% \iffalse %% File: l3fp.dtx Copyright (C) 2010 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} % %\fi \GetIdInfo$Id: l3fp.dtx 2029 2010-09-19 13:59:02Z 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} % % \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 \( 1 \le \string| x \string| < 10 \), % 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_infinity_fp } % A marker value for an infinite result from a calculation, such as % \( \tan ( \pi / 2 ) \). %\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_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_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_infinity_p:N / (EXP) | % \fp_if_infinity:N / (EXP) (TF) | %} % \begin{syntax} % \cs{fp_if_infinity_p:N} \meta{fixed-point} % \cs{fp_if_infinity:NTF} \meta{fixed-point} % ~~\Arg{true code} \Arg{false code} % \end{syntax} % Tests if \meta{floating point} is infinite (\emph{i.e}.~equal to the % special \cs{c_infinity_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_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) | % \fp_compare:NNN / (TF) | %} % \begin{syntax} % \cs{fp_compare:nNnTF} \Arg{value1} \meta{relation} \Arg{value2} % ~~\Arg{true code} \Arg{false code} % \cs{fp_compare:NNNTF} \Arg{fp1} \meta{relation} \Arg{fp2} % ~~\Arg{true code} \Arg{false code} % \end{syntax} % Compares the two \meta{values} or \meta{floating points} based on the % \meta{relation} (\texttt{=}, \verb"<" or \verb">"), and leaves % either the \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 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}. %\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}. %\end{function} % %\subsection{Trigonometric functions} % % The trigonometric functions all work in radians. They accept a maximum % input value of \( 1 000 000 000\), 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} % % 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} % % We start by ensuring that the required packages are loaded. % \begin{macrocode} %<*package> \ProvidesExplPackage {\filename}{\filedate}{\fileversion}{\filedescription} \package_check_loaded_expl: % %<*initex|package> % \end{macrocode} % %\subsection{Constants} % %\begin{macro}{\c_forty_four} %\begin{macro}{\c_one_hundred} %\begin{macro}{\c_one_thousand} %\begin{macro}{\c_one_million} %\begin{macro}{\c_one_hundred_million} %\begin{macro}{\c_five_hundred_million} %\begin{macro}{\c_one_thousand_million} % There is some speed to gain by moving numbers into fixed positions. % \begin{macrocode} \int_new:N \c_forty_four \int_set:Nn \c_forty_four { 44 } \int_new:N \c_one_hundred \int_set:Nn \c_one_hundred { 100 } \int_new:N \c_one_thousand \int_set:Nn \c_one_thousand { 1000 } \int_new:N \c_one_million \int_set:Nn \c_one_million { 1 000 000 } \int_new:N \c_one_hundred_million \int_set:Nn \c_one_hundred_million { 100 000 000 } \int_new:N \c_five_hundred_million \int_set:Nn \c_five_hundred_million { 500 000 000 } \int_new:N \c_one_thousand_million \int_set: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}{\c_fp_pi_by_four_decimal_int} %\begin{macro}{\c_fp_pi_by_four_extended_int} %\begin{macro}{\c_fp_pi_decimal_int} %\begin{macro}{\c_fp_pi_extended_int} %\begin{macro}{\c_fp_two_pi_decimal_int} %\begin{macro}{\c_fp_two_pi_extended_int} % Parts of \( \pi \) for trigonometric range reduction. % \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_infinity_fp} % Infinity is the biggest number that can be represented by \TeX's % \texttt{count} data type. % \begin{macrocode} \tl_new:N \c_infinity_fp \tl_set:Nn \c_infinity_fp { + 2147483647 . 2147483647 e 2147483647 } % \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}{\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}{\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}{\l_fp_input_a_sign_int} %\begin{macro}{\l_fp_input_a_integer_int} %\begin{macro}{\l_fp_input_a_decimal_int} %\begin{macro}{\l_fp_input_a_exponent_int} %\begin{macro}{\l_fp_input_b_sign_int} %\begin{macro}{\l_fp_input_b_integer_int} %\begin{macro}{\l_fp_input_b_decimal_int} %\begin{macro}{\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}{\l_fp_input_a_extended_int} %\begin{macro}{\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}{\l_fp_mul_a_i_int} %\begin{macro}{\l_fp_mul_a_ii_int} %\begin{macro}{\l_fp_mul_a_iii_int} %\begin{macro}{\l_fp_mul_a_iv_int} %\begin{macro}{\l_fp_mul_a_v_int} %\begin{macro}{\l_fp_mul_a_vi_int} %\begin{macro}{\l_fp_mul_b_i_int} %\begin{macro}{\l_fp_mul_b_ii_int} %\begin{macro}{\l_fp_mul_b_iii_int} %\begin{macro}{\l_fp_mul_b_iv_int} %\begin{macro}{\l_fp_mul_b_v_int} %\begin{macro}{\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}{\l_fp_mul_output_int} %\begin{macro}{\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}{\l_fp_output_sign_int} %\begin{macro}{\l_fp_output_integer_int} %\begin{macro}{\l_fp_output_decimal_int} %\begin{macro}{\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}{\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}{\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}{\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}{\l_fp_round_position_int} %\begin{macro}{\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}{\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}{\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}{\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}{\l_fp_trig_arg_tl} % A token list to store the formalised representation of the input % for trigonometry. % \begin{macrocode} \tl_new:N \l_fp_trig_arg_tl % \end{macrocode} %\end{macro} % %\begin{macro}{\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}{\l_fp_trig_sign_int} %\begin{macro}{\l_fp_trig_decimal_int} %\begin{macro}{\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 { \exp_after:wN \fp_read_aux:w #1 \q_stop } \cs_new_protected_nopar:Npn \fp_read_aux:w #1#2 . #3 e #4 \q_stop { \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} \l_fp_split_sign_int \c_one \fp_split_sign: \use:c { l_fp_input_ #1 _sign_int } \l_fp_split_sign_int \exp_after:wN \fp_split_exponent:w \l_fp_tmp_tl e e \q_stop #1 } \cs_new_protected_nopar:Npn \fp_split_sign: { \tex_ifnum:D \pdf_strcmp:D { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { - } = \c_zero \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop } \l_fp_split_sign_int -\l_fp_split_sign_int \exp_after:wN \fp_split_sign: \tex_else:D \tex_ifnum:D \pdf_strcmp:D { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { + } = \c_zero \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop } \exp_after:wN \exp_after:wN \exp_after:wN \fp_split_sign: \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: \exp_after:wN \fp_split_decimal:w \int_use:N \l_fp_tmp_int 000000000 \q_stop } \cs_new_protected_nopar:Npn \fp_split_decimal:w #1#2#3#4#5#6#7#8#9 { \fp_split_decimal_aux:w {#2#3#4#5#6#7#8#9} } \cs_new_protected_nopar:Npn \fp_split_decimal_aux:w #1#2#3 \q_stop #4 { \use:c { l_fp_input_ #4 _decimal_int } #1#2 \scan_stop: \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 \exp_after:wN \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 \exp_after:wN \use_none:nnnn \tex_else:D \exp_after:wN \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 \exp_after:wN \fp_standardise_aux:w \int_use:N #3 \q_stop \exp_after:wN \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 \exp_after:wN \use_i:nn \exp_after:wN \fp_standardise_aux:w \int_use:N #2 \exp_after:wN \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 \exp_after:wN \use_none:n \int_use:N #3 } #3 \l_fp_tmp_tl \scan_stop: \tex_advance:D #4 \c_one } \fp_standardise_aux: \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 \exp_after:wN \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 \exp_after:wN \fp_level_input_exponents_a: \tex_else:D \exp_after:wN \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 \exp_after:wN \use_i:nn \exp_after:wN \fp_level_input_exponents_a:NNNNNNNNN \int_use:N \l_fp_input_b_integer_int \exp_after:wN \fp_level_input_exponents_a: \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 \exp_after:wN \use_none:n \int_use:N \l_fp_input_b_decimal_int } \l_fp_input_b_decimal_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_input_b_exponent_int \c_one } \cs_new_protected_nopar:Npn \fp_level_input_exponents_b: { \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 \exp_after:wN \use_i:nn \exp_after:wN \fp_level_input_exponents_b:NNNNNNNNN \int_use:N \l_fp_input_a_integer_int \exp_after:wN \fp_level_input_exponents_b: \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 \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int } \l_fp_input_a_decimal_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_input_a_exponent_int \c_one } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_tmp:w} % Used for output of results, cutting down on \cs{exp_after:wN}. % This is just a place holder definition. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_tmp:w #1#2 { } % \end{macrocode} %\end{macro} % %\subsection{Operations for \texttt{fp} variables} % % The format of \texttt{fp} variables is tightly defined, so that % they can be read quickly by the internal code. The format is a single % sign token, a single number, the decimal point, nine decimal numbers, % an "e" and finally the exponent. This final part may vary in length. % When stored, floating points will always be stored with a value in % the integer position unless the number is zero. % %\begin{macro}{\fp_new:N} %\begin{macro}{\fp_new:c} % 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} %\end{macro} % %\begin{macro}{\fp_zero:N} %\begin{macro}{\fp_zero:c} %\begin{macro}{\fp_gzero:N} %\begin{macro}{\fp_gzero:c} % 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} %\end{macro} %\end{macro} % %\begin{macro}{\fp_set:Nn} %\begin{macro}{\fp_set:cn} %\begin{macro}{\fp_gset:Nn} %\begin{macro}{\fp_gset:cn} %\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 . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } \cs_generate_variant:Nn \fp_set:Nn { c } \cs_generate_variant:Nn \fp_gset:Nn { c } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_set_from_dim:Nn} %\begin{macro}{\fp_set_from_dim:cn} %\begin{macro}{\fp_gset_from_dim:Nn} %\begin{macro}{\fp_gset_from_dim:cn} %\begin{macro}[aux]{\fp_set_from_dim_aux:NNn} %\begin{macro}[aux]{\fp_set_from_dim_aux:w} %\begin{macro}{\l_fp_tmp_dim} %\begin{macro}{\l_fp_tmp_skip} % Here, dimensions are converted to fixed-points \emph{via} a % temporary variable. This ensures that they always convert as points. % The code is then essentially the same as for \cs{fp_set:Nn}, but with % the dimension passed so that it will be striped of the "pt" on the % way through. The passage through a skip is used to remove any rubber % part. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_set_from_dim:Nn { \fp_set_from_dim_aux:NNn \tl_set:Nx } \cs_new_protected_nopar:Npn \fp_gset_from_dim:Nn { \fp_set_from_dim_aux:NNn \tl_gset:Nx } \cs_new_protected_nopar:Npn \fp_set_from_dim_aux:NNn #1#2#3 { \group_begin: \l_fp_tmp_skip \etex_glueexpr:D #3 \scan_stop: \l_fp_tmp_dim \l_fp_tmp_skip \fp_split:Nn a { \exp_after:wN \fp_set_from_dim_aux:w \dim_use:N \l_fp_tmp_dim } \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \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 . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } \cs_set_protected_nopar:Npx \fp_set_from_dim_aux:w { \cs_set_nopar:Npn \exp_not:N \fp_set_from_dim_aux:w ##1 \tl_to_str:n { pt } {##1} } \fp_set_from_dim_aux:w \cs_generate_variant:Nn \fp_set_from_dim:Nn { c } \cs_generate_variant:Nn \fp_gset_from_dim:Nn { c } \dim_new:N \l_fp_tmp_dim \skip_new:N \l_fp_tmp_skip % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_set_eq:NN} %\begin{macro}{\fp_set_eq:cN} %\begin{macro}{\fp_set_eq:Nc} %\begin{macro}{\fp_set_eq:cc} %\begin{macro}{\fp_gset_eq:NN} %\begin{macro}{\fp_gset_eq:cN} %\begin{macro}{\fp_gset_eq:Nc} %\begin{macro}{\fp_gset_eq:cc} % 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} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_show:N} %\begin{macro}{\fp_show:c} % 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} %\end{macro} % %\begin{macro}{\fp_use:N} %\begin{macro}{\fp_use:c} %\begin{macro}[aux]{\fp_use_aux:w} %\begin{macro}[aux]{\fp_use_none:w} %\begin{macro}[aux]{\fp_use_small:w} %\begin{macro}[aux]{\fp_use_large:w} %\begin{macro}[aux]{\fp_use_large_aux_i:w} %\begin{macro}[aux]{\fp_use_large_aux_1:w} %\begin{macro}[aux]{\fp_use_large_aux_2:w} %\begin{macro}[aux]{\fp_use_large_aux_3:w} %\begin{macro}[aux]{\fp_use_large_aux_4:w} %\begin{macro}[aux]{\fp_use_large_aux_5:w} %\begin{macro}[aux]{\fp_use_large_aux_6:w} %\begin{macro}[aux]{\fp_use_large_aux_7:w} %\begin{macro}[aux]{\fp_use_large_aux_8:w} %\begin{macro}[aux]{\fp_use_large_aux_i:w} %\begin{macro}[aux]{\fp_use_large_aux_ii:w} % The idea of the \cs{fp_use:N} function to convert the stored % value into something suitable for \TeX\ to use as a number in an % expandable manner. The first step is to deal with the sign, then % work out how big the input is. % \begin{macrocode} \cs_new_nopar:Npn \fp_use:N #1 { \exp_after:wN \fp_use_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_use:N { c } \cs_new_nopar:Npn \fp_use_aux:w #1#2 e #3 \q_stop { \tex_if:D #1 - - \tex_fi:D \tex_ifnum:D #3 > \c_zero \exp_after:wN \fp_use_large:w \tex_else:D \tex_ifnum:D #3 < \c_zero \exp_after:wN \exp_after:wN \exp_after:wN \fp_use_small:w \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \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 \exp_after:wN \fp_use_large_aux_i:w \tex_else:D \exp_after:wN \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} %\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_int:N} %\begin{macro}{\fp_to_int:c} %\begin{macro}[aux]{\fp_to_int_aux:w} %\begin{macro}[aux]{\fp_to_int_none:w} %\begin{macro}[aux]{\fp_to_int_small:w} %\begin{macro}[aux]{\fp_to_int_large:w} %\begin{macro}[aux]{\fp_to_int_large_aux_i:w} %\begin{macro}[aux]{\fp_to_int_large_aux_1:w} %\begin{macro}[aux]{\fp_to_int_large_aux_2:w} %\begin{macro}[aux]{\fp_to_int_large_aux_3:w} %\begin{macro}[aux]{\fp_to_int_large_aux_4:w} %\begin{macro}[aux]{\fp_to_int_large_aux_5:w} %\begin{macro}[aux]{\fp_to_int_large_aux_6:w} %\begin{macro}[aux]{\fp_to_int_large_aux_7:w} %\begin{macro}[aux]{\fp_to_int_large_aux_8:w} %\begin{macro}[aux]{\fp_to_int_large_aux_i:w} %\begin{macro}[aux]{\fp_to_int_large_aux:nnn} %\begin{macro}[aux]{\fp_to_int_large_aux_ii:w} % Converting to integers in an expandable manner is very similar to % simply using floating point variables, particularly in the lead-off. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_int:N #1 { \exp_after:wN \fp_to_int_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_to_int:N { c } \cs_new_nopar:Npn \fp_to_int_aux:w #1#2 e #3 \q_stop { \tex_if:D #1 - - \tex_fi:D \tex_ifnum:D #3 < \c_zero \exp_after:wN \fp_to_int_small:w \tex_else:D \exp_after:wN \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 \exp_after:wN \fp_to_int_large_aux_i:w \tex_else:D \exp_after:wN \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} %\end{macro} % %\begin{macro}{\fp_to_tl:N} %\begin{macro}{\fp_to_tl:c} %\begin{macro}[aux]{\fp_to_tl_aux:w} %\begin{macro}[aux]{\fp_to_tl_large:w} %\begin{macro}[aux]{\fp_to_tl_large_aux_i:w} %\begin{macro}[aux]{\fp_to_tl_large_aux_ii:w} %\begin{macro}[aux]{\fp_to_tl_large_0:w} %\begin{macro}[aux]{\fp_to_tl_large_1:w} %\begin{macro}[aux]{\fp_to_tl_large_2:w} %\begin{macro}[aux]{\fp_to_tl_large_3:w} %\begin{macro}[aux]{\fp_to_tl_large_4:w} %\begin{macro}[aux]{\fp_to_tl_large_5:w} %\begin{macro}[aux]{\fp_to_tl_large_6:w} %\begin{macro}[aux]{\fp_to_tl_large_7:w} %\begin{macro}[aux]{\fp_to_tl_large_8:w} %\begin{macro}[aux]{\fp_to_tl_large_8_aux:w} %\begin{macro}[aux]{\fp_to_tl_large_9:w} %\begin{macro}[aux]{\fp_to_tl_small:w} %\begin{macro}[aux]{\fp_to_tl_small_one:w} %\begin{macro}[aux]{\fp_to_tl_small_two:w} %\begin{macro}[aux]{\fp_to_tl_small_aux:w} %\begin{macro}[aux]{\fp_to_tl_large_zeros:NNNNNNNNN} %\begin{macro}[aux]{\fp_to_tl_small_zeros:NNNNNNNNN} %\begin{macro}[aux]{\fp_use_iix_ix:NNNNNNNNN} %\begin{macro}[aux]{\fp_use_ix:NNNNNNNNN} %\begin{macro}[aux]{\fp_use_i_to_vii:NNNNNNNNN} %\begin{macro}[aux]{\fp_use_i_to_iix:NNNNNNNNN} % Converting to integers in an expandable manner is very similar to % simply using floating point variables, particularly in the lead-off. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_tl:N #1 { \exp_after:wN \fp_to_tl_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_to_tl:N { c } \cs_new_nopar:Npn \fp_to_tl_aux:w #1#2 e #3 \q_stop { \tex_if:D #1 - - \tex_fi:D \tex_ifnum:D #3 < \c_zero \exp_after:wN \fp_to_tl_small:w \tex_else:D \exp_after:wN \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 \exp_after:wN \fp_to_tl_large_aux_i:w \tex_else:D \exp_after:wN \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 \exp_after:wN \fp_to_tl_small_one:w \tex_else:D \tex_ifnum:D #2 = -\c_two \exp_after:wN \exp_after:wN \exp_after:wN \fp_to_tl_small_two:w \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \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. \exp_after:wN \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 \exp_after:wN \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} %\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} %\begin{macro}{\fp_round_figures:cn} %\begin{macro}{\fp_ground_figures:Nn} %\begin{macro}{\fp_ground_figures:cn} %\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 \exp_after:wN \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 . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_round_places:Nn} %\begin{macro}{\fp_round_places:cn} %\begin{macro}{\fp_ground_places:Nn} %\begin{macro}{\fp_ground_places:cn} %\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 \exp_after:wN \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 . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_round:} %\begin{macro}{\fp_round_aux:NNNNNNNNN} %\begin{macro}{\fp_round_loop:N} % The rounding approach is the same for decimal places and significant % figures. There are always nine decimal digits to round, so the code % can be written to account for this. The basic logic is simply to % find the rounding, track any carry digit and move along. At the end % of the loop there is a possible shuffle if the integer part has % become \( 10 \). % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_round: { \bool_set_false:N \l_fp_round_carry_bool \l_fp_round_position_int \c_eight \tl_clear:N \l_fp_round_decimal_tl \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_round_aux:NNNNNNNNN \int_use:N \l_fp_input_a_decimal_int } \cs_new_protected_nopar:Npn \fp_round_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \fp_round_loop:N #9#8#7#6#5#4#3#2#1 \bool_if:NT \l_fp_round_carry_bool { \tex_advance:D \l_fp_input_a_integer_int \c_one } \l_fp_input_a_decimal_int \l_fp_round_decimal_tl \scan_stop: \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 \exp_after:wN \fp_round_loop:N \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} % %\subsection{Unary functions} % %\begin{macro}{\fp_abs:N} %\begin{macro}{\fp_abs:c} %\begin{macro}{\fp_gabs:N} %\begin{macro}{\fp_gabs:c} %\begin{macro}[aux]{\fp_abs_aux:NN} % Setting the absolute value is easy: read the value, ignore the sign, % return the result. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_abs:N { \fp_abs_aux:NN \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gabs:N { \fp_abs_aux:NN \tl_gset:Nn } \cs_generate_variant:Nn \fp_abs:N { c } \cs_generate_variant:Nn \fp_gabs:N { c } \cs_new_protected_nopar:Npn \fp_abs_aux:NN #1#2 { \group_begin: \fp_read:N #2 \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { + \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_neg:N} %\begin{macro}{\fp_neg:c} %\begin{macro}{\fp_gneg:N} %\begin{macro}{\fp_gneg:c} %\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 . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } \exp_after:wN \group_end: \exp_after:wN #1 \exp_after:wN #2 \exp_after:wN { \l_fp_tmp_tl } } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\subsection{Basic arithmetic} % %\begin{macro}{\fp_add:Nn} %\begin{macro}{\fp_add:cn} %\begin{macro}{\fp_gadd:Nn} %\begin{macro}{\fp_gadd:cn} %\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 \exp_after:wN \fp_add_sum: \tex_else:D \exp_after:wN \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 . \exp_after:wN \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} %\end{macro} %\end{macro} % %\begin{macro}{\fp_sub:Nn} %\begin{macro}{\fp_sub:cn} %\begin{macro}{\fp_gsub:Nn} %\begin{macro}{\fp_gsub:cn} %\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} %\end{macro} %\end{macro} % %\begin{macro}{\fp_mul:Nn} %\begin{macro}{\fp_mul:cn} %\begin{macro}{\fp_gmul:Nn} %\begin{macro}{\fp_gmul:cn} %\begin{macro}[aux]{\fp_mul_aux:NNn} %\begin{macro}[aux]{\fp_mul_int:} %\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_int: \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 . \exp_after:wN \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_int: { \fp_mul_split:NNNN \l_fp_input_a_decimal_int \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int \fp_mul_split:NNNN \l_fp_input_b_decimal_int \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int \l_fp_mul_output_int \c_zero \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int \tex_divide:D \l_fp_mul_output_int \c_one_thousand \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_input_b_integer_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_input_b_integer_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_input_b_integer_int \fp_mul_end_level: \l_fp_output_decimal_int 0 \l_fp_mul_output_tl \scan_stop: \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_input_b_integer_int \fp_mul_end_level: \l_fp_output_integer_int 0 \l_fp_mul_output_tl \scan_stop: } % \end{macrocode} % The split works by making a \( 10 \) digit number, from which % the first digit can then be dropped using a delimited argument. The % groups of three digits are then assigned to the various parts of % the input: notice that "##9" contains the last two digits of the % smallest part of the input. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul_split:NNNN #1#2#3#4 { \tex_advance:D #1 \c_one_thousand_million \cs_set_protected_nopar:Npn \fp_mul_split_aux:w ##1##2##3##4##5##6##7##8##9 \q_stop { #2 ##2##3##4 \scan_stop: #3 ##5##6##7 \scan_stop: #4 ##8##9 \scan_stop: } \exp_after:wN \fp_mul_split_aux:w \int_use:N #1 \q_stop \tex_advance:D #1 -\c_one_thousand_million } \cs_new_protected_nopar:Npn \fp_mul_product:NN #1#2 { \l_fp_mul_output_int \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 \exp_after:wN \use_i:nn \exp_after:wN \fp_mul_end_level:NNNNNNNNN \int_use:N \l_fp_mul_output_int } \cs_new_protected_nopar:Npn \fp_mul_end_level:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \tl_set:Nx \l_fp_mul_output_tl { #7#8#9 \l_fp_mul_output_tl } \l_fp_mul_output_int #1#2#3#4#5#6 \scan_stop: } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_div:Nn} %\begin{macro}{\fp_div:cn} %\begin{macro}{\fp_gdiv:Nn} %\begin{macro}{\fp_gdiv:cn} %\begin{macro}[aux]{\fp_div_aux:NNn} %\begin{macro}[aux]{\fp_div_aux:} %\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 \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_aux: \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_aux: { \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 . \exp_after:wN \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 \exp_after:wN \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 \exp_after:wN \exp_after:wN \exp_after:wN \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 \exp_after:wN \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 \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_divide_aux: \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} %\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:NNNNNNNNNN} % 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. % \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_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_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} % %\subsection{Trigonometric functions} % %\begin{macro}{\fp_trig_normalise:} %\begin{macro}[aux]{\fp_trig_normalise_aux_i:} %\begin{macro}[aux]{\fp_trig_normalise_aux:w} %\begin{macro}[aux]{\fp_trig_normalise_aux_ii:} %\begin{macro}[aux]{\fp_trig_normalise_aux:NNNNNNNNN} %\begin{macro}[aux]{\fp_trig_normalise_aux_iii:} % 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_trig_normalise_aux_i: \fp_trig_normalise_aux_ii: \fp_trig_normalise_aux_iii: \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 \exp_after:wN \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 \exp_after:wN \fp_trig_overflow_msg: \tex_fi:D } \cs_new_protected_nopar:Npn \fp_trig_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 \exp_after:wN \fp_trig_normalise_aux:w \int_use:N \l_fp_input_a_decimal_int \q_stop \exp_after:wN \fp_trig_normalise_aux_i: \tex_fi:D } \cs_new_protected_nopar:Npn \fp_trig_normalise_aux: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_exponent_int \c_minus_one } \cs_new_protected_nopar:Npn \fp_trig_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 \exp_after:wN \use_i:nn \exp_after:wN \fp_trig_normalise_aux:NNNNNNNNN \int_use:N \l_fp_input_a_decimal_int \exp_after:wN \fp_trig_normalise_aux_ii: \tex_fi:D } \cs_new_protected_nopar:Npn \fp_trig_normalise_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 } \cs_new_protected_nopar:Npn \fp_trig_normalise_aux_iii: { \tex_ifnum:D \l_fp_input_a_integer_int > \c_three \fp_sub:NNNNNNNNN \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \exp_after:wN \fp_trig_normalise_aux_iii: \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_sub:NNNNNNNNN \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_trig_normalise_aux_iii: \tex_fi:D \tex_fi:D \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\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 \exp_after:wN \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 \exp_after:wN \exp_after:wN \exp_after:wN \fp_trig_octant_aux: \tex_fi:D \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} % %\begin{macro}{\fp_sin:Nn} %\begin{macro}{\fp_sin:cn} %\begin{macro}{\fp_gsin:Nn} %\begin{macro}{\fp_gsin:cn} %\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_trig_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 . \exp_after:wN \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_trig_arg_tl } } \tex_else:D \etex_ifcsname:D c_fp_sin ( \l_fp_trig_arg_tl ) _tl \tex_endcsname:D \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \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_trig_arg_tl ) _tl } } } \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_trig_arg_tl ) _tl } \tl_set:cx { c_fp_sin ( \l_fp_trig_arg_tl ) _tl } { \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 . \exp_after:wN \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 \exp_after:wN \fp_trig_calc_sin: \tex_or:D \exp_after:wN \fp_trig_calc_cos: \tex_or:D \exp_after:wN \fp_trig_calc_cos: \tex_or:D \exp_after:wN \fp_trig_calc_sin: \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_cos:Nn} %\begin{macro}{\fp_cos:cn} %\begin{macro}{\fp_gcos:Nn} %\begin{macro}{\fp_gcos:cn} %\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_trig_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 . \exp_after:wN \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_trig_arg_tl ) _tl \tex_endcsname:D \tex_else:D \exp_after:wN \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_trig_arg_tl ) _tl } } } \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_trig_arg_tl ) _tl } \tl_set:cx { c_fp_cos ( \l_fp_trig_arg_tl ) _tl } { \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 . \exp_after:wN \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 \exp_after:wN \fp_trig_calc_cos: \tex_or:D \exp_after:wN \fp_trig_calc_sin: \tex_or:D \exp_after:wN \fp_trig_calc_sin: \tex_or:D \exp_after:wN \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} %\end{macro} %\end{macro} % %\begin{macro}{\fp_trig_calc_cos:} %\begin{macro}{\fp_trig_calc_sin:} %\begin{macro}[aux]{\fp_trig_calc_aux:} % 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 \exp_after:wN \fp_trig_calc_aux: \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 \exp_after:wN \fp_trig_calc_aux: \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_aux: { \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 \exp_after:wN \fp_trig_calc_aux: \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} %\end{macro} % %\begin{macro}{\fp_tan:Nn} %\begin{macro}{\fp_tan:cn} %\begin{macro}{\fp_gtan:Nn} %\begin{macro}{\fp_gtan:cn} %\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_trig_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 . \exp_after:wN \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_trig_arg_tl } } \tex_else:D \etex_ifcsname:D c_fp_tan ( \l_fp_trig_arg_tl ) _tl \tex_endcsname:D \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \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_trig_arg_tl ) _tl } } } \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 \exp_after:wN \fp_tan_aux_ii: \tex_else:D \cs_new_eq:cN { c_fp_tan ( \l_fp_trig_arg_tl ) _tl } \c_zero_fp \exp_after:wN \fp_trig_overflow_msg: \tex_fi:D } \cs_new_protected_nopar:Npn \fp_tan_aux_ii: { \fp_trig_normalise: \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_trig_arg_tl ) _tl } \c_undefined_fp \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_iii: \tex_fi:D \tex_else:D \exp_after:wN \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_trig_arg_tl ) _tl } \c_zero_fp \tex_else:D \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_iv: \tex_fi:D \tex_else:D \exp_after:wN \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: \tex_ifnum:D \l_fp_trig_octant_int < \c_three \l_fp_output_sign_int \c_one \tex_else:D \l_fp_output_sign_int \c_minus_one \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_tan ( \l_fp_trig_arg_tl ) _tl } \tl_set:cx { c_fp_tan ( \l_fp_trig_arg_tl ) _tl } { \tex_ifnum:D \l_fp_output_sign_int > \c_zero + \tex_else:D - \tex_fi:D \int_use:N \l_fp_output_integer_int . \exp_after:wN \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} %\end{macro} %\end{macro} % %\subsection{Tests for special values} % %\begin{macro}{\fp_if_infinity_p:N} %\begin{macro}[TF]{\fp_if_infinity:N} % Testing for infinity is easy. % \begin{macrocode} \prg_new_conditional:Npnn \fp_if_infinity:N #1 { p , T , F , TF } { \tex_ifx:D #1 \c_infinity_fp \prg_return_true: \tex_else:D \prg_return_false: \tex_fi:D } % \end{macrocode} %\end{macro} %\end{macro} % %\begin{macro}{\fp_if_undefined_p:N} %\begin{macro}[TF]{\fp_if_undefined:N} % Testing for an undefined value is easy. % \begin{macrocode} \prg_new_conditional:Npnn \fp_if_undefined:N #1 { p , T , F , TF } { \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} % Testing for a zero fixed-point is also easy. % \begin{macrocode} \prg_new_conditional:Npnn \fp_if_zero:N #1 { p , T , F , TF } { \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} %\begin{macro}[aux]{\fp_compare_aux:N} %\begin{macro}[aux]{\fp_compare_=:} %\begin{macro}[aux]{\fp_compare_<:} %\begin{macro}[aux]{\fp_compare_>:} %\begin{macro}[aux]{\fp_compare_absolute_a>b:} %\begin{macro}[aux]{\fp_compare_absolute_a: } { \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 } \cs_new_protected_nopar:cpn { fp_compare_<: } { \tex_ifnum:D \l_fp_input_b_sign_int > \l_fp_input_a_sign_int \group_end: \prg_return_true: \tex_else:D \tex_ifnum:D \l_fp_input_b_sign_int < \l_fp_input_a_sign_int \group_end: \prg_return_false: \tex_else:D \tex_ifnum:D \l_fp_input_b_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 } \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 \tex_ifnum:D \etex_numexpr:D \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int = \c_zero \group_end: \prg_return_true: \tex_else:D \group_end: \prg_return_false: \tex_fi:D \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 \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 \group_end: \prg_return_true: \tex_fi:D \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} %\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_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} % % \end{macrocode} % %\end{implementation} % %\PrintChanges % %\PrintIndex