% \iffalse meta-comment % %% File: l3fp.dtx Copyright (C) 2010,2011 The LaTeX3 Project %% %% It may be distributed and/or modified under the conditions of the %% LaTeX Project Public License (LPPL), either version 1.3c of this %% license or (at your option) any later version. The latest version %% of this license is in the file %% %% http://www.latex-project.org/lppl.txt %% %% This file is part of the "expl3 bundle" (The Work in LPPL) %% and all files in that bundle must be distributed together. %% %% The released version of this bundle is available from CTAN. %% %% ----------------------------------------------------------------------- %% %% The development version of the bundle can be found at %% %% http://www.latex-project.org/svnroot/experimental/trunk/ %% %% for those people who are interested. %% %%%%%%%%%%% %% NOTE: %% %%%%%%%%%%% %% %% Snapshots taken from the repository represent work in progress and may %% not work or may contain conflicting material! We therefore ask %% people _not_ to put them into distributions, archives, etc. without %% prior consultation with the LaTeX3 Project. %% %% ----------------------------------------------------------------------- % %<*driver|package> \RequirePackage{l3names} \GetIdInfo$Id: l3fp.dtx 2665 2011-08-25 22:15:27Z joseph $ {L3 Experimental floating-point operations} % %<*driver> \documentclass[full]{l3doc} \begin{document} \DocInput{\jobname.dtx} \end{document} % % \fi % % \title{^^A % The \pkg{l3fp} package\\ Floating-point operations^^A % \thanks{This file describes v\ExplFileVersion, % last revised \ExplFileDate.}^^A % } % % \author{^^A % The \LaTeX3 Project\thanks % {^^A % E-mail: % \href{mailto:latex-team@latex-project.org} % {latex-team@latex-project.org}^^A % }^^A % } % % \date{Released \ExplFileDate} % % \maketitle % % \begin{documentation} % % A floating point number is one which is stored as a mantissa and % a separate exponent. This module implements arithmetic using radix % $10$ floating point numbers. This means that the mantissa should % be a real number in the range $1 \le \expandafter\mathopen\string| % x \expandafter\mathclose\string| < 10$, with the % exponent given as an integer between $-99$ and $99$. In the % input, the exponent part is represented starting with an \texttt{e}. % As this is a low-level module, error-checking is minimal. Numbers % which are too large for the floating point unit to handle will result % in errors, either from \TeX{} or from \LaTeX{}. The \LaTeX{} code does not % check that the input will not overflow, hence the possibility of a % \TeX{} error. On the other hand, numbers which are too small will be % dropped, which will mean that extra decimal digits will simply be % lost. % % When parsing numbers, any missing parts will be interpreted as % zero. So for example %\begin{verbatim} % \fp_set:Nn \l_my_fp { } % \fp_set:Nn \l_my_fp { . } % \fp_set:Nn \l_my_fp { - } % \end{verbatim} % will all be interpreted as zero values without raising an error. % % Operations which give an undefined result (such as division by % $0$) will not lead to errors. Instead special marker values are % returned, which can be tested for using fr example % \cs{fp_if_undefined:N(TF)}. In this way it is possible to work with % asymptotic functions without first checking the input. If these % special values are carried forward in calculations they will be % treated as $0$. % % Floating point numbers are stored in the \texttt{fp} floating point % variable type. This has a standard range of functions for % variable management. % % \section{Floating-point variables} % % \begin{function}{\fp_new:N, \fp_new:c} % \begin{syntax} % \cs{fp_new:N} \meta{floating point variable} % \end{syntax} % Creates a new \meta{floating point variable} or raises an error if % the name is already taken. The declaration global. The % \meta{floating point} will initially be set to |+0.000000000e0| % (the zero floating point). % \end{function} % % \begin{function}{\fp_const:Nn, \fp_const:cn} % \begin{syntax} % \cs{fp_const:Nn} \meta{floating point variable} \Arg{value} % \end{syntax} % Creates a new constant \meta{floating point variable} or raises an % error if the name is already taken. The value of the % \meta{floating point variable} will be set globally to the % \meta{value}. % \end{function} % % \begin{function}{\fp_set_eq:NN, \fp_set_eq:cN, \fp_set_eq:Nc, \fp_set_eq:cc} % \begin{syntax} % \cs{fp_set_eq:NN} \meta{fp var1} \meta{fp var2} % \end{syntax} % Sets the value of \meta{floating point variable1} equal to that of % \meta{floating point variable2}. This assignment is restricted to the % current \TeX{} group level. % \end{function} % % \begin{function} % {\fp_gset_eq:NN, \fp_gset_eq:cN, \fp_gset_eq:Nc, \fp_gset_eq:cc} % \begin{syntax} % \cs{fp_gset_eq:NN} \meta{fp var1} \meta{fp var2} % \end{syntax} % Sets the value of \meta{floating point variable1} equal to that of % \meta{floating point variable2}. This assignment is global and so is % not limited by the current \TeX{} group level. % \end{function} % % \begin{function}{\fp_zero:N, \fp_zero:c} % \begin{syntax} % \cs{fp_zero:N} \meta{floating point variable} % \end{syntax} % Sets the \meta{floating point variable} to |+0.000000000e0| within % the current scope. % \end{function} % % \begin{function}{\fp_gzero:N, \fp_gzero:c} % \begin{syntax} % \cs{fp_gzero:N} \meta{floating point variable} % \end{syntax} % Sets the \meta{floating point variable} to |+0.000000000e0| globally. % \end{function} % % \begin{function}{\fp_set:Nn, \fp_set:cn} % \begin{syntax} % \cs{fp_set:Nn} \meta{floating point variable} \Arg{value} % \end{syntax} % Sets the \meta{floating point variable} variable to \meta{value} % within the scope of the current \TeX{} group. % \end{function} % % \begin{function}{\fp_gset:Nn, \fp_gset:cn} % \begin{syntax} % \cs{fp_gset:Nn} \meta{floating point variable} \Arg{value} % \end{syntax} % Sets the \meta{floating point variable} variable to \meta{value} % globally. % \end{function} % % \begin{function}{\fp_set_from_dim:Nn, \fp_set_from_dim:cn} % \begin{syntax} % \cs{fp_set_from_dim:Nn} \meta{floating point variable} \Arg{dimexpr} % \end{syntax} % Sets the \meta{floating point variable} to the distance represented % by the \meta{dimension expression} in the units points. This means % that distances given in other units are first converted to points % before being assigned to the \meta{floating point variable}. The % assignment is local. % \end{function} % % \begin{function}{\fp_gset_from_dim:Nn, \fp_gset_from_dim:cn} % \begin{syntax} % \cs{fp_gset_from_dim:Nn} \meta{floating point variable} \Arg{dimexpr} % \end{syntax} % Sets the \meta{floating point variable} to the distance represented % by the \meta{dimension expression} in the units points. This means % that distances given in other units are first converted to points % before being assigned to the \meta{floating point variable}. The % assignment is global. % \end{function} % % \begin{function}[EXP]{\fp_use:N, \fp_use:c} % \begin{syntax} % \cs{fp_use:N} \meta{floating point variable} % \end{syntax} % Inserts the value of the \meta{floating point variable} into the % input stream. The value will be given as a real number without any % exponent part, and will always include a decimal point. For example, % \begin{verbatim} % \fp_new:Nn \test % \fp_set:Nn \test { 1.234 e 5 } % \fp_use:N \test % \end{verbatim} % will insert |12345.00000| into the input stream. % As illustrated, a floating point will always be inserted with ten % significant digits given. Very large and very small values will % include additional zeros for place value. % \end{function} % % \begin{function}{\fp_show:N, \fp_show:c} % \begin{syntax} % \cs{fp_show:N} \meta{floating point variable} % \end{syntax} % Displays the content of the \meta{floating point variable} on the % terminal. % \end{function} % % \section{Conversion of floating point values to other formats} % % It is useful to be able to convert floating point variables to % other forms. These functions are expandable, so that the material % can be used in a variety of contexts. The \cs{fp_use:N} function % should also be consulted in this context, as it will insert the % value of the floating point variable as a real number. % % \begin{function}[EXP]{\fp_to_dim:N, \fp_to_dim:c} % \begin{syntax} % \cs{fp_to_dim:N} \meta{floating point variable} % \end{syntax} % Inserts the value of the \meta{floating point variable} % into the input stream converted into a dimension in points. % \end{function} % % \begin{function}[EXP]{\fp_to_int:N, \fp_to_int:c} % \begin{syntax} % \cs{fp_to_int:N} \meta{floating point variable} % \end{syntax} % Inserts the integer value of the \meta{floating point variable} % into the input stream. The decimal part of the number will not be % included, but will be used to round the integer. % \end{function} % % \begin{function}[EXP]{\fp_to_tl:N, \fp_to_tl:c} % \begin{syntax} % \cs{fp_to_tl:N} \meta{floating point variable} % \end{syntax} % Inserts a representation of the \meta{floating point variable} into % the input stream as a token list. The representation follows the % conventions of a pocket calculator: % \begin{center} % \ttfamily % \begin{tabular}{r@{.}lr@{.}l} % \toprule % \multicolumn{2}{l}{\rmfamily{Floating point value}} & % \multicolumn{2}{l}{\rmfamily{Representation}} \\ % \midrule % 1 & 234000000000e0 & 1 & 234 \\ % -1 & 234000000000e0 & -1 & 234 \\ % 1 & 234000000000e3 & \multicolumn{2}{l}{1234} \\ % 1 & 234000000000e13 & \multicolumn{2}{l}{1234e13} \\ % 1 & 234000000000e-1 & 0 & 1234 \\ % 1 & 234000000000e-2 & 0 & 01234 \\ % 1 & 234000000000e-3 & 1 & 234e-3 \\ % \bottomrule % \end{tabular} % \end{center} % Notice that trailing zeros are removed in this process, and that % numbers which do not require a decimal part do \emph{not} include % a decimal marker. % \end{function} % % \section{Rounding floating point values} % % The module can round floating point values to either decimal places % or significant figures using the usual method in which exact halves % are rounded up. % % \begin{function}{\fp_round_figures:Nn, \fp_round_figures:cn} % \begin{syntax} % \cs{fp_round_figures:Nn} \meta{floating point variable} \Arg{target} % \end{syntax} % Rounds the \meta{floating point variable} to the \meta{target} number % of significant figures (an integer expression). The rounding is % carried out locally. % \end{function} % % \begin{function}{\fp_ground_figures:Nn, \fp_ground_figures:cn} % \begin{syntax} % \cs{fp_ground_figures:Nn} \meta{floating point variable} \Arg{target} % \end{syntax} % Rounds the \meta{floating point variable} to the \meta{target} number % of significant figures (an integer expression). The rounding is % carried out globally. % \end{function} % % \begin{function}{\fp_round_places:Nn, \fp_round_places:cn} % \begin{syntax} % \cs{fp_round_places:Nn} \meta{floating point variable} \Arg{target} % \end{syntax} % Rounds the \meta{floating point variable} to the \meta{target} number % of decimal places (an integer expression). The rounding is % carried out locally. % \end{function} % % \begin{function}{\fp_ground_places:Nn, \fp_ground_places:cn} % \begin{syntax} % \cs{fp_ground_places:Nn} \meta{floating point variable} \Arg{target} % \end{syntax} % Rounds the \meta{floating point variable} to the \meta{target} number % of decimal places (an integer expression). The rounding is % carried out globally. % \end{function} % % \section{Floating-point conditionals} % % \begin{function}[EXP,pTF]{\fp_if_undefined:N} % \begin{syntax} % \cs{fp_if_undefined_p:N} \meta{fixed-point} % \cs{fp_if_undefined:NTF} \meta{fixed-point} % ~~\Arg{true code} \Arg{false code} % \end{syntax} % Tests if \meta{floating point} is undefined (\emph{i.e.}~equal to the % special \cs{c_undefined_fp} variable). % \end{function} % % \begin{function}[EXP]{\fp_if_zero:N} % \begin{syntax} % \cs{fp_if_zero_p:N} \meta{fixed-point} % \cs{fp_if_zero:NTF} \meta{fixed-point} \Arg{true code} \Arg{false code} % \end{syntax} % Tests if \meta{floating point} is equal to zero (\emph{i.e.}~equal to % the special \cs{c_zero_fp} variable). % \end{function} % % \begin{function}[TF]{\fp_compare:nNn} % \begin{syntax} % \cs{fp_compare:nNnTF} % ~~\Arg{floating point1} \meta{relation} \Arg{floating point2} % ~~\Arg{true code} \Arg{false code} % \end{syntax} % This function compared the two \meta{floating point} values, which % may be stored as \texttt{fp} variables, using the \meta{relation}: % \begin{center} % \begin{tabular}{ll} % Equal & |=| \\ % Greater than & |>| \\ % Less than & |<| \\ % \end{tabular} % \end{center} % The tests treat undefined floating points as % zero as the comparison is intended for real numbers only. % \end{function} % % \begin{function}[TF]{\fp_compare:n} % \begin{syntax} % \cs{fp_compare:nTF} % ~~\{ \meta{floating point1} \meta{relation} \meta{floating point2} \} % ~~\Arg{true code} \Arg{false code} % \end{syntax} % This function compared the two \meta{floating point} values, which % may be stored as \texttt{fp} variables, using the \meta{relation}: % \begin{center} % \begin{tabular}{ll} % Equal & |=| or |==| \\ % Greater than & |>| \\ % Greater than or equal & |>=| \\ % Less than & |<| \\ % Less than or equal & |<=| \\ % Not equal & |!=| \\ % \end{tabular} % \end{center} % The tests treat undefined floating points as % zero as the comparison is intended for real numbers only. % \end{function} % % \section{Unary floating-point operations} % % The unary operations alter the value stored within an \texttt{fp} % variable. % % \begin{function}{\fp_abs:N, \fp_abs:c} % \begin{syntax} % \cs{fp_abs:N} \meta{floating point variable} % \end{syntax} % Converts the \meta{floating point variable} to its absolute value, % assigning the result within the current \TeX\ group. % \end{function} % % \begin{function}{\fp_gabs:N, \fp_gabs:c} % \begin{syntax} % \cs{fp_gabs:N} \meta{floating point variable} % \end{syntax} % Converts the \meta{floating point variable} to its absolute value, % assigning the result globally. % \end{function} % % \begin{function}{\fp_neg:N, \fp_neg:c} % \begin{syntax} % \cs{fp_neg:N} \meta{floating point variable} % \end{syntax} % Reverse the sign of the \meta{floating point variable}, assigning the % result within the current \TeX\ group. % \end{function} % % \begin{function}{\fp_gneg:N, \fp_gneg:c} % \begin{syntax} % \cs{fp_gneg:N} \meta{floating point variable} % \end{syntax} % Reverse the sign of the \meta{floating point variable}, assigning the % result globally. % \end{function} % % \section{Floating-point arithmetic} % % Binary arithmetic operations act on the value stored in an % \texttt{fp}, so for example % \begin{verbatim} % \fp_set:Nn \l_my_fp { 1.234 } % \fp_sub:Nn \l_my_fp { 5.678 } % \end{verbatim} % sets \cs{l_my_fp} to the result of $1.234 - 5.678$ % (\emph{i.e.}~$-4.444$). % % \begin{function}{\fp_add:Nn, \fp_add:cn} % \begin{syntax} % \cs{fp_add:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Adds the \meta{value} to the \meta{floating point}, making the % assignment within the current \TeX{} group level. % \end{function} % % \begin{function}{\fp_gadd:Nn, \fp_gadd:cn} % \begin{syntax} % \cs{fp_gadd:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Adds the \meta{value} to the \meta{floating point}, making the % assignment globally. % \end{function} % % \begin{function}{\fp_sub:Nn, \fp_sub:cn} % \begin{syntax} % \cs{fp_sub:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Subtracts the \meta{value} from the \meta{floating point}, making the % assignment within the current \TeX{} group level. % \end{function} % % \begin{function}{\fp_gsub:Nn, \fp_gsub:cn} % \begin{syntax} % \cs{fp_gsub:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Subtracts the \meta{value} from the \meta{floating point}, making the % assignment globally. % \end{function} % % \begin{function}{\fp_mul:Nn, \fp_mul:cn} % \begin{syntax} % \cs{fp_mul:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Multiples the \meta{floating point} by the \meta{value}, making the % assignment within the current \TeX{} group level. % \end{function} % % \begin{function}{\fp_gmul:Nn, \fp_gmul:cn} % \begin{syntax} % \cs{fp_gmul:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Multiples the \meta{floating point} by the \meta{value}, making the % assignment globally. % \end{function} % % \begin{function}{\fp_div:Nn, \fp_div:cn} % \begin{syntax} % \cs{fp_div:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Divides the \meta{floating point} by the \meta{value}, making the % assignment within the current \TeX{} group level. If the \meta{value} % is zero, the \meta{floating point} will be set to % \cs{c_undefined_fp}. The assignment is local. % \end{function} % % \begin{function}{\fp_gdiv:Nn, \fp_gdiv:cn} % \begin{syntax} % \cs{fp_gdiv:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Divides the \meta{floating point} by the \meta{value}, making the % assignment globally. If the \meta{value} is zero, the % \meta{floating point} will be set to \cs{c_undefined_fp}. % The assignment is global. % \end{function} % % \section{Floating-point power operations} % % \begin{function}{\fp_pow:Nn, \fp_pow:cn} % \begin{syntax} % \cs{fp_pow:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Raises the \meta{floating point} to the given \meta{value}. If the % \meta{floating point} is negative, then the \meta{value} should be % either a positive real number or a negative integer. If the % \meta{floating point} is positive, then the \meta{value} may be any % real value. Mathematically invalid operations such as $0^{0}$ % will give set the \meta{floating point} to to \cs{c_undefined_fp}. % The assignment is local. % \end{function} % % \begin{function}{\fp_gpow:Nn, \fp_gpow:cn} % \begin{syntax} % \cs{fp_gpow:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Raises the \meta{floating point} to the given \meta{value}. If the % \meta{floating point} is negative, then the \meta{value} should be % either a positive real number or a negative integer. If the % \meta{floating point} is positive, then the \meta{value} may be any % real value. Mathematically invalid operations such as $0^{0}$ % will give set the \meta{floating point} to to \cs{c_undefined_fp}. % The assignment is global. % \end{function} % % \section{Exponential and logarithm functions} % % \begin{function}{\fp_exp:Nn, \fp_exp:cn} % \begin{syntax} % \cs{fp_exp:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Calculates the exponential of the \meta{value} and assigns this % to the \meta{floating point}. The assignment is local. % \end{function} % % \begin{function}{\fp_gexp:Nn, \fp_gexp:cn} % \begin{syntax} % \cs{fp_gexp:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Calculates the exponential of the \meta{value} and assigns this % to the \meta{floating point}. The assignment is global. % \end{function} % % \begin{function}{\fp_ln:Nn, \fp_ln:cn} % \begin{syntax} % \cs{fp_ln:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Calculates the natural logarithm of the \meta{value} and assigns % this to the \meta{floating point}. The assignment is local. % \end{function} % % \begin{function}{\fp_gln:Nn, \fp_gln:cn} % \begin{syntax} % \cs{fp_gln:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Calculates the natural logarithm of the \meta{value} and assigns % this to the \meta{floating point}. The assignment is global. % \end{function} % % \section{Trigonometric functions} % % The trigonometric functions all work in radians. They accept a maximum % input value of $100\,000\,000$, as there are issues with range % reduction and very large input values. % % \begin{function}{\fp_sin:Nn, \fp_sin:cn} % \begin{syntax} % \cs{fp_sin:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the sine of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % local. % \end{function} % % \begin{function}{\fp_gsin:Nn, \fp_gsin:cn} % \begin{syntax} % \cs{fp_gsin:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the sine of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % global. % \end{function} % % \begin{function}{\fp_cos:Nn, \fp_cos:cn} % \begin{syntax} % \cs{fp_cos:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the cosine of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % local. % \end{function} % % \begin{function}{\fp_gcos:Nn, \fp_gcos:cn} % \begin{syntax} % \cs{fp_gcos:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the cosine of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % global. % \end{function} % % \begin{function}{\fp_tan:Nn, \fp_tan:cn} % \begin{syntax} % \cs{fp_tan:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the tangent of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % local. % \end{function} % % \begin{function}{\fp_gtan:Nn, \fp_gtan:cn} % \begin{syntax} % \cs{fp_gtan:Nn} \meta{floating point} \Arg{value} % \end{syntax} % Assigns the tangent of the \meta{value} to the \meta{floating point}. % The \meta{value} should be given in radians. The assignment is % global. % \end{function} % % \section{Constant floating point values} % % \begin{variable}{\c_e_fp} % The value of the base of natural numbers, $\mathrm{e}$. % \end{variable} % % \begin{variable}{\c_one_fp} % A floating point variable with permanent value $1$: used for % speeding up some comparisons. % \end{variable} % % \begin{variable}{\c_pi_fp} % The value of $\pi$. % \end{variable} % % \begin{variable}{\c_undefined_fp} % A special marker floating point variable representing the result of % an operation which does not give a defined result (such as division % by $0$). % \end{variable} % % \begin{variable}{\c_zero_fp} % A permanently zero floating point variable. % \end{variable} % % \section{Notes on the floating point unit} % % As calculation of the elemental transcendental functions is % computationally expensive compared to storage of results, after % calculating a trigonometric function, exponent, \emph{etc.}~the module % stored the result for reuse. Thus the performance of the module for % repeated operations, most probably trigonometric functions, should be % much higher than if the values were re-calculated every time they % were needed. % % Anyone with experience of programming floating point calculations will % know that this is a complex area. The aim of the unit is to be % accurate enough for the likely applications in a typesetting context. % The arithmetic operations are therefore intended to provide ten digit % accuracy with the last digit accurate to $\pm 1$. The elemental % transcendental functions may not provide such high accuracy in every % case, although the design aim has been to provide $10$ digit % accuracy for cases likely to be relevant in typesetting situations. % A good overview of the challenges in this area can be found in % J.-M.~Muller, \emph{Elementary functions: algorithms and % implementation}, 2nd edition, Birkh{\"a}uer Boston, New York, USA, % 2006. % % The internal representation of numbers is tuned to the needs of the % underlying \TeX{} system. This means that the format is somewhat % different from that used in, for example, computer floating point % units. Programming in \TeX{} makes it most convenient to use a % radix $10$ system, using \TeX{} \texttt{count} registers for % storage and taking advantage where possible of delimited arguments. % % \end{documentation} % % \begin{implementation} % % \section{\pkg{l3fp} Implementation} % % \TestFiles{m3fp003.lvt} % % \begin{macrocode} %<*initex|package> % \end{macrocode} % % \begin{macrocode} %<*package> \ProvidesExplPackage {\ExplFileName}{\ExplFileDate}{\ExplFileVersion}{\ExplFileDescription} \package_check_loaded_expl: % % \end{macrocode} % % \subsection{Constants} % % \begin{variable}{\c_forty_four} % \begin{variable}{\c_one_million} % \begin{variable}{\c_one_hundred_million} % \begin{variable}{\c_five_hundred_million} % \begin{variable}{\c_one_thousand_million} % There is some speed to gain by moving numbers into fixed positions. % \begin{macrocode} \int_const:Nn \c_forty_four { 44 } \int_const:Nn \c_one_million { 1 000 000 } \int_const:Nn \c_one_hundred_million { 100 000 000 } \int_const:Nn \c_five_hundred_million { 500 000 000 } \int_const:Nn \c_one_thousand_million { 1 000 000 000 } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\c_fp_pi_by_four_decimal_int} % \begin{variable}{\c_fp_pi_by_four_extended_int} % \begin{variable}{\c_fp_pi_decimal_int} % \begin{variable}{\c_fp_pi_extended_int} % \begin{variable}{\c_fp_two_pi_decimal_int} % \begin{variable}{\c_fp_two_pi_extended_int} % Parts of $\pi$ for trigonometric range reduction, implemented % as \texttt{int} variables for speed. % \begin{macrocode} \int_new:N \c_fp_pi_by_four_decimal_int \int_set:Nn \c_fp_pi_by_four_decimal_int { 785 398 158 } \int_new:N \c_fp_pi_by_four_extended_int \int_set:Nn \c_fp_pi_by_four_extended_int { 897 448 310 } \int_new:N \c_fp_pi_decimal_int \int_set:Nn \c_fp_pi_decimal_int { 141 592 653 } \int_new:N \c_fp_pi_extended_int \int_set:Nn \c_fp_pi_extended_int { 589 793 238 } \int_new:N \c_fp_two_pi_decimal_int \int_set:Nn \c_fp_two_pi_decimal_int { 283 185 307 } \int_new:N \c_fp_two_pi_extended_int \int_set:Nn \c_fp_two_pi_extended_int { 179 586 477 } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\c_e_fp} % The value $\mathrm{e}$ as a \enquote{machine number}. % \begin{macrocode} \tl_const:Nn \c_e_fp { + 2.718281828 e 0 } % \end{macrocode} % \end{variable} % % \begin{variable}{\c_one_fp} % The constant value $1$: used for fast comparisons. % \begin{macrocode} \tl_const:Nn \c_one_fp { + 1.000000000 e 0 } % \end{macrocode} % \end{variable} % % \begin{variable}{\c_pi_fp} % The value $\pi$ as a \enquote{machine number}. % \begin{macrocode} \tl_const:Nn \c_pi_fp { + 3.141592654 e 0 } % \end{macrocode} % \end{variable} % % \begin{variable}{\c_undefined_fp} % A marker for undefined values. % \begin{macrocode} \tl_const:Nn \c_undefined_fp { X 0.000000000 e 0 } % \end{macrocode} % \end{variable} % % \begin{variable}{\c_zero_fp} % The constant zero value. % \begin{macrocode} \tl_const:Nn \c_zero_fp { + 0.000000000 e 0 } % \end{macrocode} % \end{variable} % % \subsection{Variables} % % \begin{variable}{\l_fp_arg_tl} % A token list to store the formalised representation of the input % for transcendental functions. % \begin{macrocode} \tl_new:N \l_fp_arg_tl % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_count_int} % A counter for things like the number of divisions possible. % \begin{macrocode} \int_new:N \l_fp_count_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_div_offset_int} % When carrying out division, an offset is used for the results to % get the decimal part correct. % \begin{macrocode} \int_new:N \l_fp_div_offset_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_exp_integer_int} % \begin{variable}{\l_fp_exp_decimal_int} % \begin{variable}{\l_fp_exp_extended_int} % \begin{variable}{\l_fp_exp_exponent_int} % Used for the calculation of exponent values. % \begin{macrocode} \int_new:N \l_fp_exp_integer_int \int_new:N \l_fp_exp_decimal_int \int_new:N \l_fp_exp_extended_int \int_new:N \l_fp_exp_exponent_int % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_input_a_sign_int} % \begin{variable}{\l_fp_input_a_integer_int} % \begin{variable}{\l_fp_input_a_decimal_int} % \begin{variable}{\l_fp_input_a_exponent_int} % \begin{variable}{\l_fp_input_b_sign_int} % \begin{variable}{\l_fp_input_b_integer_int} % \begin{variable}{\l_fp_input_b_decimal_int} % \begin{variable}{\l_fp_input_b_exponent_int} % Storage for the input: two storage areas as there are at most two % inputs. % \begin{macrocode} \int_new:N \l_fp_input_a_sign_int \int_new:N \l_fp_input_a_integer_int \int_new:N \l_fp_input_a_decimal_int \int_new:N \l_fp_input_a_exponent_int \int_new:N \l_fp_input_b_sign_int \int_new:N \l_fp_input_b_integer_int \int_new:N \l_fp_input_b_decimal_int \int_new:N \l_fp_input_b_exponent_int % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_input_a_extended_int} % \begin{variable}{\l_fp_input_b_extended_int} % For internal use, \enquote{extended} floating point numbers are % needed. % \begin{macrocode} \int_new:N \l_fp_input_a_extended_int \int_new:N \l_fp_input_b_extended_int % \end{macrocode} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_mul_a_i_int} % \begin{variable}{\l_fp_mul_a_ii_int} % \begin{variable}{\l_fp_mul_a_iii_int} % \begin{variable}{\l_fp_mul_a_iv_int} % \begin{variable}{\l_fp_mul_a_v_int} % \begin{variable}{\l_fp_mul_a_vi_int} % \begin{variable}{\l_fp_mul_b_i_int} % \begin{variable}{\l_fp_mul_b_ii_int} % \begin{variable}{\l_fp_mul_b_iii_int} % \begin{variable}{\l_fp_mul_b_iv_int} % \begin{variable}{\l_fp_mul_b_v_int} % \begin{variable}{\l_fp_mul_b_vi_int} % Multiplication requires that the decimal part is split into parts % so that there are no overflows. % \begin{macrocode} \int_new:N \l_fp_mul_a_i_int \int_new:N \l_fp_mul_a_ii_int \int_new:N \l_fp_mul_a_iii_int \int_new:N \l_fp_mul_a_iv_int \int_new:N \l_fp_mul_a_v_int \int_new:N \l_fp_mul_a_vi_int \int_new:N \l_fp_mul_b_i_int \int_new:N \l_fp_mul_b_ii_int \int_new:N \l_fp_mul_b_iii_int \int_new:N \l_fp_mul_b_iv_int \int_new:N \l_fp_mul_b_v_int \int_new:N \l_fp_mul_b_vi_int % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_mul_output_int} % \begin{variable}{\l_fp_mul_output_tl} % Space for multiplication results. % \begin{macrocode} \int_new:N \l_fp_mul_output_int \tl_new:N \l_fp_mul_output_tl % \end{macrocode} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_output_sign_int} % \begin{variable}{\l_fp_output_integer_int} % \begin{variable}{\l_fp_output_decimal_int} % \begin{variable}{\l_fp_output_exponent_int} % Output is stored in the same way as input. % \begin{macrocode} \int_new:N \l_fp_output_sign_int \int_new:N \l_fp_output_integer_int \int_new:N \l_fp_output_decimal_int \int_new:N \l_fp_output_exponent_int % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_output_extended_int} % Again, for calculations an extended part. % \begin{macrocode} \int_new:N \l_fp_output_extended_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_round_carry_bool} % To indicate that a digit needs to be carried forward. % \begin{macrocode} \bool_new:N \l_fp_round_carry_bool % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_round_decimal_tl} % A temporary store when rounding, to build up the decimal part without % needing to do any maths. % \begin{macrocode} \tl_new:N \l_fp_round_decimal_tl % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_round_position_int} % \begin{variable}{\l_fp_round_target_int} % Used to check the position for rounding. % \begin{macrocode} \int_new:N \l_fp_round_position_int \int_new:N \l_fp_round_target_int % \end{macrocode} % \end{variable} % \end{variable} % % \begin{variable}{\l_fp_sign_tl} % There are places where the sign needs to be set up \enquote{early}, % so that the registers can be re-used. % \begin{macrocode} \tl_new:N \l_fp_sign_tl % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_split_sign_int} % When splitting the input it is fastest to use a fixed name for the % sign part, and to transfer it after the split is complete. % \begin{macrocode} \int_new:N \l_fp_split_sign_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_tmp_int} % A scratch \texttt{int}: used only where the value is not carried % forward. % \begin{macrocode} \int_new:N \l_fp_tmp_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_tmp_tl} % A scratch token list variable for expanding material. % \begin{macrocode} \tl_new:N \l_fp_tmp_tl % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_trig_octant_int} % To track which octant the trigonometric input is in. % \begin{macrocode} \int_new:N \l_fp_trig_octant_int % \end{macrocode} % \end{variable} % % \begin{variable}{\l_fp_trig_sign_int} % \begin{variable}{\l_fp_trig_decimal_int} % \begin{variable}{\l_fp_trig_extended_int} % Used for the calculation of trigonometric values. % \begin{macrocode} \int_new:N \l_fp_trig_sign_int \int_new:N \l_fp_trig_decimal_int \int_new:N \l_fp_trig_extended_int % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % % \subsection{Parsing numbers} % % \begin{macro}{\fp_read:N} % \begin{macro}[aux]{\fp_read_aux:w} % Reading a stored value is made easier as the format is designed to % match the delimited function. This is always used to read the first % value (register |a|). % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_read:N #1 { \exp_after:wN \fp_read_aux:w #1 \q_stop } \cs_new_protected_nopar:Npn \fp_read_aux:w #1#2 . #3 e #4 \q_stop { \if:w #1 - \l_fp_input_a_sign_int \c_minus_one \else: \l_fp_input_a_sign_int \c_one \fi: \l_fp_input_a_integer_int #2 \scan_stop: \l_fp_input_a_decimal_int #3 \scan_stop: \l_fp_input_a_exponent_int #4 \scan_stop: } % \end{macrocode} % \end{macro} % \end{macro} % % \begin{macro}{\fp_split:Nn} % \begin{macro}[aux]{\fp_split_sign:} % \begin{macro}[aux]{\fp_split_exponent:} % \begin{macro}[aux]{\fp_split_aux_i:w} % \begin{macro}[aux]{\fp_split_aux_ii:w} % \begin{macro}[aux]{\fp_split_aux_iii:w} % \begin{macro}[aux]{\fp_split_decimal:w} % \begin{macro}[aux]{\fp_split_decimal_aux:w} % The aim here is to use as much of \TeX{}'s mechanism as possible to pick % up the numerical input without any mistakes. In particular, negative % numbers have to be filtered out first in case the integer part is % $0$ (in which case \TeX{} would drop the |-| sign). That process % has to be done in a loop for cases where the sign is repeated. % Finding an exponent is relatively easy, after which the next phase is % to find the integer part, which will terminate with a |.|, and trigger % the decimal-finding code. The later will allow the decimal to be too % long, truncating the result. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_split:Nn #1#2 { \tl_set:Nx \l_fp_tmp_tl {#2} \tl_set_rescan:Nno \l_fp_tmp_tl { \char_set_catcode_ignore:n { 32 } } { \l_fp_tmp_tl } \l_fp_split_sign_int \c_one \fp_split_sign: \use:c { l_fp_input_ #1 _sign_int } \l_fp_split_sign_int \exp_after:wN \fp_split_exponent:w \l_fp_tmp_tl e e \q_stop #1 } \cs_new_protected_nopar:Npn \fp_split_sign: { \if_int_compare:w \pdftex_strcmp:D { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { - } = \c_zero \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop } \l_fp_split_sign_int -\l_fp_split_sign_int \exp_after:wN \fp_split_sign: \else: \if_int_compare:w \pdftex_strcmp:D { \exp_after:wN \tl_head:w \l_fp_tmp_tl ? \q_stop } { + } = \c_zero \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \tl_tail:w \l_fp_tmp_tl \prg_do_nothing: \q_stop } \exp_after:wN \exp_after:wN \exp_after:wN \fp_split_sign: \fi: \fi: } \cs_new_protected_nopar:Npn \fp_split_exponent:w #1 e #2 e #3 \q_stop #4 { \use:c { l_fp_input_ #4 _exponent_int } \int_eval:w 0 #2 \scan_stop: \tex_afterassignment:D \fp_split_aux_i:w \use:c { l_fp_input_ #4 _integer_int } \int_eval:w 0 #1 . . \q_stop #4 } \cs_new_protected_nopar:Npn \fp_split_aux_i:w #1 . #2 . #3 \q_stop { \fp_split_aux_ii:w #2 000000000 \q_stop } \cs_new_protected_nopar:Npn \fp_split_aux_ii:w #1#2#3#4#5#6#7#8#9 { \fp_split_aux_iii:w {#1#2#3#4#5#6#7#8#9} } \cs_new_protected_nopar:Npn \fp_split_aux_iii:w #1#2 \q_stop { \l_fp_tmp_int 1 #1 \scan_stop: \exp_after:wN \fp_split_decimal:w \int_use:N \l_fp_tmp_int 000000000 \q_stop } \cs_new_protected_nopar:Npn \fp_split_decimal:w #1#2#3#4#5#6#7#8#9 { \fp_split_decimal_aux:w {#2#3#4#5#6#7#8#9} } \cs_new_protected_nopar:Npn \fp_split_decimal_aux:w #1#2#3 \q_stop #4 { \use:c { l_fp_input_ #4 _decimal_int } #1#2 \scan_stop: \if_int_compare:w \int_eval:w \use:c { l_fp_input_ #4 _integer_int } + \use:c { l_fp_input_ #4 _decimal_int } \scan_stop: = \c_zero \use:c { l_fp_input_ #4 _sign_int } \c_one \fi: \if_int_compare:w \use:c { l_fp_input_ #4 _integer_int } < \c_one_thousand_million \else: \exp_after:wN \fp_overflow_msg: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_standardise:NNNN} % \begin{macro}[aux]{\fp_standardise_aux:NNNN} % \begin{macro}[aux]{\fp_standardise_aux:} % \begin{macro}[aux]{\fp_standardise_aux:w} % The idea here is to shift the input into a known exponent range. This % is done using \TeX{} tokens where possible, as this is faster than % arithmetic. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_standardise:NNNN #1#2#3#4 { \if_int_compare:w \int_eval:w #2 + #3 = \c_zero #1 \c_one #4 \c_zero \exp_after:wN \use_none:nnnn \else: \exp_after:wN \fp_standardise_aux:NNNN \fi: #1#2#3#4 } \cs_new_protected_nopar:Npn \fp_standardise_aux:NNNN #1#2#3#4 { \cs_set_protected_nopar:Npn \fp_standardise_aux: { \if_int_compare:w #2 = \c_zero \tex_advance:D #3 \c_one_thousand_million \exp_after:wN \fp_standardise_aux:w \int_use:N #3 \q_stop \exp_after:wN \fp_standardise_aux: \fi: } \cs_set_protected_nopar:Npn \fp_standardise_aux:w ##1##2##3##4##5##6##7##8##9 \q_stop { #2 ##2 \scan_stop: #3 ##3##4##5##6##7##8##9 0 \scan_stop: \tex_advance:D #4 \c_minus_one } \fp_standardise_aux: \cs_set_protected_nopar:Npn \fp_standardise_aux: { \if_int_compare:w #2 > \c_nine \tex_advance:D #2 \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_standardise_aux:w \int_use:N #2 \exp_after:wN \fp_standardise_aux: \fi: } \cs_set_protected_nopar:Npn \fp_standardise_aux:w ##1##2##3##4##5##6##7##8##9 { #2 ##1##2##3##4##5##6##7##8 \scan_stop: \tex_advance:D #3 \c_one_thousand_million \tex_divide:D #3 \c_ten \tl_set:Nx \l_fp_tmp_tl { ##9 \exp_after:wN \use_none:n \int_use:N #3 } #3 \l_fp_tmp_tl \scan_stop: \tex_advance:D #4 \c_one } \fp_standardise_aux: \if_int_compare:w #4 < \c_one_hundred \if_int_compare:w #4 > -\c_one_hundred \else: #1 \c_one #2 \c_zero #3 \c_zero #4 \c_zero \fi: \else: \exp_after:wN \fp_overflow_msg: \fi: } \cs_new_protected_nopar:Npn \fp_standardise_aux: { } \cs_new_protected_nopar:Npn \fp_standardise_aux:w { } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Internal utilities} % % \begin{macro}{\fp_level_input_exponents:} % \begin{macro}[aux]{\fp_level_input_exponents_a:} % \begin{macro}[aux]{\fp_level_input_exponents_a:NNNNNNNNN} % \begin{macro}[aux]{\fp_level_input_exponents_b:} % \begin{macro}[aux]{\fp_level_input_exponents_b:NNNNNNNNN} % The routines here are similar to those used to standardise the % exponent. However, the aim here is different: the two exponents need % to end up the same. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_level_input_exponents: { \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int \exp_after:wN \fp_level_input_exponents_a: \else: \exp_after:wN \fp_level_input_exponents_b: \fi: } \cs_new_protected_nopar:Npn \fp_level_input_exponents_a: { \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int \tex_advance:D \l_fp_input_b_integer_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_level_input_exponents_a:NNNNNNNNN \int_use:N \l_fp_input_b_integer_int \exp_after:wN \fp_level_input_exponents_a: \fi: } \cs_new_protected_nopar:Npn \fp_level_input_exponents_a:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \l_fp_input_b_integer_int #1#2#3#4#5#6#7#8 \scan_stop: \tex_advance:D \l_fp_input_b_decimal_int \c_one_thousand_million \tex_divide:D \l_fp_input_b_decimal_int \c_ten \tl_set:Nx \l_fp_tmp_tl { #9 \exp_after:wN \use_none:n \int_use:N \l_fp_input_b_decimal_int } \l_fp_input_b_decimal_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_input_b_exponent_int \c_one } \cs_new_protected_nopar:Npn \fp_level_input_exponents_b: { \if_int_compare:w \l_fp_input_b_exponent_int > \l_fp_input_a_exponent_int \tex_advance:D \l_fp_input_a_integer_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_level_input_exponents_b:NNNNNNNNN \int_use:N \l_fp_input_a_integer_int \exp_after:wN \fp_level_input_exponents_b: \fi: } \cs_new_protected_nopar:Npn \fp_level_input_exponents_b:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \l_fp_input_a_integer_int #1#2#3#4#5#6#7#8 \scan_stop: \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \tex_divide:D \l_fp_input_a_decimal_int \c_ten \tl_set:Nx \l_fp_tmp_tl { #9 \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int } \l_fp_input_a_decimal_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_input_a_exponent_int \c_one } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}[aux]{\fp_tmp:w} % Used for output of results, cutting down on \cs{exp_after:wN}. % This is just a place holder definition. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_tmp:w #1#2 { } % \end{macrocode} % \end{macro} % % \subsection{Operations for \texttt{fp} variables} % % The format of \texttt{fp} variables is tightly defined, so that % they can be read quickly by the internal code. The format is a single % sign token, a single number, the decimal point, nine decimal numbers, % an |e| and finally the exponent. This final part may vary in length. % When stored, floating points will always be stored with a value in % the integer position unless the number is zero. % % \begin{macro}{\fp_new:N, \fp_new:c} % \UnitTested % Fixed-points always have a value, and of course this has to be % initialised globally. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_new:N #1 { \tl_new:N #1 \tl_gset_eq:NN #1 \c_zero_fp } \cs_generate_variant:Nn \fp_new:N { c } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_const:Nn, \fp_const:cn} % A simple wrapper. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_const:Nn #1#2 { \fp_new:N #1 \fp_gset:Nn #1 {#2} } \cs_generate_variant:Nn \fp_const:Nn { c } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_zero:N, \fp_zero:c} % \UnitTested % \begin{macro}{\fp_gzero:N, \fp_gzero:c} % \UnitTested % Zeroing fixed-points is pretty obvious. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_zero:N #1 { \tl_set_eq:NN #1 \c_zero_fp } \cs_new_protected_nopar:Npn \fp_gzero:N #1 { \tl_gset_eq:NN #1 \c_zero_fp } \cs_generate_variant:Nn \fp_zero:N { c } \cs_generate_variant:Nn \fp_gzero:N { c } % \end{macrocode} % \end{macro} % \end{macro} % % \begin{macro}{\fp_set:Nn, \fp_set:cn} % \UnitTested % \begin{macro}{\fp_gset:Nn, \fp_gset:cn} % \UnitTested % \begin{macro}[aux]{\fp_set_aux:NNn} % To trap any input errors, a very simple version of the parser is run % here. This will pick up any invalid characters at this stage, saving % issues later. The splitting approach is the same as the more % advanced function later. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_set:Nn { \fp_set_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gset:Nn { \fp_set_aux:NNn \tl_gset:Nn } \cs_new_protected_nopar:Npn \fp_set_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } \cs_generate_variant:Nn \fp_set:Nn { c } \cs_generate_variant:Nn \fp_gset:Nn { c } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % % % \begin{macro}{\fp_set_from_dim:Nn, \fp_set_from_dim:cn} % \UnitTested % \begin{macro}{\fp_gset_from_dim:Nn, \fp_gset_from_dim:cn} % \UnitTested % \begin{macro}[aux]{\fp_set_from_dim_aux:NNn} % \begin{macro}[aux]{\fp_set_from_dim_aux:w} % \begin{variable}{\l_fp_tmp_dim} % \begin{variable}{\l_fp_tmp_skip} % Here, dimensions are converted to fixed-points \emph{via} a % temporary variable. This ensures that they always convert as points. % The code is then essentially the same as for \cs{fp_set:Nn}, but with % the dimension passed so that it will be striped of the |pt| on the % way through. The passage through a skip is used to remove any rubber % part. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_set_from_dim:Nn { \fp_set_from_dim_aux:NNn \tl_set:Nx } \cs_new_protected_nopar:Npn \fp_gset_from_dim:Nn { \fp_set_from_dim_aux:NNn \tl_gset:Nx } \cs_new_protected_nopar:Npn \fp_set_from_dim_aux:NNn #1#2#3 { \group_begin: \l_fp_tmp_skip \etex_glueexpr:D #3 \scan_stop: \l_fp_tmp_dim \l_fp_tmp_skip \fp_split:Nn a { \exp_after:wN \fp_set_from_dim_aux:w \dim_use:N \l_fp_tmp_dim } \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } \cs_set_protected_nopar:Npx \fp_set_from_dim_aux:w { \cs_set_nopar:Npn \exp_not:N \fp_set_from_dim_aux:w ##1 \tl_to_str:n { pt } {##1} } \fp_set_from_dim_aux:w \cs_generate_variant:Nn \fp_set_from_dim:Nn { c } \cs_generate_variant:Nn \fp_gset_from_dim:Nn { c } \dim_new:N \l_fp_tmp_dim \skip_new:N \l_fp_tmp_skip % \end{macrocode} % \end{variable} % \end{variable} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_set_eq:NN, \fp_set_eq:cN, \fp_set_eq:Nc, \fp_set_eq:cc} % \UnitTested % \begin{macro}{\fp_gset_eq:NN, \fp_gset_eq:cN, \fp_gset_eq:Nc, \fp_gset_eq:cc} % \UnitTested % Pretty simple, really. % \begin{macrocode} \cs_new_eq:NN \fp_set_eq:NN \tl_set_eq:NN \cs_new_eq:NN \fp_set_eq:cN \tl_set_eq:cN \cs_new_eq:NN \fp_set_eq:Nc \tl_set_eq:Nc \cs_new_eq:NN \fp_set_eq:cc \tl_set_eq:cc \cs_new_eq:NN \fp_gset_eq:NN \tl_gset_eq:NN \cs_new_eq:NN \fp_gset_eq:cN \tl_gset_eq:cN \cs_new_eq:NN \fp_gset_eq:Nc \tl_gset_eq:Nc \cs_new_eq:NN \fp_gset_eq:cc \tl_gset_eq:cc % \end{macrocode} % \end{macro} % \end{macro} % % \begin{macro}{\fp_show:N, \fp_show:c} % \UnitTested % Simple showing of the underlying variable. % \begin{macrocode} \cs_new_eq:NN \fp_show:N \tl_show:N \cs_new_eq:NN \fp_show:c \tl_show:c % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_use:N, \fp_use:c} % \UnitTested % \begin{macro}[aux]{\fp_use_aux:w} % \begin{macro}[aux]{\fp_use_none:w} % \begin{macro}[aux]{\fp_use_small:w} % \begin{macro}[aux]{\fp_use_large:w} % \begin{macro}[aux]{\fp_use_large_aux_i:w} % \begin{macro}[aux]{\fp_use_large_aux_1:w} % \begin{macro}[aux]{\fp_use_large_aux_2:w} % \begin{macro}[aux]{\fp_use_large_aux_3:w} % \begin{macro}[aux]{\fp_use_large_aux_4:w} % \begin{macro}[aux]{\fp_use_large_aux_5:w} % \begin{macro}[aux]{\fp_use_large_aux_6:w} % \begin{macro}[aux]{\fp_use_large_aux_7:w} % \begin{macro}[aux]{\fp_use_large_aux_8:w} % \begin{macro}[aux]{\fp_use_large_aux_i:w} % \begin{macro}[aux]{\fp_use_large_aux_ii:w} % The idea of the \cs{fp_use:N} function to convert the stored % value into something suitable for \TeX{} to use as a number in an % expandable manner. The first step is to deal with the sign, then % work out how big the input is. % \begin{macrocode} \cs_new_nopar:Npn \fp_use:N #1 { \exp_after:wN \fp_use_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_use:N { c } \cs_new_nopar:Npn \fp_use_aux:w #1#2 e #3 \q_stop { \if:w #1 - - \fi: \if_int_compare:w #3 > \c_zero \exp_after:wN \fp_use_large:w \else: \if_int_compare:w #3 < \c_zero \exp_after:wN \exp_after:wN \exp_after:wN \fp_use_small:w \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_use_none:w \fi: \fi: #2 e #3 \q_stop } % \end{macrocode} % When the exponent is zero, the input is simply returned as output. % \begin{macrocode} \cs_new_nopar:Npn \fp_use_none:w #1 e #2 \q_stop {#1} % \end{macrocode} % For small numbers (less than $1$) the correct number of zeros % have to be inserted, but the decimal point is easy. % \begin{macrocode} \cs_new_nopar:Npn \fp_use_small:w #1 . #2 e #3 \q_stop { 0 . \prg_replicate:nn { -#3 - 1 } { 0 } #1#2 } % \end{macrocode} % Life is more complex for large numbers. The decimal point needs to % be shuffled, with potentially some zero-filling for very large values. % \begin{macrocode} \cs_new_nopar:Npn \fp_use_large:w #1 . #2 e #3 \q_stop { \if_int_compare:w #3 < \c_ten \exp_after:wN \fp_use_large_aux_i:w \else: \exp_after:wN \fp_use_large_aux_ii:w \fi: #1#2 e #3 \q_stop } \cs_new_nopar:Npn \fp_use_large_aux_i:w #1#2 e #3 \q_stop { #1 \use:c { fp_use_large_aux_ #3 :w } #2 \q_stop } \cs_new_nopar:cpn { fp_use_large_aux_1:w } #1#2 \q_stop { #1 . #2 } \cs_new_nopar:cpn { fp_use_large_aux_2:w } #1#2#3 \q_stop { #1#2 . #3 } \cs_new_nopar:cpn { fp_use_large_aux_3:w } #1#2#3#4 \q_stop { #1#2#3 . #4 } \cs_new_nopar:cpn { fp_use_large_aux_4:w } #1#2#3#4#5 \q_stop { #1#2#3#4 . #5 } \cs_new_nopar:cpn { fp_use_large_aux_5:w } #1#2#3#4#5#6 \q_stop { #1#2#3#4#5 . #6 } \cs_new_nopar:cpn { fp_use_large_aux_6:w } #1#2#3#4#5#6#7 \q_stop { #1#2#3#4#5#6 . #7 } \cs_new_nopar:cpn { fp_use_large_aux_7:w } #1#2#3#4#5#6#7#8 \q_stop { #1#2#3#4#6#7 . #8 } \cs_new_nopar:cpn { fp_use_large_aux_8:w } #1#2#3#4#5#6#7#8#9 \q_stop { #1#2#3#4#5#6#7#8 . #9 } \cs_new_nopar:cpn { fp_use_large_aux_9:w } #1 \q_stop { #1 . } \cs_new_nopar:Npn \fp_use_large_aux_ii:w #1 e #2 \q_stop { #1 \prg_replicate:nn { #2 - 9 } { 0 } . } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Transferring to other types} % % The \cs{fp_use:N} function converts a floating point variable to % a form that can be used by \TeX{}. Here, the functions are slightly % different, as some information may be discarded. % % \begin{macro}{\fp_to_dim:N, \fp_to_dim:c} % A very simple wrapper. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_dim:N #1 { \fp_use:N #1 pt } \cs_generate_variant:Nn \fp_to_dim:N { c } % \end{macrocode} % \end{macro} % % % \begin{macro}{\fp_to_int:N, \fp_to_int:c} % \UnitTested % \begin{macro}[aux]{\fp_to_int_aux:w} % \begin{macro}[aux]{\fp_to_int_none:w} % \begin{macro}[aux]{\fp_to_int_small:w} % \begin{macro}[aux]{\fp_to_int_large:w} % \begin{macro}[aux]{\fp_to_int_large_aux_i:w} % \begin{macro}[aux]{\fp_to_int_large_aux_1:w} % \begin{macro}[aux]{\fp_to_int_large_aux_2:w} % \begin{macro}[aux]{\fp_to_int_large_aux_3:w} % \begin{macro}[aux]{\fp_to_int_large_aux_4:w} % \begin{macro}[aux]{\fp_to_int_large_aux_5:w} % \begin{macro}[aux]{\fp_to_int_large_aux_6:w} % \begin{macro}[aux]{\fp_to_int_large_aux_7:w} % \begin{macro}[aux]{\fp_to_int_large_aux_8:w} % \begin{macro}[aux]{\fp_to_int_large_aux_i:w} % \begin{macro}[aux]{\fp_to_int_large_aux:nnn} % \begin{macro}[aux]{\fp_to_int_large_aux_ii:w} % Converting to integers in an expandable manner is very similar to % simply using floating point variables, particularly in the lead-off. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_int:N #1 { \exp_after:wN \fp_to_int_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_to_int:N { c } \cs_new_nopar:Npn \fp_to_int_aux:w #1#2 e #3 \q_stop { \if:w #1 - - \fi: \if_int_compare:w #3 < \c_zero \exp_after:wN \fp_to_int_small:w \else: \exp_after:wN \fp_to_int_large:w \fi: #2 e #3 \q_stop } % \end{macrocode} % For small numbers, if the decimal part is greater than a half then % there is rounding up to do. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_int_small:w #1 . #2 e #3 \q_stop { \if_int_compare:w #3 > \c_one \else: \if_int_compare:w #1 < \c_five 0 \else: 1 \fi: \fi: } % \end{macrocode} % For large numbers, the idea is to split off the part for rounding, % do the rounding and fill if needed. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_int_large:w #1 . #2 e #3 \q_stop { \if_int_compare:w #3 < \c_ten \exp_after:wN \fp_to_int_large_aux_i:w \else: \exp_after:wN \fp_to_int_large_aux_ii:w \fi: #1#2 e #3 \q_stop } \cs_new_nopar:Npn \fp_to_int_large_aux_i:w #1#2 e #3 \q_stop { \use:c { fp_to_int_large_aux_ #3 :w } #2 \q_stop {#1} } \cs_new_nopar:cpn { fp_to_int_large_aux_1:w } #1#2 \q_stop { \fp_to_int_large_aux:nnn { #2 0 } {#1} } \cs_new_nopar:cpn { fp_to_int_large_aux_2:w } #1#2#3 \q_stop { \fp_to_int_large_aux:nnn { #3 00 } {#1#2} } \cs_new_nopar:cpn { fp_to_int_large_aux_3:w } #1#2#3#4 \q_stop { \fp_to_int_large_aux:nnn { #4 000 } {#1#2#3} } \cs_new_nopar:cpn { fp_to_int_large_aux_4:w } #1#2#3#4#5 \q_stop { \fp_to_int_large_aux:nnn { #5 0000 } {#1#2#3#4} } \cs_new_nopar:cpn { fp_to_int_large_aux_5:w } #1#2#3#4#5#6 \q_stop { \fp_to_int_large_aux:nnn { #6 00000 } {#1#2#3#4#5} } \cs_new_nopar:cpn { fp_to_int_large_aux_6:w } #1#2#3#4#5#6#7 \q_stop { \fp_to_int_large_aux:nnn { #7 000000 } {#1#2#3#4#5#6} } \cs_new_nopar:cpn { fp_to_int_large_aux_7:w } #1#2#3#4#5#6#7#8 \q_stop { \fp_to_int_large_aux:nnn { #8 0000000 } {#1#2#3#4#5#6#7} } \cs_new_nopar:cpn { fp_to_int_large_aux_8:w } #1#2#3#4#5#6#7#8#9 \q_stop { \fp_to_int_large_aux:nnn { #9 00000000 } {#1#2#3#4#5#6#7#8} } \cs_new_nopar:cpn { fp_to_int_large_aux_9:w } #1 \q_stop {#1} \cs_new_nopar:Npn \fp_to_int_large_aux:nnn #1#2#3 { \if_int_compare:w #1 < \c_five_hundred_million #3#2 \else: \int_value:w \int_eval:w #3#2 + 1 \int_eval_end: \fi: } \cs_new_nopar:Npn \fp_to_int_large_aux_ii:w #1 e #2 \q_stop { #1 \prg_replicate:nn { #2 - 9 } { 0 } } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_to_tl:N, \fp_to_tl:c} % \UnitTested % \begin{macro}[aux]{\fp_to_tl_aux:w} % \begin{macro}[aux]{\fp_to_tl_large:w} % \begin{macro}[aux]{\fp_to_tl_large_aux_i:w} % \begin{macro}[aux]{\fp_to_tl_large_aux_ii:w} % \begin{macro}[aux]{\fp_to_tl_large_0:w} % \begin{macro}[aux]{\fp_to_tl_large_1:w} % \begin{macro}[aux]{\fp_to_tl_large_2:w} % \begin{macro}[aux]{\fp_to_tl_large_3:w} % \begin{macro}[aux]{\fp_to_tl_large_4:w} % \begin{macro}[aux]{\fp_to_tl_large_5:w} % \begin{macro}[aux]{\fp_to_tl_large_6:w} % \begin{macro}[aux]{\fp_to_tl_large_7:w} % \begin{macro}[aux]{\fp_to_tl_large_8:w} % \begin{macro}[aux]{\fp_to_tl_large_8_aux:w} % \begin{macro}[aux]{\fp_to_tl_large_9:w} % \begin{macro}[aux]{\fp_to_tl_small:w} % \begin{macro}[aux]{\fp_to_tl_small_one:w} % \begin{macro}[aux]{\fp_to_tl_small_two:w} % \begin{macro}[aux]{\fp_to_tl_small_aux:w} % \begin{macro}[aux]{\fp_to_tl_large_zeros:NNNNNNNNN} % \begin{macro}[aux]{\fp_to_tl_small_zeros:NNNNNNNNN} % \begin{macro}[aux]{\fp_use_iix_ix:NNNNNNNNN} % \begin{macro}[aux]{\fp_use_ix:NNNNNNNNN} % \begin{macro}[aux]{\fp_use_i_to_vii:NNNNNNNNN} % \begin{macro}[aux]{\fp_use_i_to_iix:NNNNNNNNN} % Converting to integers in an expandable manner is very similar to % simply using floating point variables, particularly in the lead-off. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_tl:N #1 { \exp_after:wN \fp_to_tl_aux:w #1 \q_stop } \cs_generate_variant:Nn \fp_to_tl:N { c } \cs_new_nopar:Npn \fp_to_tl_aux:w #1#2 e #3 \q_stop { \if:w #1 - - \fi: \if_int_compare:w #3 < \c_zero \exp_after:wN \fp_to_tl_small:w \else: \exp_after:wN \fp_to_tl_large:w \fi: #2 e #3 \q_stop } % \end{macrocode} % For \enquote{large} numbers (exponent $\ge 0$) there are two % cases. For very large exponents ($ \ge 10 $) life is easy: apart % from dropping extra zeros there is no work to do. On the other hand, % for intermediate exponent values the decimal needs to be moved, then % zeros can be dropped. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_tl_large:w #1 e #2 \q_stop { \if_int_compare:w #2 < \c_ten \exp_after:wN \fp_to_tl_large_aux_i:w \else: \exp_after:wN \fp_to_tl_large_aux_ii:w \fi: #1 e #2 \q_stop } \cs_new_nopar:Npn \fp_to_tl_large_aux_i:w #1 e #2 \q_stop { \use:c { fp_to_tl_large_ #2 :w } #1 \q_stop } \cs_new_nopar:Npn \fp_to_tl_large_aux_ii:w #1 . #2 e #3 \q_stop { #1 \fp_to_tl_large_zeros:NNNNNNNNN #2 e #3 } \cs_new_nopar:cpn { fp_to_tl_large_0:w } #1 . #2 \q_stop { #1 \fp_to_tl_large_zeros:NNNNNNNNN #2 } \cs_new_nopar:cpn { fp_to_tl_large_1:w } #1 . #2#3 \q_stop { #1#2 \fp_to_tl_large_zeros:NNNNNNNNN #3 0 } \cs_new_nopar:cpn { fp_to_tl_large_2:w } #1 . #2#3#4 \q_stop { #1#2#3 \fp_to_tl_large_zeros:NNNNNNNNN #4 00 } \cs_new_nopar:cpn { fp_to_tl_large_3:w } #1 . #2#3#4#5 \q_stop { #1#2#3#4 \fp_to_tl_large_zeros:NNNNNNNNN #5 000 } \cs_new_nopar:cpn { fp_to_tl_large_4:w } #1 . #2#3#4#5#6 \q_stop { #1#2#3#4#5 \fp_to_tl_large_zeros:NNNNNNNNN #6 0000 } \cs_new_nopar:cpn { fp_to_tl_large_5:w } #1 . #2#3#4#5#6#7 \q_stop { #1#2#3#4#5#6 \fp_to_tl_large_zeros:NNNNNNNNN #7 00000 } \cs_new_nopar:cpn { fp_to_tl_large_6:w } #1 . #2#3#4#5#6#7#8 \q_stop { #1#2#3#4#5#6#7 \fp_to_tl_large_zeros:NNNNNNNNN #8 000000 } \cs_new_nopar:cpn { fp_to_tl_large_7:w } #1 . #2#3#4#5#6#7#8#9 \q_stop { #1#2#3#4#5#6#7#8 \fp_to_tl_large_zeros:NNNNNNNNN #9 0000000 } \cs_new_nopar:cpn { fp_to_tl_large_8:w } #1 . { #1 \use:c { fp_to_tl_large_8_aux:w } } \cs_new_nopar:cpn { fp_to_tl_large_8_aux:w } #1#2#3#4#5#6#7#8#9 \q_stop { #1#2#3#4#5#6#7#8 \fp_to_tl_large_zeros:NNNNNNNNN #9 00000000 } \cs_new_nopar:cpn { fp_to_tl_large_9:w } #1 . #2 \q_stop {#1#2} % \end{macrocode} % Dealing with small numbers is a bit more complex as there has to be % rounding. This makes life rather awkward, as there need to be a series % of tests and calculations, as things cannot be stored in an % expandable system. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_tl_small:w #1 e #2 \q_stop { \if_int_compare:w #2 = \c_minus_one \exp_after:wN \fp_to_tl_small_one:w \else: \if_int_compare:w #2 = -\c_two \exp_after:wN \exp_after:wN \exp_after:wN \fp_to_tl_small_two:w \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_to_tl_small_aux:w \fi: \fi: #1 e #2 \q_stop } \cs_new_nopar:Npn \fp_to_tl_small_one:w #1 . #2 e #3 \q_stop { \if_int_compare:w \fp_use_ix:NNNNNNNNN #2 > \c_four \if_int_compare:w \int_eval:w #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1 < \c_one_thousand_million 0. \exp_after:wN \fp_to_tl_small_zeros:NNNNNNNNN \int_value:w \int_eval:w #1 \fp_use_i_to_iix:NNNNNNNNN #2 + 1 \int_eval_end: \else: 1 \fi: \else: 0. #1 \fp_to_tl_small_zeros:NNNNNNNNN #2 \fi: } \cs_new_nopar:Npn \fp_to_tl_small_two:w #1 . #2 e #3 \q_stop { \if_int_compare:w \fp_use_iix_ix:NNNNNNNNN #2 > \c_forty_four \if_int_compare:w \int_eval:w #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten < \c_one_thousand_million 0.0 \exp_after:wN \fp_to_tl_small_zeros:NNNNNNNNN \int_value:w \int_eval:w #1 \fp_use_i_to_vii:NNNNNNNNN #2 0 + \c_ten \int_eval_end: \else: 0.1 \fi: \else: 0.0 #1 \fp_to_tl_small_zeros:NNNNNNNNN #2 \fi: } \cs_new_nopar:Npn \fp_to_tl_small_aux:w #1 . #2 e #3 \q_stop { #1 \fp_to_tl_large_zeros:NNNNNNNNN #2 e #3 } % \end{macrocode} % Rather than a complex recursion, the tests for finding trailing zeros % are written out long-hand. The difference between the two is only the % need for a decimal marker. % \begin{macrocode} \cs_new_nopar:Npn \fp_to_tl_large_zeros:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \if_int_compare:w #9 = \c_zero \if_int_compare:w #8 = \c_zero \if_int_compare:w #7 = \c_zero \if_int_compare:w #6 = \c_zero \if_int_compare:w #5 = \c_zero \if_int_compare:w #4 = \c_zero \if_int_compare:w #3 = \c_zero \if_int_compare:w #2 = \c_zero \if_int_compare:w #1 = \c_zero \else: . #1 \fi: \else: . #1#2 \fi: \else: . #1#2#3 \fi: \else: . #1#2#3#4 \fi: \else: . #1#2#3#4#5 \fi: \else: . #1#2#3#4#5#6 \fi: \else: . #1#2#3#4#5#6#7 \fi: \else: . #1#2#3#4#5#6#7#8 \fi: \else: . #1#2#3#4#5#6#7#8#9 \fi: } \cs_new_nopar:Npn \fp_to_tl_small_zeros:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \if_int_compare:w #9 = \c_zero \if_int_compare:w #8 = \c_zero \if_int_compare:w #7 = \c_zero \if_int_compare:w #6 = \c_zero \if_int_compare:w #5 = \c_zero \if_int_compare:w #4 = \c_zero \if_int_compare:w #3 = \c_zero \if_int_compare:w #2 = \c_zero \if_int_compare:w #1 = \c_zero \else: #1 \fi: \else: #1#2 \fi: \else: #1#2#3 \fi: \else: #1#2#3#4 \fi: \else: #1#2#3#4#5 \fi: \else: #1#2#3#4#5#6 \fi: \else: #1#2#3#4#5#6#7 \fi: \else: #1#2#3#4#5#6#7#8 \fi: \else: #1#2#3#4#5#6#7#8#9 \fi: } % \end{macrocode} % Some quick \enquote{return a few} functions. % \begin{macrocode} \cs_new_nopar:Npn \fp_use_iix_ix:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#8#9} \cs_new_nopar:Npn \fp_use_ix:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#9} \cs_new_nopar:Npn \fp_use_i_to_vii:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#1#2#3#4#5#6#7} \cs_new_nopar:Npn \fp_use_i_to_iix:NNNNNNNNN #1#2#3#4#5#6#7#8#9 {#1#2#3#4#5#6#7#8} % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Rounding numbers} % % The results may well need to be rounded. A couple of related functions % to do this for a stored value. % % \begin{macro}{\fp_round_figures:Nn, \fp_round_figures:cn} % \UnitTested % \begin{macro}{\fp_ground_figures:Nn, \fp_ground_figures:cn} % \UnitTested % \begin{macro}[aux]{\fp_round_figures_aux:NNn} % Rounding to figures needs only an adjustment to the target by one % (as the target is in decimal places). % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_round_figures:Nn { \fp_round_figures_aux:NNn \tl_set:Nn } \cs_generate_variant:Nn \fp_round_figures:Nn { c } \cs_new_protected_nopar:Npn \fp_ground_figures:Nn { \fp_round_figures_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_ground_figures:Nn { c } \cs_new_protected_nopar:Npn \fp_round_figures_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \int_set:Nn \l_fp_round_target_int { #3 - 1 } \if_int_compare:w \l_fp_round_target_int < \c_ten \exp_after:wN \fp_round: \fi: \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_round_places:Nn, \fp_round_places:cn} % \UnitTested % \begin{macro}{\fp_ground_places:Nn, \fp_ground_places:cn} % \UnitTested % \begin{macro}[aux]{\fp_round_places_aux:NNn} % Rounding to places needs an adjustment for the exponent value, which % will mean that everything should be correct. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_round_places:Nn { \fp_round_places_aux:NNn \tl_set:Nn } \cs_generate_variant:Nn \fp_round_places:Nn { c } \cs_new_protected_nopar:Npn \fp_ground_places:Nn { \fp_round_places_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_ground_places:Nn { c } \cs_new_protected_nopar:Npn \fp_round_places_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \int_set:Nn \l_fp_round_target_int { #3 + \l_fp_input_a_exponent_int } \if_int_compare:w \l_fp_round_target_int < \c_ten \exp_after:wN \fp_round: \fi: \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_round:} % \begin{macro}[aux]{\fp_round_aux:NNNNNNNNN} % \begin{macro}{\fp_round_loop:N} % The rounding approach is the same for decimal places and significant % figures. There are always nine decimal digits to round, so the code % can be written to account for this. The basic logic is simply to % find the rounding, track any carry digit and move along. At the end % of the loop there is a possible shuffle if the integer part has % become $10$. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_round: { \bool_set_false:N \l_fp_round_carry_bool \l_fp_round_position_int \c_eight \tl_clear:N \l_fp_round_decimal_tl \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_round_aux:NNNNNNNNN \int_use:N \l_fp_input_a_decimal_int } \cs_new_protected_nopar:Npn \fp_round_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \fp_round_loop:N #9#8#7#6#5#4#3#2#1 \bool_if:NT \l_fp_round_carry_bool { \tex_advance:D \l_fp_input_a_integer_int \c_one } \l_fp_input_a_decimal_int \l_fp_round_decimal_tl \scan_stop: \if_int_compare:w \l_fp_input_a_integer_int < \c_ten \else: \l_fp_input_a_integer_int \c_one \tex_divide:D \l_fp_input_a_decimal_int \c_ten \tex_advance:D \l_fp_input_a_exponent_int \c_one \fi: } \cs_new_protected_nopar:Npn \fp_round_loop:N #1 { \if_int_compare:w \l_fp_round_position_int < \l_fp_round_target_int \bool_if:NTF \l_fp_round_carry_bool { \l_fp_tmp_int \int_eval:w #1 + \c_one \scan_stop: } { \l_fp_tmp_int \int_eval:w #1 \scan_stop: } \if_int_compare:w \l_fp_tmp_int = \c_ten \l_fp_tmp_int \c_zero \else: \bool_set_false:N \l_fp_round_carry_bool \fi: \tl_set:Nx \l_fp_round_decimal_tl { \int_use:N \l_fp_tmp_int \l_fp_round_decimal_tl } \else: \tl_set:Nx \l_fp_round_decimal_tl { 0 \l_fp_round_decimal_tl } \if_int_compare:w \l_fp_round_position_int = \l_fp_round_target_int \if_int_compare:w #1 > \c_four \bool_set_true:N \l_fp_round_carry_bool \fi: \fi: \fi: \tex_advance:D \l_fp_round_position_int \c_minus_one \if_int_compare:w \l_fp_round_position_int > \c_minus_one \exp_after:wN \fp_round_loop:N \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Unary functions} % % \begin{macro}{\fp_abs:N, \fp_abs:c} % \UnitTested % \begin{macro}{\fp_gabs:N, \fp_gabs:c} % \UnitTested % \begin{macro}[aux]{\fp_abs_aux:NN} % Setting the absolute value is easy: read the value, ignore the sign, % return the result. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_abs:N { \fp_abs_aux:NN \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gabs:N { \fp_abs_aux:NN \tl_gset:Nn } \cs_generate_variant:Nn \fp_abs:N { c } \cs_generate_variant:Nn \fp_gabs:N { c } \cs_new_protected_nopar:Npn \fp_abs_aux:NN #1#2 { \group_begin: \fp_read:N #2 \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { + \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } } \fp_tmp:w } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_neg:N, \fp_neg:c} % \UnitTested % \begin{macro}{\fp_gneg:N, \fp_gneg:c} % \UnitTested % \begin{macro}[aux]{\fp_neg:NN} % Just a bit more complex: read the input, reverse the sign and % output the result. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_neg:N { \fp_neg_aux:NN \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gneg:N { \fp_neg_aux:NN \tl_gset:Nn } \cs_generate_variant:Nn \fp_neg:N { c } \cs_generate_variant:Nn \fp_gneg:N { c } \cs_new_protected_nopar:Npn \fp_neg_aux:NN #1#2 { \group_begin: \fp_read:N #2 \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \tl_set:Nx \l_fp_tmp_tl { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero + \else: - \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_use:N \l_fp_input_a_decimal_int e \int_use:N \l_fp_input_a_exponent_int } \exp_after:wN \group_end: \exp_after:wN #1 \exp_after:wN #2 \exp_after:wN { \l_fp_tmp_tl } } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Basic arithmetic} % % \begin{macro}{\fp_add:Nn, \fp_add:cn} % \UnitTested % \begin{macro}{\fp_gadd:Nn,\fp_gadd:cn} % \UnitTested % \begin{macro}[aux]{\fp_add_aux:NNn} % \begin{macro}[aux]{\fp_add_core:} % \begin{macro}[aux]{\fp_add_sum:} % \begin{macro}[aux]{\fp_add_difference:} % The various addition functions are simply different ways to call the % single master function below. This pattern is repeated for the % other arithmetic functions. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_add:Nn { \fp_add_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gadd:Nn { \fp_add_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_add:Nn { c } \cs_generate_variant:Nn \fp_gadd:Nn { c } % \end{macrocode} % Addition takes place using one of two paths. If the signs of the % two parts are the same, they are simply combined. On the other % hand, if the signs are different the calculation finds this % difference. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_add_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \fp_split:Nn b {#3} \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \fp_add_core: \fp_tmp:w #1#2 } \cs_new_protected_nopar:Npn \fp_add_core: { \fp_level_input_exponents: \if_int_compare:w \int_eval:w \l_fp_input_a_sign_int * \l_fp_input_b_sign_int > \c_zero \exp_after:wN \fp_add_sum: \else: \exp_after:wN \fp_add_difference: \fi: \l_fp_output_exponent_int \l_fp_input_a_exponent_int \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \if_int_compare:w \l_fp_output_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } } % \end{macrocode} % Finding the sum of two numbers is trivially easy. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_add_sum: { \l_fp_output_sign_int \l_fp_input_a_sign_int \l_fp_output_integer_int \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_b_integer_int \scan_stop: \l_fp_output_decimal_int \int_eval:w \l_fp_input_a_decimal_int + \l_fp_input_b_decimal_int \scan_stop: \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_output_integer_int \c_one \tex_advance:D \l_fp_output_decimal_int -\c_one_thousand_million \fi: } % \end{macrocode} % When the signs of the two parts of the input are different, the % absolute difference is worked out first. There is then a calculation % to see which way around everything has worked out, so that the final % sign is correct. The difference might also give a zero result with % a negative sign, which is reversed as zero is regarded as positive. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_add_difference: { \l_fp_output_integer_int \int_eval:w \l_fp_input_a_integer_int - \l_fp_input_b_integer_int \scan_stop: \l_fp_output_decimal_int \int_eval:w \l_fp_input_a_decimal_int - \l_fp_input_b_decimal_int \scan_stop: \if_int_compare:w \l_fp_output_decimal_int < \c_zero \tex_advance:D \l_fp_output_integer_int \c_minus_one \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million \fi: \if_int_compare:w \l_fp_output_integer_int < \c_zero \l_fp_output_sign_int \l_fp_input_b_sign_int \if_int_compare:w \l_fp_output_decimal_int = \c_zero \l_fp_output_integer_int -\l_fp_output_integer_int \else: \l_fp_output_decimal_int \int_eval:w \c_one_thousand_million - \l_fp_output_decimal_int \scan_stop: \l_fp_output_integer_int \int_eval:w - \l_fp_output_integer_int - \c_one \scan_stop: \fi: \else: \l_fp_output_sign_int \l_fp_input_a_sign_int \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_sub:Nn, \fp_sub:cn} % \UnitTested % \begin{macro}{\fp_gsub:Nn,\fp_gsub:cn} % \UnitTested % \begin{macro}[aux]{\fp_sub_aux:NNn} % Subtraction is essentially the same as addition, but with the sign % of the second component reversed. Thus the core of the two function % groups is the same, with just a little set up here. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_sub:Nn { \fp_sub_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gsub:Nn { \fp_sub_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_sub:Nn { c } \cs_generate_variant:Nn \fp_gsub:Nn { c } \cs_new_protected_nopar:Npn \fp_sub_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \fp_split:Nn b {#3} \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \tex_multiply:D \l_fp_input_b_sign_int \c_minus_one \fp_add_core: \fp_tmp:w #1#2 } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_mul:Nn, \fp_mul:cn} % \UnitTested % \begin{macro}{\fp_gmul:Nn,\fp_gmul:cn} % \UnitTested % \begin{macro}[aux]{\fp_mul_aux:NNn} % \begin{macro}[aux]{\fp_mul_internal:} % \begin{macro}[aux]{\fp_mul_split:NNNN} % \begin{macro}[aux]{\fp_mul_split:w} % \begin{macro}[aux]{\fp_mul_end_level:} % \begin{macro}[aux]{\fp_mul_end_level:NNNNNNNNN} % The pattern is much the same for multiplication. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul:Nn { \fp_mul_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gmul:Nn { \fp_mul_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_mul:Nn { c } \cs_generate_variant:Nn \fp_gmul:Nn { c } % \end{macrocode} % The approach to multiplication is as follows. First, the two numbers % are split into blocks of three digits. These are then multiplied % together to find products for each group of three output digits. This % is al written out in full for speed reasons. Between each block of % three digits in the output, there is a carry step. The very lowest % digits are not calculated, while % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \fp_split:Nn b {#3} \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \fp_mul_internal: \l_fp_output_exponent_int \int_eval:w \l_fp_input_a_exponent_int + \l_fp_input_b_exponent_int \scan_stop: \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_int_compare:w \int_eval:w \l_fp_input_a_sign_int * \l_fp_input_b_sign_int < \c_zero \if_int_compare:w \int_eval:w \l_fp_output_integer_int + \l_fp_output_decimal_int = \c_zero + \else: - \fi: \else: + \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } \fp_tmp:w } % \end{macrocode} % Done separately so that the internal use is a bit easier. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul_internal: { \fp_mul_split:NNNN \l_fp_input_a_decimal_int \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int \fp_mul_split:NNNN \l_fp_input_b_decimal_int \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int \l_fp_mul_output_int \c_zero \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int \tex_divide:D \l_fp_mul_output_int \c_one_thousand \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_input_b_integer_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_input_b_integer_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_input_b_integer_int \fp_mul_end_level: \l_fp_output_decimal_int 0 \l_fp_mul_output_tl \scan_stop: \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_input_a_integer_int \l_fp_input_b_integer_int \fp_mul_end_level: \l_fp_output_integer_int 0 \l_fp_mul_output_tl \scan_stop: } % \end{macrocode} % The split works by making a $10$ digit number, from which % the first digit can then be dropped using a delimited argument. The % groups of three digits are then assigned to the various parts of % the input: notice that |##9| contains the last two digits of the % smallest part of the input. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul_split:NNNN #1#2#3#4 { \tex_advance:D #1 \c_one_thousand_million \cs_set_protected_nopar:Npn \fp_mul_split_aux:w ##1##2##3##4##5##6##7##8##9 \q_stop { #2 ##2##3##4 \scan_stop: #3 ##5##6##7 \scan_stop: #4 ##8##9 \scan_stop: } \exp_after:wN \fp_mul_split_aux:w \int_use:N #1 \q_stop \tex_advance:D #1 -\c_one_thousand_million } \cs_new_protected_nopar:Npn \fp_mul_product:NN #1#2 { \l_fp_mul_output_int \int_eval:w \l_fp_mul_output_int + #1 * #2 \scan_stop: } % \end{macrocode} % At the end of each output group of three, there is a transfer of % information so that there is no danger of an overflow. This is done by % expansion to keep the number of calculations down. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul_end_level: { \tex_advance:D \l_fp_mul_output_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_mul_end_level:NNNNNNNNN \int_use:N \l_fp_mul_output_int } \cs_new_protected_nopar:Npn \fp_mul_end_level:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \tl_set:Nx \l_fp_mul_output_tl { #7#8#9 \l_fp_mul_output_tl } \l_fp_mul_output_int #1#2#3#4#5#6 \scan_stop: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_div:Nn, \fp_div:cn} % \UnitTested % \begin{macro}{\fp_gdiv:Nn,\fp_gdiv:cn} % \UnitTested % \begin{macro}[aux]{\fp_div_aux:NNn} % \begin{macro}{\fp_div_internal:} % \begin{macro}[aux]{\fp_div_loop:} % \begin{macro}[aux]{\fp_div_divide:} % \begin{macro}[aux]{\fp_div_divide_aux:} % \begin{macro}[aux]{\fp_div_store:} % \begin{macro}[aux]{\fp_div_store_integer:} % \begin{macro}[aux]{\fp_div_store_decimal:} % The pattern is much the same for multiplication. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div:Nn { \fp_div_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gdiv:Nn { \fp_div_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_div:Nn { c } \cs_generate_variant:Nn \fp_gdiv:Nn { c } % \end{macrocode} % Division proper starts with a couple of tests. If the denominator is % zero then a error is issued. On the other hand, if the numerator is % zero then the result must be $0.0$ and can be given with no % further work. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \fp_split:Nn b {#3} \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \if_int_compare:w \int_eval:w \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int = \c_zero \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: #1 \exp_not:N #2 { \c_undefined_fp } } \else: \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int = \c_zero \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: #1 \exp_not:N #2 { \c_zero_fp } } \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_internal: \fi: \fi: \fp_tmp:w #1#2 } % \end{macrocode} % The main division algorithm works by finding how many times |b| can % be removed from |a|, storing the result and doing the subtraction. % Input |a| is then multiplied by $10$, and the process is repeated. % The looping ends either when there is nothing left of |a| % (\emph{i.e.}~an exact result) or when the code reaches the ninth % decimal place. Most of the process takes place in the loop function % below. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_internal: { \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_zero \cs_set_eq:NN \fp_div_store: \fp_div_store_integer: \l_fp_div_offset_int \c_one_hundred_million \fp_div_loop: \l_fp_output_exponent_int \int_eval:w \l_fp_input_a_exponent_int - \l_fp_input_b_exponent_int \scan_stop: \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \if_int_compare:w \int_eval:w \l_fp_input_a_sign_int * \l_fp_input_b_sign_int < \c_zero \if_int_compare:w \int_eval:w \l_fp_output_integer_int + \l_fp_output_decimal_int = \c_zero + \else: - \fi: \else: + \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million \int_eval_end: e \int_use:N \l_fp_output_exponent_int } } } % \end{macrocode} % The main loop implements the approach described above. The storing % function is done as a function so that the integer and decimal parts % can be done separately but rapidly. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_loop: { \l_fp_count_int \c_zero \fp_div_divide: \fp_div_store: \tex_multiply:D \l_fp_input_a_integer_int \c_ten \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \exp_after:wN \fp_div_loop_step:w \int_use:N \l_fp_input_a_decimal_int \q_stop \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int > \c_zero \if_int_compare:w \l_fp_div_offset_int > \c_zero \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_loop: \fi: \fi: } % \end{macrocode} % Checking to see if the numerator can be divides needs quite an % involved check. Either the integer part has to be bigger for the % numerator or, if it is not smaller then the decimal part of the % numerator must not be smaller than that of the denominator. Once % the test is right the rest is much as elsewhere. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_divide: { \if_int_compare:w \l_fp_input_a_integer_int > \l_fp_input_b_integer_int \exp_after:wN \fp_div_divide_aux: \else: \if_int_compare:w \l_fp_input_a_integer_int < \l_fp_input_b_integer_int \else: \if_int_compare:w \l_fp_input_a_decimal_int < \l_fp_input_b_decimal_int \else: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_div_divide_aux: \fi: \fi: \fi: } \cs_new_protected_nopar:Npn \fp_div_divide_aux: { \tex_advance:D \l_fp_count_int \c_one \tex_advance:D \l_fp_input_a_integer_int -\l_fp_input_b_integer_int \tex_advance:D \l_fp_input_a_decimal_int -\l_fp_input_b_decimal_int \if_int_compare:w \l_fp_input_a_decimal_int < \c_zero \tex_advance:D \l_fp_input_a_integer_int \c_minus_one \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \fi: \fp_div_divide: } % \end{macrocode} % Storing the number of each division is done differently for the % integer and decimal. The integer is easy and a one-off, while the % decimal also needs to account for the position of the digit to store. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_store: { } \cs_new_protected_nopar:Npn \fp_div_store_integer: { \l_fp_output_integer_int \l_fp_count_int \cs_set_eq:NN \fp_div_store: \fp_div_store_decimal: } \cs_new_protected_nopar:Npn \fp_div_store_decimal: { \l_fp_output_decimal_int \int_eval:w \l_fp_output_decimal_int + \l_fp_count_int * \l_fp_div_offset_int \int_eval_end: \tex_divide:D \l_fp_div_offset_int \c_ten } \cs_new_protected_nopar:Npn \fp_div_loop_step:w #1#2#3#4#5#6#7#8#9 \q_stop { \l_fp_input_a_integer_int \int_eval:w #2 + \l_fp_input_a_integer_int \int_eval_end: \l_fp_input_a_decimal_int #3#4#5#6#7#8#9 0 \scan_stop: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Arithmetic for internal use} % % For the more complex functions, it is only possible to deliver % reliable $10$ digit accuracy if the internal calculations are % carried out to a higher degree of precision. This is done using a % second set of functions so that the `user' versions are not % slowed down. These versions are also focussed on the needs of internal % calculations. No error checking, sign checking or exponent levelling % is done. For addition and subtraction, the arguments are: % \begin{itemize} % \item Integer part of input |a|. % \item Decimal part of input |a|. % \item Additional decimal part of input |a|. % \item Integer part of input |b|. % \item Decimal part of input |b|. % \item Additional decimal part of input |b|. % \item Integer part of output. % \item Decimal part of output. % \item Additional decimal part of output. % \end{itemize} % The situation for multiplication and division is a little different as % they only deal with the decimal part. % % \begin{macro}{\fp_add:NNNNNNNNN} % The internal sum is always exactly that: it is always a sum and there % is no sign check. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_add:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { #7 \int_eval:w #1 + #4 \int_eval_end: #8 \int_eval:w #2 + #5 \int_eval_end: #9 \int_eval:w #3 + #6 \int_eval_end: \if_int_compare:w #9 < \c_one_thousand_million \else: \tex_advance:D #8 \c_one \tex_advance:D #9 -\c_one_thousand_million \fi: \if_int_compare:w #8 < \c_one_thousand_million \else: \tex_advance:D #7 \c_one \tex_advance:D #8 -\c_one_thousand_million \fi: } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_sub:NNNNNNNNN} % Internal subtraction is needed only when the first number is bigger % than the second, so there is no need to worry about the sign. This is % a good job as there are no arguments left. The flipping flag is % used in the rare case where a sign change is possible. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_sub:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { #7 \int_eval:w #1 - #4 \int_eval_end: #8 \int_eval:w #2 - #5 \int_eval_end: #9 \int_eval:w #3 - #6 \int_eval_end: \if_int_compare:w #9 < \c_zero \tex_advance:D #8 \c_minus_one \tex_advance:D #9 \c_one_thousand_million \fi: \if_int_compare:w #8 < \c_zero \tex_advance:D #7 \c_minus_one \tex_advance:D #8 \c_one_thousand_million \fi: \if_int_compare:w #7 < \c_zero \if_int_compare:w \int_eval:w #8 + #9 = \c_zero #7 -#7 \else: \tex_advance:D #7 \c_one #8 \int_eval:w \c_one_thousand_million - #8 \int_eval_end: #9 \int_eval:w \c_one_thousand_million - #9 \int_eval_end: \fi: \fi: } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_mul:NNNNNN} % Decimal-part only multiplication but with higher accuracy than the % user version. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul:NNNNNN #1#2#3#4#5#6 { \fp_mul_split:NNNN #1 \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int \fp_mul_split:NNNN #2 \l_fp_mul_a_iv_int \l_fp_mul_a_v_int \l_fp_mul_a_vi_int \fp_mul_split:NNNN #3 \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int \fp_mul_split:NNNN #4 \l_fp_mul_b_iv_int \l_fp_mul_b_v_int \l_fp_mul_b_vi_int \l_fp_mul_output_int \c_zero \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_vi_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_v_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_vi_int \l_fp_mul_b_i_int \tex_divide:D \l_fp_mul_output_int \c_one_thousand \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_v_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_i_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_i_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int \fp_mul_end_level: #6 0 \l_fp_mul_output_tl \scan_stop: \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int \fp_mul_end_level: \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int \fp_mul_end_level: \fp_mul_end_level: #5 0 \l_fp_mul_output_tl \scan_stop: } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_mul:NNNNNNNNN} % For internal multiplication where the integer does need to be % retained. This means of course that this code is quite slow, and so % is only used when necessary. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_mul:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \fp_mul_split:NNNN #2 \l_fp_mul_a_i_int \l_fp_mul_a_ii_int \l_fp_mul_a_iii_int \fp_mul_split:NNNN #3 \l_fp_mul_a_iv_int \l_fp_mul_a_v_int \l_fp_mul_a_vi_int \fp_mul_split:NNNN #5 \l_fp_mul_b_i_int \l_fp_mul_b_ii_int \l_fp_mul_b_iii_int \fp_mul_split:NNNN #6 \l_fp_mul_b_iv_int \l_fp_mul_b_v_int \l_fp_mul_b_vi_int \l_fp_mul_output_int \c_zero \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_vi_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_v_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_vi_int \l_fp_mul_b_i_int \tex_divide:D \l_fp_mul_output_int \c_one_thousand \fp_mul_product:NN #1 \l_fp_mul_b_vi_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_v_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_v_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_vi_int #4 \fp_mul_end_level: \fp_mul_product:NN #1 \l_fp_mul_b_v_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iv_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_v_int #4 \fp_mul_end_level: \fp_mul_product:NN #1 \l_fp_mul_b_iv_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_iii_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_iv_int #4 \fp_mul_end_level: #9 0 \l_fp_mul_output_tl \scan_stop: \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN #1 \l_fp_mul_b_iii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_ii_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_iii_int #4 \fp_mul_end_level: \fp_mul_product:NN #1 \l_fp_mul_b_ii_int \fp_mul_product:NN \l_fp_mul_a_i_int \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_ii_int #4 \fp_mul_end_level: \fp_mul_product:NN #1 \l_fp_mul_b_i_int \fp_mul_product:NN \l_fp_mul_a_i_int #4 \fp_mul_end_level: #8 0 \l_fp_mul_output_tl \scan_stop: \tl_clear:N \l_fp_mul_output_tl \fp_mul_product:NN #1 #4 \fp_mul_end_level: #7 0 \l_fp_mul_output_tl \scan_stop: } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_div_integer:NNNNN} % Here, division is always by an integer, and so it is possible to % use \TeX{}'s native calculations rather than doing it in macros. % The idea here is to divide the decimal part, find any remainder, % then do the real division of the two parts before adding in what % is needed for the remainder. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_div_integer:NNNNN #1#2#3#4#5 { \l_fp_tmp_int #1 \tex_divide:D \l_fp_tmp_int #3 \l_fp_tmp_int \int_eval:w #1 - \l_fp_tmp_int * #3 \int_eval_end: #4 #1 \tex_divide:D #4 #3 #5 #2 \tex_divide:D #5 #3 \tex_multiply:D \l_fp_tmp_int \c_one_thousand \tex_divide:D \l_fp_tmp_int #3 #5 \int_eval:w #5 + \l_fp_tmp_int * \c_one_million \int_eval_end: \if_int_compare:w #5 > \c_one_thousand_million \tex_advance:D #4 \c_one \tex_advance:D #5 -\c_one_thousand_million \fi: } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_extended_normalise:} % \begin{macro}[aux]{\fp_extended_normalise_aux_i:} % \begin{macro}[aux]{\fp_extended_normalise_aux_i:w} % \begin{macro}[aux]{\fp_extended_normalise_aux_ii:w} % \begin{macro}[aux]{\fp_extended_normalise_aux_ii:} % \begin{macro}[aux]{\fp_extended_normalise_aux:NNNNNNNNN} % The \enquote{extended} integers for internal use are mainly used in % fixed-point mode. This comes up in a few places, so a generalised % utility is made available to carry out the change. This function % simply calls the two loops to shift the input to the point of % having a zero exponent. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_extended_normalise: { \fp_extended_normalise_aux_i: \fp_extended_normalise_aux_ii: } \cs_new_protected_nopar:Npn \fp_extended_normalise_aux_i: { \if_int_compare:w \l_fp_input_a_exponent_int > \c_zero \tex_multiply:D \l_fp_input_a_integer_int \c_ten \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \exp_after:wN \fp_extended_normalise_aux_i:w \int_use:N \l_fp_input_a_decimal_int \q_stop \exp_after:wN \fp_extended_normalise_aux_i: \fi: } \cs_new_protected_nopar:Npn \fp_extended_normalise_aux_i:w #1#2#3#4#5#6#7#8#9 \q_stop { \l_fp_input_a_integer_int \int_eval:w \l_fp_input_a_integer_int + #2 \scan_stop: \l_fp_input_a_decimal_int #3#4#5#6#7#8#9 0 \scan_stop: \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million \exp_after:wN \fp_extended_normalise_aux_ii:w \int_use:N \l_fp_input_a_extended_int \q_stop } \cs_new_protected_nopar:Npn \fp_extended_normalise_aux_ii:w #1#2#3#4#5#6#7#8#9 \q_stop { \l_fp_input_a_decimal_int \int_eval:w \l_fp_input_a_decimal_int + #2 \scan_stop: \l_fp_input_a_extended_int #3#4#5#6#7#8#9 0 \scan_stop: \tex_advance:D \l_fp_input_a_exponent_int \c_minus_one } \cs_new_protected_nopar:Npn \fp_extended_normalise_aux_ii: { \if_int_compare:w \l_fp_input_a_exponent_int < \c_zero \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_extended_normalise_ii_aux:NNNNNNNNN \int_use:N \l_fp_input_a_decimal_int \exp_after:wN \fp_extended_normalise_aux_ii: \fi: } \cs_new_protected_nopar:Npn \fp_extended_normalise_ii_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \if_int_compare:w \l_fp_input_a_integer_int = \c_zero \l_fp_input_a_decimal_int #1#2#3#4#5#6#7#8 \scan_stop: \else: \tl_set:Nx \l_fp_tmp_tl { \int_use:N \l_fp_input_a_integer_int #1#2#3#4#5#6#7#8 } \l_fp_input_a_integer_int \c_zero \l_fp_input_a_decimal_int \l_fp_tmp_tl \scan_stop: \fi: \tex_divide:D \l_fp_input_a_extended_int \c_ten \tl_set:Nx \l_fp_tmp_tl { #9 \int_use:N \l_fp_input_a_extended_int } \l_fp_input_a_extended_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_input_a_exponent_int \c_one } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_extended_normalise_output:} % \begin{macro}[aux]{\fp_extended_normalise_output_aux_i:NNNNNNNNN} % \begin{macro}[aux]{\fp_extended_normalise_output_aux_ii:NNNNNNNNN} % \begin{macro}[aux]{\fp_extended_normalise_output_aux:N} % At some stages in working out extended output, it is possible for the % value to need shifting to keep the integer part in range. This only % ever happens such that the integer needs to be made smaller. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_extended_normalise_output: { \if_int_compare:w \l_fp_output_integer_int > \c_nine \tex_advance:D \l_fp_output_integer_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_extended_normalise_output_aux_i:NNNNNNNNN \int_use:N \l_fp_output_integer_int \exp_after:wN \fp_extended_normalise_output: \fi: } \cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux_i:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \l_fp_output_integer_int #1#2#3#4#5#6#7#8 \scan_stop: \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million \tl_set:Nx \l_fp_tmp_tl { #9 \exp_after:wN \use_none:n \int_use:N \l_fp_output_decimal_int } \exp_after:wN \fp_extended_normalise_output_aux_ii:NNNNNNNNN \l_fp_tmp_tl } \cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux_ii:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \l_fp_output_decimal_int #1#2#3#4#5#6#7#8#9 \scan_stop: \fp_extended_normalise_output_aux:N } \cs_new_protected_nopar:Npn \fp_extended_normalise_output_aux:N #1 { \tex_advance:D \l_fp_output_extended_int \c_one_thousand_million \tex_divide:D \l_fp_output_extended_int \c_ten \tl_set:Nx \l_fp_tmp_tl { #1 \exp_after:wN \use_none:n \int_use:N \l_fp_output_extended_int } \l_fp_output_extended_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_output_exponent_int \c_one } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Trigonometric functions} % % \begin{macro}{\fp_trig_normalise:} % \begin{macro}[aux]{\fp_trig_normalise_aux:} % \begin{macro}[aux]{\fp_trig_sub:NNN} % For normalisation, the code essentially switches to fixed-point % arithmetic. There is a shift of the exponent, then repeated % subtractions. The end result is a number in the range % $ -\pi < x \le \pi $. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_trig_normalise: { \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten \l_fp_input_a_extended_int \c_zero \fp_extended_normalise: \fp_trig_normalise_aux: \if_int_compare:w \l_fp_input_a_integer_int < \c_zero \l_fp_input_a_sign_int -\l_fp_input_a_sign_int \l_fp_input_a_integer_int -\l_fp_input_a_integer_int \fi: \exp_after:wN \fp_trig_octant: \else: \l_fp_input_a_sign_int \c_one \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_zero \l_fp_output_exponent_int \c_zero \exp_after:wN \fp_trig_overflow_msg: \fi: } \cs_new_protected_nopar:Npn \fp_trig_normalise_aux: { \if_int_compare:w \l_fp_input_a_integer_int > \c_three \fp_trig_sub:NNN \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int \exp_after:wN \fp_trig_normalise_aux: \else: \if_int_compare:w \l_fp_input_a_integer_int > \c_two \if_int_compare:w \l_fp_input_a_decimal_int > \c_fp_pi_decimal_int \fp_trig_sub:NNN \c_six \c_fp_two_pi_decimal_int \c_fp_two_pi_extended_int \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_trig_normalise_aux: \fi: \fi: \fi: } % \end{macrocode} % Here, there may be a sign change but there will never be any % variation in the input. So a dedicated function can be used. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_trig_sub:NNN #1#2#3 { \l_fp_input_a_integer_int \int_eval:w \l_fp_input_a_integer_int - #1 \int_eval_end: \l_fp_input_a_decimal_int \int_eval:w \l_fp_input_a_decimal_int - #2 \int_eval_end: \l_fp_input_a_extended_int \int_eval:w \l_fp_input_a_extended_int - #3 \int_eval_end: \if_int_compare:w \l_fp_input_a_extended_int < \c_zero \tex_advance:D \l_fp_input_a_decimal_int \c_minus_one \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million \fi: \if_int_compare:w \l_fp_input_a_decimal_int < \c_zero \tex_advance:D \l_fp_input_a_integer_int \c_minus_one \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \fi: \if_int_compare:w \l_fp_input_a_integer_int < \c_zero \l_fp_input_a_sign_int -\l_fp_input_a_sign_int \if_int_compare:w \int_eval:w \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int = \c_zero \l_fp_input_a_integer_int -\l_fp_input_a_integer_int \else: \l_fp_input_a_integer_int \int_eval:w - \l_fp_input_a_integer_int - \c_one \int_eval_end: \l_fp_input_a_decimal_int \int_eval:w \c_one_thousand_million - \l_fp_input_a_decimal_int \int_eval_end: \l_fp_input_a_extended_int \int_eval:w \c_one_thousand_million - \l_fp_input_a_extended_int \int_eval_end: \fi: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_trig_octant:} % \begin{macro}[aux]{\fp_trig_octant_aux:} % Here, the input is further reduced into the range % $ 0 \le x < \pi / 4 $. This is pretty simple: check if % $ \pi / 4 $ can be taken off and if it can do it and loop. The % check at the end is to \enquote{mop up} values which are so close to % $ \pi / 4 $ that they should be treated as such. The test for % an even octant is needed as the `remainder' needed is from % the nearest $ \pi / 2 $. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_trig_octant: { \l_fp_trig_octant_int \c_one \fp_trig_octant_aux: \if_int_compare:w \l_fp_input_a_decimal_int < \c_ten \l_fp_input_a_decimal_int \c_zero \l_fp_input_a_extended_int \c_zero \fi: \if_int_odd:w \l_fp_trig_octant_int \else: \fp_sub:NNNNNNNNN \c_zero \c_fp_pi_by_four_decimal_int \c_fp_pi_by_four_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \fi: } \cs_new_protected_nopar:Npn \fp_trig_octant_aux: { \if_int_compare:w \l_fp_input_a_integer_int > \c_zero \fp_sub:NNNNNNNNN \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \c_zero \c_fp_pi_by_four_decimal_int \c_fp_pi_by_four_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \tex_advance:D \l_fp_trig_octant_int \c_one \exp_after:wN \fp_trig_octant_aux: \else: \if_int_compare:w \l_fp_input_a_decimal_int > \c_fp_pi_by_four_decimal_int \fp_sub:NNNNNNNNN \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \c_zero \c_fp_pi_by_four_decimal_int \c_fp_pi_by_four_extended_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \tex_advance:D \l_fp_trig_octant_int \c_one \exp_after:wN \exp_after:wN \exp_after:wN \fp_trig_octant_aux: \fi: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % % \begin{macro}{\fp_sin:Nn, \fp_sin:cn} % \UnitTested % \begin{macro}{\fp_gsin:Nn,\fp_gsin:cn} % \UnitTested % \begin{macro}[aux]{\fp_sin_aux:NNn} % \begin{macro}[aux]{\fp_sin_aux_i:} % \begin{macro}[aux]{\fp_sin_aux_ii:} % Calculating the sine starts off in the usual way. There is a check % to see if the value has already been worked out before proceeding % further. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_sin:Nn { \fp_sin_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gsin:Nn { \fp_sin_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_sin:Nn { c } \cs_generate_variant:Nn \fp_gsin:Nn { c } % \end{macrocode} % The internal routine for sines does a check to see if the value is % already known. This saves a lot of repetition when doing rotations. % For very small values it is best to simply return the input as the % sine: the cut-off is $ 1 \times 10^{-5} $. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_sin_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tl_set:Nx \l_fp_arg_tl { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_input_a_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_input_a_exponent_int } \if_int_compare:w \l_fp_input_a_exponent_int < -\c_five \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \l_fp_arg_tl } } \else: \if_cs_exist:w c_fp_sin ( \l_fp_arg_tl ) _fp \cs_end: \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_sin_aux_i: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \use:c { c_fp_sin ( \l_fp_arg_tl ) _fp } } } \fi: \fp_tmp:w } % \end{macrocode} % The internals for sine first normalise the input into an octant, then % choose the correct set up for the Taylor series. The sign for the sine % function is easy, so there is no worry about it. So the only thing to % do is to get the output standardised. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_sin_aux_i: { \fp_trig_normalise: \fp_sin_aux_ii: \if_int_compare:w \l_fp_output_integer_int = \c_one \l_fp_output_exponent_int \c_zero \else: \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_exponent_int -\c_nine \fi: \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \tl_new:c { c_fp_sin ( \l_fp_arg_tl ) _fp } \tl_gset:cx { c_fp_sin ( \l_fp_arg_tl ) _fp } { \if_int_compare:w \l_fp_input_a_sign_int > \c_zero + \else: - \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } \cs_new_protected_nopar:Npn \fp_sin_aux_ii: { \if_case:w \l_fp_trig_octant_int \or: \exp_after:wN \fp_trig_calc_sin: \or: \exp_after:wN \fp_trig_calc_cos: \or: \exp_after:wN \fp_trig_calc_cos: \or: \exp_after:wN \fp_trig_calc_sin: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_cos:Nn, \fp_cos:cn} % \UnitTested % \begin{macro}{\fp_gcos:Nn,\fp_gcos:cn} % \UnitTested % \begin{macro}[aux]{\fp_cos_aux:NNn} % \begin{macro}[aux]{\fp_cos_aux_i:} % \begin{macro}[aux]{\fp_cos_aux_ii:} % Cosine is almost identical, but there is no short cut code here. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_cos:Nn { \fp_cos_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gcos:Nn { \fp_cos_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_cos:Nn { c } \cs_generate_variant:Nn \fp_gcos:Nn { c } \cs_new_protected_nopar:Npn \fp_cos_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tl_set:Nx \l_fp_arg_tl { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_input_a_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_input_a_exponent_int } \if_cs_exist:w c_fp_cos ( \l_fp_arg_tl ) _fp \cs_end: \else: \exp_after:wN \fp_cos_aux_i: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \use:c { c_fp_cos ( \l_fp_arg_tl ) _fp } } } \fp_tmp:w } % \end{macrocode} % Almost the same as for sine: just a bit of correction for the sign % of the output. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_cos_aux_i: { \fp_trig_normalise: \fp_cos_aux_ii: \if_int_compare:w \l_fp_output_integer_int = \c_one \l_fp_output_exponent_int \c_zero \else: \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_exponent_int -\c_nine \fi: \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \tl_new:c { c_fp_cos ( \l_fp_arg_tl ) _fp } \tl_gset:cx { c_fp_cos ( \l_fp_arg_tl ) _fp } { \if_int_compare:w \l_fp_input_a_sign_int > \c_zero + \else: - \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } \cs_new_protected_nopar:Npn \fp_cos_aux_ii: { \if_case:w \l_fp_trig_octant_int \or: \exp_after:wN \fp_trig_calc_cos: \or: \exp_after:wN \fp_trig_calc_sin: \or: \exp_after:wN \fp_trig_calc_sin: \or: \exp_after:wN \fp_trig_calc_cos: \fi: \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \if_int_compare:w \l_fp_trig_octant_int > \c_two \l_fp_input_a_sign_int \c_minus_one \fi: \else: \if_int_compare:w \l_fp_trig_octant_int > \c_two \else: \l_fp_input_a_sign_int \c_one \fi: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_trig_calc_cos:} % \begin{macro}{\fp_trig_calc_sin:} % \begin{macro}[aux]{\fp_trig_calc_Taylor:} % These functions actually do the calculation for sine and cosine. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_trig_calc_cos: { \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero \l_fp_output_integer_int \c_one \l_fp_output_decimal_int \c_zero \else: \l_fp_trig_sign_int \c_minus_one \fp_mul:NNNNNN \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_trig_decimal_int \l_fp_trig_extended_int \fp_div_integer:NNNNN \l_fp_trig_decimal_int \l_fp_trig_extended_int \c_two \l_fp_trig_decimal_int \l_fp_trig_extended_int \l_fp_count_int \c_three \if_int_compare:w \l_fp_trig_extended_int = \c_zero \if_int_compare:w \l_fp_trig_decimal_int = \c_zero \l_fp_output_integer_int \c_one \l_fp_output_decimal_int \c_zero \l_fp_output_extended_int \c_zero \else: \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_one_thousand_million \l_fp_output_extended_int \c_zero \fi: \else: \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int 999999999 \scan_stop: \l_fp_output_extended_int \c_one_thousand_million \fi: \tex_advance:D \l_fp_output_extended_int -\l_fp_trig_extended_int \tex_advance:D \l_fp_output_decimal_int -\l_fp_trig_decimal_int \exp_after:wN \fp_trig_calc_Taylor: \fi: } \cs_new_protected_nopar:Npn \fp_trig_calc_sin: { \l_fp_output_integer_int \c_zero \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero \l_fp_output_decimal_int \c_zero \else: \l_fp_output_decimal_int \l_fp_input_a_decimal_int \l_fp_output_extended_int \l_fp_input_a_extended_int \l_fp_trig_sign_int \c_one \l_fp_trig_decimal_int \l_fp_input_a_decimal_int \l_fp_trig_extended_int \l_fp_input_a_extended_int \l_fp_count_int \c_two \exp_after:wN \fp_trig_calc_Taylor: \fi: } % \end{macrocode} % This implements a Taylor series calculation for the trigonometric % functions. Lots of shuffling about as \TeX\ is not exactly a natural % choice for this sort of thing. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_trig_calc_Taylor: { \l_fp_trig_sign_int -\l_fp_trig_sign_int \fp_mul:NNNNNN \l_fp_trig_decimal_int \l_fp_trig_extended_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_trig_decimal_int \l_fp_trig_extended_int \fp_mul:NNNNNN \l_fp_trig_decimal_int \l_fp_trig_extended_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_trig_decimal_int \l_fp_trig_extended_int \fp_div_integer:NNNNN \l_fp_trig_decimal_int \l_fp_trig_extended_int \l_fp_count_int \l_fp_trig_decimal_int \l_fp_trig_extended_int \tex_advance:D \l_fp_count_int \c_one \fp_div_integer:NNNNN \l_fp_trig_decimal_int \l_fp_trig_extended_int \l_fp_count_int \l_fp_trig_decimal_int \l_fp_trig_extended_int \tex_advance:D \l_fp_count_int \c_one \if_int_compare:w \l_fp_trig_decimal_int > \c_zero \if_int_compare:w \l_fp_trig_sign_int > \c_zero \tex_advance:D \l_fp_output_decimal_int \l_fp_trig_decimal_int \tex_advance:D \l_fp_output_extended_int \l_fp_trig_extended_int \if_int_compare:w \l_fp_output_extended_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_output_decimal_int \c_one \tex_advance:D \l_fp_output_extended_int -\c_one_thousand_million \fi: \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_output_integer_int \c_one \tex_advance:D \l_fp_output_decimal_int -\c_one_thousand_million \fi: \else: \tex_advance:D \l_fp_output_decimal_int -\l_fp_trig_decimal_int \tex_advance:D \l_fp_output_extended_int -\l_fp_input_a_extended_int \if_int_compare:w \l_fp_output_extended_int < \c_zero \tex_advance:D \l_fp_output_decimal_int \c_minus_one \tex_advance:D \l_fp_output_extended_int \c_one_thousand_million \fi: \if_int_compare:w \l_fp_output_decimal_int < \c_zero \tex_advance:D \l_fp_output_integer_int \c_minus_one \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million \fi: \fi: \exp_after:wN \fp_trig_calc_Taylor: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_tan:Nn, \fp_tan:cn} % \UnitTested % \begin{macro}{\fp_gtan:Nn,\fp_gtan:cn} % \UnitTested % \begin{macro}[aux]{\fp_tan_aux:NNn} % \begin{macro}[aux]{\fp_tan_aux_i:} % \begin{macro}[aux]{\fp_tan_aux_ii:} % \begin{macro}[aux]{\fp_tan_aux_iii:} % \begin{macro}[aux]{\fp_tan_aux_iv:} % As might be expected, tangents are calculated from the sine and cosine % by division. So there is a bit of set up, the two subsidiary pieces % of work are done and then a division takes place. For small numbers, % the same approach is used as for sines, with the input value simply % returned as is. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_tan:Nn { \fp_tan_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gtan:Nn { \fp_tan_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_tan:Nn { c } \cs_generate_variant:Nn \fp_gtan:Nn { c } \cs_new_protected_nopar:Npn \fp_tan_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \tl_set:Nx \l_fp_arg_tl { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_input_a_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_input_a_exponent_int } \if_int_compare:w \l_fp_input_a_exponent_int < -\c_five \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \l_fp_arg_tl } } \else: \if_cs_exist:w c_fp_tan ( \l_fp_arg_tl ) _fp \cs_end: \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_i: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \use:c { c_fp_tan ( \l_fp_arg_tl ) _fp } } } \fi: \fp_tmp:w } % \end{macrocode} % The business of the calculation does not check for stored sines or % cosines as there would then be an overhead to reading them back in. % There is also no need to worry about \enquote{small} sine values as % these will have been dealt with earlier. There is a two-step lead off % so that undefined division is not even attempted. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_tan_aux_i: { \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten \exp_after:wN \fp_tan_aux_ii: \else: \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp } \c_zero_fp \exp_after:wN \fp_trig_overflow_msg: \fi: } \cs_new_protected_nopar:Npn \fp_tan_aux_ii: { \fp_trig_normalise: \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \if_int_compare:w \l_fp_trig_octant_int > \c_two \l_fp_output_sign_int \c_minus_one \else: \l_fp_output_sign_int \c_one \fi: \else: \if_int_compare:w \l_fp_trig_octant_int > \c_two \l_fp_output_sign_int \c_one \else: \l_fp_output_sign_int \c_minus_one \fi: \fi: \fp_cos_aux_ii: \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero \if_int_compare:w \l_fp_input_a_integer_int = \c_zero \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp } \c_undefined_fp \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_iii: \fi: \else: \exp_after:wN \fp_tan_aux_iii: \fi: } % \end{macrocode} % The division is done here using the same code as the standard division % unit, shifting the digits in the calculated sine and cosine to % maintain accuracy. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_tan_aux_iii: { \l_fp_input_b_integer_int \l_fp_output_decimal_int \l_fp_input_b_decimal_int \l_fp_output_extended_int \l_fp_input_b_exponent_int -\c_nine \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \fp_sin_aux_ii: \l_fp_input_a_integer_int \l_fp_output_decimal_int \l_fp_input_a_decimal_int \l_fp_output_extended_int \l_fp_input_a_exponent_int -\c_nine \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero \if_int_compare:w \l_fp_input_a_integer_int = \c_zero \cs_new_eq:cN { c_fp_tan ( \l_fp_arg_tl ) _fp } \c_zero_fp \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_tan_aux_iv: \fi: \else: \exp_after:wN \fp_tan_aux_iv: \fi: } \cs_new_protected_nopar:Npn \fp_tan_aux_iv: { \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_zero \cs_set_eq:NN \fp_div_store: \fp_div_store_integer: \l_fp_div_offset_int \c_one_hundred_million \fp_div_loop: \l_fp_output_exponent_int \int_eval:w \l_fp_input_a_exponent_int - \l_fp_input_b_exponent_int \int_eval_end: \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \tl_new:c { c_fp_tan ( \l_fp_arg_tl ) _fp } \tl_gset:cx { c_fp_tan ( \l_fp_arg_tl ) _fp } { \if_int_compare:w \l_fp_output_sign_int > \c_zero + \else: - \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Exponent and logarithm functions} % % \begin{variable}{\c_fp_exp_1_tl} % \begin{variable}{\c_fp_exp_2_tl} % \begin{variable}{\c_fp_exp_3_tl} % \begin{variable}{\c_fp_exp_4_tl} % \begin{variable}{\c_fp_exp_5_tl} % \begin{variable}{\c_fp_exp_6_tl} % \begin{variable}{\c_fp_exp_7_tl} % \begin{variable}{\c_fp_exp_8_tl} % \begin{variable}{\c_fp_exp_9_tl} % \begin{variable}{\c_fp_exp_10_tl} % \begin{variable}{\c_fp_exp_20_tl} % \begin{variable}{\c_fp_exp_30_tl} % \begin{variable}{\c_fp_exp_40_tl} % \begin{variable}{\c_fp_exp_50_tl} % \begin{variable}{\c_fp_exp_60_tl} % \begin{variable}{\c_fp_exp_70_tl} % \begin{variable}{\c_fp_exp_80_tl} % \begin{variable}{\c_fp_exp_90_tl} % \begin{variable}{\c_fp_exp_100_tl} % \begin{variable}{\c_fp_exp_200_tl} % Calculation of exponentials requires a number of precomputed values: % first the positive integers. % \begin{macrocode} \tl_const:cn { c_fp_exp_1_tl } { { 2 } { 718281828 } { 459045235 } { 0 } } \tl_const:cn { c_fp_exp_2_tl } { { 7 } { 389056098 } { 930650227 } { 0 } } \tl_const:cn { c_fp_exp_3_tl } { { 2 } { 008553692 } { 318766774 } { 1 } } \tl_const:cn { c_fp_exp_4_tl } { { 5 } { 459815003 } { 314423908 } { 1 } } \tl_const:cn { c_fp_exp_5_tl } { { 1 } { 484131591 } { 025766034 } { 2 } } \tl_const:cn { c_fp_exp_6_tl } { { 4 } { 034287934 } { 927351226 } { 2 } } \tl_const:cn { c_fp_exp_7_tl } { { 1 } { 096633158 } { 428458599 } { 3 } } \tl_const:cn { c_fp_exp_8_tl } { { 2 } { 980957987 } { 041728275 } { 3 } } \tl_const:cn { c_fp_exp_9_tl } { { 8 } { 103083927 } { 575384008 } { 3 } } \tl_const:cn { c_fp_exp_10_tl } { { 2 } { 202646579 } { 480671652 } { 4 } } \tl_const:cn { c_fp_exp_20_tl } { { 4 } { 851651954 } { 097902280 } { 8 } } \tl_const:cn { c_fp_exp_30_tl } { { 1 } { 068647458 } { 152446215 } { 13 } } \tl_const:cn { c_fp_exp_40_tl } { { 2 } { 353852668 } { 370199854 } { 17 } } \tl_const:cn { c_fp_exp_50_tl } { { 5 } { 184705528 } { 587072464 } { 21 } } \tl_const:cn { c_fp_exp_60_tl } { { 1 } { 142007389 } { 815684284 } { 26 } } \tl_const:cn { c_fp_exp_70_tl } { { 2 } { 515438670 } { 919167006 } { 30 } } \tl_const:cn { c_fp_exp_80_tl } { { 5 } { 540622384 } { 393510053 } { 34 } } \tl_const:cn { c_fp_exp_90_tl } { { 1 } { 220403294 } { 317840802 } { 39 } } \tl_const:cn { c_fp_exp_100_tl } { { 2 } { 688117141 } { 816135448 } { 43 } } \tl_const:cn { c_fp_exp_200_tl } { { 7 } { 225973768 } { 125749258 } { 86 } } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{variable}{\c_fp_exp_-1_tl} % \begin{variable}{\c_fp_exp_-2_tl} % \begin{variable}{\c_fp_exp_-3_tl} % \begin{variable}{\c_fp_exp_-4_tl} % \begin{variable}{\c_fp_exp_-5_tl} % \begin{variable}{\c_fp_exp_-6_tl} % \begin{variable}{\c_fp_exp_-7_tl} % \begin{variable}{\c_fp_exp_-8_tl} % \begin{variable}{\c_fp_exp_-9_tl} % \begin{variable}{\c_fp_exp_-10_tl} % \begin{variable}{\c_fp_exp_-20_tl} % \begin{variable}{\c_fp_exp_-30_tl} % \begin{variable}{\c_fp_exp_-40_tl} % \begin{variable}{\c_fp_exp_-50_tl} % \begin{variable}{\c_fp_exp_-60_tl} % \begin{variable}{\c_fp_exp_-70_tl} % \begin{variable}{\c_fp_exp_-80_tl} % \begin{variable}{\c_fp_exp_-90_tl} % \begin{variable}{\c_fp_exp_-100_tl} % \begin{variable}{\c_fp_exp_-200_tl} % Now the negative integers. % \begin{macrocode} \tl_const:cn { c_fp_exp_-1_tl } { { 3 } { 678794411 } { 71442322 } { -1 } } \tl_const:cn { c_fp_exp_-2_tl } { { 1 } { 353352832 } { 366132692 } { -1 } } \tl_const:cn { c_fp_exp_-3_tl } { { 4 } { 978706836 } { 786394298 } { -2 } } \tl_const:cn { c_fp_exp_-4_tl } { { 1 } { 831563888 } { 873418029 } { -2 } } \tl_const:cn { c_fp_exp_-5_tl } { { 6 } { 737946999 } { 085467097 } { -3 } } \tl_const:cn { c_fp_exp_-6_tl } { { 2 } { 478752176 } { 666358423 } { -3 } } \tl_const:cn { c_fp_exp_-7_tl } { { 9 } { 118819655 } { 545162080 } { -4 } } \tl_const:cn { c_fp_exp_-8_tl } { { 3 } { 354626279 } { 025118388 } { -4 } } \tl_const:cn { c_fp_exp_-9_tl } { { 1 } { 234098040 } { 866795495 } { -4 } } \tl_const:cn { c_fp_exp_-10_tl } { { 4 } { 539992976 } { 248451536 } { -5 } } \tl_const:cn { c_fp_exp_-20_tl } { { 2 } { 061153622 } { 438557828 } { -9 } } \tl_const:cn { c_fp_exp_-30_tl } { { 9 } { 357622968 } { 840174605 } { -14 } } \tl_const:cn { c_fp_exp_-40_tl } { { 4 } { 248354255 } { 291588995 } { -18 } } \tl_const:cn { c_fp_exp_-50_tl } { { 1 } { 928749847 } { 963917783 } { -22 } } \tl_const:cn { c_fp_exp_-60_tl } { { 8 } { 756510762 } { 696520338 } { -27 } } \tl_const:cn { c_fp_exp_-70_tl } { { 3 } { 975449735 } { 908646808 } { -31 } } \tl_const:cn { c_fp_exp_-80_tl } { { 1 } { 804851387 } { 845415172 } { -35 } } \tl_const:cn { c_fp_exp_-90_tl } { { 8 } { 194012623 } { 990515430 } { -40 } } \tl_const:cn { c_fp_exp_-100_tl } { { 3 } { 720075976 } { 020835963 } { -44 } } \tl_const:cn { c_fp_exp_-200_tl } { { 1 } { 383896526 } { 736737530 } { -87 } } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % % \begin{macro}{\fp_exp:Nn, \fp_exp:cn} % \UnitTested % \begin{macro}{\fp_gexp:Nn,\fp_gexp:cn} % \UnitTested % \begin{macro}[aux]{\fp_exp_aux:NNn} % \begin{macro}[aux]{\fp_exp_internal:} % \begin{macro}[aux]{\fp_exp_aux:} % \begin{macro}[aux]{\fp_exp_integer:} % \begin{macro}[aux]{\fp_exp_integer_tens:} % \begin{macro}[aux]{\fp_exp_integer_units:} % \begin{macro}[aux]{\fp_exp_integer_const:n} % \begin{macro}[aux]{\fp_exp_integer_const:nnnn} % \begin{macro}[aux]{\fp_exp_decimal:} % \begin{macro}[aux]{\fp_exp_Taylor:} % \begin{macro}[aux]{\fp_exp_const:Nx} % \begin{macro}[aux]{\fp_exp_const:cx} % The calculation of an exponent starts off starts in much the same % way as the trigonometric functions: normalise the input, look for % a pre-defined value and if one is not found hand off to the real % workhorse function. The test for a definition of the result is used % so that overflows do not result in any outcome being defined. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp:Nn { \fp_exp_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gexp:Nn { \fp_exp_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_exp:Nn { c } \cs_generate_variant:Nn \fp_gexp:Nn { c } \cs_new_protected_nopar:Npn \fp_exp_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \l_fp_input_a_extended_int \c_zero \tl_set:Nx \l_fp_arg_tl { \if_int_compare:w \l_fp_input_a_sign_int < \c_zero - \else: + \fi: \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_input_a_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_input_a_exponent_int } \if_cs_exist:w c_fp_exp ( \l_fp_arg_tl ) _fp \cs_end: \else: \exp_after:wN \fp_exp_internal: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: #1 \exp_not:N #2 { \if_cs_exist:w c_fp_exp ( \l_fp_arg_tl ) _fp \cs_end: \use:c { c_fp_exp ( \l_fp_arg_tl ) _fp } \else: \c_zero_fp \fi: } } \fp_tmp:w } % \end{macrocode} % The first real step is to convert the input into a fixed-point % representation for further calculation: anything which is dropped % here as too small would not influence the output in any case. There % are a couple of overflow tests: the maximum % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_internal: { \if_int_compare:w \l_fp_input_a_exponent_int < \c_three \fp_extended_normalise: \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \if_int_compare:w \l_fp_input_a_integer_int < 230 \scan_stop: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_aux: \else: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_overflow_msg: \fi: \else: \if_int_compare:w \l_fp_input_a_integer_int < 230 \scan_stop: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_aux: \else: \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp } { \c_zero_fp } \fi: \fi: \else: \exp_after:wN \fp_exp_overflow_msg: \fi: } % \end{macrocode} % The main algorithm makes use of the fact that % \[ % \mathrm{e}^{nmp.q} = % \mathrm{e}^{n} % \mathrm{e}^{m} % \mathrm{e}^{p} % \mathrm{e}^{0.q} % \] % and that there is a Taylor series that can be used to calculate % $ \mathrm{e}^{0.q} $. Thus the approach needed is in three parts. % First, the exponent of the integer part of the input is found % using the pre-calculated constants. Second, the Taylor series is % used to find the exponent for the decimal part of the input. Finally, % the two parts are multiplied together to give the result. As the % normalisation code will already have dealt with any overflowing % values, there are no further checks needed. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_aux: { \if_int_compare:w \l_fp_input_a_integer_int > \c_zero \exp_after:wN \fp_exp_integer: \else: \l_fp_output_integer_int \c_one \l_fp_output_decimal_int \c_zero \l_fp_output_extended_int \c_zero \l_fp_output_exponent_int \c_zero \exp_after:wN \fp_exp_decimal: \fi: } % \end{macrocode} % The integer part calculation starts with the hundreds. This is % set up such that very large negative numbers can short-cut the entire % procedure and simply return zero. In other cases, the code either % recovers the exponent of the hundreds value or sets the appropriate % storage to one (so that multiplication works correctly). % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_integer: { \if_int_compare:w \l_fp_input_a_integer_int < \c_one_hundred \l_fp_exp_integer_int \c_one \l_fp_exp_decimal_int \c_zero \l_fp_exp_extended_int \c_zero \l_fp_exp_exponent_int \c_zero \exp_after:wN \fp_exp_integer_tens: \else: \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \use_i:nnn \int_use:N \l_fp_input_a_integer_int } \l_fp_input_a_integer_int \int_eval:w \l_fp_input_a_integer_int - \l_fp_tmp_tl 00 \int_eval_end: \if_int_compare:w \l_fp_input_a_sign_int < \c_zero \if_int_compare:w \l_fp_output_integer_int > 200 \scan_stop: \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp } { \c_zero_fp } \else: \fp_exp_integer_const:n { - \l_fp_tmp_tl 00 } \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_integer_tens: \fi: \else: \fp_exp_integer_const:n { \l_fp_tmp_tl 00 } \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_integer_tens: \fi: \fi: } % \end{macrocode} % The tens and units parts are handled in a similar way, with a % multiplication step to build up the final value. That also includes a % correction step to avoid an overflow of the integer part. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_integer_tens: { \l_fp_output_integer_int \l_fp_exp_integer_int \l_fp_output_decimal_int \l_fp_exp_decimal_int \l_fp_output_extended_int \l_fp_exp_extended_int \l_fp_output_exponent_int \l_fp_exp_exponent_int \if_int_compare:w \l_fp_input_a_integer_int > \c_nine \tl_set:Nx \l_fp_tmp_tl { \exp_after:wN \use_i:nn \int_use:N \l_fp_input_a_integer_int } \l_fp_input_a_integer_int \int_eval:w \l_fp_input_a_integer_int - \l_fp_tmp_tl 0 \int_eval_end: \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \fp_exp_integer_const:n { \l_fp_tmp_tl 0 } \else: \fp_exp_integer_const:n { - \l_fp_tmp_tl 0 } \fi: \fp_mul:NNNNNNNNN \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \tex_advance:D \l_fp_output_exponent_int \l_fp_exp_exponent_int \fp_extended_normalise_output: \fi: \fp_exp_integer_units: } \cs_new_protected_nopar:Npn \fp_exp_integer_units: { \if_int_compare:w \l_fp_input_a_integer_int > \c_zero \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \fp_exp_integer_const:n { \int_use:N \l_fp_input_a_integer_int } \else: \fp_exp_integer_const:n { - \int_use:N \l_fp_input_a_integer_int } \fi: \fp_mul:NNNNNNNNN \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \tex_advance:D \l_fp_output_exponent_int \l_fp_exp_exponent_int \fp_extended_normalise_output: \fi: \fp_exp_decimal: } % \end{macrocode} % Recovery of the stored constant values into the separate registers % is done with a simple expansion then assignment. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_integer_const:n #1 { \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_integer_const:nnnn \cs:w c_fp_exp_ #1 _tl \cs_end: } \cs_new_protected_nopar:Npn \fp_exp_integer_const:nnnn #1#2#3#4 { \l_fp_exp_integer_int #1 \scan_stop: \l_fp_exp_decimal_int #2 \scan_stop: \l_fp_exp_extended_int #3 \scan_stop: \l_fp_exp_exponent_int #4 \scan_stop: } % \end{macrocode} % Finding the exponential for the decimal part of the number requires % a Taylor series calculation. The set up is done here with the loop % itself a separate function. Once the decimal part is available this % is multiplied by the integer part already worked out to give % the final result. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_decimal: { \if_int_compare:w \l_fp_input_a_decimal_int > \c_zero \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \l_fp_exp_integer_int \c_one \l_fp_exp_decimal_int \l_fp_input_a_decimal_int \l_fp_exp_extended_int \l_fp_input_a_extended_int \else: \l_fp_exp_integer_int \c_zero \if_int_compare:w \l_fp_exp_extended_int = \c_zero \l_fp_exp_decimal_int \int_eval:w \c_one_thousand_million - \l_fp_input_a_decimal_int \int_eval_end: \l_fp_exp_extended_int \c_zero \else: \l_fp_exp_decimal_int \int_eval:w 999999999 - \l_fp_input_a_decimal_int \scan_stop: \l_fp_exp_extended_int \int_eval:w \c_one_thousand_million - \l_fp_input_a_extended_int \int_eval_end: \fi: \fi: \l_fp_input_b_sign_int \l_fp_input_a_sign_int \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int \l_fp_input_b_extended_int \l_fp_input_a_extended_int \l_fp_count_int \c_one \fp_exp_Taylor: \fp_mul:NNNNNNNNN \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \fi: \if_int_compare:w \l_fp_output_extended_int < \c_five_hundred_million \else: \tex_advance:D \l_fp_output_decimal_int \c_one \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million \else: \l_fp_output_decimal_int \c_zero \tex_advance:D \l_fp_output_integer_int \c_one \fi: \fi: \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \fp_exp_const:cx { c_fp_exp ( \l_fp_arg_tl ) _fp } { + \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } % \end{macrocode} % The Taylor series for $ \exp(x) $ is % \[ % 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots % \] % which converges for $ -1 < x < 1 $. The code above sets up % the $ x $ part, leaving the loop to multiply the running % value by $ x / n $ and add it onto the sum. The way that this is % done is that the running total is stored in the \texttt{exp} set of % registers, while the current item is stored as \texttt{input_b}. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_Taylor: { \tex_advance:D \l_fp_count_int \c_one \tex_multiply:D \l_fp_input_b_sign_int \l_fp_input_a_sign_int \fp_mul:NNNNNN \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \fp_div_integer:NNNNN \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \l_fp_count_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \if_int_compare:w \int_eval:w \l_fp_input_b_decimal_int + \l_fp_input_b_extended_int > \c_zero \if_int_compare:w \l_fp_input_b_sign_int > \c_zero \tex_advance:D \l_fp_exp_decimal_int \l_fp_input_b_decimal_int \tex_advance:D \l_fp_exp_extended_int \l_fp_input_b_extended_int \if_int_compare:w \l_fp_exp_extended_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_exp_decimal_int \c_one \tex_advance:D \l_fp_exp_extended_int -\c_one_thousand_million \fi: \if_int_compare:w \l_fp_exp_decimal_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_exp_integer_int \c_one \tex_advance:D \l_fp_exp_decimal_int -\c_one_thousand_million \fi: \else: \tex_advance:D \l_fp_exp_decimal_int -\l_fp_input_b_decimal_int \tex_advance:D \l_fp_exp_extended_int -\l_fp_input_a_extended_int \if_int_compare:w \l_fp_exp_extended_int < \c_zero \tex_advance:D \l_fp_exp_decimal_int \c_minus_one \tex_advance:D \l_fp_exp_extended_int \c_one_thousand_million \fi: \if_int_compare:w \l_fp_exp_decimal_int < \c_zero \tex_advance:D \l_fp_exp_integer_int \c_minus_one \tex_advance:D \l_fp_exp_decimal_int \c_one_thousand_million \fi: \fi: \exp_after:wN \fp_exp_Taylor: \fi: } % \end{macrocode} % This is set up as a function so that the power code can redirect % the effect. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_exp_const:Nx #1#2 { \tl_new:N #1 \tl_gset:Nx #1 {#2} } \cs_generate_variant:Nn \fp_exp_const:Nx { c } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{variable}{\c_fp_ln_10_1_tl} % \begin{variable}{\c_fp_ln_10_2_tl} % \begin{variable}{\c_fp_ln_10_3_tl} % \begin{variable}{\c_fp_ln_10_4_tl} % \begin{variable}{\c_fp_ln_10_5_tl} % \begin{variable}{\c_fp_ln_10_6_tl} % \begin{variable}{\c_fp_ln_10_7_tl} % \begin{variable}{\c_fp_ln_10_8_tl} % \begin{variable}{\c_fp_ln_10_9_tl} % Constants for working out logarithms: first those for the powers of % ten. % \begin{macrocode} \tl_const:cn { c_fp_ln_10_1_tl } { { 2 } { 302585092 } { 994045684 } { 0 } } \tl_const:cn { c_fp_ln_10_2_tl } { { 4 } { 605170185 } { 988091368 } { 0 } } \tl_const:cn { c_fp_ln_10_3_tl } { { 6 } { 907755278 } { 982137052 } { 0 } } \tl_const:cn { c_fp_ln_10_4_tl } { { 9 } { 210340371 } { 976182736 } { 0 } } \tl_const:cn { c_fp_ln_10_5_tl } { { 1 } { 151292546 } { 497022842 } { 1 } } \tl_const:cn { c_fp_ln_10_6_tl } { { 1 } { 381551055 } { 796427410 } { 1 } } \tl_const:cn { c_fp_ln_10_7_tl } { { 1 } { 611809565 } { 095831979 } { 1 } } \tl_const:cn { c_fp_ln_10_8_tl } { { 1 } { 842068074 } { 395226547 } { 1 } } \tl_const:cn { c_fp_ln_10_9_tl } { { 2 } { 072326583 } { 694641116 } { 1 } } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} % \end{variable} %^^A 0.69...309 4. % \begin{variable}{\c_fp_ln_2_1_tl } % \begin{variable}{\c_fp_ln_2_2_tl } % \begin{variable}{\c_fp_ln_2_3_tl } % The smaller set for powers of two. % \begin{macrocode} \tl_const:cn { c_fp_ln_2_1_tl } { { 0 } { 693147180 } { 559945309 } { 0 } } \tl_const:cn { c_fp_ln_2_2_tl } { { 1 } { 386294361 } { 119890618 } { 0 } } \tl_const:cn { c_fp_ln_2_3_tl } { { 2 } { 079441541 } { 679835928 } { 0 } } % \end{macrocode} % \end{variable} % \end{variable} % \end{variable} % % \begin{macro}{\fp_ln:Nn, \fp_ln:cn} % \UnitTested % \begin{macro}{\fp_gln:Nn,\fp_gln:cn} % \UnitTested % \begin{macro}[aux]{\fp_ln_aux:NNn} % \begin{macro}[aux]{\fp_ln_aux:} % \begin{macro}[aux]{\fp_ln_exponent:} % \begin{macro}[aux]{\fp_ln_internal:} % \begin{macro}[aux]{\fp_ln_exponent_tens:} % \begin{macro}[aux]{\fp_ln_exponent_units:} % \begin{macro}[aux]{\fp_ln_normalise:} % \begin{macro}[aux]{\fp_ln_nornalise_aux:NNNNNNNNN} % \begin{macro}[aux]{\fp_ln_mantissa:} % \begin{macro}[aux]{\fp_ln_mantissa_aux:} % \begin{macro}[aux]{\fp_ln_mantissa_divide_two:} % \begin{macro}[aux]{\fp_ln_integer_const:nn} % \begin{macro}[aux]{\fp_ln_Taylor:} % \begin{macro}[aux]{\fp_ln_fixed:} % \begin{macro}[aux]{\fp_ln_fixed_aux:NNNNNNNNN} % \begin{macro}[aux]{\fp_ln_Taylor_aux:} % The approach for logarithms is again based on a mix of tables and % Taylor series. Here, the initial validation is a bit easier and so it % is set up earlier, meaning less need to escape later on. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln:Nn { \fp_ln_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gln:Nn { \fp_ln_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_ln:Nn { c } \cs_generate_variant:Nn \fp_gln:Nn { c } \cs_new_protected_nopar:Npn \fp_ln_aux:NNn #1#2#3 { \group_begin: \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int > \c_zero \exp_after:wN \exp_after:wN \exp_after:wN \fp_ln_aux: \else: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 \exp_not:N ##2 { \c_zero_fp } } \exp_after:wN \exp_after:wN \exp_after:wN \fp_ln_error_msg: \fi: \else: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 \exp_not:N ##2 { \c_zero_fp } } \exp_after:wN \fp_ln_error_msg: \fi: \fp_tmp:w #1 #2 } % \end{macrocode} % As the input at this stage meets the validity criteria above, the % argument can now be saved for further processing. There is no need % to look at the sign of the input as it must be positive. The function % here simply sets up to either do the full calculation or recover % the stored value, as appropriate. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_aux: { \tl_set:Nx \l_fp_arg_tl { + \int_use:N \l_fp_input_a_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_input_a_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_input_a_exponent_int } \if_cs_exist:w c_fp_ln ( \l_fp_arg_tl ) _fp \cs_end: \else: \exp_after:wN \fp_ln_exponent: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 \exp_not:N ##2 { \use:c { c_fp_ln ( \l_fp_arg_tl ) _fp } } } } % \end{macrocode} % The main algorithm here uses the fact the logarithm can be divided % up, first taking out the powers of ten, then powers of two and finally % using a Taylor series for the remainder. % \[ % \ln ( 10^{n} \times 2^{m} \times x ) % = \ln ( 10^{n} ) + \ln ( 2^{m} ) + \ln ( x ) % \] % The second point to remember is that % \[ % \ln ( x^{-1} ) = - \ln ( x ) % \] % which means that for the powers of $ 10 $ and $ 2 $ constants % are only needed for positive powers. % % The first step is to set up the sign for the output functions and % work out the powers of ten in the exponent. First the larger powers % are sorted out. The values for the constants are the same as those % for the smaller ones, just with a shift in the exponent. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_exponent: { \fp_ln_internal: \if_int_compare:w \l_fp_output_extended_int < \c_five_hundred_million \else: \tex_advance:D \l_fp_output_decimal_int \c_one \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million \else: \l_fp_output_decimal_int \c_zero \tex_advance:D \l_fp_output_integer_int \c_one \fi: \fi: \fp_standardise:NNNN \l_fp_output_sign_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_exponent_int \tl_const:cx { c_fp_ln ( \l_fp_arg_tl ) _fp } { \if_int_compare:w \l_fp_output_sign_int > \c_zero + \else: - \fi: \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million \scan_stop: e \int_use:N \l_fp_output_exponent_int } } \cs_new_protected_nopar:Npn \fp_ln_internal: { \if_int_compare:w \l_fp_input_a_exponent_int < \c_zero \l_fp_input_a_exponent_int -\l_fp_input_a_exponent_int \l_fp_output_sign_int \c_minus_one \else: \l_fp_output_sign_int \c_one \fi: \if_int_compare:w \l_fp_input_a_exponent_int > \c_nine \exp_after:wN \fp_ln_exponent_tens:NN \int_use:N \l_fp_input_a_exponent_int \else: \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_zero \l_fp_output_extended_int \c_zero \l_fp_output_exponent_int \c_zero \fi: \fp_ln_exponent_units: } \cs_new_protected_nopar:Npn \fp_ln_exponent_tens:NN #1 #2 { \l_fp_input_a_exponent_int #2 \scan_stop: \fp_ln_const:nn { 10 } { #1 } \tex_advance:D \l_fp_exp_exponent_int \c_one \l_fp_output_integer_int \l_fp_exp_integer_int \l_fp_output_decimal_int \l_fp_exp_decimal_int \l_fp_output_extended_int \l_fp_exp_extended_int \l_fp_output_exponent_int \l_fp_exp_exponent_int } % \end{macrocode} % Next the smaller powers of ten, which will need to be combined % with the above: always an additive process. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_exponent_units: { \if_int_compare:w \l_fp_input_a_exponent_int > \c_zero \fp_ln_const:nn { 10 } { \int_use:N \l_fp_input_a_exponent_int } \fp_ln_normalise: \fp_add:NNNNNNNNN \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \fi: \fp_ln_mantissa: } % \end{macrocode} % The smaller table-based parts may need to be exponent shifted so that % they stay in line with the larger parts. This is similar to the % approach in other places, but here there is a need to watch the % extended part of the number. The only case where the new exponent is % larger than the old is if there was no previous part. Then simply set % the exponent. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_normalise: { \if_int_compare:w \l_fp_exp_exponent_int < \l_fp_output_exponent_int \tex_advance:D \l_fp_exp_decimal_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_ln_normalise_aux:NNNNNNNNN \int_use:N \l_fp_exp_decimal_int \exp_after:wN \fp_ln_normalise: \else: \l_fp_output_exponent_int \l_fp_exp_exponent_int \fi: } \cs_new_protected_nopar:Npn \fp_ln_normalise_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \if_int_compare:w \l_fp_exp_integer_int = \c_zero \l_fp_exp_decimal_int #1#2#3#4#5#6#7#8 \scan_stop: \else: \tl_set:Nx \l_fp_tmp_tl { \int_use:N \l_fp_exp_integer_int #1#2#3#4#5#6#7#8 } \l_fp_exp_integer_int \c_zero \l_fp_exp_decimal_int \l_fp_tmp_tl \scan_stop: \fi: \tex_divide:D \l_fp_exp_extended_int \c_ten \tl_set:Nx \l_fp_tmp_tl { #9 \int_use:N \l_fp_exp_extended_int } \l_fp_exp_extended_int \l_fp_tmp_tl \scan_stop: \tex_advance:D \l_fp_exp_exponent_int \c_one } % \end{macrocode} % The next phase is to decompose the mantissa by division by two to % leave a value which is in the range $ 1 \le x < 2 $. The sum of the % two powers needs to take account of the sign of the output: if it % is negative then the result gets \emph{smaller} as the mantissa gets % \emph{bigger}. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_mantissa: { \l_fp_count_int \c_zero \l_fp_input_a_extended_int \c_zero \fp_ln_mantissa_aux: \if_int_compare:w \l_fp_count_int > \c_zero \fp_ln_const:nn { 2 } { \int_use:N \l_fp_count_int } \fp_ln_normalise: \if_int_compare:w \l_fp_output_sign_int > \c_zero \exp_after:wN \fp_add:NNNNNNNNN \else: \exp_after:wN \fp_sub:NNNNNNNNN \fi: \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_exp_integer_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \fi: \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int > \c_one \exp_after:wN \fp_ln_Taylor: \fi: } \cs_new_protected_nopar:Npn \fp_ln_mantissa_aux: { \if_int_compare:w \l_fp_input_a_integer_int > \c_one \tex_advance:D \l_fp_count_int \c_one \fp_ln_mantissa_divide_two: \exp_after:wN \fp_ln_mantissa_aux: \fi: } % \end{macrocode} % A fast one-shot division by two. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_mantissa_divide_two: { \if_int_odd:w \l_fp_input_a_decimal_int \tex_advance:D \l_fp_input_a_extended_int \c_one_thousand_million \fi: \if_int_odd:w \l_fp_input_a_integer_int \tex_advance:D \l_fp_input_a_decimal_int \c_one_thousand_million \fi: \tex_divide:D \l_fp_input_a_integer_int \c_two \tex_divide:D \l_fp_input_a_decimal_int \c_two \tex_divide:D \l_fp_input_a_extended_int \c_two } % \end{macrocode} % Recovering constants makes use of the same auxiliary code as for % exponents. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_const:nn #1#2 { \exp_after:wN \exp_after:wN \exp_after:wN \fp_exp_integer_const:nnnn \cs:w c_fp_ln_ #1 _ #2 _tl \cs_end: } % \end{macrocode} % The Taylor series for the logarithm function is best implemented using % the identity % \[ % \ln(x) = \ln\left( \frac{y + 1}{y - 1} \right) % \] % with % \[ % y = \frac{x - 1}{x + 1} % \] % This leads to the series % \[ % \ln(x) % = 2y % \left( % 1 + y^{2} % \left( % \frac{1}{3} + y^{2} % \left( % \frac{1}{5} + y^{2} % \left( % \frac{1}{7} + y^{2} % \left( % \frac{1}{9} + \cdots % \right) % \right) % \right) % \right) % \right) % \] % This expansion has the advantage that a lot of the work can be % loaded up early by finding $ y^{2} $ before the loop itself starts. % (In practice, the implementation does the multiplication by two at the % end of the loop, and expands out the brackets as this is an overall % more efficient approach.) % % At the implementation level, the code starts by calculating $ y $ % and storing that in input \texttt{a} (which is no longer needed % for other purposes). That is done using the full division system % avoiding the parsing step. The value is then switched to a fixed-point % representation. There is then some shuffling to get all of the working % space set up. At this stage, a lot of registers are in use and so % the Taylor series is calculated within a group so that the % \texttt{output} variables can be used to hold the result. The value % of $ y^{2} $ is held in input \texttt{b} (there are a few % assignments saved by choosing this over \texttt{a}), while input % \texttt{a} is used for the \enquote{loop value}. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_Taylor: { \group_begin: \l_fp_input_a_integer_int \c_zero \l_fp_input_a_exponent_int \c_zero \l_fp_input_b_integer_int \c_two \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int \l_fp_input_b_exponent_int \c_zero \fp_div_internal: \fp_ln_fixed: \l_fp_input_a_integer_int \l_fp_output_integer_int \l_fp_input_a_decimal_int \l_fp_output_decimal_int \l_fp_input_a_extended_int \c_zero \l_fp_input_a_exponent_int \l_fp_output_exponent_int \l_fp_output_decimal_int \c_zero %^^A Bug? \l_fp_output_decimal_int \l_fp_input_a_decimal_int \l_fp_output_extended_int \l_fp_input_a_extended_int \fp_mul:NNNNNN \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \l_fp_count_int \c_one \fp_ln_Taylor_aux: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: \l_fp_exp_integer_int \c_zero \exp_not:N \l_fp_exp_decimal_int \int_use:N \l_fp_output_decimal_int \scan_stop: \exp_not:N \l_fp_exp_extended_int \int_use:N \l_fp_output_extended_int \scan_stop: \exp_not:N \l_fp_exp_exponent_int \int_use:N \l_fp_output_exponent_int \scan_stop: } \fp_tmp:w % \end{macrocode} % After the loop part of the Taylor series, the factor of $ 2 $ needs % to be included. The total for the result can then be constructed. % \begin{macrocode} \tex_advance:D \l_fp_exp_decimal_int \l_fp_exp_decimal_int \if_int_compare:w \l_fp_exp_extended_int < \c_five_hundred_million \else: \tex_advance:D \l_fp_exp_extended_int -\c_five_hundred_million \tex_advance:D \l_fp_exp_decimal_int \c_one \fi: \tex_advance:D \l_fp_exp_extended_int \l_fp_exp_extended_int \fp_ln_normalise: \if_int_compare:w \l_fp_output_sign_int > \c_zero \exp_after:wN \fp_add:NNNNNNNNN \else: \exp_after:wN \fp_sub:NNNNNNNNN \fi: \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \c_zero \l_fp_exp_decimal_int \l_fp_exp_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int } % \end{macrocode} % The usual shifts to move to fixed-point working. This is done using % the \texttt{output} registers as this saves a reassignment here. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_fixed: { \if_int_compare:w \l_fp_output_exponent_int < \c_zero \tex_advance:D \l_fp_output_decimal_int \c_one_thousand_million \exp_after:wN \use_i:nn \exp_after:wN \fp_ln_fixed_aux:NNNNNNNNN \int_use:N \l_fp_output_decimal_int \exp_after:wN \fp_ln_fixed: \fi: } \cs_new_protected_nopar:Npn \fp_ln_fixed_aux:NNNNNNNNN #1#2#3#4#5#6#7#8#9 { \if_int_compare:w \l_fp_output_integer_int = \c_zero \l_fp_output_decimal_int #1#2#3#4#5#6#7#8 \scan_stop: \else: \tl_set:Nx \l_fp_tmp_tl { \int_use:N \l_fp_output_integer_int #1#2#3#4#5#6#7#8 } \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \l_fp_tmp_tl \scan_stop: \fi: \tex_advance:D \l_fp_output_exponent_int \c_one } % \end{macrocode} % The main loop for the Taylor series: unlike some of the other similar % functions, the result here is not the final value and is therefore % subject to further manipulation outside of the loop. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_ln_Taylor_aux: { \tex_advance:D \l_fp_count_int \c_two \fp_mul:NNNNNN \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \if_int_compare:w \int_eval:w \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int > \c_zero \fp_div_integer:NNNNN \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_count_int \l_fp_exp_decimal_int \l_fp_exp_extended_int \tex_advance:D \l_fp_output_decimal_int \l_fp_exp_decimal_int \tex_advance:D \l_fp_output_extended_int \l_fp_exp_extended_int \if_int_compare:w \l_fp_output_extended_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_output_decimal_int \c_one \tex_advance:D \l_fp_output_extended_int -\c_one_thousand_million \fi: \if_int_compare:w \l_fp_output_decimal_int < \c_one_thousand_million \else: \tex_advance:D \l_fp_output_integer_int \c_one \tex_advance:D \l_fp_output_decimal_int -\c_one_thousand_million \fi: \exp_after:wN \fp_ln_Taylor_aux: \fi: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}{\fp_pow:Nn, \fp_pow:cn} % \UnitTested % \begin{macro}{\fp_gpow:Nn,\fp_gpow:cn} % \UnitTested % \begin{macro}[aux]{\fp_pow_aux:NNn} % \begin{macro}[aux]{\fp_pow_aux_i:} % \begin{macro}[aux]{\fp_pow_positive:} % \begin{macro}[aux]{\fp_pow_negative:} % \begin{macro}[aux]{\fp_pow_aux_ii:} % \begin{macro}[aux]{\fp_pow_aux_iii:} % \begin{macro}[aux]{\fp_pow_aux_iv:} % The approach used for working out powers is to first filter out the % various special cases and then do most of the work using the % logarithm and exponent functions. The two storage areas are used % in the reverse of the `natural' logic as this avoids some % re-assignment in the sanity checking code. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_pow:Nn { \fp_pow_aux:NNn \tl_set:Nn } \cs_new_protected_nopar:Npn \fp_gpow:Nn { \fp_pow_aux:NNn \tl_gset:Nn } \cs_generate_variant:Nn \fp_pow:Nn { c } \cs_generate_variant:Nn \fp_gpow:Nn { c } \cs_new_protected_nopar:Npn \fp_pow_aux:NNn #1#2#3 { \group_begin: \fp_read:N #2 \l_fp_input_b_sign_int \l_fp_input_a_sign_int \l_fp_input_b_integer_int \l_fp_input_a_integer_int \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int \l_fp_input_b_exponent_int \l_fp_input_a_exponent_int \fp_split:Nn a {#3} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \if_int_compare:w \int_eval:w \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int = \c_zero \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int = \c_zero \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \c_undefined_fp } } \else: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \c_zero_fp } } \fi: \else: \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int = \c_zero \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \c_one_fp } } \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_i: \fi: \fi: \fp_tmp:w #1 #2 } % \end{macrocode} % Simply using the logarithm function directly will fail when negative % numbers are raised to integer powers, which is a mathematically valid % operation. So there are some more tests to make, after forcing the % power into an integer and decimal parts, if necessary. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_pow_aux_i: { \if_int_compare:w \l_fp_input_b_sign_int > \c_zero \tl_set:Nn \l_fp_sign_tl { + } \exp_after:wN \fp_pow_aux_ii: \else: \l_fp_input_a_extended_int \c_zero \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten \group_begin: \fp_extended_normalise: \if_int_compare:w \int_eval:w \l_fp_input_a_decimal_int + \l_fp_input_a_extended_int = \c_zero \group_end: \tl_set:Nn \l_fp_sign_tl { - } \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_ii: \else: \group_end: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \c_undefined_fp } } \fi: \else: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \c_undefined_fp } } \fi: \fi: } % \end{macrocode} % The approach used here for powers works well in most cases but gives % poorer results for negative integer powers, which often have exact % values. So there is some filtering to do. For negative powers where % the power is small, an alternative approach is used in which the % positive value is worked out and the reciprocal is then taken. The % filtering is unfortunately rather long. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_pow_aux_ii: { \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \exp_after:wN \fp_pow_aux_iv: \else: \if_int_compare:w \l_fp_input_a_exponent_int < \c_ten \group_begin: \l_fp_input_a_extended_int \c_zero \fp_extended_normalise: \if_int_compare:w \l_fp_input_a_decimal_int = \c_zero \if_int_compare:w \l_fp_input_a_integer_int > \c_ten \group_end: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_iv: \else: \group_end: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_iii: \fi: \else: \group_end: \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_iv: \fi: \else: \exp_after:wN \exp_after:wN \exp_after:wN \fp_pow_aux_iv: \fi: \fi: \cs_set_protected_nopar:Npx \fp_tmp:w ##1##2 { \group_end: ##1 ##2 { \l_fp_sign_tl \int_use:N \l_fp_output_integer_int . \exp_after:wN \use_none:n \int_value:w \int_eval:w \l_fp_output_decimal_int + \c_one_thousand_million e \int_use:N \l_fp_output_exponent_int } } } % \end{macrocode} % For the small negative integer powers, the calculation is done for % the positive power and the reciprocal is then taken. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_pow_aux_iii: { \l_fp_input_a_sign_int \c_one \fp_pow_aux_iv: \l_fp_input_a_integer_int \c_one \l_fp_input_a_decimal_int \c_zero \l_fp_input_a_exponent_int \c_zero \l_fp_input_b_integer_int \l_fp_output_integer_int \l_fp_input_b_decimal_int \l_fp_output_decimal_int \l_fp_input_b_exponent_int \l_fp_output_exponent_int \fp_div_internal: } % \end{macrocode} % The business end of the code starts by finding the logarithm of the % given base. There is a bit of a shuffle so that this does not have % to be re-parsed and so that the output ends up in the correct place. % There is also a need to enable using the short-cut for a % pre-calculated result. The internal part of the multiplication % function can then be used to do the second part of the calculation % directly. There is some more set up before doing the exponential: % the idea here is to deactivate some internals so that everything works % smoothly. % \begin{macrocode} \cs_new_protected_nopar:Npn \fp_pow_aux_iv: { \group_begin: \l_fp_input_a_integer_int \l_fp_input_b_integer_int \l_fp_input_a_decimal_int \l_fp_input_b_decimal_int \l_fp_input_a_exponent_int \l_fp_input_b_exponent_int \fp_ln_internal: \cs_set_protected_nopar:Npx \fp_tmp:w { \group_end: \exp_not:N \l_fp_input_b_sign_int \int_use:N \l_fp_output_sign_int \scan_stop: \exp_not:N \l_fp_input_b_integer_int \int_use:N \l_fp_output_integer_int \scan_stop: \exp_not:N \l_fp_input_b_decimal_int \int_use:N \l_fp_output_decimal_int \scan_stop: \exp_not:N \l_fp_input_b_extended_int \int_use:N \l_fp_output_extended_int \scan_stop: \exp_not:N \l_fp_input_b_exponent_int \int_use:N \l_fp_output_exponent_int \scan_stop: } \fp_tmp:w \l_fp_input_a_extended_int \c_zero \fp_mul:NNNNNNNNN \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_extended_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_extended_int \l_fp_output_integer_int \l_fp_output_decimal_int \l_fp_output_extended_int \l_fp_output_exponent_int \int_eval:w \l_fp_input_a_exponent_int + \l_fp_input_b_exponent_int \scan_stop: \fp_extended_normalise_output: \tex_multiply:D \l_fp_input_a_sign_int \l_fp_input_b_sign_int \l_fp_input_a_integer_int \l_fp_output_integer_int \l_fp_input_a_decimal_int \l_fp_output_decimal_int \l_fp_input_a_extended_int \l_fp_output_extended_int \l_fp_input_a_exponent_int \l_fp_output_exponent_int \l_fp_output_integer_int \c_zero \l_fp_output_decimal_int \c_zero \l_fp_output_extended_int \c_zero \l_fp_output_exponent_int \c_zero \cs_set_eq:NN \fp_exp_const:Nx \use_none:nn \fp_exp_internal: } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \subsection{Tests for special values} % % \begin{macro}[pTF]{\fp_if_undefined:N} % \UnitTested % Testing for an undefined value is easy. % \begin{macrocode} \prg_new_conditional:Npnn \fp_if_undefined:N #1 { p , T , F , TF } { \if_meaning:w #1 \c_undefined_fp \prg_return_true: \else: \prg_return_false: \fi: } % \end{macrocode} % \end{macro} % % \begin{macro}[pTF]{\fp_if_zero:N} % \UnitTested % Testing for a zero fixed-point is also easy. % \begin{macrocode} \prg_new_conditional:Npnn \fp_if_zero:N #1 { p , T , F , TF } { \if_meaning:w #1 \c_zero_fp \prg_return_true: \else: \prg_return_false: \fi: } % \end{macrocode} % \end{macro} % % \subsection{Floating-point conditionals} % % \begin{macro}[TF]{\fp_compare:nNn} % \begin{macro}[TF]{\fp_compare:NNN} % \UnitTested % \begin{macro}[aux]{\fp_compare_aux:N} % \begin{macro}[aux]{\fp_compare_=:} % \begin{macro}[aux]{\fp_compare_<:} % \begin{macro}[aux]{\fp_compare_<_aux:} % \begin{macro}[aux]{\fp_compare_absolute_a>b:} % \begin{macro}[aux]{\fp_compare_absolute_a:} % The idea for the comparisons is to provide two versions: slower and % faster. The lead off for both is the same: get the two numbers % read and then look for a function to handle the comparison. % \begin{macrocode} \prg_new_protected_conditional:Npnn \fp_compare:nNn #1#2#3 { T , F , TF } { \group_begin: \fp_split:Nn a {#1} \fp_standardise:NNNN \l_fp_input_a_sign_int \l_fp_input_a_integer_int \l_fp_input_a_decimal_int \l_fp_input_a_exponent_int \fp_split:Nn b {#3} \fp_standardise:NNNN \l_fp_input_b_sign_int \l_fp_input_b_integer_int \l_fp_input_b_decimal_int \l_fp_input_b_exponent_int \fp_compare_aux:N #2 } \prg_new_protected_conditional:Npnn \fp_compare:NNN #1#2#3 { T , F , TF } { \group_begin: \fp_read:N #3 \l_fp_input_b_sign_int \l_fp_input_a_sign_int \l_fp_input_b_integer_int \l_fp_input_a_integer_int \l_fp_input_b_decimal_int \l_fp_input_a_decimal_int \l_fp_input_b_exponent_int \l_fp_input_a_exponent_int \fp_read:N #1 \fp_compare_aux:N #2 } \cs_new_protected_nopar:Npn \fp_compare_aux:N #1 { \cs_if_exist:cTF { fp_compare_#1: } { \use:c { fp_compare_#1: } } { \group_end: \prg_return_false: } } % \end{macrocode} % For equality, the test is pretty easy as things are either equal or % they are not. % \begin{macrocode} \cs_new_protected_nopar:cpn { fp_compare_=: } { \if_int_compare:w \l_fp_input_a_sign_int = \l_fp_input_b_sign_int \if_int_compare:w \l_fp_input_a_integer_int = \l_fp_input_b_integer_int \if_int_compare:w \l_fp_input_a_decimal_int = \l_fp_input_b_decimal_int \if_int_compare:w \l_fp_input_a_exponent_int = \l_fp_input_b_exponent_int \group_end: \prg_return_true: \else: \group_end: \prg_return_false: \fi: \else: \group_end: \prg_return_false: \fi: \else: \group_end: \prg_return_false: \fi: \else: \group_end: \prg_return_false: \fi: } % \end{macrocode} % Comparing two values is quite complex. First, there is a filter step % to check if one or other of the given values is zero. If it is then % the result is relatively easy to determine. % \begin{macrocode} \cs_new_protected_nopar:cpn { fp_compare_>: } { \if_int_compare:w \int_eval:w \l_fp_input_a_integer_int + \l_fp_input_a_decimal_int = \c_zero \if_int_compare:w \int_eval:w \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int = \c_zero \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_b_sign_int > \c_zero \group_end: \prg_return_false: \else: \group_end: \prg_return_true: \fi: \fi: \else: \if_int_compare:w \int_eval:w \l_fp_input_b_integer_int + \l_fp_input_b_decimal_int = \c_zero \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \group_end: \prg_return_true: \else: \group_end: \prg_return_false: \fi: \else: \use:c { fp_compare_>_aux: } \fi: \fi: } % \end{macrocode} % Next, check the sign of the input: this again may give an obvious % result. If both signs are the same, then hand off to comparing the % absolute values. % \begin{macrocode} \cs_new_protected_nopar:cpn { fp_compare_>_aux: } { \if_int_compare:w \l_fp_input_a_sign_int > \l_fp_input_b_sign_int \group_end: \prg_return_true: \else: \if_int_compare:w \l_fp_input_a_sign_int < \l_fp_input_b_sign_int \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_a_sign_int > \c_zero \use:c { fp_compare_absolute_a>b: } \else: \use:c { fp_compare_absolute_ab: } { \if_int_compare:w \l_fp_input_a_exponent_int > \l_fp_input_b_exponent_int \group_end: \prg_return_true: \else: \if_int_compare:w \l_fp_input_a_exponent_int < \l_fp_input_b_exponent_int \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_a_integer_int > \l_fp_input_b_integer_int \group_end: \prg_return_true: \else: \if_int_compare:w \l_fp_input_a_integer_int < \l_fp_input_b_integer_int \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_a_decimal_int > \l_fp_input_b_decimal_int \group_end: \prg_return_true: \else: \group_end: \prg_return_false: \fi: \fi: \fi: \fi: \fi: } \cs_new_protected_nopar:cpn { fp_compare_absolute_a \l_fp_input_a_exponent_int \group_end: \prg_return_true: \else: \if_int_compare:w \l_fp_input_b_exponent_int < \l_fp_input_a_exponent_int \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_b_integer_int > \l_fp_input_a_integer_int \group_end: \prg_return_true: \else: \if_int_compare:w \l_fp_input_b_integer_int < \l_fp_input_a_integer_int \group_end: \prg_return_false: \else: \if_int_compare:w \l_fp_input_b_decimal_int > \l_fp_input_a_decimal_int \group_end: \prg_return_true: \else: \group_end: \prg_return_false: \fi: \fi: \fi: \fi: \fi: } % \end{macrocode} % This is just a case of reversing the two input values and then % running the tests already defined. % \begin{macrocode} \cs_new_protected_nopar:cpn { fp_compare_<: } { \tl_set:Nx \l_fp_tmp_tl { \int_set:Nn \exp_not:N \l_fp_input_a_sign_int { \int_use:N \l_fp_input_b_sign_int } \int_set:Nn \exp_not:N \l_fp_input_a_integer_int { \int_use:N \l_fp_input_b_integer_int } \int_set:Nn \exp_not:N \l_fp_input_a_decimal_int { \int_use:N \l_fp_input_b_decimal_int } \int_set:Nn \exp_not:N \l_fp_input_a_exponent_int { \int_use:N \l_fp_input_b_exponent_int } \int_set:Nn \exp_not:N \l_fp_input_b_sign_int { \int_use:N \l_fp_input_a_sign_int } \int_set:Nn \exp_not:N \l_fp_input_b_integer_int { \int_use:N \l_fp_input_a_integer_int } \int_set:Nn \exp_not:N \l_fp_input_b_decimal_int { \int_use:N \l_fp_input_a_decimal_int } \int_set:Nn \exp_not:N \l_fp_input_b_exponent_int { \int_use:N \l_fp_input_a_exponent_int } } \l_fp_tmp_tl \use:c { fp_compare_>: } } % \end{macrocode} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % \end{macro} % % \begin{macro}[TF]{\fp_compare:n} % \begin{macro}[aux] % { % \fp_compare_aux_i:w, \fp_compare_aux_ii:w, \fp_compare_aux_iii:w, % \fp_compare_aux_iv:w, \fp_compare_aux_v:w, \fp_compare_aux_vi:w, % \fp_compare_aux_vii:w % } % As \TeX{} cannot help out here, a daisy-chain of delimited functions % are used. This is very much a first-generation approach: revision will % be needed if these functions are really useful. % \begin{macrocode} \prg_new_protected_conditional:Npnn \fp_compare:n #1 { T , F , TF } { \group_begin: \tl_set:Nx \l_fp_tmp_tl { \group_end: \fp_compare_aux_i:w #1 \exp_not:n { == \q_nil == \q_stop } } \l_fp_tmp_tl } \cs_new_protected_nopar:Npn \fp_compare_aux_i:w #1 == #2 == #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_ii:w #1 != \q_nil != \q_stop } { \fp_compare:nNnTF {#1} = {#2} \prg_return_true: \prg_return_false: } } \cs_new_protected_nopar:Npn \fp_compare_aux_ii:w #1 != #2 != #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_iii:w #1 <= \q_nil <= \q_stop } { \fp_compare:nNnTF {#1} = {#2} \prg_return_false: \prg_return_true: } } \cs_new_protected_nopar:Npn \fp_compare_aux_iii:w #1 <= #2 <= #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_iv:w #1 >= \q_nil >= \q_stop } { \fp_compare:nNnTF {#1} > {#2} \prg_return_false: \prg_return_true: } } \cs_new_protected_nopar:Npn \fp_compare_aux_iv:w #1 >= #2 >= #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_v:w #1 = \q_nil \q_stop } { \fp_compare:nNnTF {#1} < {#2} \prg_return_false: \prg_return_true: } } \cs_new_protected_nopar:Npn \fp_compare_aux_v:w #1 = #2 = #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_vi:w #1 < \q_nil < \q_stop } { \fp_compare:nNnTF {#1} = {#2} \prg_return_true: \prg_return_false: } } \cs_new_protected_nopar:Npn \fp_compare_aux_vi:w #1 < #2 < #3 \q_stop { \quark_if_nil:nTF {#2} { \fp_compare_aux_vii:w #1 > \q_nil > \q_stop } { \fp_compare:nNnTF {#1} < {#2} \prg_return_true: \prg_return_false: } } \cs_new_protected_nopar:Npn \fp_compare_aux_vii:w #1 > #2 > #3 \q_stop { \quark_if_nil:nTF {#2} { \prg_return_false: } { \fp_compare:nNnTF {#1} > {#2} \prg_return_true: \prg_return_false: } } % \end{macrocode} % \end{macro} % \end{macro} % % \subsection{Messages} % % \begin{macro}{\fp_overflow_msg:} % A generic overflow message, used whenever there is a possible % overflow. % \begin{macrocode} \msg_kernel_new:nnnn { fpu } { overflow } { Number~too~big. } { The~input~given~is~too~big~for~the~LaTeX~floating~point~unit. \\ Further~errors~may~well~occur! } \cs_new_protected_nopar:Npn \fp_overflow_msg: { \msg_kernel_error:nn { fpu } { overflow } } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_exp_overflow_msg:} % A slightly more helpful message for exponent overflows. % \begin{macrocode} \msg_kernel_new:nnnn { fpu } { exponent-overflow } { Number~too~big~for~exponent~unit. } { The~exponent~of~the~input~given~is~too~big~for~the~floating~point~ unit:~the~maximum~input~value~for~an~exponent~is~230. } \cs_new_protected_nopar:Npn \fp_exp_overflow_msg: { \msg_kernel_error:nn { fpu } { exponent-overflow } } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_ln_error_msg:} % Logarithms are only valid for positive number % \begin{macrocode} \msg_kernel_new:nnnn { fpu } { logarithm-input-error } { Invalid~input~to~ln~function. } { Logarithms~can~only~be~calculated~for~positive~numbers. } \cs_new_protected_nopar:Npn \fp_ln_error_msg: { \msg_kernel_error:nn { fpu } { logarithm-input-error } } % \end{macrocode} % \end{macro} % % \begin{macro}{\fp_trig_overflow_msg:} % A slightly more helpful message for trigonometric overflows. % \begin{macrocode} \msg_kernel_new:nnnn { fpu } { trigonometric-overflow } { Number~too~big~for~trigonometry~unit. } { The~trigonometry~code~can~only~work~with~numbers~smaller~ than~1000000000. } \cs_new_protected_nopar:Npn \fp_trig_overflow_msg: { \msg_kernel_error:nn { fpu } { trigonometric-overflow } } % \end{macrocode} % \end{macro} % % \begin{macrocode} % % \end{macrocode} % % \end{implementation} % %\PrintIndex