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-% \iffalse
-%% File: l3fp.dtx Copyright (C) 2010-2011 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 LaTeX Project Team.
-%%
-%% -----------------------------------------------------------------------
-%<*driver|package>
-\RequirePackage{l3names}
-%</driver|package>
-%\fi
-\GetIdInfo$Id: l3fp.dtx 2201 2011-03-19 17:16:13Z joseph $
- {L3 Experimental floating-point operations}
-%\iffalse
-%<*driver>
-%\fi
-\ProvidesFile{\filename.\filenameext}
- [\filedate\space v\fileversion\space\filedescription]
-%\iffalse
-\documentclass[full]{l3doc}
-\begin{document}
- \DocInput{l3fp.dtx}
-\end{document}
-%</driver>
-% \fi
-%
-% \title{The \textsf{l3fp} package\thanks{This file
-% has version number \fileversion, last
-% revised \filedate.}\\
-% Floating point arithmetic}
-% \author{\Team}
-% \date{\filedate}
-% \maketitle
-%
-%\begin{documentation}
-%
-%\section{Floating point numbers}
-%
-% 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.
-%
-%\subsection{Constants}
-%
-%\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}
-%
-%\subsection{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}{
-% \fp_use:N / (EXP) |
-% \fp_use:c / (EXP) |
-%}
-% \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 `\texttt{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}
-%
-%\subsection{Conversion 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}{
-% \fp_to_dim:N / (EXP) |
-% \fp_to_dim:c / (EXP) |
-%}
-% \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}{
-% \fp_to_int:N / (EXP) |
-% \fp_to_int:c / (EXP) |
-%}
-% \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}{
-% \fp_to_tl:N / (EXP) |
-% \fp_to_tl:c / (EXP) |
-%}
-% \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}
-%
-%\subsection{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}
-%
-%\subsection{Tests on floating-point values}
-%
-%\begin{function}{
-% \fp_if_undefined_p:N / (EXP) |
-% \fp_if_undefined:N / (EXP) (TF) |
-%}
-% \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}{
-% \fp_if_zero_p:N / (EXP) |
-% \fp_if_zero:N / (EXP) (TF) |
-%}
-% \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}{ \fp_compare:nNn / (TF) }
-% \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}
-%
-%\subsection{Unary 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}
-%
-%\subsection{Arithmetic operations}
-%
-% 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}
-%
-%\subsection{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}
-%
-%\subsection{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}
-%
-%\subsection{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}
-%
-%\subsection{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{Implementation}
-%
-% \TestFiles{m3fp003.lvt}
-%
-% We start by ensuring that the required packages are loaded.
-% \begin{macrocode}
-%<*package>
-\ProvidesExplPackage
- {\filename}{\filedate}{\fileversion}{\filedescription}
-\package_check_loaded_expl:
-%</package>
-%<*initex|package>
-% \end{macrocode}
-%
-%\subsection{Constants}
-%
-%\begin{macro}[aux]{\c_forty_four}
-%\begin{macro}[aux]{\c_one_hundred}
-%\begin{macro}[aux]{\c_one_thousand}
-%\begin{macro}[aux]{\c_one_million}
-%\begin{macro}[aux]{\c_one_hundred_million}
-%\begin{macro}[aux]{\c_five_hundred_million}
-%\begin{macro}[aux]{\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_hundred { 100 }
-\int_const:Nn \c_one_thousand { 1000 }
-\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{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\c_fp_pi_by_four_decimal_int}
-%\begin{macro}[aux]{\c_fp_pi_by_four_extended_int}
-%\begin{macro}[aux]{\c_fp_pi_decimal_int}
-%\begin{macro}[aux]{\c_fp_pi_extended_int}
-%\begin{macro}[aux]{\c_fp_two_pi_decimal_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}{\c_e_fp}
-% The value \( \mathrm{e} \) as a `machine number'.
-% \begin{macrocode}
-\tl_new:N \c_e_fp
-\tl_set:Nn \c_e_fp { + 2.718281828 e 0 }
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}{\c_one_fp}
-% The constant value \( 1 \): used for fast comparisons.
-% \begin{macrocode}
-\tl_new:N \c_one_fp
-\tl_set:Nn \c_one_fp { + 1.000000000 e 0 }
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}{\c_pi_fp}
-% The value \( \pi \) as a `machine number'.
-% \begin{macrocode}
-\tl_new:N \c_pi_fp
-\tl_set:Nn \c_pi_fp { + 3.141592654 e 0 }
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}{\c_undefined_fp}
-% A marker for undefined values.
-% \begin{macrocode}
-\tl_new:N \c_undefined_fp
-\tl_set:Nn \c_undefined_fp { X 0.000000000 e 0 }
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}{\c_zero_fp}
-% The constant zero value.
-% \begin{macrocode}
-\tl_new:N \c_zero_fp
-\tl_set:Nn \c_zero_fp { + 0.000000000 e 0 }
-% \end{macrocode}
-%\end{macro}
-%
-%\subsection{Variables}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\l_fp_exp_integer_int}
-%\begin{macro}[aux]{\l_fp_exp_decimal_int}
-%\begin{macro}[aux]{\l_fp_exp_extended_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_input_a_sign_int}
-%\begin{macro}[aux]{\l_fp_input_a_integer_int}
-%\begin{macro}[aux]{\l_fp_input_a_decimal_int}
-%\begin{macro}[aux]{\l_fp_input_a_exponent_int}
-%\begin{macro}[aux]{\l_fp_input_b_sign_int}
-%\begin{macro}[aux]{\l_fp_input_b_integer_int}
-%\begin{macro}[aux]{\l_fp_input_b_decimal_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_input_a_extended_int}
-%\begin{macro}[aux]{\l_fp_input_b_extended_int}
-% For internal use, `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{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_mul_a_i_int}
-%\begin{macro}[aux]{\l_fp_mul_a_ii_int}
-%\begin{macro}[aux]{\l_fp_mul_a_iii_int}
-%\begin{macro}[aux]{\l_fp_mul_a_iv_int}
-%\begin{macro}[aux]{\l_fp_mul_a_v_int}
-%\begin{macro}[aux]{\l_fp_mul_a_vi_int}
-%\begin{macro}[aux]{\l_fp_mul_b_i_int}
-%\begin{macro}[aux]{\l_fp_mul_b_ii_int}
-%\begin{macro}[aux]{\l_fp_mul_b_iii_int}
-%\begin{macro}[aux]{\l_fp_mul_b_iv_int}
-%\begin{macro}[aux]{\l_fp_mul_b_v_int}
-%\begin{macro}[aux]{\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{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}[aux]{\l_fp_mul_output_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_output_sign_int}
-%\begin{macro}[aux]{\l_fp_output_integer_int}
-%\begin{macro}[aux]{\l_fp_output_decimal_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_output_extended_int}
-% Again, for calculations an extended part.
-% \begin{macrocode}
-\int_new:N \l_fp_output_extended_int
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\l_fp_round_position_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\l_fp_sign_tl}
-% There are places where the sign needs to be set up `early',
-% so that the registers can be re-used.
-% \begin{macrocode}
-\tl_new:N \l_fp_sign_tl
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\l_fp_tmp_tl}
-% A scratch token list variable for expanding material.
-% \begin{macrocode}
-\tl_new:N \l_fp_tmp_tl
-% \end{macrocode}
-%\end{macro}
-%
-%\begin{macro}[aux]{\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{macro}
-%
-%\begin{macro}[aux]{\l_fp_trig_sign_int}
-%\begin{macro}[aux]{\l_fp_trig_decimal_int}
-%\begin{macro}[aux]{\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{macro}
-%\end{macro}
-%\end{macro}
-%
-%\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 {
- \tex_expandafter:D \fp_read_aux:w #1 \q_stop
-}
-\cs_new_protected_nopar:Npn \fp_read_aux:w #1#2 . #3 e #4 \q_stop {
- \tex_if:D #1 -
- \l_fp_input_a_sign_int \c_minus_one
- \tex_else:D
- \l_fp_input_a_sign_int \c_one
- \tex_fi:D
- \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_make_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
- \tex_expandafter:D \fp_split_exponent:w \l_fp_tmp_tl e e \q_stop #1
-}
-\cs_new_protected_nopar:Npn \fp_split_sign: {
- \tex_ifnum:D \pdf_strcmp:D
- { \tex_expandafter:D \tl_head:w \l_fp_tmp_tl ? \q_stop } { - }
- = \c_zero
- \tl_set:Nx \l_fp_tmp_tl
- {
- \tex_expandafter:D
- \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop
- }
- \l_fp_split_sign_int -\l_fp_split_sign_int
- \tex_expandafter:D \fp_split_sign:
- \tex_else:D
- \tex_ifnum:D \pdf_strcmp:D
- { \tex_expandafter:D \tl_head:w \l_fp_tmp_tl ? \q_stop } { + }
- = \c_zero
- \tl_set:Nx \l_fp_tmp_tl
- {
- \tex_expandafter:D
- \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop
- }
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_split_sign:
- \tex_fi:D
- \tex_fi:D
-}
-\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 }
- \etex_numexpr:D 0 #2 \scan_stop:
- \tex_afterassignment:D \fp_split_aux_i:w
- \use:c { l_fp_input_ #4 _integer_int }
- \etex_numexpr:D 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:
- \tex_expandafter:D \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:
- \tex_ifnum:D
- \etex_numexpr:D
- \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
- \tex_fi:D
- \tex_ifnum:D
- \use:c { l_fp_input_ #4 _integer_int } < \c_one_thousand_million
- \tex_else:D
- \tex_expandafter:D \fp_overflow_msg:
- \tex_fi:D
-}
-% \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 {
- \tex_ifnum:D
- \etex_numexpr:D #2 + #3 = \c_zero
- #1 \c_one
- #4 \c_zero
- \tex_expandafter:D \use_none:nnnn
- \tex_else:D
- \tex_expandafter:D \fp_standardise_aux:NNNN
- \tex_fi:D
- #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:
- {
- \tex_ifnum:D #2 = \c_zero
- \tex_advance:D #3 \c_one_thousand_million
- \tex_expandafter:D \fp_standardise_aux:w
- \int_use:N #3 \q_stop
- \tex_expandafter:D \fp_standardise_aux:
- \tex_fi:D
- }
- \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:
- {
- \tex_ifnum:D #2 > \c_nine
- \tex_advance:D #2 \c_one_thousand_million
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_standardise_aux:w \int_use:N #2
- \tex_expandafter:D \fp_standardise_aux:
- \tex_fi:D
- }
- \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
- \tex_expandafter:D \use_none:n \int_use:N #3
- }
- #3 \l_fp_tmp_tl \scan_stop:
- \tex_advance:D #4 \c_one
- }
- \fp_standardise_aux:
- \tex_ifnum:D #4 < \c_one_hundred
- \tex_ifnum:D #4 > -\c_one_hundred
- \tex_else:D
- #1 \c_one
- #2 \c_zero
- #3 \c_zero
- #4 \c_zero
- \tex_fi:D
- \tex_else:D
- \tex_expandafter:D \fp_overflow_msg:
- \tex_fi:D
-}
-\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: {
- \tex_ifnum:D \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int
- \tex_expandafter:D \fp_level_input_exponents_a:
- \tex_else:D
- \tex_expandafter:D \fp_level_input_exponents_b:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_level_input_exponents_a: {
- \tex_ifnum:D \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
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_level_input_exponents_a:NNNNNNNNN
- \int_use:N \l_fp_input_b_integer_int
- \tex_expandafter:D \fp_level_input_exponents_a:
- \tex_fi:D
-}
-\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
- \tex_expandafter:D \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: {
- \tex_ifnum:D \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
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_level_input_exponents_b:NNNNNNNNN
- \int_use:N \l_fp_input_a_integer_int
- \tex_expandafter:D \fp_level_input_exponents_b:
- \tex_fi:D
-}
-\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
- \tex_expandafter:D \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{tex_expandafter:D}.
-% 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 {
- \cs_if_free:NTF #1
- {
- \fp_new:N #1
- \fp_gset:Nn #1 {#2}
- }
- {
- \msg_kernel_error:nx { variable-already-defined }
- { \token_to_str:N #1 }
- }
-}
-\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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \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{macro}[aux]{\l_fp_tmp_dim}
-%\begin{macro}[aux]{\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
- {
- \tex_expandafter:D \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \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{macro}
-%\end{macro}
-%\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 {
- \tex_expandafter:D \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 {
- \tex_if:D #1 -
- -
- \tex_fi:D
- \tex_ifnum:D #3 > \c_zero
- \tex_expandafter:D \fp_use_large:w
- \tex_else:D
- \tex_ifnum:D #3 < \c_zero
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_use_small:w
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_use_none:w
- \tex_fi:D
- \tex_fi:D
- #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 {
- \tex_ifnum:D #3 < \c_ten
- \tex_expandafter:D \fp_use_large_aux_i:w
- \tex_else:D
- \tex_expandafter:D \fp_use_large_aux_ii:w
- \tex_fi:D
- #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 {
- \tex_expandafter:D \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 {
- \tex_if:D #1 -
- -
- \tex_fi:D
- \tex_ifnum:D #3 < \c_zero
- \tex_expandafter:D \fp_to_int_small:w
- \tex_else:D
- \tex_expandafter:D \fp_to_int_large:w
- \tex_fi:D
- #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 {
- \tex_ifnum:D #3 > \c_one
- \tex_else:D
- \tex_ifnum:D #1 < \c_five
- 0
- \tex_else:D
- 1
- \tex_fi:D
- \tex_fi:D
-}
-% \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 {
- \tex_ifnum:D #3 < \c_ten
- \tex_expandafter:D \fp_to_int_large_aux_i:w
- \tex_else:D
- \tex_expandafter:D \fp_to_int_large_aux_ii:w
- \tex_fi:D
- #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 {
- \tex_ifnum:D #1 < \c_five_hundred_million
- #3#2
- \tex_else:D
- \tex_number:D \etex_numexpr:D #3#2 + 1 \scan_stop:
- \tex_fi:D
-}
-\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 {
- \tex_expandafter:D \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 {
- \tex_if:D #1 -
- -
- \tex_fi:D
- \tex_ifnum:D #3 < \c_zero
- \tex_expandafter:D \fp_to_tl_small:w
- \tex_else:D
- \tex_expandafter:D \fp_to_tl_large:w
- \tex_fi:D
- #2 e #3 \q_stop
-}
-% \end{macrocode}
-% For `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 {
- \tex_ifnum:D #2 < \c_ten
- \tex_expandafter:D \fp_to_tl_large_aux_i:w
- \tex_else:D
- \tex_expandafter:D \fp_to_tl_large_aux_ii:w
- \tex_fi:D
- #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 {
- \tex_ifnum:D #2 = \c_minus_one
- \tex_expandafter:D \fp_to_tl_small_one:w
- \tex_else:D
- \tex_ifnum:D #2 = -\c_two
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_to_tl_small_two:w
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_to_tl_small_aux:w
- \tex_fi:D
- \tex_fi:D
- #1 e #2 \q_stop
-}
-\cs_new_nopar:Npn \fp_to_tl_small_one:w #1 . #2 e #3 \q_stop {
- \tex_ifnum:D \fp_use_ix:NNNNNNNNN #2 > \c_four
- \tex_ifnum:D
- \etex_numexpr:D #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1
- < \c_one_thousand_million
- 0.
- \tex_expandafter:D \fp_to_tl_small_zeros:NNNNNNNNN
- \tex_number:D
- \etex_numexpr:D
- #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1
- \scan_stop:
- \tex_else:D
- 1
- \tex_fi:D
- \tex_else:D
- 0. #1
- \fp_to_tl_small_zeros:NNNNNNNNN #2
- \tex_fi:D
-}
-\cs_new_nopar:Npn \fp_to_tl_small_two:w #1 . #2 e #3 \q_stop {
- \tex_ifnum:D \fp_use_iix_ix:NNNNNNNNN #2 > \c_forty_four
- \tex_ifnum:D
- \etex_numexpr:D #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten
- < \c_one_thousand_million
- 0.0
- \tex_expandafter:D \fp_to_tl_small_zeros:NNNNNNNNN
- \tex_number:D
- \etex_numexpr:D
- #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten
- \scan_stop:
- \tex_else:D
- 0.1
- \tex_fi:D
- \tex_else:D
- 0.0
- #1
- \fp_to_tl_small_zeros:NNNNNNNNN #2
- \tex_fi:D
-}
-\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 {
- \tex_ifnum:D #9 = \c_zero
- \tex_ifnum:D #8 = \c_zero
- \tex_ifnum:D #7 = \c_zero
- \tex_ifnum:D #6 = \c_zero
- \tex_ifnum:D #5 = \c_zero
- \tex_ifnum:D #4 = \c_zero
- \tex_ifnum:D #3 = \c_zero
- \tex_ifnum:D #2 = \c_zero
- \tex_ifnum:D #1 = \c_zero
- \tex_else:D
- . #1
- \tex_fi:D
- \tex_else:D
- . #1#2
- \tex_fi:D
- \tex_else:D
- . #1#2#3
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4#5
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4#5#6
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4#5#6#7
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4#5#6#7#8
- \tex_fi:D
- \tex_else:D
- . #1#2#3#4#5#6#7#8#9
- \tex_fi:D
-}
-\cs_new_nopar:Npn \fp_to_tl_small_zeros:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {
- \tex_ifnum:D #9 = \c_zero
- \tex_ifnum:D #8 = \c_zero
- \tex_ifnum:D #7 = \c_zero
- \tex_ifnum:D #6 = \c_zero
- \tex_ifnum:D #5 = \c_zero
- \tex_ifnum:D #4 = \c_zero
- \tex_ifnum:D #3 = \c_zero
- \tex_ifnum:D #2 = \c_zero
- \tex_ifnum:D #1 = \c_zero
- \tex_else:D
- #1
- \tex_fi:D
- \tex_else:D
- #1#2
- \tex_fi:D
- \tex_else:D
- #1#2#3
- \tex_fi:D
- \tex_else:D
- #1#2#3#4
- \tex_fi:D
- \tex_else:D
- #1#2#3#4#5
- \tex_fi:D
- \tex_else:D
- #1#2#3#4#5#6
- \tex_fi:D
- \tex_else:D
- #1#2#3#4#5#6#7
- \tex_fi:D
- \tex_else:D
- #1#2#3#4#5#6#7#8
- \tex_fi:D
- \tex_else:D
- #1#2#3#4#5#6#7#8#9
- \tex_fi:D
-}
-% \end{macrocode}
-% Some quick `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 }
- \tex_ifnum:D \l_fp_round_target_int < \c_ten
- \tex_expandafter:D \fp_round:
- \tex_fi:D
- \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \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 }
- \tex_ifnum:D \l_fp_round_target_int < \c_ten
- \tex_expandafter:D \fp_round:
- \tex_fi:D
- \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \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
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \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:
- \tex_ifnum:D \l_fp_input_a_integer_int < \c_ten
- \tex_else:D
- \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
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_round_loop:N #1 {
- \tex_ifnum:D \l_fp_round_position_int < \l_fp_round_target_int
- \bool_if:NTF \l_fp_round_carry_bool
- { \l_fp_tmp_int \etex_numexpr:D #1 + \c_one \scan_stop: }
- { \l_fp_tmp_int \etex_numexpr:D #1 \scan_stop: }
- \tex_ifnum:D \l_fp_tmp_int = \c_ten
- \l_fp_tmp_int \c_zero
- \tex_else:D
- \bool_set_false:N \l_fp_round_carry_bool
- \tex_fi:D
- \tl_set:Nx \l_fp_round_decimal_tl
- { \int_use:N \l_fp_tmp_int \l_fp_round_decimal_tl }
- \tex_else:D
- \tl_set:Nx \l_fp_round_decimal_tl { 0 \l_fp_round_decimal_tl }
- \tex_ifnum:D \l_fp_round_position_int = \l_fp_round_target_int
- \tex_ifnum:D #1 > \c_four
- \bool_set_true:N \l_fp_round_carry_bool
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
- \tex_advance:D \l_fp_round_position_int \c_minus_one
- \tex_ifnum:D \l_fp_round_position_int > \c_minus_one
- \tex_expandafter:D \fp_round_loop:N
- \tex_fi:D
-}
-% \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
- .
- \tex_expandafter:D \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \use_none:n
- \int_use:N \l_fp_input_a_decimal_int
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \tex_expandafter:D \group_end: \tex_expandafter:D
- #1 \tex_expandafter:D #2 \tex_expandafter:D { \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:
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
- \scan_stop:
- > \c_zero
- \tex_expandafter:D \fp_add_sum:
- \tex_else:D
- \tex_expandafter:D \fp_add_difference:
- \tex_fi:D
- \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
- {
- \tex_ifnum:D \l_fp_output_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \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
- \etex_numexpr:D
- \l_fp_input_a_integer_int + \l_fp_input_b_integer_int
- \scan_stop:
- \l_fp_output_decimal_int
- \etex_numexpr:D
- \l_fp_input_a_decimal_int + \l_fp_input_b_decimal_int
- \scan_stop:
- \tex_ifnum:D \l_fp_output_decimal_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_output_integer_int \c_one
- \tex_advance:D \l_fp_output_decimal_int -\c_one_thousand_million
- \tex_fi:D
-}
-% \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
- \etex_numexpr:D
- \l_fp_input_a_integer_int - \l_fp_input_b_integer_int
- \scan_stop:
- \l_fp_output_decimal_int
- \etex_numexpr:D
- \l_fp_input_a_decimal_int - \l_fp_input_b_decimal_int
- \scan_stop:
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_ifnum:D \l_fp_output_integer_int < \c_zero
- \l_fp_output_sign_int \l_fp_input_b_sign_int
- \tex_ifnum:D \l_fp_output_decimal_int = \c_zero
- \l_fp_output_integer_int -\l_fp_output_integer_int
- \tex_else:D
- \l_fp_output_decimal_int
- \etex_numexpr:D
- \c_one_thousand_million - \l_fp_output_decimal_int
- \scan_stop:
- \l_fp_output_integer_int
- \etex_numexpr:D
- - \l_fp_output_integer_int - \c_one
- \scan_stop:
- \tex_fi:D
- \tex_else:D
- \l_fp_output_sign_int \l_fp_input_a_sign_int
- \tex_fi:D
-}
-% \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
- \etex_numexpr:D
- \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
- {
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
- < \c_zero
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_output_integer_int + \l_fp_output_decimal_int
- = \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \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:
- }
- \tex_expandafter:D \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
- \etex_numexpr:D \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
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \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
- \tex_ifnum:D
- \etex_numexpr:D
- \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 }
- }
- \tex_else:D
- \tex_ifnum:D
- \etex_numexpr:D
- \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 }
- }
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_div_internal:
- \tex_fi:D
- \tex_fi:D
- \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
- \etex_numexpr:D
- \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
- {
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_sign_int * \l_fp_input_b_sign_int
- < \c_zero
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_output_integer_int + \l_fp_output_decimal_int
- = \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_output_decimal_int + \c_one_thousand_million
- \scan_stop:
- 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
- \tex_expandafter:D \fp_div_loop_step:w
- \int_use:N \l_fp_input_a_decimal_int \q_stop
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
- > \c_zero
- \tex_ifnum:D \l_fp_div_offset_int > \c_zero
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_div_loop:
- \tex_fi:D
- \tex_fi:D
-}
-% \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: {
- \tex_ifnum:D \l_fp_input_a_integer_int > \l_fp_input_b_integer_int
- \tex_expandafter:D \fp_div_divide_aux:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_integer_int < \l_fp_input_b_integer_int
- \tex_else:D
- \tex_ifnum:D
- \l_fp_input_a_decimal_int < \l_fp_input_b_decimal_int
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_div_divide_aux:
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
-}
-\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
- \tex_ifnum:D \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
- \tex_fi:D
- \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
- \etex_numexpr:D
- \l_fp_output_decimal_int +
- \l_fp_count_int * \l_fp_div_offset_int
- \scan_stop:
- \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
- \etex_numexpr:D
- #2 + \l_fp_input_a_integer_int
- \scan_stop:
- \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 \etex_numexpr:D #1 + #4 \scan_stop:
- #8 \etex_numexpr:D #2 + #5 \scan_stop:
- #9 \etex_numexpr:D #3 + #6 \scan_stop:
- \tex_ifnum:D #9 < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D #8 \c_one
- \tex_advance:D #9 -\c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D #8 < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D #7 \c_one
- \tex_advance:D #8 -\c_one_thousand_million
- \tex_fi:D
-}
-% \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 \etex_numexpr:D #1 - #4 \scan_stop:
- #8 \etex_numexpr:D #2 - #5 \scan_stop:
- #9 \etex_numexpr:D #3 - #6 \scan_stop:
- \tex_ifnum:D #9 < \c_zero
- \tex_advance:D #8 \c_minus_one
- \tex_advance:D #9 \c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D #8 < \c_zero
- \tex_advance:D #7 \c_minus_one
- \tex_advance:D #8 \c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D #7 < \c_zero
- \tex_ifnum:D \etex_numexpr:D #8 + #9 = \c_zero
- #7 -#7
- \tex_else:D
- \tex_advance:D #7 \c_one
- #8 \etex_numexpr:D \c_one_thousand_million - #8 \scan_stop:
- #9 \etex_numexpr:D \c_one_thousand_million - #9 \scan_stop:
- \tex_fi:D
- \tex_fi:D
-}
-% \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 \etex_numexpr:D #1 - \l_fp_tmp_int * #3 \scan_stop:
- #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 \etex_numexpr:D #5 + \l_fp_tmp_int * \c_one_million \scan_stop:
- \tex_ifnum:D #5 > \c_one_thousand_million
- \tex_advance:D #4 \c_one
- \tex_advancd:D #5 -\c_one_thousand_million
- \tex_fi:D
-}
-% \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 `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: {
- \tex_ifnum:D \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
- \tex_expandafter:D \fp_extended_normalise_aux_i:w
- \int_use:N \l_fp_input_a_decimal_int \q_stop
- \tex_expandafter:D \fp_extended_normalise_aux_i:
- \tex_fi:D
-}
-\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
- \etex_numexpr:D \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
- \tex_expandafter:D \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
- \etex_numexpr:D \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: {
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_zero
- \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_extended_normalise_ii_aux:NNNNNNNNN
- \int_use:N \l_fp_input_a_decimal_int
- \tex_expandafter:D \fp_extended_normalise_aux_ii:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn
- \fp_extended_normalise_ii_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {
- \tex_ifnum:D \l_fp_input_a_integer_int = \c_zero
- \l_fp_input_a_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
- \tex_else:D
- \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:
- \tex_fi:D
- \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: {
- \tex_ifnum:D \l_fp_output_integer_int > \c_nine
- \tex_advance:D \l_fp_output_integer_int \c_one_thousand_million
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_extended_normalise_output_aux_i:NNNNNNNNN
- \int_use:N \l_fp_output_integer_int
- \tex_expandafter:D \fp_extended_normalise_output:
- \tex_fi:D
-}
-\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
- \tex_expandafter:D \use_none:n
- \int_use:N \l_fp_output_decimal_int
- }
- \tex_expandafter:D \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
- \tex_expandafter:D \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: {
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_ten
- \l_fp_input_a_extended_int \c_zero
- \fp_extended_normalise:
- \fp_trig_normalise_aux:
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_expandafter:D \fp_trig_octant:
- \tex_else:D
- \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
- \tex_expandafter:D \fp_trig_overflow_msg:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_trig_normalise_aux: {
- \tex_ifnum:D \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
- \tex_expandafter:D \fp_trig_normalise_aux:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_two
- \tex_ifnum:D \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
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_trig_normalise_aux:
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
-}
-% \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
- \etex_numexpr:D \l_fp_input_a_integer_int - #1 \scan_stop:
- \l_fp_input_a_decimal_int
- \etex_numexpr:D \l_fp_input_a_decimal_int - #2 \scan_stop:
- \l_fp_input_a_extended_int
- \etex_numexpr:D \l_fp_input_a_extended_int - #3 \scan_stop:
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_ifnum:D \l_fp_input_a_integer_int < \c_zero
- \l_fp_input_a_sign_int -\l_fp_input_a_sign_int
- \tex_ifnum:D
- \etex_numexpr:D
- \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
- \tex_else:D
- \l_fp_input_a_integer_int
- \etex_numexpr:D
- - \l_fp_input_a_integer_int - \c_one
- \scan_stop:
- \l_fp_input_a_decimal_int
- \etex_numexpr:D
- \c_one_thousand_million - \l_fp_input_a_decimal_int
- \scan_stop:
- \l_fp_input_a_extended_int
- \etex_numexpr:D
- \c_one_thousand_million - \l_fp_input_a_extended_int
- \scan_stop:
- \tex_fi:D
- \tex_fi:D
-}
-% \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 `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:
- \tex_ifnum:D \l_fp_input_a_decimal_int < \c_ten
- \l_fp_input_a_decimal_int \c_zero
- \l_fp_input_a_extended_int \c_zero
- \tex_fi:D
- \tex_ifodd:D \l_fp_trig_octant_int
- \tex_else:D
- \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
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_trig_octant_aux: {
- \tex_ifnum:D \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
- \tex_expandafter:D \fp_trig_octant_aux:
- \tex_else:D
- \tex_ifnum:D
- \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
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_trig_octant_aux:
- \tex_fi:D
- \tex_fi:D
-}
-% \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_input_a_decimal_int + \c_one_thousand_million
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \tex_ifnum:D \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 }
- }
- \tex_else:D
- \etex_ifcsname:D
- c_fp_sin ( \l_fp_arg_tl ) _fp
- \tex_endcsname:D
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_sin_aux_i:
- \tex_fi:D
- \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 } }
- }
- \tex_fi:D
- \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:
- \tex_ifnum:D \l_fp_output_integer_int = \c_one
- \l_fp_output_exponent_int \c_zero
- \tex_else:D
- \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
- \tex_fi:D
- \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 }
- {
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \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_sin_aux_ii: {
- \tex_ifcase:D \l_fp_trig_octant_int
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_sin:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_cos:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_cos:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_sin:
- \tex_fi:D
-}
-% \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_input_a_decimal_int + \c_one_thousand_million
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \etex_ifcsname:D c_fp_cos ( \l_fp_arg_tl ) _fp \tex_endcsname:D
- \tex_else:D
- \tex_expandafter:D \fp_cos_aux_i:
- \tex_fi:D
- \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:
- \tex_ifnum:D \l_fp_output_integer_int = \c_one
- \l_fp_output_exponent_int \c_zero
- \tex_else:D
- \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
- \tex_fi:D
- \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 }
- {
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \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_cos_aux_ii: {
- \tex_ifcase:D \l_fp_trig_octant_int
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_cos:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_sin:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_sin:
- \tex_or:D
- \tex_expandafter:D \fp_trig_calc_cos:
- \tex_fi:D
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \tex_ifnum:D \l_fp_trig_octant_int > \c_two
- \l_fp_input_a_sign_int \c_minus_one
- \tex_fi:D
- \tex_else:D
- \tex_ifnum:D \l_fp_trig_octant_int > \c_two
- \tex_else:D
- \l_fp_input_a_sign_int \c_one
- \tex_fi:D
- \tex_fi:D
-}
-% \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: {
- \tex_ifnum:D \l_fp_input_a_decimal_int = \c_zero
- \l_fp_output_integer_int \c_one
- \l_fp_output_decimal_int \c_zero
- \tex_else:D
- \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
- \tex_ifnum:D \l_fp_trig_extended_int = \c_zero
- \tex_ifnum:D \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
- \tex_else:D
- \l_fp_output_integer_int \c_zero
- \l_fp_output_decimal_int \c_one_thousand_million
- \l_fp_output_extended_int \c_zero
- \tex_fi:D
- \tex_else:D
- \l_fp_output_integer_int \c_zero
- \l_fp_output_decimal_int 999999999 \scan_stop:
- \l_fp_output_extended_int \c_one_thousand_million
- \tex_fi:D
- \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
- \tex_expandafter:D \fp_trig_calc_Taylor:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_trig_calc_sin: {
- \l_fp_output_integer_int \c_zero
- \tex_ifnum:D \l_fp_input_a_decimal_int = \c_zero
- \l_fp_output_decimal_int \c_zero
- \tex_else:D
- \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
- \tex_expandafter:D \fp_trig_calc_Taylor:
- \tex_fi:D
-}
-% \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
- \tex_ifnum:D \l_fp_trig_decimal_int > \c_zero
- \tex_ifnum:D \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
- \tex_ifnum:D \l_fp_output_extended_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_output_decimal_int \c_one
- \tex_advance:D \l_fp_output_extended_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D \l_fp_output_decimal_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_output_integer_int \c_one
- \tex_advance:D \l_fp_output_decimal_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_else:D
- \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
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_fi:D
- \tex_expandafter:D \fp_trig_calc_Taylor:
- \tex_fi:D
-}
-% \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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_input_a_decimal_int + \c_one_thousand_million
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \tex_ifnum:D \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 }
- }
- \tex_else:D
- \etex_ifcsname:D
- c_fp_tan ( \l_fp_arg_tl ) _fp
- \tex_endcsname:D
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_tan_aux_i:
- \tex_fi:D
- \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 } }
- }
- \tex_fi:D
- \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 `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: {
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_ten
- \tex_expandafter:D \fp_tan_aux_ii:
- \tex_else:D
- \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
- \c_zero_fp
- \tex_expandafter:D \fp_trig_overflow_msg:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_tan_aux_ii: {
- \fp_trig_normalise:
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \tex_ifnum:D \l_fp_trig_octant_int > \c_two
- \l_fp_output_sign_int \c_minus_one
- \tex_else:D
- \l_fp_output_sign_int \c_one
- \tex_fi:D
- \tex_else:D
- \tex_ifnum:D \l_fp_trig_octant_int > \c_two
- \l_fp_output_sign_int \c_one
- \tex_else:D
- \l_fp_output_sign_int \c_minus_one
- \tex_fi:D
- \tex_fi:D
- \fp_cos_aux_ii:
- \tex_ifnum:D \l_fp_input_a_decimal_int = \c_zero
- \tex_ifnum:D \l_fp_input_a_integer_int = \c_zero
- \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
- \c_undefined_fp
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_tan_aux_iii:
- \tex_fi:D
- \tex_else:D
- \tex_expandafter:D \fp_tan_aux_iii:
- \tex_fi:D
-}
-% \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
- \tex_ifnum:D \l_fp_input_a_decimal_int = \c_zero
- \tex_ifnum:D \l_fp_input_a_integer_int = \c_zero
- \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp }
- \c_zero_fp
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_tan_aux_iv:
- \tex_fi:D
- \tex_else:D
- \tex_expandafter:D \fp_tan_aux_iv:
- \tex_fi:D
-}
- \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
- \etex_numexpr:D
- \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
- \tl_new:c { c_fp_tan ( \l_fp_arg_tl ) _fp }
- \tl_gset:cx { c_fp_tan ( \l_fp_arg_tl ) _fp }
- {
- \tex_ifnum:D \l_fp_output_sign_int > \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_output_decimal_int + \c_one_thousand_million
- \scan_stop:
- 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{macro}[aux]{\c_fp_exp_1_tl}
-%\begin{macro}[aux]{\c_fp_exp_2_tl}
-%\begin{macro}[aux]{\c_fp_exp_3_tl}
-%\begin{macro}[aux]{\c_fp_exp_4_tl}
-%\begin{macro}[aux]{\c_fp_exp_5_tl}
-%\begin{macro}[aux]{\c_fp_exp_6_tl}
-%\begin{macro}[aux]{\c_fp_exp_7_tl}
-%\begin{macro}[aux]{\c_fp_exp_8_tl}
-%\begin{macro}[aux]{\c_fp_exp_9_tl}
-%\begin{macro}[aux]{\c_fp_exp_10_tl}
-%\begin{macro}[aux]{\c_fp_exp_20_tl}
-%\begin{macro}[aux]{\c_fp_exp_30_tl}
-%\begin{macro}[aux]{\c_fp_exp_40_tl}
-%\begin{macro}[aux]{\c_fp_exp_50_tl}
-%\begin{macro}[aux]{\c_fp_exp_60_tl}
-%\begin{macro}[aux]{\c_fp_exp_70_tl}
-%\begin{macro}[aux]{\c_fp_exp_80_tl}
-%\begin{macro}[aux]{\c_fp_exp_90_tl}
-%\begin{macro}[aux]{\c_fp_exp_100_tl}
-%\begin{macro}[aux]{\c_fp_exp_200_tl}
-% Calculation of exponentials requires a number of precomputed values:
-% first the positive integers.
-% \begin{macrocode}
-\tl_new:c { c_fp_exp_1_tl }
-\tl_set:cn { c_fp_exp_1_tl }
- { { 2 } { 718281828 } { 459045235 } { 0 } }
-\tl_new:c { c_fp_exp_2_tl }
-\tl_set:cn { c_fp_exp_2_tl }
- { { 7 } { 389056098 } { 930650227 } { 0 } }
-\tl_new:c { c_fp_exp_3_tl }
-\tl_set:cn { c_fp_exp_3_tl }
- { { 2 } { 008553692 } { 318766774 } { 1 } }
-\tl_new:c { c_fp_exp_4_tl }
-\tl_set:cn { c_fp_exp_4_tl }
- { { 5 } { 459815003 } { 314423908 } { 1 } }
-\tl_new:c { c_fp_exp_5_tl }
-\tl_set:cn { c_fp_exp_5_tl }
- { { 1 } { 484131591 } { 025766034 } { 2 } }
-\tl_new:c { c_fp_exp_6_tl }
-\tl_set:cn { c_fp_exp_6_tl }
- { { 4 } { 034287934 } { 927351226 } { 2 } }
-\tl_new:c { c_fp_exp_7_tl }
-\tl_set:cn { c_fp_exp_7_tl }
- { { 1 } { 096633158 } { 428458599 } { 3 } }
-\tl_new:c { c_fp_exp_8_tl }
-\tl_set:cn { c_fp_exp_8_tl }
- { { 2 } { 980957987 } { 041728275 } { 3 } }
-\tl_new:c { c_fp_exp_9_tl }
-\tl_set:cn { c_fp_exp_9_tl }
- { { 8 } { 103083927 } { 575384008 } { 3 } }
-\tl_new:c { c_fp_exp_10_tl }
-\tl_set:cn { c_fp_exp_10_tl }
- { { 2 } { 202646579 } { 480671652 } { 4 } }
-\tl_new:c { c_fp_exp_20_tl }
-\tl_set:cn { c_fp_exp_20_tl }
- { { 4 } { 851651954 } { 097902280 } { 8 } }
-\tl_new:c { c_fp_exp_30_tl }
-\tl_set:cn { c_fp_exp_30_tl }
- { { 1 } { 068647458 } { 152446215 } { 13 } }
-\tl_new:c { c_fp_exp_40_tl }
-\tl_set:cn { c_fp_exp_40_tl }
- { { 2 } { 353852668 } { 370199854 } { 17 } }
-\tl_new:c { c_fp_exp_50_tl }
-\tl_set:cn { c_fp_exp_50_tl }
- { { 5 } { 184705528 } { 587072464 } { 21 } }
-\tl_new:c { c_fp_exp_60_tl }
-\tl_set:cn { c_fp_exp_60_tl }
- { { 1 } { 142007389 } { 815684284 } { 26 } }
-\tl_new:c { c_fp_exp_70_tl }
-\tl_set:cn { c_fp_exp_70_tl }
- { { 2 } { 515438670 } { 919167006 } { 30 } }
-\tl_new:c { c_fp_exp_80_tl }
-\tl_set:cn { c_fp_exp_80_tl }
- { { 5 } { 540622384 } { 393510053 } { 34 } }
-\tl_new:c { c_fp_exp_90_tl }
-\tl_set:cn { c_fp_exp_90_tl }
- { { 1 } { 220403294 } { 317840802 } { 39 } }
-\tl_new:c { c_fp_exp_100_tl }
-\tl_set:cn { c_fp_exp_100_tl }
- { { 2 } { 688117141 } { 816135448 } { 43 } }
-\tl_new:c { c_fp_exp_200_tl }
-\tl_set:cn { c_fp_exp_200_tl }
- { { 7 } { 225973768 } { 125749258 } { 86 } }
-% \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}
-%
-%
-%
-%
-%\begin{macro}[aux]{\c_fp_exp_-1_tl}
-%\begin{macro}[aux]{\c_fp_exp_-2_tl}
-%\begin{macro}[aux]{\c_fp_exp_-3_tl}
-%\begin{macro}[aux]{\c_fp_exp_-4_tl}
-%\begin{macro}[aux]{\c_fp_exp_-5_tl}
-%\begin{macro}[aux]{\c_fp_exp_-6_tl}
-%\begin{macro}[aux]{\c_fp_exp_-7_tl}
-%\begin{macro}[aux]{\c_fp_exp_-8_tl}
-%\begin{macro}[aux]{\c_fp_exp_-9_tl}
-%\begin{macro}[aux]{\c_fp_exp_-10_tl}
-%\begin{macro}[aux]{\c_fp_exp_-20_tl}
-%\begin{macro}[aux]{\c_fp_exp_-30_tl}
-%\begin{macro}[aux]{\c_fp_exp_-40_tl}
-%\begin{macro}[aux]{\c_fp_exp_-50_tl}
-%\begin{macro}[aux]{\c_fp_exp_-60_tl}
-%\begin{macro}[aux]{\c_fp_exp_-70_tl}
-%\begin{macro}[aux]{\c_fp_exp_-80_tl}
-%\begin{macro}[aux]{\c_fp_exp_-90_tl}
-%\begin{macro}[aux]{\c_fp_exp_-100_tl}
-%\begin{macro}[aux]{\c_fp_exp_-200_tl}
-% Now the negative integers.
-% \begin{macrocode}
-\tl_new:c { c_fp_exp_-1_tl }
-\tl_set:cn { c_fp_exp_-1_tl }
- { { 3 } { 678794411 } { 71442322 } { -1 } }
-\tl_new:c { c_fp_exp_-2_tl }
-\tl_set:cn { c_fp_exp_-2_tl }
- { { 1 } { 353352832 } { 366132692 } { -1 } }
-\tl_new:c { c_fp_exp_-3_tl }
-\tl_set:cn { c_fp_exp_-3_tl }
- { { 4 } { 978706836 } { 786394298 } { -2 } }
-\tl_new:c { c_fp_exp_-4_tl }
-\tl_set:cn { c_fp_exp_-4_tl }
- { { 1 } { 831563888 } { 873418029 } { -2 } }
-\tl_new:c { c_fp_exp_-5_tl }
-\tl_set:cn { c_fp_exp_-5_tl }
- { { 6 } { 737946999 } { 085467097 } { -3 } }
-\tl_new:c { c_fp_exp_-6_tl }
-\tl_set:cn { c_fp_exp_-6_tl }
- { { 2 } { 478752176 } { 666358423 } { -3 } }
-\tl_new:c { c_fp_exp_-7_tl }
-\tl_set:cn { c_fp_exp_-7_tl }
- { { 9 } { 118819655 } { 545162080 } { -4 } }
-\tl_new:c { c_fp_exp_-8_tl }
-\tl_set:cn { c_fp_exp_-8_tl }
- { { 3 } { 354626279 } { 025118388 } { -4 } }
-\tl_new:c { c_fp_exp_-9_tl }
-\tl_set:cn { c_fp_exp_-9_tl }
- { { 1 } { 234098040 } { 866795495 } { -4 } }
-\tl_new:c { c_fp_exp_-10_tl }
-\tl_set:cn { c_fp_exp_-10_tl }
- { { 4 } { 539992976 } { 248451536 } { -5 } }
-\tl_new:c { c_fp_exp_-20_tl }
-\tl_set:cn { c_fp_exp_-20_tl }
- { { 2 } { 061153622 } { 438557828 } { -9 } }
-\tl_new:c { c_fp_exp_-30_tl }
-\tl_set:cn { c_fp_exp_-30_tl }
- { { 9 } { 357622968 } { 840174605 } { -14 } }
-\tl_new:c { c_fp_exp_-40_tl }
-\tl_set:cn { c_fp_exp_-40_tl }
- { { 4 } { 248354255 } { 291588995 } { -18 } }
-\tl_new:c { c_fp_exp_-50_tl }
-\tl_set:cn { c_fp_exp_-50_tl }
- { { 1 } { 928749847 } { 963917783 } { -22 } }
-\tl_new:c { c_fp_exp_-60_tl }
-\tl_set:cn { c_fp_exp_-60_tl }
- { { 8 } { 756510762 } { 696520338 } { -27 } }
-\tl_new:c { c_fp_exp_-70_tl }
-\tl_set:cn { c_fp_exp_-70_tl }
- { { 3 } { 975449735 } { 908646808 } { -31 } }
-\tl_new:c { c_fp_exp_-80_tl }
-\tl_set:cn { c_fp_exp_-80_tl }
- { { 1 } { 804851387 } { 845415172 } { -35 } }
-\tl_new:c { c_fp_exp_-90_tl }
-\tl_set:cn { c_fp_exp_-90_tl }
- { { 8 } { 194012623 } { 990515430 } { -40 } }
-\tl_new:c { c_fp_exp_-100_tl }
-\tl_set:cn { c_fp_exp_-100_tl }
- { { 3 } { 720075976 } { 020835963 } { -44 } }
-\tl_new:c { c_fp_exp_-200_tl }
-\tl_set:cn { c_fp_exp_-200_tl }
- { { 1 } { 383896526 } { 736737530 } { -87 } }
-% \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}
-%
-%
-%
-%
-%
-%
-%
-%\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
- {
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- -
- \tex_else:D
- +
- \tex_fi:D
- \int_use:N \l_fp_input_a_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_input_a_decimal_int + \c_one_thousand_million
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \etex_ifcsname:D c_fp_exp ( \l_fp_arg_tl ) _fp \tex_endcsname:D
- \tex_else:D
- \tex_expandafter:D \fp_exp_internal:
- \tex_fi:D
- \cs_set_protected_nopar:Npx \fp_tmp:w
- {
- \group_end:
- #1 \exp_not:N #2
- {
- \etex_ifcsname:D c_fp_exp ( \l_fp_arg_tl ) _fp
- \tex_endcsname:D
- \use:c { c_fp_exp ( \l_fp_arg_tl ) _fp }
- \tex_else:D
- \c_zero_fp
- \tex_fi:D
- }
- }
- \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: {
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_three
- \fp_extended_normalise:
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \tex_ifnum:D \l_fp_input_a_integer_int < 230 \scan_stop:
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_exp_aux:
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_exp_overflow_msg:
- \tex_fi:D
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_integer_int < 230 \scan_stop:
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_exp_aux:
- \tex_else:D
- \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp }
- { \c_zero_fp }
- \tex_fi:D
- \tex_fi:D
- \tex_else:D
- \tex_expandafter:D \fp_exp_overflow_msg:
- \tex_fi:D
-}
-% \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: {
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_zero
- \tex_expandafter:D \fp_exp_integer:
- \tex_else:D
- \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
- \tex_expandafter:D \fp_exp_decimal:
- \tex_fi:D
-}
-% \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: {
- \tex_ifnum:D \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
- \tex_expandafter:D \fp_exp_integer_tens:
- \tex_else:D
- \tl_set:Nx \l_fp_tmp_tl
- {
- \tex_expandafter:D \use_i:nnn
- \int_use:N \l_fp_input_a_integer_int
- }
- \l_fp_input_a_integer_int
- \etex_numexpr:D
- \l_fp_input_a_integer_int - \l_fp_tmp_tl 00
- \scan_stop:
- \tex_ifnum:D \l_fp_input_a_sign_int < \c_zero
- \tex_ifnum:D \l_fp_output_integer_int > 200 \scan_stop:
- \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp }
- { \c_zero_fp }
- \tex_else:D
- \fp_exp_integer_const:n { - \l_fp_tmp_tl 00 }
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_exp_integer_tens:
- \tex_fi:D
- \tex_else:D
- \fp_exp_integer_const:n { \l_fp_tmp_tl 00 }
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_exp_integer_tens:
- \tex_fi:D
- \tex_fi:D
-}
-% \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
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_nine
- \tl_set:Nx \l_fp_tmp_tl
- {
- \tex_expandafter:D \use_i:nn
- \int_use:N \l_fp_input_a_integer_int
- }
- \l_fp_input_a_integer_int
- \etex_numexpr:D
- \l_fp_input_a_integer_int - \l_fp_tmp_tl 0
- \scan_stop:
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \fp_exp_integer_const:n { \l_fp_tmp_tl 0 }
- \tex_else:D
- \fp_exp_integer_const:n { - \l_fp_tmp_tl 0 }
- \tex_fi:D
- \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:
- \tex_fi:D
- \fp_exp_integer_units:
-}
-\cs_new_protected_nopar:Npn \fp_exp_integer_units: {
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_zero
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \fp_exp_integer_const:n { \int_use:N \l_fp_input_a_integer_int }
- \tex_else:D
- \fp_exp_integer_const:n
- { - \int_use:N \l_fp_input_a_integer_int }
- \tex_fi:D
- \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:
- \tex_fi:D
- \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 {
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_exp_integer_const:nnnn
- \tex_csname:D c_fp_exp_ #1 _tl \tex_endcsname:D
-}
-\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: {
- \tex_ifnum:D \l_fp_input_a_decimal_int > \c_zero
- \tex_ifnum:D \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
- \tex_else:D
- \l_fp_exp_integer_int \c_zero
- \tex_ifnum:D \l_fp_exp_extended_int = \c_zero
- \l_fp_exp_decimal_int
- \etex_numexpr:D
- \c_one_thousand_million - \l_fp_input_a_decimal_int
- \scan_stop:
- \l_fp_exp_extended_int \c_zero
- \tex_else:D
- \l_fp_exp_decimal_int
- \etex_numexpr:D
- 999999999 - \l_fp_input_a_decimal_int
- \scan_stop:
- \l_fp_exp_extended_int
- \etex_numexpr:D
- \c_one_thousand_million - \l_fp_input_a_extended_int
- \scan_stop:
- \tex_fi:D
- \tex_fi:D
- \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
- \tex_fi:D
- \tex_ifnum:D \l_fp_output_extended_int < \c_five_hundred_million
- \tex_else:D
- \tex_advance:D \l_fp_output_decimal_int \c_one
- \tex_ifnum:D \l_fp_output_decimal_int < \c_one_thousand_million
- \tex_else:D
- \l_fp_output_decimal_int \c_zero
- \tex_advance:D \l_fp_output_integer_int \c_one
- \tex_fi:D
- \tex_fi:D
- \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
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_output_decimal_int + \c_one_thousand_million
- \scan_stop:
- 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
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_b_decimal_int + \l_fp_input_b_extended_int
- > \c_zero
- \tex_ifnum:D \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
- \tex_ifnum:D \l_fp_exp_extended_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_exp_decimal_int \c_one
- \tex_advance:D \l_fp_exp_extended_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D \l_fp_exp_decimal_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_exp_integer_int \c_one
- \tex_advance:D \l_fp_exp_decimal_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_else:D
- \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
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_ifnum:D \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
- \tex_fi:D
- \tex_fi:D
- \tex_expandafter:D \fp_exp_Taylor:
- \tex_fi:D
-}
-% \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{macro}[aux]{\c_fp_ln_10_1_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_2_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_3_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_4_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_5_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_6_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_7_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_8_tl}
-%\begin{macro}[aux]{\c_fp_ln_10_9_tl}
-% Constants for working out logarithms: first those for the powers of
-% ten.
-% \begin{macrocode}
-\tl_new:c { c_fp_ln_10_1_tl }
-\tl_set:cn { c_fp_ln_10_1_tl }
- { { 2 } { 302585092 } { 994045684 } { 0 } }
-\tl_new:c { c_fp_ln_10_2_tl }
-\tl_set:cn { c_fp_ln_10_2_tl }
- { { 4 } { 605170185 } { 988091368 } { 0 } }
-\tl_new:c { c_fp_ln_10_3_tl }
-\tl_set:cn { c_fp_ln_10_3_tl }
- { { 6 } { 907755278 } { 982137052 } { 0 } }
-\tl_new:c { c_fp_ln_10_4_tl }
-\tl_set:cn { c_fp_ln_10_4_tl }
- { { 9 } { 210340371 } { 976182736 } { 0 } }
-\tl_new:c { c_fp_ln_10_5_tl }
-\tl_set:cn { c_fp_ln_10_5_tl }
- { { 1 } { 151292546 } { 497022842 } { 1 } }
-\tl_new:c { c_fp_ln_10_6_tl }
-\tl_set:cn { c_fp_ln_10_6_tl }
- { { 1 } { 381551055 } { 796427410 } { 1 } }
-\tl_new:c { c_fp_ln_10_7_tl }
-\tl_set:cn { c_fp_ln_10_7_tl }
- { { 1 } { 611809565 } { 095831979 } { 1 } }
-\tl_new:c { c_fp_ln_10_8_tl }
-\tl_set:cn { c_fp_ln_10_8_tl }
- { { 1 } { 842068074 } { 395226547 } { 1 } }
-\tl_new:c { c_fp_ln_10_9_tl }
-\tl_set:cn { c_fp_ln_10_9_tl }
- { { 2 } { 072326583 } { 694641116 } { 1 } }
-% \end{macrocode}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%\begin{macro}[aux]{\c_fp_ln_2_1_tl }
-%\begin{macro}[aux]{\c_fp_ln_2_2_tl }
-%\begin{macro}[aux]{\c_fp_ln_2_3_tl }
-% The smaller set for powers of two.
-% \begin{macrocode}
-\tl_new:c { c_fp_ln_2_1_tl }
-\tl_set:cn { c_fp_ln_2_1_tl }
- { { 0 } { 693147180 } { 559945309 } { 0 } }
-\tl_new:c { c_fp_ln_2_2_tl }
-\tl_set:cn { c_fp_ln_2_2_tl }
- { { 1 } { 386294361 } { 119890618 } { 0 } }
-\tl_new:c { c_fp_ln_2_3_tl }
-\tl_set:cn { c_fp_ln_2_3_tl }
- { { 2 } { 079441541 } { 679835928 } { 0 } }
-% \end{macrocode}
-%\end{macro}
-%\end{macro}
-%\end{macro}
-%
-%
-%
-%
-%\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_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
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
- > \c_zero
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_ln_aux:
- \tex_else:D
- \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
- {
- \group_end:
- ##1 \exp_not:N ##2 { \c_zero_fp }
- }
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_ln_error_msg:
- \tex_fi:D
- \tex_else:D
- \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
- {
- \group_end:
- ##1 \exp_not:N ##2 { \c_zero_fp }
- }
- \tex_expandafter:D \fp_ln_error_msg:
- \tex_fi:D
- \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
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_input_a_decimal_int + \c_one_thousand_million
- e
- \int_use:N \l_fp_input_a_exponent_int
- }
- \etex_ifcsname:D c_fp_ln ( \l_fp_arg_tl ) _fp \tex_endcsname:D
- \tex_else:D
- \tex_expandafter:D \fp_ln_exponent:
- \tex_fi:D
- \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.
-%\[
-% \ln ( 10^{n} \times 2^{m} \times x )
-% = \ln ( 10^{n} ) \times \ln ( 2^{m} ) \times \ln ( x )
-%\]
-% 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:
- \tex_ifnum:D \l_fp_output_extended_int < \c_five_hundred_million
- \tex_else:D
- \tex_advance:D \l_fp_output_decimal_int \c_one
- \tex_ifnum:D \l_fp_output_decimal_int < \c_one_thousand_million
- \tex_else:D
- \l_fp_output_decimal_int \c_zero
- \tex_advance:D \l_fp_output_integer_int \c_one
- \tex_fi:D
- \tex_fi:D
- \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_ln ( \l_fp_arg_tl ) _fp }
- \tl_gset:cx { c_fp_ln ( \l_fp_arg_tl ) _fp }
- {
- \tex_ifnum:D \l_fp_output_sign_int > \c_zero
- +
- \tex_else:D
- -
- \tex_fi:D
- \int_use:N \l_fp_output_integer_int
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \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: {
- \tex_ifnum:D \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
- \tex_else:D
- \l_fp_output_sign_int \c_one
- \tex_fi:D
- \tex_ifnum:D \l_fp_input_a_exponent_int > \c_nine
- \tl_set:Nx \l_fp_tmp_tl
- {
- \tex_expandafter:D \use_i:nn
- \int_use:N \l_fp_input_a_exponent_int
- }
- \l_fp_input_a_exponent_int
- \etex_numexpr:D
- \l_fp_input_a_exponent_int - \l_fp_tmp_tl 0
- \scan_stop:
- \fp_ln_const:nn { 10 } { \l_fp_tmp_tl }
- \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
- \tex_else:D
- \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
- \tex_fi:D
- \fp_ln_exponent_units:
-}
-% \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: {
- \tex_ifnum:D \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
- \tex_fi:D
- \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.
-% \begin{macrocode}
-\cs_new_protected_nopar:Npn \fp_ln_normalise: {
- \tex_ifnum:D \l_fp_exp_exponent_int < \l_fp_output_exponent_int
- \tex_advance:D \l_fp_exp_decimal_int \c_one_thousand_million
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_ln_normalise_aux:NNNNNNNNN
- \int_use:N \l_fp_exp_decimal_int
- \tex_expandafter:D \fp_ln_normalise:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn
- \fp_ln_normalise_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {
- \tex_ifnum:D \l_fp_exp_integer_int = \c_zero
- \l_fp_exp_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
- \tex_else:D
- \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:
- \tex_fi:D
- \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:
- \tex_ifnum:D \l_fp_count_int > \c_zero
- \fp_ln_const:nn { 2 } { \int_use:N \l_fp_count_int }
- \fp_ln_normalise:
- \tex_ifnum:D \l_fp_output_sign_int > \c_zero
- \tex_expandafter:D \fp_add:NNNNNNNNN
- \tex_else:D
- \tex_expandafter:D \fp_sub:NNNNNNNNN
- \tex_fi:D
- \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
- \tex_fi:D
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int > \c_one
- \scan_stop:
- \tex_expandafter:D \fp_ln_Taylor:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn \fp_ln_mantissa_aux: {
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_one
- \tex_advance:D \l_fp_count_int \c_one
- \fp_ln_mantissa_divide_two:
- \tex_expandafter:D \fp_ln_mantissa_aux:
- \tex_fi:D
-}
-% \end{macrocode}
-% A fast one-shot division by two.
-% \begin{macrocode}
-\cs_new_protected_nopar:Npn \fp_ln_mantissa_divide_two: {
- \tex_ifodd:D \l_fp_input_a_decimal_int
- \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million
- \tex_fi:D
- \tex_ifodd:D \l_fp_input_a_integer_int
- \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million
- \tex_fi:D
- \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 {
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_exp_integer_const:nnnn
- \tex_csname:D c_fp_ln_ #1 _ #2 _tl \tex_endcsname:D
-}
-% \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 `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_exponent_int \l_fp_output_exponent_int
- \l_fp_input_a_extended_int \c_zero
- \l_fp_output_decimal_int \c_zero
- \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:
- \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
- \tex_ifnum:D \l_fp_exp_extended_int < \c_five_hundred_million
- \tex_else:D
- \tex_advance:D \l_fp_exp_extended_int -\c_five_hundred_million
- \tex_advance:D \l_fp_exp_decimal_int \c_one
- \tex_fi:D
- \tex_advance:D \l_fp_exp_extended_int \l_fp_exp_extended_int
- \tex_ifnum:D \l_fp_output_sign_int > \c_zero
- \tex_expandafter:D \fp_add:NNNNNNNNN
- \tex_else:D
- \tex_expandafter:D \fp_sub:NNNNNNNNN
- \tex_fi:D
- \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: {
- \tex_ifnum:D \l_fp_output_exponent_int < \c_zero
- \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million
- \tex_expandafter:D \use_i:nn \tex_expandafter:D
- \fp_ln_fixed_aux:NNNNNNNNN
- \int_use:N \l_fp_output_decimal_int
- \tex_expandafter:D \fp_ln_fixed:
- \tex_fi:D
-}
-\cs_new_protected_nopar:Npn
- \fp_ln_fixed_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {
- \tex_ifnum:D \l_fp_output_integer_int = \c_zero
- \l_fp_output_decimal_int #1#2#3#4#5#6#7#8 \scan_stop:
- \tex_else:D
- \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:
- \tex_fi:D
- \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
- \tex_ifnum:D
- \etex_numexpr:D
- \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
- \tex_ifnum:D \l_fp_output_extended_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_output_decimal_int \c_one
- \tex_advance:D \l_fp_output_extended_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_ifnum:D \l_fp_output_decimal_int < \c_one_thousand_million
- \tex_else:D
- \tex_advance:D \l_fp_output_integer_int \c_one
- \tex_advance:D \l_fp_output_decimal_int
- -\c_one_thousand_million
- \tex_fi:D
- \tex_expandafter:D \fp_ln_Taylor_aux:
- \tex_fi:D
-}
-% \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_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
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
- = \c_zero
- \tex_ifnum:D
- \etex_numexpr:D
- \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 }
- }
- \tex_else:D
- \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
- {
- \group_end:
- ##1 ##2 { \c_zero_fp }
- }
- \tex_fi:D
- \tex_else:D
- \tex_ifnum:D
- \etex_numexpr:D
- \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 }
- }
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_pow_aux_i:
- \tex_fi:D
- \tex_fi:D
- \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: {
- \tex_ifnum:D \l_fp_input_b_sign_int > \c_zero
- \tl_set:Nn \l_fp_sign_tl { + }
- \tex_expandafter:D \fp_pow_aux_ii:
- \tex_else:D
- \l_fp_input_a_extended_int \c_zero
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_ten
- \group_begin:
- \fp_extended_normalise:
- \tex_ifnum:D
- \etex_numexpr:D
- \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int
- = \c_zero
- \group_end:
- \tl_set:Nn \l_fp_sign_tl { - }
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_pow_aux_ii:
- \tex_else:D
- \group_end:
- \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
- {
- \group_end:
- ##1 ##2 { \c_undefined_fp }
- }
- \tex_fi:D
- \tex_else:D
- \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2
- {
- \group_end:
- ##1 ##2 { \c_undefined_fp }
- }
- \tex_fi:D
- \tex_fi:D
-}
-% \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: {
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \tex_expandafter:D \fp_pow_aux_iv:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_exponent_int < \c_ten
- \group_begin:
- \l_fp_input_a_extended_int \c_zero
- \fp_extended_normalise:
- \tex_ifnum:D \l_fp_input_a_decimal_int = \c_zero
- \tex_ifnum:D \l_fp_input_a_integer_int > \c_ten
- \group_end:
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_pow_aux_iv:
- \tex_else:D
- \group_end:
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_pow_aux_iii:
- \tex_fi:D
- \tex_else:D
- \group_end:
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \tex_expandafter:D \fp_pow_aux_iv:
- \tex_fi:D
- \tex_else:D
- \tex_expandafter:D \tex_expandafter:D \tex_expandafter:D
- \fp_pow_aux_iv:
- \tex_fi:D
- \tex_fi:D
- \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
- .
- \tex_expandafter:D \use_none:n
- \tex_number:D \etex_numexpr:D
- \l_fp_output_decimal_int + \c_one_thousand_million
- \scan_stop:
- 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_input_a_integer_int \l_fp_input_a_decimal_int
- \l_fp_input_a_extended_int
- \tex_advance:D \l_fp_input_a_exponent_int \l_fp_input_b_exponent_int
- \l_fp_output_integer_int \c_zero
- \l_fp_output_decimal_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}{\fp_if_undefined_p:N}
-%\begin{macro}[TF]{\fp_if_undefined:N}
-%\UnitTested
-% Testing for an undefined value is easy.
-% \begin{macrocode}
-\prg_new_conditional:Npnn \fp_if_undefined:N #1 { T , F , TF , p } {
- \tex_ifx:D #1 \c_undefined_fp
- \prg_return_true:
- \tex_else:D
- \prg_return_false:
- \tex_fi:D
-}
-% \end{macrocode}
-%\end{macro}
-%\end{macro}
-%
-%
-%
-%
-%\begin{macro}{\fp_if_zero_p:N}
-%\begin{macro}[TF]{\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 { T , F , TF , p } {
- \tex_ifx:D #1 \c_zero_fp
- \prg_return_true:
- \tex_else:D
- \prg_return_false:
- \tex_fi:D
-}
-% \end{macrocode}
-%\end{macro}
-%\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_=: } {
- \tex_ifnum:D \l_fp_input_a_sign_int = \l_fp_input_b_sign_int
- \tex_ifnum:D \l_fp_input_a_integer_int = \l_fp_input_b_integer_int
- \tex_ifnum:D \l_fp_input_a_decimal_int = \l_fp_input_b_decimal_int
- \tex_ifnum:D
- \l_fp_input_a_exponent_int = \l_fp_input_b_exponent_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
-}
-% \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_>: } {
- \tex_ifnum:D \etex_numexpr:D
- \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int
- = \c_zero
- \tex_ifnum:D \etex_numexpr:D
- \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
- = \c_zero
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_b_sign_int > \c_zero
- \group_end:
- \prg_return_false:
- \tex_else:D
- \group_end:
- \prg_return_true:
- \tex_fi:D
- \tex_fi:D
- \tex_else:D
- \tex_ifnum:D \etex_numexpr:D
- \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int
- = \c_zero
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \group_end:
- \prg_return_true:
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_else:D
- \use:c { fp_compare_>_aux: }
- \tex_fi:D
- \tex_fi:D
-}
-% \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: } {
- \tex_ifnum:D \l_fp_input_a_sign_int > \l_fp_input_b_sign_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_sign_int < \l_fp_input_b_sign_int
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_sign_int > \c_zero
- \use:c { fp_compare_absolute_a>b: }
- \tex_else:D
- \use:c { fp_compare_absolute_a<b: }
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
-}
-% \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: } {
- \tex_ifnum:D \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_exponent_int < \l_fp_input_b_exponent_int
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_a_integer_int > \l_fp_input_b_integer_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \tex_ifnum:D
- \l_fp_input_a_integer_int < \l_fp_input_b_integer_int
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D
- \l_fp_input_a_decimal_int > \l_fp_input_b_decimal_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
-}
-\cs_new_protected_nopar:cpn { fp_compare_absolute_a<b: } {
- \tex_ifnum:D \l_fp_input_b_exponent_int > \l_fp_input_a_exponent_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_b_exponent_int < \l_fp_input_a_exponent_int
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D \l_fp_input_b_integer_int > \l_fp_input_a_integer_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \tex_ifnum:D
- \l_fp_input_b_integer_int < \l_fp_input_a_integer_int
- \group_end:
- \prg_return_false:
- \tex_else:D
- \tex_ifnum:D
- \l_fp_input_b_decimal_int > \l_fp_input_a_decimal_int
- \group_end:
- \prg_return_true:
- \tex_else:D
- \group_end:
- \prg_return_false:
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
- \tex_fi:D
-}
-% \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}
-%
-%\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}
-%
-%\PrintChanges
-%
-%\PrintIndex