fp_input_a_integer_int \c_minus_one
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_input_a_integer_int < \c_zero
+ \l_fp_input_a_sign_int -\l_fp_input_a_sign_int
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int
+ = \c_zero
+ \l_fp_input_a_integer_int -\l_fp_input_a_integer_int
+ \else:
+ \l_fp_input_a_integer_int
+ \int_eval:w
+ - \l_fp_input_a_integer_int - \c_one
+ \int_eval_end:
+ \l_fp_input_a_decimal_int
+ \int_eval:w
+ \c_one_thousand_million - \l_fp_input_a_decimal_int
+ \int_eval_end:
+ \l_fp_input_a_extended_int
+ \int_eval:w
+ \c_one_thousand_million - \l_fp_input_a_extended_int
+ \int_eval_end:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_trig_octant:}
+% \begin{macro}[aux]{\fp_trig_octant_aux:}
+% Here, the input is further reduced into the range
+% $ 0 \le x < \pi / 4 $. This is pretty simple: check if
+% $ \pi / 4 $ can be taken off and if it can do it and loop. The
+% check at the end is to \enquote{mop up} values which are so close to
+% $ \pi / 4 $ that they should be treated as such. The test for
+% an even octant is needed as the `remainder' needed is from
+% the nearest $ \pi / 2 $.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_trig_octant:
+ {
+ \l_fp_trig_octant_int \c_one
+ \fp_trig_octant_aux:
+ \if_int_compare:w \l_fp_input_a_decimal_int < \c_ten
+ \l_fp_input_a_decimal_int \c_zero
+ \l_fp_input_a_extended_int \c_zero
+ \fi:
+ \if_int_odd:w \l_fp_trig_octant_int
+ \else:
+ \fp_sub:NNNNNNNNN
+ \c_zero \c_fp_pi_by_four_decimal_int \c_fp_pi_by_four_extended_int
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_trig_octant_aux:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_zero
+ \fp_sub:NNNNNNNNN
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \c_zero \c_fp_pi_by_four_decimal_int \c_fp_pi_by_four_extended_int
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \tex_advance:D \l_fp_trig_octant_int \c_one
+ \exp_after:wN \fp_trig_octant_aux:
+ \else:
+ \if_int_compare:w
+ \l_fp_input_a_decimal_int > \c_fp_pi_by_four_decimal_int
+ \fp_sub:NNNNNNNNN
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \c_zero \c_fp_pi_by_four_decimal_int
+ \c_fp_pi_by_four_extended_int
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \tex_advance:D \l_fp_trig_octant_int \c_one
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_trig_octant_aux:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_sin:Nn, \fp_sin:cn}
+% \UnitTested
+% \begin{macro}{\fp_gsin:Nn,\fp_gsin:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_sin_aux:NNn}
+% \begin{macro}[aux]{\fp_sin_aux_i:}
+% \begin{macro}[aux]{\fp_sin_aux_ii:}
+% Calculating the sine starts off in the usual way. There is a check
+% to see if the value has already been worked out before proceeding
+% further.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_sin:Nn { \fp_sin_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gsin:Nn { \fp_sin_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_sin:Nn { c }
+\cs_generate_variant:Nn \fp_gsin:Nn { c }
+% \end{macrocode}
+% The internal routine for sines does a check to see if the value is
+% already known. This saves a lot of repetition when doing rotations.
+% For very small values it is best to simply return the input as the
+% sine: the cut-off is $ 1 \times 10^{-5} $.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_sin_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \tl_set:Nx \l_fp_arg_tl
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_input_a_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \if_int_compare:w \l_fp_input_a_exponent_int < -\c_five
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2 { \l_fp_arg_tl }
+ }
+ \else:
+ \if_cs_exist:w
+ c_fp_sin ( \l_fp_arg_tl ) _fp
+ \cs_end:
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_sin_aux_i:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ { \use:c { c_fp_sin ( \l_fp_arg_tl ) _fp } }
+ }
+ \fi:
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% The internals for sine first normalise the input into an octant, then
+% choose the correct set up for the Taylor series. The sign for the sine
+% function is easy, so there is no worry about it. So the only thing to
+% do is to get the output standardised.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_sin_aux_i:
+ {
+ \fp_trig_normalise:
+ \fp_sin_aux_ii:
+ \if_int_compare:w \l_fp_output_integer_int = \c_one
+ \l_fp_output_exponent_int \c_zero
+ \else:
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_decimal_int \l_fp_output_extended_int
+ \l_fp_output_exponent_int -\c_nine
+ \fi:
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \tl_new:c { c_fp_sin ( \l_fp_arg_tl ) _fp }
+ \tl_gset:cx { c_fp_sin ( \l_fp_arg_tl ) _fp }
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+\cs_new_protected_nopar:Npn \fp_sin_aux_ii:
+ {
+ \if_case:w \l_fp_trig_octant_int
+ \or:
+ \exp_after:wN \fp_trig_calc_sin:
+ \or:
+ \exp_after:wN \fp_trig_calc_cos:
+ \or:
+ \exp_after:wN \fp_trig_calc_cos:
+ \or:
+ \exp_after:wN \fp_trig_calc_sin:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_cos:Nn, \fp_cos:cn}
+% \UnitTested
+% \begin{macro}{\fp_gcos:Nn,\fp_gcos:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_cos_aux:NNn}
+% \begin{macro}[aux]{\fp_cos_aux_i:}
+% \begin{macro}[aux]{\fp_cos_aux_ii:}
+% Cosine is almost identical, but there is no short cut code here.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_cos:Nn { \fp_cos_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gcos:Nn { \fp_cos_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_cos:Nn { c }
+\cs_generate_variant:Nn \fp_gcos:Nn { c }
+\cs_new_protected_nopar:Npn \fp_cos_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \tl_set:Nx \l_fp_arg_tl
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_input_a_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \if_cs_exist:w c_fp_cos ( \l_fp_arg_tl ) _fp \cs_end:
+ \else:
+ \exp_after:wN \fp_cos_aux_i:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ { \use:c { c_fp_cos ( \l_fp_arg_tl ) _fp } }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% Almost the same as for sine: just a bit of correction for the sign
+% of the output.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_cos_aux_i:
+ {
+ \fp_trig_normalise:
+ \fp_cos_aux_ii:
+ \if_int_compare:w \l_fp_output_integer_int = \c_one
+ \l_fp_output_exponent_int \c_zero
+ \else:
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_decimal_int \l_fp_output_extended_int
+ \l_fp_output_exponent_int -\c_nine
+ \fi:
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \tl_new:c { c_fp_cos ( \l_fp_arg_tl ) _fp }
+ \tl_gset:cx { c_fp_cos ( \l_fp_arg_tl ) _fp }
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+\cs_new_protected_nopar:Npn \fp_cos_aux_ii:
+ {
+ \if_case:w \l_fp_trig_octant_int
+ \or:
+ \exp_after:wN \fp_trig_calc_cos:
+ \or:
+ \exp_after:wN \fp_trig_calc_sin:
+ \or:
+ \exp_after:wN \fp_trig_calc_sin:
+ \or:
+ \exp_after:wN \fp_trig_calc_cos:
+ \fi:
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \if_int_compare:w \l_fp_trig_octant_int > \c_two
+ \l_fp_input_a_sign_int \c_minus_one
+ \fi:
+ \else:
+ \if_int_compare:w \l_fp_trig_octant_int > \c_two
+ \else:
+ \l_fp_input_a_sign_int \c_one
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_trig_calc_cos:}
+% \begin{macro}{\fp_trig_calc_sin:}
+% \begin{macro}[aux]{\fp_trig_calc_Taylor:}
+% These functions actually do the calculation for sine and cosine.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_trig_calc_cos:
+ {
+ \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero
+ \l_fp_output_integer_int \c_one
+ \l_fp_output_decimal_int \c_zero
+ \else:
+ \l_fp_trig_sign_int \c_minus_one
+ \fp_mul:NNNNNN
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \fp_div_integer:NNNNN
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \c_two
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \l_fp_count_int \c_three
+ \if_int_compare:w \l_fp_trig_extended_int = \c_zero
+ \if_int_compare:w \l_fp_trig_decimal_int = \c_zero
+ \l_fp_output_integer_int \c_one
+ \l_fp_output_decimal_int \c_zero
+ \l_fp_output_extended_int \c_zero
+ \else:
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_one_thousand_million
+ \l_fp_output_extended_int \c_zero
+ \fi:
+ \else:
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int 999999999 \scan_stop:
+ \l_fp_output_extended_int \c_one_thousand_million
+ \fi:
+ \tex_advance:D \l_fp_output_extended_int -\l_fp_trig_extended_int
+ \tex_advance:D \l_fp_output_decimal_int -\l_fp_trig_decimal_int
+ \exp_after:wN \fp_trig_calc_Taylor:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_trig_calc_sin:
+ {
+ \l_fp_output_integer_int \c_zero
+ \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \else:
+ \l_fp_output_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_output_extended_int \l_fp_input_a_extended_int
+ \l_fp_trig_sign_int \c_one
+ \l_fp_trig_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_trig_extended_int \l_fp_input_a_extended_int
+ \l_fp_count_int \c_two
+ \exp_after:wN \fp_trig_calc_Taylor:
+ \fi:
+ }
+% \end{macrocode}
+% This implements a Taylor series calculation for the trigonometric
+% functions. Lots of shuffling about as \TeX\ is not exactly a natural
+% choice for this sort of thing.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_trig_calc_Taylor:
+ {
+ \l_fp_trig_sign_int -\l_fp_trig_sign_int
+ \fp_mul:NNNNNN
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \fp_mul:NNNNNN
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \fp_div_integer:NNNNN
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \l_fp_count_int
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \tex_advance:D \l_fp_count_int \c_one
+ \fp_div_integer:NNNNN
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \l_fp_count_int
+ \l_fp_trig_decimal_int \l_fp_trig_extended_int
+ \tex_advance:D \l_fp_count_int \c_one
+ \if_int_compare:w \l_fp_trig_decimal_int > \c_zero
+ \if_int_compare:w \l_fp_trig_sign_int > \c_zero
+ \tex_advance:D \l_fp_output_decimal_int \l_fp_trig_decimal_int
+ \tex_advance:D \l_fp_output_extended_int
+ \l_fp_trig_extended_int
+ \if_int_compare:w \l_fp_output_extended_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_output_decimal_int \c_one
+ \tex_advance:D \l_fp_output_extended_int
+ -\c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_output_integer_int \c_one
+ \tex_advance:D \l_fp_output_decimal_int
+ -\c_one_thousand_million
+ \fi:
+ \else:
+ \tex_advance:D \l_fp_output_decimal_int -\l_fp_trig_decimal_int
+ \tex_advance:D \l_fp_output_extended_int
+ -\l_fp_input_a_extended_int
+ \if_int_compare:w \l_fp_output_extended_int < \c_zero
+ \tex_advance:D \l_fp_output_decimal_int \c_minus_one
+ \tex_advance:D \l_fp_output_extended_int \c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_output_decimal_int < \c_zero
+ \tex_advance:D \l_fp_output_integer_int \c_minus_one
+ \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million
+ \fi:
+ \fi:
+ \exp_after:wN \fp_trig_calc_Taylor:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_tan:Nn, \fp_tan:cn}
+% \UnitTested
+% \begin{macro}{\fp_gtan:Nn,\fp_gtan:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_tan_aux:NNn}
+% \begin{macro}[aux]{\fp_tan_aux_i:}
+% \begin{macro}[aux]{\fp_tan_aux_ii:}
+% \begin{macro}[aux]{\fp_tan_aux_iii:}
+% \begin{macro}[aux]{\fp_tan_aux_iv:}
+% As might be expected, tangents are calculated from the sine and cosine
+% by division. So there is a bit of set up, the two subsidiary pieces
+% of work are done and then a division takes place. For small numbers,
+% the same approach is used as for sines, with the input value simply
+% returned as is.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_tan:Nn { \fp_tan_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gtan:Nn { \fp_tan_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_tan:Nn { c }
+\cs_generate_variant:Nn \fp_gtan:Nn { c }
+\cs_new_protected_nopar:Npn \fp_tan_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \tl_set:Nx \l_fp_arg_tl
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_input_a_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \if_int_compare:w \l_fp_input_a_exponent_int < -\c_five
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2 { \l_fp_arg_tl }
+ }
+ \else:
+ \if_cs_exist:w
+ c_fp_tan ( \l_fp_arg_tl ) _fp
+ \cs_end:
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_tan_aux_i:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ { \use:c { c_fp_tan ( \l_fp_arg_tl ) _fp } }
+ }
+ \fi:
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% The business of the calculation does not check for stored sines or
+% cosines as there would then be an overhead to reading them back in.
+% There is also no need to worry about \enquote{small} sine values as
+% these will have been dealt with earlier. There is a two-step lead off
+% so that undefined division is not even attempted.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_tan_aux_i:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten
+ \exp_after:wN \fp_tan_aux_ii:
+ \else:
+ \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
+ \c_zero_fp
+ \exp_after:wN \fp_trig_overflow_msg:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_tan_aux_ii:
+ {
+ \fp_trig_normalise:
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \if_int_compare:w \l_fp_trig_octant_int > \c_two
+ \l_fp_output_sign_int \c_minus_one
+ \else:
+ \l_fp_output_sign_int \c_one
+ \fi:
+ \else:
+ \if_int_compare:w \l_fp_trig_octant_int > \c_two
+ \l_fp_output_sign_int \c_one
+ \else:
+ \l_fp_output_sign_int \c_minus_one
+ \fi:
+ \fi:
+ \fp_cos_aux_ii:
+ \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero
+ \if_int_compare:w \l_fp_input_a_integer_int = \c_zero
+ \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
+ \c_undefined_fp
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_tan_aux_iii:
+ \fi:
+ \else:
+ \exp_after:wN \fp_tan_aux_iii:
+ \fi:
+ }
+% \end{macrocode}
+% The division is done here using the same code as the standard division
+% unit, shifting the digits in the calculated sine and cosine to
+% maintain accuracy.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_tan_aux_iii:
+ {
+ \l_fp_input_b_integer_int \l_fp_output_decimal_int
+ \l_fp_input_b_decimal_int \l_fp_output_extended_int
+ \l_fp_input_b_exponent_int -\c_nine
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \fp_sin_aux_ii:
+ \l_fp_input_a_integer_int \l_fp_output_decimal_int
+ \l_fp_input_a_decimal_int \l_fp_output_extended_int
+ \l_fp_input_a_exponent_int -\c_nine
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero
+ \if_int_compare:w \l_fp_input_a_integer_int = \c_zero
+ \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
+ \c_zero_fp
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_iv:
+ \fi:
+ \else:
+ \exp_after:wN \fp_tan_aux_iv:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_tan_aux_iv:
+ {
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \cs_set_eq:NN \fp_div_store: \fp_div_store_integer:
+ \l_fp_div_offset_int \c_one_hundred_million
+ \fp_div_loop:
+ \l_fp_output_exponent_int
+ \int_eval:w
+ \l_fp_input_a_exponent_int - \l_fp_input_b_exponent_int
+ \int_eval_end:
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \tl_new:c { c_fp_tan ( \l_fp_arg_tl ) _fp }
+ \tl_gset:cx { c_fp_tan ( \l_fp_arg_tl ) _fp }
+ {
+ \if_int_compare:w \l_fp_output_sign_int > \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Exponent and logarithm functions}
+%
+% \begin{variable}{\c_fp_exp_1_tl}
+% \begin{variable}{\c_fp_exp_2_tl}
+% \begin{variable}{\c_fp_exp_3_tl}
+% \begin{variable}{\c_fp_exp_4_tl}
+% \begin{variable}{\c_fp_exp_5_tl}
+% \begin{variable}{\c_fp_exp_6_tl}
+% \begin{variable}{\c_fp_exp_7_tl}
+% \begin{variable}{\c_fp_exp_8_tl}
+% \begin{variable}{\c_fp_exp_9_tl}
+% \begin{variable}{\c_fp_exp_10_tl}
+% \begin{variable}{\c_fp_exp_20_tl}
+% \begin{variable}{\c_fp_exp_30_tl}
+% \begin{variable}{\c_fp_exp_40_tl}
+% \begin{variable}{\c_fp_exp_50_tl}
+% \begin{variable}{\c_fp_exp_60_tl}
+% \begin{variable}{\c_fp_exp_70_tl}
+% \begin{variable}{\c_fp_exp_80_tl}
+% \begin{variable}{\c_fp_exp_90_tl}
+% \begin{variable}{\c_fp_exp_100_tl}
+% \begin{variable}{\c_fp_exp_200_tl}
+% Calculation of exponentials requires a number of precomputed values:
+% first the positive integers.
+% \begin{macrocode}
+\tl_const:cn { c_fp_exp_1_tl } { { 2 } { 718281828 } { 459045235 } { 0 } }
+\tl_const:cn { c_fp_exp_2_tl } { { 7 } { 389056098 } { 930650227 } { 0 } }
+\tl_const:cn { c_fp_exp_3_tl } { { 2 } { 008553692 } { 318766774 } { 1 } }
+\tl_const:cn { c_fp_exp_4_tl } { { 5 } { 459815003 } { 314423908 } { 1 } }
+\tl_const:cn { c_fp_exp_5_tl } { { 1 } { 484131591 } { 025766034 } { 2 } }
+\tl_const:cn { c_fp_exp_6_tl } { { 4 } { 034287934 } { 927351226 } { 2 } }
+\tl_const:cn { c_fp_exp_7_tl } { { 1 } { 096633158 } { 428458599 } { 3 } }
+\tl_const:cn { c_fp_exp_8_tl } { { 2 } { 980957987 } { 041728275 } { 3 } }
+\tl_const:cn { c_fp_exp_9_tl } { { 8 } { 103083927 } { 575384008 } { 3 } }
+\tl_const:cn { c_fp_exp_10_tl } { { 2 } { 202646579 } { 480671652 } { 4 } }
+\tl_const:cn { c_fp_exp_20_tl } { { 4 } { 851651954 } { 097902280 } { 8 } }
+\tl_const:cn { c_fp_exp_30_tl } { { 1 } { 068647458 } { 152446215 } { 13 } }
+\tl_const:cn { c_fp_exp_40_tl } { { 2 } { 353852668 } { 370199854 } { 17 } }
+\tl_const:cn { c_fp_exp_50_tl } { { 5 } { 184705528 } { 587072464 } { 21 } }
+\tl_const:cn { c_fp_exp_60_tl } { { 1 } { 142007389 } { 815684284 } { 26 } }
+\tl_const:cn { c_fp_exp_70_tl } { { 2 } { 515438670 } { 919167006 } { 30 } }
+\tl_const:cn { c_fp_exp_80_tl } { { 5 } { 540622384 } { 393510053 } { 34 } }
+\tl_const:cn { c_fp_exp_90_tl } { { 1 } { 220403294 } { 317840802 } { 39 } }
+\tl_const:cn { c_fp_exp_100_tl } { { 2 } { 688117141 } { 816135448 } { 43 } }
+\tl_const:cn { c_fp_exp_200_tl } { { 7 } { 225973768 } { 125749258 } { 86 } }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{variable}{\c_fp_exp_-1_tl}
+% \begin{variable}{\c_fp_exp_-2_tl}
+% \begin{variable}{\c_fp_exp_-3_tl}
+% \begin{variable}{\c_fp_exp_-4_tl}
+% \begin{variable}{\c_fp_exp_-5_tl}
+% \begin{variable}{\c_fp_exp_-6_tl}
+% \begin{variable}{\c_fp_exp_-7_tl}
+% \begin{variable}{\c_fp_exp_-8_tl}
+% \begin{variable}{\c_fp_exp_-9_tl}
+% \begin{variable}{\c_fp_exp_-10_tl}
+% \begin{variable}{\c_fp_exp_-20_tl}
+% \begin{variable}{\c_fp_exp_-30_tl}
+% \begin{variable}{\c_fp_exp_-40_tl}
+% \begin{variable}{\c_fp_exp_-50_tl}
+% \begin{variable}{\c_fp_exp_-60_tl}
+% \begin{variable}{\c_fp_exp_-70_tl}
+% \begin{variable}{\c_fp_exp_-80_tl}
+% \begin{variable}{\c_fp_exp_-90_tl}
+% \begin{variable}{\c_fp_exp_-100_tl}
+% \begin{variable}{\c_fp_exp_-200_tl}
+% Now the negative integers.
+% \begin{macrocode}
+\tl_const:cn { c_fp_exp_-1_tl } { { 3 } { 678794411 } { 71442322 } { -1 } }
+\tl_const:cn { c_fp_exp_-2_tl } { { 1 } { 353352832 } { 366132692 } { -1 } }
+\tl_const:cn { c_fp_exp_-3_tl } { { 4 } { 978706836 } { 786394298 } { -2 } }
+\tl_const:cn { c_fp_exp_-4_tl } { { 1 } { 831563888 } { 873418029 } { -2 } }
+\tl_const:cn { c_fp_exp_-5_tl } { { 6 } { 737946999 } { 085467097 } { -3 } }
+\tl_const:cn { c_fp_exp_-6_tl } { { 2 } { 478752176 } { 666358423 } { -3 } }
+\tl_const:cn { c_fp_exp_-7_tl } { { 9 } { 118819655 } { 545162080 } { -4 } }
+\tl_const:cn { c_fp_exp_-8_tl } { { 3 } { 354626279 } { 025118388 } { -4 } }
+\tl_const:cn { c_fp_exp_-9_tl } { { 1 } { 234098040 } { 866795495 } { -4 } }
+\tl_const:cn { c_fp_exp_-10_tl } { { 4 } { 539992976 } { 248451536 } { -5 } }
+\tl_const:cn { c_fp_exp_-20_tl } { { 2 } { 061153622 } { 438557828 } { -9 } }
+\tl_const:cn { c_fp_exp_-30_tl } { { 9 } { 357622968 } { 840174605 } { -14 } }
+\tl_const:cn { c_fp_exp_-40_tl } { { 4 } { 248354255 } { 291588995 } { -18 } }
+\tl_const:cn { c_fp_exp_-50_tl } { { 1 } { 928749847 } { 963917783 } { -22 } }
+\tl_const:cn { c_fp_exp_-60_tl } { { 8 } { 756510762 } { 696520338 } { -27 } }
+\tl_const:cn { c_fp_exp_-70_tl } { { 3 } { 975449735 } { 908646808 } { -31 } }
+\tl_const:cn { c_fp_exp_-80_tl } { { 1 } { 804851387 } { 845415172 } { -35 } }
+\tl_const:cn { c_fp_exp_-90_tl } { { 8 } { 194012623 } { 990515430 } { -40 } }
+\tl_const:cn { c_fp_exp_-100_tl } { { 3 } { 720075976 } { 020835963 } { -44 } }
+\tl_const:cn { c_fp_exp_-200_tl } { { 1 } { 383896526 } { 736737530 } { -87 } }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{macro}{\fp_exp:Nn, \fp_exp:cn}
+% \UnitTested
+% \begin{macro}{\fp_gexp:Nn,\fp_gexp:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_exp_aux:NNn}
+% \begin{macro}[aux]{\fp_exp_internal:}
+% \begin{macro}[aux]{\fp_exp_aux:}
+% \begin{macro}[aux]{\fp_exp_integer:}
+% \begin{macro}[aux]{\fp_exp_integer_tens:}
+% \begin{macro}[aux]{\fp_exp_integer_units:}
+% \begin{macro}[aux]{\fp_exp_integer_const:n}
+% \begin{macro}[aux]{\fp_exp_integer_const:nnnn}
+% \begin{macro}[aux]{\fp_exp_decimal:}
+% \begin{macro}[aux]{\fp_exp_Taylor:}
+% \begin{macro}[aux]{\fp_exp_const:Nx}
+% \begin{macro}[aux]{\fp_exp_const:cx}
+% The calculation of an exponent starts off starts in much the same
+% way as the trigonometric functions: normalise the input, look for
+% a pre-defined value and if one is not found hand off to the real
+% workhorse function. The test for a definition of the result is used
+% so that overflows do not result in any outcome being defined.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp:Nn { \fp_exp_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gexp:Nn { \fp_exp_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_exp:Nn { c }
+\cs_generate_variant:Nn \fp_gexp:Nn { c }
+\cs_new_protected_nopar:Npn \fp_exp_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \l_fp_input_a_extended_int \c_zero
+ \tl_set:Nx \l_fp_arg_tl
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ -
+ \else:
+ +
+ \fi:
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_input_a_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \if_cs_exist:w c_fp_exp ( \l_fp_arg_tl ) _fp \cs_end:
+ \else:
+ \exp_after:wN \fp_exp_internal:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ #1 \exp_not:N #2
+ {
+ \if_cs_exist:w c_fp_exp ( \l_fp_arg_tl ) _fp
+ \cs_end:
+ \use:c { c_fp_exp ( \l_fp_arg_tl ) _fp }
+ \else:
+ \c_zero_fp
+ \fi:
+ }
+ }
+ \fp_tmp:w
+ }
+% \end{macrocode}
+% The first real step is to convert the input into a fixed-point
+% representation for further calculation: anything which is dropped
+% here as too small would not influence the output in any case. There
+% are a couple of overflow tests: the maximum
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_internal:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_three
+ \fp_extended_normalise:
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \if_int_compare:w \l_fp_input_a_integer_int < 230 \scan_stop:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_exp_aux:
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_exp_overflow_msg:
+ \fi:
+ \else:
+ \if_int_compare:w \l_fp_input_a_integer_int < 230 \scan_stop:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_exp_aux:
+ \else:
+ \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp }
+ { \c_zero_fp }
+ \fi:
+ \fi:
+ \else:
+ \exp_after:wN \fp_exp_overflow_msg:
+ \fi:
+ }
+% \end{macrocode}
+% The main algorithm makes use of the fact that
+% \[
+% \mathrm{e}^{nmp.q} =
+% \mathrm{e}^{n}
+% \mathrm{e}^{m}
+% \mathrm{e}^{p}
+% \mathrm{e}^{0.q}
+% \]
+% and that there is a Taylor series that can be used to calculate
+% $ \mathrm{e}^{0.q} $. Thus the approach needed is in three parts.
+% First, the exponent of the integer part of the input is found
+% using the pre-calculated constants. Second, the Taylor series is
+% used to find the exponent for the decimal part of the input. Finally,
+% the two parts are multiplied together to give the result. As the
+% normalisation code will already have dealt with any overflowing
+% values, there are no further checks needed.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_aux:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_zero
+ \exp_after:wN \fp_exp_integer:
+ \else:
+ \l_fp_output_integer_int \c_one
+ \l_fp_output_decimal_int \c_zero
+ \l_fp_output_extended_int \c_zero
+ \l_fp_output_exponent_int \c_zero
+ \exp_after:wN \fp_exp_decimal:
+ \fi:
+ }
+% \end{macrocode}
+% The integer part calculation starts with the hundreds. This is
+% set up such that very large negative numbers can short-cut the entire
+% procedure and simply return zero. In other cases, the code either
+% recovers the exponent of the hundreds value or sets the appropriate
+% storage to one (so that multiplication works correctly).
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_integer:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int < \c_one_hundred
+ \l_fp_exp_integer_int \c_one
+ \l_fp_exp_decimal_int \c_zero
+ \l_fp_exp_extended_int \c_zero
+ \l_fp_exp_exponent_int \c_zero
+ \exp_after:wN \fp_exp_integer_tens:
+ \else:
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \exp_after:wN \use_i:nnn
+ \int_use:N \l_fp_input_a_integer_int
+ }
+ \l_fp_input_a_integer_int
+ \int_eval:w
+ \l_fp_input_a_integer_int - \l_fp_tmp_tl 00
+ \int_eval_end:
+ \if_int_compare:w \l_fp_input_a_sign_int < \c_zero
+ \if_int_compare:w \l_fp_output_integer_int > 200 \scan_stop:
+ \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp }
+ { \c_zero_fp }
+ \else:
+ \fp_exp_integer_const:n { - \l_fp_tmp_tl 00 }
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_exp_integer_tens:
+ \fi:
+ \else:
+ \fp_exp_integer_const:n { \l_fp_tmp_tl 00 }
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_exp_integer_tens:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% The tens and units parts are handled in a similar way, with a
+% multiplication step to build up the final value. That also includes a
+% correction step to avoid an overflow of the integer part.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_integer_tens:
+ {
+ \l_fp_output_integer_int \l_fp_exp_integer_int
+ \l_fp_output_decimal_int \l_fp_exp_decimal_int
+ \l_fp_output_extended_int \l_fp_exp_extended_int
+ \l_fp_output_exponent_int \l_fp_exp_exponent_int
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_nine
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \exp_after:wN \use_i:nn
+ \int_use:N \l_fp_input_a_integer_int
+ }
+ \l_fp_input_a_integer_int
+ \int_eval:w
+ \l_fp_input_a_integer_int - \l_fp_tmp_tl 0
+ \int_eval_end:
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \fp_exp_integer_const:n { \l_fp_tmp_tl 0 }
+ \else:
+ \fp_exp_integer_const:n { - \l_fp_tmp_tl 0 }
+ \fi:
+ \fp_mul:NNNNNNNNN
+ \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \tex_advance:D \l_fp_output_exponent_int \l_fp_exp_exponent_int
+ \fp_extended_normalise_output:
+ \fi:
+ \fp_exp_integer_units:
+ }
+\cs_new_protected_nopar:Npn \fp_exp_integer_units:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_zero
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \fp_exp_integer_const:n { \int_use:N \l_fp_input_a_integer_int }
+ \else:
+ \fp_exp_integer_const:n
+ { - \int_use:N \l_fp_input_a_integer_int }
+ \fi:
+ \fp_mul:NNNNNNNNN
+ \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \tex_advance:D \l_fp_output_exponent_int \l_fp_exp_exponent_int
+ \fp_extended_normalise_output:
+ \fi:
+ \fp_exp_decimal:
+ }
+% \end{macrocode}
+% Recovery of the stored constant values into the separate registers
+% is done with a simple expansion then assignment.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_integer_const:n #1
+ {
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_exp_integer_const:nnnn
+ \cs:w c_fp_exp_ #1 _tl \cs_end:
+ }
+\cs_new_protected_nopar:Npn \fp_exp_integer_const:nnnn #1#2#3#4
+ {
+ \l_fp_exp_integer_int #1 \scan_stop:
+ \l_fp_exp_decimal_int #2 \scan_stop:
+ \l_fp_exp_extended_int #3 \scan_stop:
+ \l_fp_exp_exponent_int #4 \scan_stop:
+ }
+% \end{macrocode}
+% Finding the exponential for the decimal part of the number requires
+% a Taylor series calculation. The set up is done here with the loop
+% itself a separate function. Once the decimal part is available this
+% is multiplied by the integer part already worked out to give
+% the final result.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_decimal:
+ {
+ \if_int_compare:w \l_fp_input_a_decimal_int > \c_zero
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \l_fp_exp_integer_int \c_one
+ \l_fp_exp_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_exp_extended_int \l_fp_input_a_extended_int
+ \else:
+ \l_fp_exp_integer_int \c_zero
+ \if_int_compare:w \l_fp_exp_extended_int = \c_zero
+ \l_fp_exp_decimal_int
+ \int_eval:w
+ \c_one_thousand_million - \l_fp_input_a_decimal_int
+ \int_eval_end:
+ \l_fp_exp_extended_int \c_zero
+ \else:
+ \l_fp_exp_decimal_int
+ \int_eval:w
+ 999999999 - \l_fp_input_a_decimal_int
+ \scan_stop:
+ \l_fp_exp_extended_int
+ \int_eval:w
+ \c_one_thousand_million - \l_fp_input_a_extended_int
+ \int_eval_end:
+ \fi:
+ \fi:
+ \l_fp_input_b_sign_int \l_fp_input_a_sign_int
+ \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_input_b_extended_int \l_fp_input_a_extended_int
+ \l_fp_count_int \c_one
+ \fp_exp_Taylor:
+ \fp_mul:NNNNNNNNN
+ \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \fi:
+ \if_int_compare:w \l_fp_output_extended_int < \c_five_hundred_million
+ \else:
+ \tex_advance:D \l_fp_output_decimal_int \c_one
+ \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million
+ \else:
+ \l_fp_output_decimal_int \c_zero
+ \tex_advance:D \l_fp_output_integer_int \c_one
+ \fi:
+ \fi:
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp }
+ {
+ +
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+% \end{macrocode}
+% The Taylor series for $ \exp(x) $ is
+% \[
+% 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
+% \]
+% which converges for $ -1 < x < 1 $. The code above sets up
+% the $ x $ part, leaving the loop to multiply the running
+% value by $ x / n $ and add it onto the sum. The way that this is
+% done is that the running total is stored in the \texttt{exp} set of
+% registers, while the current item is stored as \texttt{input_b}.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_Taylor:
+ {
+ \tex_advance:D \l_fp_count_int \c_one
+ \tex_multiply:D \l_fp_input_b_sign_int \l_fp_input_a_sign_int
+ \fp_mul:NNNNNN
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \fp_div_integer:NNNNN
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \l_fp_count_int
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_b_decimal_int + \l_fp_input_b_extended_int
+ > \c_zero
+ \if_int_compare:w \l_fp_input_b_sign_int > \c_zero
+ \tex_advance:D \l_fp_exp_decimal_int \l_fp_input_b_decimal_int
+ \tex_advance:D \l_fp_exp_extended_int
+ \l_fp_input_b_extended_int
+ \if_int_compare:w \l_fp_exp_extended_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_exp_decimal_int \c_one
+ \tex_advance:D \l_fp_exp_extended_int
+ -\c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_exp_decimal_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_exp_integer_int \c_one
+ \tex_advance:D \l_fp_exp_decimal_int
+ -\c_one_thousand_million
+ \fi:
+ \else:
+ \tex_advance:D \l_fp_exp_decimal_int -\l_fp_input_b_decimal_int
+ \tex_advance:D \l_fp_exp_extended_int
+ -\l_fp_input_a_extended_int
+ \if_int_compare:w \l_fp_exp_extended_int < \c_zero
+ \tex_advance:D \l_fp_exp_decimal_int \c_minus_one
+ \tex_advance:D \l_fp_exp_extended_int \c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_exp_decimal_int < \c_zero
+ \tex_advance:D \l_fp_exp_integer_int \c_minus_one
+ \tex_advance:D \l_fp_exp_decimal_int \c_one_thousand_million
+ \fi:
+ \fi:
+ \exp_after:wN \fp_exp_Taylor:
+ \fi:
+ }
+% \end{macrocode}
+% This is set up as a function so that the power code can redirect
+% the effect.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_exp_const:Nx #1#2
+ {
+ \tl_new:N #1
+ \tl_gset:Nx #1 {#2}
+ }
+\cs_generate_variant:Nn \fp_exp_const:Nx { c }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{variable}{\c_fp_ln_10_1_tl}
+% \begin{variable}{\c_fp_ln_10_2_tl}
+% \begin{variable}{\c_fp_ln_10_3_tl}
+% \begin{variable}{\c_fp_ln_10_4_tl}
+% \begin{variable}{\c_fp_ln_10_5_tl}
+% \begin{variable}{\c_fp_ln_10_6_tl}
+% \begin{variable}{\c_fp_ln_10_7_tl}
+% \begin{variable}{\c_fp_ln_10_8_tl}
+% \begin{variable}{\c_fp_ln_10_9_tl}
+% Constants for working out logarithms: first those for the powers of
+% ten.
+% \begin{macrocode}
+\tl_const:cn { c_fp_ln_10_1_tl } { { 2 } { 302585092 } { 994045684 } { 0 } }
+\tl_const:cn { c_fp_ln_10_2_tl } { { 4 } { 605170185 } { 988091368 } { 0 } }
+\tl_const:cn { c_fp_ln_10_3_tl } { { 6 } { 907755278 } { 982137052 } { 0 } }
+\tl_const:cn { c_fp_ln_10_4_tl } { { 9 } { 210340371 } { 976182736 } { 0 } }
+\tl_const:cn { c_fp_ln_10_5_tl } { { 1 } { 151292546 } { 497022842 } { 1 } }
+\tl_const:cn { c_fp_ln_10_6_tl } { { 1 } { 381551055 } { 796427410 } { 1 } }
+\tl_const:cn { c_fp_ln_10_7_tl } { { 1 } { 611809565 } { 095831979 } { 1 } }
+\tl_const:cn { c_fp_ln_10_8_tl } { { 1 } { 842068074 } { 395226547 } { 1 } }
+\tl_const:cn { c_fp_ln_10_9_tl } { { 2 } { 072326583 } { 694641116 } { 1 } }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%^^A 0.69...309 4.
+% \begin{variable}{\c_fp_ln_2_1_tl }
+% \begin{variable}{\c_fp_ln_2_2_tl }
+% \begin{variable}{\c_fp_ln_2_3_tl }
+% The smaller set for powers of two.
+% \begin{macrocode}
+\tl_const:cn { c_fp_ln_2_1_tl } { { 0 } { 693147180 } { 559945309 } { 0 } }
+\tl_const:cn { c_fp_ln_2_2_tl } { { 1 } { 386294361 } { 119890618 } { 0 } }
+\tl_const:cn { c_fp_ln_2_3_tl } { { 2 } { 079441541 } { 679835928 } { 0 } }
+% \end{macrocode}
+% \end{variable}
+% \end{variable}
+% \end{variable}
+%
+% \begin{macro}{\fp_ln:Nn, \fp_ln:cn}
+% \UnitTested
+% \begin{macro}{\fp_gln:Nn,\fp_gln:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_ln_aux:NNn}
+% \begin{macro}[aux]{\fp_ln_aux:}
+% \begin{macro}[aux]{\fp_ln_exponent:}
+% \begin{macro}[aux]{\fp_ln_internal:}
+% \begin{macro}[aux]{\fp_ln_exponent_tens:}
+% \begin{macro}[aux]{\fp_ln_exponent_units:}
+% \begin{macro}[aux]{\fp_ln_normalise:}
+% \begin{macro}[aux]{\fp_ln_nornalise_aux:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_ln_mantissa:}
+% \begin{macro}[aux]{\fp_ln_mantissa_aux:}
+% \begin{macro}[aux]{\fp_ln_mantissa_divide_two:}
+% \begin{macro}[aux]{\fp_ln_integer_const:nn}
+% \begin{macro}[aux]{\fp_ln_Taylor:}
+% \begin{macro}[aux]{\fp_ln_fixed:}
+% \begin{macro}[aux]{\fp_ln_fixed_aux:NNNNNNNNN}
+% \begin{macro}[aux]{\fp_ln_Taylor_aux:}
+% The approach for logarithms is again based on a mix of tables and
+% Taylor series. Here, the initial validation is a bit easier and so it
+% is set up earlier, meaning less need to escape later on.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln:Nn { \fp_ln_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gln:Nn { \fp_ln_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_ln:Nn { c }
+\cs_generate_variant:Nn \fp_gln:Nn { c }
+\cs_new_protected_nopar:Npn \fp_ln_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ > \c_zero
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_ln_aux:
+ \else:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 \exp_not:N ##2 { \c_zero_fp }
+ }
+ \exp_after:wN \exp_after:wN \exp_after:wN \fp_ln_error_msg:
+ \fi:
+ \else:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 \exp_not:N ##2 { \c_zero_fp }
+ }
+ \exp_after:wN \fp_ln_error_msg:
+ \fi:
+ \fp_tmp:w #1 #2
+ }
+% \end{macrocode}
+% As the input at this stage meets the validity criteria above, the
+% argument can now be saved for further processing. There is no need
+% to look at the sign of the input as it must be positive. The function
+% here simply sets up to either do the full calculation or recover
+% the stored value, as appropriate.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_aux:
+ {
+ \tl_set:Nx \l_fp_arg_tl
+ {
+ +
+ \int_use:N \l_fp_input_a_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_input_a_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_input_a_exponent_int
+ }
+ \if_cs_exist:w c_fp_ln ( \l_fp_arg_tl ) _fp \cs_end:
+ \else:
+ \exp_after:wN \fp_ln_exponent:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 \exp_not:N ##2
+ { \use:c { c_fp_ln ( \l_fp_arg_tl ) _fp } }
+ }
+ }
+% \end{macrocode}
+% The main algorithm here uses the fact the logarithm can be divided
+% up, first taking out the powers of ten, then powers of two and finally
+% using a Taylor series for the remainder.
+% \[
+% \ln ( 10^{n} \times 2^{m} \times x )
+% = \ln ( 10^{n} ) + \ln ( 2^{m} ) + \ln ( x )
+% \]
+% The second point to remember is that
+% \[
+% \ln ( x^{-1} ) = - \ln ( x )
+% \]
+% which means that for the powers of $ 10 $ and $ 2 $ constants
+% are only needed for positive powers.
+%
+% The first step is to set up the sign for the output functions and
+% work out the powers of ten in the exponent. First the larger powers
+% are sorted out. The values for the constants are the same as those
+% for the smaller ones, just with a shift in the exponent.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_exponent:
+ {
+ \fp_ln_internal:
+ \if_int_compare:w \l_fp_output_extended_int < \c_five_hundred_million
+ \else:
+ \tex_advance:D \l_fp_output_decimal_int \c_one
+ \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million
+ \else:
+ \l_fp_output_decimal_int \c_zero
+ \tex_advance:D \l_fp_output_integer_int \c_one
+ \fi:
+ \fi:
+ \fp_standardise:NNNN
+ \l_fp_output_sign_int
+ \l_fp_output_integer_int
+ \l_fp_output_decimal_int
+ \l_fp_output_exponent_int
+ \tl_const:cx { c_fp_ln ( \l_fp_arg_tl ) _fp }
+ {
+ \if_int_compare:w \l_fp_output_sign_int > \c_zero
+ +
+ \else:
+ -
+ \fi:
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ \scan_stop:
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+\cs_new_protected_nopar:Npn \fp_ln_internal:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_zero
+ \l_fp_input_a_exponent_int -\l_fp_input_a_exponent_int
+ \l_fp_output_sign_int \c_minus_one
+ \else:
+ \l_fp_output_sign_int \c_one
+ \fi:
+ \if_int_compare:w \l_fp_input_a_exponent_int > \c_nine
+ \exp_after:wN \fp_ln_exponent_tens:NN
+ \int_use:N \l_fp_input_a_exponent_int
+ \else:
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \l_fp_output_extended_int \c_zero
+ \l_fp_output_exponent_int \c_zero
+ \fi:
+ \fp_ln_exponent_units:
+ }
+\cs_new_protected_nopar:Npn \fp_ln_exponent_tens:NN #1 #2
+ {
+ \l_fp_input_a_exponent_int #2 \scan_stop:
+ \fp_ln_const:nn { 10 } { #1 }
+ \tex_advance:D \l_fp_exp_exponent_int \c_one
+ \l_fp_output_integer_int \l_fp_exp_integer_int
+ \l_fp_output_decimal_int \l_fp_exp_decimal_int
+ \l_fp_output_extended_int \l_fp_exp_extended_int
+ \l_fp_output_exponent_int \l_fp_exp_exponent_int
+ }
+% \end{macrocode}
+% Next the smaller powers of ten, which will need to be combined
+% with the above: always an additive process.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_exponent_units:
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int > \c_zero
+ \fp_ln_const:nn { 10 } { \int_use:N \l_fp_input_a_exponent_int }
+ \fp_ln_normalise:
+ \fp_add:NNNNNNNNN
+ \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \fi:
+ \fp_ln_mantissa:
+ }
+% \end{macrocode}
+% The smaller table-based parts may need to be exponent shifted so that
+% they stay in line with the larger parts. This is similar to the
+% approach in other places, but here there is a need to watch the
+% extended part of the number. The only case where the new exponent is
+% larger than the old is if there was no previous part. Then simply set
+% the exponent.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_normalise:
+ {
+ \if_int_compare:w \l_fp_exp_exponent_int < \l_fp_output_exponent_int
+ \tex_advance:D \l_fp_exp_decimal_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_ln_normalise_aux:NNNNNNNNN
+ \int_use:N \l_fp_exp_decimal_int
+ \exp_after:wN \fp_ln_normalise:
+ \else:
+ \l_fp_output_exponent_int \l_fp_exp_exponent_int
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_ln_normalise_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \if_int_compare:w \l_fp_exp_integer_int = \c_zero
+ \l_fp_exp_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \else:
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \int_use:N \l_fp_exp_integer_int
+ #1#2#3#4#5#6#7#8
+ }
+ \l_fp_exp_integer_int \c_zero
+ \l_fp_exp_decimal_int \l_fp_tmp_tl \scan_stop:
+ \fi:
+ \tex_divide:D \l_fp_exp_extended_int \c_ten
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ #9
+ \int_use:N \l_fp_exp_extended_int
+ }
+ \l_fp_exp_extended_int \l_fp_tmp_tl \scan_stop:
+ \tex_advance:D \l_fp_exp_exponent_int \c_one
+ }
+% \end{macrocode}
+% The next phase is to decompose the mantissa by division by two to
+% leave a value which is in the range $ 1 \le x < 2 $. The sum of the
+% two powers needs to take account of the sign of the output: if it
+% is negative then the result gets \emph{smaller} as the mantissa gets
+% \emph{bigger}.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_mantissa:
+ {
+ \l_fp_count_int \c_zero
+ \l_fp_input_a_extended_int \c_zero
+ \fp_ln_mantissa_aux:
+ \if_int_compare:w \l_fp_count_int > \c_zero
+ \fp_ln_const:nn { 2 } { \int_use:N \l_fp_count_int }
+ \fp_ln_normalise:
+ \if_int_compare:w \l_fp_output_sign_int > \c_zero
+ \exp_after:wN \fp_add:NNNNNNNNN
+ \else:
+ \exp_after:wN \fp_sub:NNNNNNNNN
+ \fi:
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \fi:
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int > \c_one
+ \exp_after:wN \fp_ln_Taylor:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_ln_mantissa_aux:
+ {
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_one
+ \tex_advance:D \l_fp_count_int \c_one
+ \fp_ln_mantissa_divide_two:
+ \exp_after:wN \fp_ln_mantissa_aux:
+ \fi:
+ }
+% \end{macrocode}
+% A fast one-shot division by two.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_mantissa_divide_two:
+ {
+ \if_int_odd:w \l_fp_input_a_decimal_int
+ \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million
+ \fi:
+ \if_int_odd:w \l_fp_input_a_integer_int
+ \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
+ \fi:
+ \tex_divide:D \l_fp_input_a_integer_int \c_two
+ \tex_divide:D \l_fp_input_a_decimal_int \c_two
+ \tex_divide:D \l_fp_input_a_extended_int \c_two
+ }
+% \end{macrocode}
+% Recovering constants makes use of the same auxiliary code as for
+% exponents.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_const:nn #1#2
+ {
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_exp_integer_const:nnnn
+ \cs:w c_fp_ln_ #1 _ #2 _tl \cs_end:
+ }
+% \end{macrocode}
+% The Taylor series for the logarithm function is best implemented using
+% the identity
+% \[
+% \ln(x) = \ln\left( \frac{y + 1}{y - 1} \right)
+% \]
+% with
+% \[
+% y = \frac{x - 1}{x + 1}
+% \]
+% This leads to the series
+% \[
+% \ln(x)
+% = 2y
+% \left(
+% 1 + y^{2}
+% \left(
+% \frac{1}{3} + y^{2}
+% \left(
+% \frac{1}{5} + y^{2}
+% \left(
+% \frac{1}{7} + y^{2}
+% \left(
+% \frac{1}{9} + \cdots
+% \right)
+% \right)
+% \right)
+% \right)
+% \right)
+% \]
+% This expansion has the advantage that a lot of the work can be
+% loaded up early by finding $ y^{2} $ before the loop itself starts.
+% (In practice, the implementation does the multiplication by two at the
+% end of the loop, and expands out the brackets as this is an overall
+% more efficient approach.)
+%
+% At the implementation level, the code starts by calculating $ y $
+% and storing that in input \texttt{a} (which is no longer needed
+% for other purposes). That is done using the full division system
+% avoiding the parsing step. The value is then switched to a fixed-point
+% representation. There is then some shuffling to get all of the working
+% space set up. At this stage, a lot of registers are in use and so
+% the Taylor series is calculated within a group so that the
+% \texttt{output} variables can be used to hold the result. The value
+% of $ y^{2} $ is held in input \texttt{b} (there are a few
+% assignments saved by choosing this over \texttt{a}), while input
+% \texttt{a} is used for the \enquote{loop value}.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_Taylor:
+ {
+ \group_begin:
+ \l_fp_input_a_integer_int \c_zero
+ \l_fp_input_a_exponent_int \c_zero
+ \l_fp_input_b_integer_int \c_two
+ \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_input_b_exponent_int \c_zero
+ \fp_div_internal:
+ \fp_ln_fixed:
+ \l_fp_input_a_integer_int \l_fp_output_integer_int
+ \l_fp_input_a_decimal_int \l_fp_output_decimal_int
+ \l_fp_input_a_extended_int \c_zero
+ \l_fp_input_a_exponent_int \l_fp_output_exponent_int
+ \l_fp_output_decimal_int \c_zero %^^A Bug?
+ \l_fp_output_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_output_extended_int \l_fp_input_a_extended_int
+ \fp_mul:NNNNNN
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \l_fp_count_int \c_one
+ \fp_ln_Taylor_aux:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ \l_fp_exp_integer_int \c_zero
+ \exp_not:N \l_fp_exp_decimal_int
+ \int_use:N \l_fp_output_decimal_int \scan_stop:
+ \exp_not:N \l_fp_exp_extended_int
+ \int_use:N \l_fp_output_extended_int \scan_stop:
+ \exp_not:N \l_fp_exp_exponent_int
+ \int_use:N \l_fp_output_exponent_int \scan_stop:
+ }
+ \fp_tmp:w
+% \end{macrocode}
+% After the loop part of the Taylor series, the factor of $ 2 $ needs
+% to be included. The total for the result can then be constructed.
+% \begin{macrocode}
+ \tex_advance:D \l_fp_exp_decimal_int \l_fp_exp_decimal_int
+ \if_int_compare:w \l_fp_exp_extended_int < \c_five_hundred_million
+ \else:
+ \tex_advance:D \l_fp_exp_extended_int -\c_five_hundred_million
+ \tex_advance:D \l_fp_exp_decimal_int \c_one
+ \fi:
+ \tex_advance:D \l_fp_exp_extended_int \l_fp_exp_extended_int
+ \fp_ln_normalise:
+ \if_int_compare:w \l_fp_output_sign_int > \c_zero
+ \exp_after:wN \fp_add:NNNNNNNNN
+ \else:
+ \exp_after:wN \fp_sub:NNNNNNNNN
+ \fi:
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \c_zero \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ }
+% \end{macrocode}
+% The usual shifts to move to fixed-point working. This is done using
+% the \texttt{output} registers as this saves a reassignment here.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_fixed:
+ {
+ \if_int_compare:w \l_fp_output_exponent_int < \c_zero
+ \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million
+ \exp_after:wN \use_i:nn \exp_after:wN
+ \fp_ln_fixed_aux:NNNNNNNNN
+ \int_use:N \l_fp_output_decimal_int
+ \exp_after:wN \fp_ln_fixed:
+ \fi:
+ }
+\cs_new_protected_nopar:Npn \fp_ln_fixed_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9
+ {
+ \if_int_compare:w \l_fp_output_integer_int = \c_zero
+ \l_fp_output_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
+ \else:
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \int_use:N \l_fp_output_integer_int
+ #1#2#3#4#5#6#7#8
+ }
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \l_fp_tmp_tl \scan_stop:
+ \fi:
+ \tex_advance:D \l_fp_output_exponent_int \c_one
+ }
+% \end{macrocode}
+% The main loop for the Taylor series: unlike some of the other similar
+% functions, the result here is not the final value and is therefore
+% subject to further manipulation outside of the loop.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_ln_Taylor_aux:
+ {
+ \tex_advance:D \l_fp_count_int \c_two
+ \fp_mul:NNNNNN
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_input_b_decimal_int \l_fp_input_b_extended_int
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int
+ > \c_zero
+ \fp_div_integer:NNNNN
+ \l_fp_input_a_decimal_int \l_fp_input_a_extended_int
+ \l_fp_count_int
+ \l_fp_exp_decimal_int \l_fp_exp_extended_int
+ \tex_advance:D \l_fp_output_decimal_int \l_fp_exp_decimal_int
+ \tex_advance:D \l_fp_output_extended_int \l_fp_exp_extended_int
+ \if_int_compare:w \l_fp_output_extended_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_output_decimal_int \c_one
+ \tex_advance:D \l_fp_output_extended_int
+ -\c_one_thousand_million
+ \fi:
+ \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million
+ \else:
+ \tex_advance:D \l_fp_output_integer_int \c_one
+ \tex_advance:D \l_fp_output_decimal_int
+ -\c_one_thousand_million
+ \fi:
+ \exp_after:wN \fp_ln_Taylor_aux:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\fp_pow:Nn, \fp_pow:cn}
+% \UnitTested
+% \begin{macro}{\fp_gpow:Nn,\fp_gpow:cn}
+% \UnitTested
+% \begin{macro}[aux]{\fp_pow_aux:NNn}
+% \begin{macro}[aux]{\fp_pow_aux_i:}
+% \begin{macro}[aux]{\fp_pow_positive:}
+% \begin{macro}[aux]{\fp_pow_negative:}
+% \begin{macro}[aux]{\fp_pow_aux_ii:}
+% \begin{macro}[aux]{\fp_pow_aux_iii:}
+% \begin{macro}[aux]{\fp_pow_aux_iv:}
+% The approach used for working out powers is to first filter out the
+% various special cases and then do most of the work using the
+% logarithm and exponent functions. The two storage areas are used
+% in the reverse of the `natural' logic as this avoids some
+% re-assignment in the sanity checking code.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_pow:Nn { \fp_pow_aux:NNn \tl_set:Nn }
+\cs_new_protected_nopar:Npn \fp_gpow:Nn { \fp_pow_aux:NNn \tl_gset:Nn }
+\cs_generate_variant:Nn \fp_pow:Nn { c }
+\cs_generate_variant:Nn \fp_gpow:Nn { c }
+\cs_new_protected_nopar:Npn \fp_pow_aux:NNn #1#2#3
+ {
+ \group_begin:
+ \fp_read:N #2
+ \l_fp_input_b_sign_int \l_fp_input_a_sign_int
+ \l_fp_input_b_integer_int \l_fp_input_a_integer_int
+ \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_input_b_exponent_int \l_fp_input_a_exponent_int
+ \fp_split:Nn a {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
+ = \c_zero
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ = \c_zero
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2 { \c_undefined_fp }
+ }
+ \else:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2 { \c_zero_fp }
+ }
+ \fi:
+ \else:
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ = \c_zero
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2 { \c_one_fp }
+ }
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_pow_aux_i:
+ \fi:
+ \fi:
+ \fp_tmp:w #1 #2
+}
+% \end{macrocode}
+% Simply using the logarithm function directly will fail when negative
+% numbers are raised to integer powers, which is a mathematically valid
+% operation. So there are some more tests to make, after forcing the
+% power into an integer and decimal parts, if necessary.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_pow_aux_i:
+ {
+ \if_int_compare:w \l_fp_input_b_sign_int > \c_zero
+ \tl_set:Nn \l_fp_sign_tl { + }
+ \exp_after:wN \fp_pow_aux_ii:
+ \else:
+ \l_fp_input_a_extended_int \c_zero
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten
+ \group_begin:
+ \fp_extended_normalise:
+ \if_int_compare:w
+ \int_eval:w
+ \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int
+ = \c_zero
+ \group_end:
+ \tl_set:Nn \l_fp_sign_tl { - }
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_pow_aux_ii:
+ \else:
+ \group_end:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2 { \c_undefined_fp }
+ }
+ \fi:
+ \else:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2 { \c_undefined_fp }
+ }
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% The approach used here for powers works well in most cases but gives
+% poorer results for negative integer powers, which often have exact
+% values. So there is some filtering to do. For negative powers where
+% the power is small, an alternative approach is used in which the
+% positive value is worked out and the reciprocal is then taken. The
+% filtering is unfortunately rather long.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_pow_aux_ii:
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \exp_after:wN \fp_pow_aux_iv:
+ \else:
+ \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten
+ \group_begin:
+ \l_fp_input_a_extended_int \c_zero
+ \fp_extended_normalise:
+ \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero
+ \if_int_compare:w \l_fp_input_a_integer_int > \c_ten
+ \group_end:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_pow_aux_iv:
+ \else:
+ \group_end:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_pow_aux_iii:
+ \fi:
+ \else:
+ \group_end:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \exp_after:wN \fp_pow_aux_iv:
+ \fi:
+ \else:
+ \exp_after:wN \exp_after:wN \exp_after:wN
+ \fp_pow_aux_iv:
+ \fi:
+ \fi:
+ \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
+ {
+ \group_end:
+ ##1 ##2
+ {
+ \l_fp_sign_tl
+ \int_use:N \l_fp_output_integer_int
+ .
+ \exp_after:wN \use_none:n
+ \int_value:w \int_eval:w
+ \l_fp_output_decimal_int + \c_one_thousand_million
+ e
+ \int_use:N \l_fp_output_exponent_int
+ }
+ }
+ }
+% \end{macrocode}
+% For the small negative integer powers, the calculation is done for
+% the positive power and the reciprocal is then taken.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_pow_aux_iii:
+ {
+ \l_fp_input_a_sign_int \c_one
+ \fp_pow_aux_iv:
+ \l_fp_input_a_integer_int \c_one
+ \l_fp_input_a_decimal_int \c_zero
+ \l_fp_input_a_exponent_int \c_zero
+ \l_fp_input_b_integer_int \l_fp_output_integer_int
+ \l_fp_input_b_decimal_int \l_fp_output_decimal_int
+ \l_fp_input_b_exponent_int \l_fp_output_exponent_int
+ \fp_div_internal:
+ }
+% \end{macrocode}
+% The business end of the code starts by finding the logarithm of the
+% given base. There is a bit of a shuffle so that this does not have
+% to be re-parsed and so that the output ends up in the correct place.
+% There is also a need to enable using the short-cut for a
+% pre-calculated result. The internal part of the multiplication
+% function can then be used to do the second part of the calculation
+% directly. There is some more set up before doing the exponential:
+% the idea here is to deactivate some internals so that everything works
+% smoothly.
+% \begin{macrocode}
+\cs_new_protected_nopar:Npn \fp_pow_aux_iv:
+ {
+ \group_begin:
+ \l_fp_input_a_integer_int \l_fp_input_b_integer_int
+ \l_fp_input_a_decimal_int \l_fp_input_b_decimal_int
+ \l_fp_input_a_exponent_int \l_fp_input_b_exponent_int
+ \fp_ln_internal:
+ \cs_set_protected_nopar:Npx \fp_tmp:w
+ {
+ \group_end:
+ \exp_not:N \l_fp_input_b_sign_int
+ \int_use:N \l_fp_output_sign_int \scan_stop:
+ \exp_not:N \l_fp_input_b_integer_int
+ \int_use:N \l_fp_output_integer_int \scan_stop:
+ \exp_not:N \l_fp_input_b_decimal_int
+ \int_use:N \l_fp_output_decimal_int \scan_stop:
+ \exp_not:N \l_fp_input_b_extended_int
+ \int_use:N \l_fp_output_extended_int \scan_stop:
+ \exp_not:N \l_fp_input_b_exponent_int
+ \int_use:N \l_fp_output_exponent_int \scan_stop:
+ }
+ \fp_tmp:w
+ \l_fp_input_a_extended_int \c_zero
+ \fp_mul:NNNNNNNNN
+ \l_fp_input_a_integer_int \l_fp_input_a_decimal_int
+ \l_fp_input_a_extended_int
+ \l_fp_input_b_integer_int \l_fp_input_b_decimal_int
+ \l_fp_input_b_extended_int
+ \l_fp_output_integer_int \l_fp_output_decimal_int
+ \l_fp_output_extended_int
+ \l_fp_output_exponent_int
+ \int_eval:w
+ \l_fp_input_a_exponent_int + \l_fp_input_b_exponent_int
+ \scan_stop:
+ \fp_extended_normalise_output:
+ \tex_multiply:D \l_fp_input_a_sign_int \l_fp_input_b_sign_int
+ \l_fp_input_a_integer_int \l_fp_output_integer_int
+ \l_fp_input_a_decimal_int \l_fp_output_decimal_int
+ \l_fp_input_a_extended_int \l_fp_output_extended_int
+ \l_fp_input_a_exponent_int \l_fp_output_exponent_int
+ \l_fp_output_integer_int \c_zero
+ \l_fp_output_decimal_int \c_zero
+ \l_fp_output_extended_int \c_zero
+ \l_fp_output_exponent_int \c_zero
+ \cs_set_eq:NN \fp_exp_const:Nx \use_none:nn
+ \fp_exp_internal:
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Tests for special values}
+%
+% \begin{macro}[pTF]{\fp_if_undefined:N}
+% \UnitTested
+% Testing for an undefined value is easy.
+% \begin{macrocode}
+\prg_new_conditional:Npnn \fp_if_undefined:N #1 { p , T , F , TF }
+ {
+ \if_meaning:w #1 \c_undefined_fp
+ \prg_return_true:
+ \else:
+ \prg_return_false:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}[pTF]{\fp_if_zero:N}
+% \UnitTested
+% Testing for a zero fixed-point is also easy.
+% \begin{macrocode}
+\prg_new_conditional:Npnn \fp_if_zero:N #1 { p , T , F , TF }
+ {
+ \if_meaning:w #1 \c_zero_fp
+ \prg_return_true:
+ \else:
+ \prg_return_false:
+ \fi:
+ }
+% \end{macrocode}
+% \end{macro}
+%
+% \subsection{Floating-point conditionals}
+%
+% \begin{macro}[TF]{\fp_compare:nNn}
+% \begin{macro}[TF]{\fp_compare:NNN}
+% \UnitTested
+% \begin{macro}[aux]{\fp_compare_aux:N}
+% \begin{macro}[aux]{\fp_compare_=:}
+% \begin{macro}[aux]{\fp_compare_<:}
+% \begin{macro}[aux]{\fp_compare_<_aux:}
+% \begin{macro}[aux]{\fp_compare_absolute_a>b:}
+% \begin{macro}[aux]{\fp_compare_absolute_a<b:}
+% \begin{macro}[aux]{\fp_compare_>:}
+% The idea for the comparisons is to provide two versions: slower and
+% faster. The lead off for both is the same: get the two numbers
+% read and then look for a function to handle the comparison.
+% \begin{macrocode}
+\prg_new_protected_conditional:Npnn \fp_compare:nNn #1#2#3 { T , F , TF }
+ {
+ \group_begin:
+ \fp_split:Nn a {#1}
+ \fp_standardise:NNNN
+ \l_fp_input_a_sign_int
+ \l_fp_input_a_integer_int
+ \l_fp_input_a_decimal_int
+ \l_fp_input_a_exponent_int
+ \fp_split:Nn b {#3}
+ \fp_standardise:NNNN
+ \l_fp_input_b_sign_int
+ \l_fp_input_b_integer_int
+ \l_fp_input_b_decimal_int
+ \l_fp_input_b_exponent_int
+ \fp_compare_aux:N #2
+ }
+\prg_new_protected_conditional:Npnn \fp_compare:NNN #1#2#3 { T , F , TF }
+ {
+ \group_begin:
+ \fp_read:N #3
+ \l_fp_input_b_sign_int \l_fp_input_a_sign_int
+ \l_fp_input_b_integer_int \l_fp_input_a_integer_int
+ \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int
+ \l_fp_input_b_exponent_int \l_fp_input_a_exponent_int
+ \fp_read:N #1
+ \fp_compare_aux:N #2
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux:N #1
+ {
+ \cs_if_exist:cTF { fp_compare_#1: }
+ { \use:c { fp_compare_#1: } }
+ {
+ \group_end:
+ \prg_return_false:
+ }
+ }
+% \end{macrocode}
+% For equality, the test is pretty easy as things are either equal or
+% they are not.
+% \begin{macrocode}
+\cs_new_protected_nopar:cpn { fp_compare_=: }
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int = \l_fp_input_b_sign_int
+ \if_int_compare:w \l_fp_input_a_integer_int = \l_fp_input_b_integer_int
+ \if_int_compare:w \l_fp_input_a_decimal_int = \l_fp_input_b_decimal_int
+ \if_int_compare:w
+ \l_fp_input_a_exponent_int = \l_fp_input_b_exponent_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ }
+% \end{macrocode}
+% Comparing two values is quite complex. First, there is a filter step
+% to check if one or other of the given values is zero. If it is then
+% the result is relatively easy to determine.
+% \begin{macrocode}
+\cs_new_protected_nopar:cpn { fp_compare_>: }
+ {
+ \if_int_compare:w \int_eval:w
+ \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
+ = \c_zero
+ \if_int_compare:w \int_eval:w
+ \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
+ = \c_zero
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w \l_fp_input_b_sign_int > \c_zero
+ \group_end:
+ \prg_return_false:
+ \else:
+ \group_end:
+ \prg_return_true:
+ \fi:
+ \fi:
+ \else:
+ \if_int_compare:w \int_eval:w
+ \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
+ = \c_zero
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \group_end:
+ \prg_return_true:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \else:
+ \use:c { fp_compare_>_aux: }
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% Next, check the sign of the input: this again may give an obvious
+% result. If both signs are the same, then hand off to comparing the
+% absolute values.
+% \begin{macrocode}
+\cs_new_protected_nopar:cpn { fp_compare_>_aux: }
+ {
+ \if_int_compare:w \l_fp_input_a_sign_int > \l_fp_input_b_sign_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \if_int_compare:w \l_fp_input_a_sign_int < \l_fp_input_b_sign_int
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w \l_fp_input_a_sign_int > \c_zero
+ \use:c { fp_compare_absolute_a>b: }
+ \else:
+ \use:c { fp_compare_absolute_a<b: }
+ \fi:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% Rather long runs of checks, as there is the need to go through each
+% layer of the input and do the comparison. There is also the need to
+% avoid messing up with equal inputs at each stage.
+% \begin{macrocode}
+\cs_new_protected_nopar:cpn { fp_compare_absolute_a>b: }
+ {
+ \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \if_int_compare:w \l_fp_input_a_exponent_int < \l_fp_input_b_exponent_int
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w \l_fp_input_a_integer_int > \l_fp_input_b_integer_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \if_int_compare:w
+ \l_fp_input_a_integer_int < \l_fp_input_b_integer_int
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w
+ \l_fp_input_a_decimal_int > \l_fp_input_b_decimal_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \fi:
+ \fi:
+ \fi:
+ \fi:
+ }
+\cs_new_protected_nopar:cpn { fp_compare_absolute_a<b: }
+ {
+ \if_int_compare:w \l_fp_input_b_exponent_int > \l_fp_input_a_exponent_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \if_int_compare:w \l_fp_input_b_exponent_int < \l_fp_input_a_exponent_int
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w \l_fp_input_b_integer_int > \l_fp_input_a_integer_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \if_int_compare:w
+ \l_fp_input_b_integer_int < \l_fp_input_a_integer_int
+ \group_end:
+ \prg_return_false:
+ \else:
+ \if_int_compare:w
+ \l_fp_input_b_decimal_int > \l_fp_input_a_decimal_int
+ \group_end:
+ \prg_return_true:
+ \else:
+ \group_end:
+ \prg_return_false:
+ \fi:
+ \fi:
+ \fi:
+ \fi:
+ \fi:
+ }
+% \end{macrocode}
+% This is just a case of reversing the two input values and then
+% running the tests already defined.
+% \begin{macrocode}
+\cs_new_protected_nopar:cpn { fp_compare_<: }
+ {
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \int_set:Nn \exp_not:N \l_fp_input_a_sign_int
+ { \int_use:N \l_fp_input_b_sign_int }
+ \int_set:Nn \exp_not:N \l_fp_input_a_integer_int
+ { \int_use:N \l_fp_input_b_integer_int }
+ \int_set:Nn \exp_not:N \l_fp_input_a_decimal_int
+ { \int_use:N \l_fp_input_b_decimal_int }
+ \int_set:Nn \exp_not:N \l_fp_input_a_exponent_int
+ { \int_use:N \l_fp_input_b_exponent_int }
+ \int_set:Nn \exp_not:N \l_fp_input_b_sign_int
+ { \int_use:N \l_fp_input_a_sign_int }
+ \int_set:Nn \exp_not:N \l_fp_input_b_integer_int
+ { \int_use:N \l_fp_input_a_integer_int }
+ \int_set:Nn \exp_not:N \l_fp_input_b_decimal_int
+ { \int_use:N \l_fp_input_a_decimal_int }
+ \int_set:Nn \exp_not:N \l_fp_input_b_exponent_int
+ { \int_use:N \l_fp_input_a_exponent_int }
+ }
+ \l_fp_tmp_tl
+ \use:c { fp_compare_>: }
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}[TF]{\fp_compare:n}
+% \begin{macro}[aux]
+% {
+% \fp_compare_aux_i:w, \fp_compare_aux_ii:w, \fp_compare_aux_iii:w,
+% \fp_compare_aux_iv:w, \fp_compare_aux_v:w, \fp_compare_aux_vi:w,
+% \fp_compare_aux_vii:w
+% }
+% As \TeX{} cannot help out here, a daisy-chain of delimited functions
+% are used. This is very much a first-generation approach: revision will
+% be needed if these functions are really useful.
+% \begin{macrocode}
+\prg_new_protected_conditional:Npnn \fp_compare:n #1 { T , F , TF }
+ {
+ \group_begin:
+ \tl_set:Nx \l_fp_tmp_tl
+ {
+ \group_end:
+ \fp_compare_aux_i:w #1 \exp_not:n { == \q_nil == \q_stop }
+ }
+ \l_fp_tmp_tl
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_i:w #1 == #2 == #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_ii:w #1 != \q_nil != \q_stop }
+ { \fp_compare:nNnTF {#1} = {#2} \prg_return_true: \prg_return_false: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_ii:w #1 != #2 != #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_iii:w #1 <= \q_nil <= \q_stop }
+ { \fp_compare:nNnTF {#1} = {#2} \prg_return_false: \prg_return_true: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_iii:w #1 <= #2 <= #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_iv:w #1 >= \q_nil >= \q_stop }
+ { \fp_compare:nNnTF {#1} > {#2} \prg_return_false: \prg_return_true: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_iv:w #1 >= #2 >= #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_v:w #1 = \q_nil \q_stop }
+ { \fp_compare:nNnTF {#1} < {#2} \prg_return_false: \prg_return_true: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_v:w #1 = #2 = #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_vi:w #1 < \q_nil < \q_stop }
+ { \fp_compare:nNnTF {#1} = {#2} \prg_return_true: \prg_return_false: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_vi:w #1 < #2 < #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \fp_compare_aux_vii:w #1 > \q_nil > \q_stop }
+ { \fp_compare:nNnTF {#1} < {#2} \prg_return_true: \prg_return_false: }
+ }
+\cs_new_protected_nopar:Npn \fp_compare_aux_vii:w #1 > #2 > #3 \q_stop
+ {
+ \quark_if_nil:nTF {#2}
+ { \prg_return_false: }
+ { \fp_compare:nNnTF {#1} > {#2} \prg_return_true: \prg_return_false: }
+ }
+% \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \subsection{Messages}
+%
+% \begin{macro}{\fp_overflow_msg:}
+% A generic overflow message, used whenever there is a possible
+% overflow.
+% \begin{macrocode}
+\msg_kernel_new:nnnn { fpu } { overflow }
+ { Number~too~big. }
+ {
+ The~input~given~is~too~big~for~the~LaTeX~floating~point~unit. \\
+ Further~errors~may~well~occur!
+ }
+\cs_new_protected_nopar:Npn \fp_overflow_msg:
+ { \msg_kernel_error:nn { fpu } { overflow } }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_exp_overflow_msg:}
+% A slightly more helpful message for exponent overflows.
+% \begin{macrocode}
+\msg_kernel_new:nnnn { fpu } { exponent-overflow }
+ { Number~too~big~for~exponent~unit. }
+ {
+ The~exponent~of~the~input~given~is~too~big~for~the~floating~point~
+ unit:~the~maximum~input~value~for~an~exponent~is~230.
+ }
+\cs_new_protected_nopar:Npn \fp_exp_overflow_msg:
+ { \msg_kernel_error:nn { fpu } { exponent-overflow } }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_ln_error_msg:}
+% Logarithms are only valid for positive number
+% \begin{macrocode}
+\msg_kernel_new:nnnn { fpu } { logarithm-input-error }
+ { Invalid~input~to~ln~function. }
+ { Logarithms~can~only~be~calculated~for~positive~numbers. }
+\cs_new_protected_nopar:Npn \fp_ln_error_msg: {
+ \msg_kernel_error:nn { fpu } { logarithm-input-error }
+}
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\fp_trig_overflow_msg:}
+% A slightly more helpful message for trigonometric overflows.
+% \begin{macrocode}
+\msg_kernel_new:nnnn { fpu } { trigonometric-overflow }
+ { Number~too~big~for~trigonometry~unit. }
+ {
+ The~trigonometry~code~can~only~work~with~numbers~smaller~
+ than~1000000000.
+ }
+\cs_new_protected_nopar:Npn \fp_trig_overflow_msg:
+ { \msg_kernel_error:nn { fpu } { trigonometric-overflow } }
+% \end{macrocode}
+% \end{macro}
+%
+% \begin{macrocode}
+%</initex|package>
+% \end{macrocode}
+%
+% \end{implementation}
+%
+%\PrintIndex