diff options
Diffstat (limited to 'Master/texmf-dist/source/latex/expl3/l3fp.dtx')
-rw-r--r-- | Master/texmf-dist/source/latex/expl3/l3fp.dtx | 5893 |
1 files changed, 0 insertions, 5893 deletions
diff --git a/Master/texmf-dist/source/latex/expl3/l3fp.dtx b/Master/texmf-dist/source/latex/expl3/l3fp.dtx deleted file mode 100644 index 8e592fe9a88..00000000000 --- a/Master/texmf-dist/source/latex/expl3/l3fp.dtx +++ /dev/null @@ -1,5893 +0,0 @@ -% \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 |