diff options
Diffstat (limited to 'macros/latex/required/l3kernel/l3fp-aux.dtx')
-rw-r--r-- | macros/latex/required/l3kernel/l3fp-aux.dtx | 1291 |
1 files changed, 1291 insertions, 0 deletions
diff --git a/macros/latex/required/l3kernel/l3fp-aux.dtx b/macros/latex/required/l3kernel/l3fp-aux.dtx new file mode 100644 index 0000000000..c9d0c228f5 --- /dev/null +++ b/macros/latex/required/l3kernel/l3fp-aux.dtx @@ -0,0 +1,1291 @@ +% \iffalse meta-comment +% +%% File: l3fp-aux.dtx +% +% Copyright (C) 2011-2024 The LaTeX 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 +% +% https://www.latex-project.org/lppl.txt +% +% This file is part of the "l3kernel bundle" (The Work in LPPL) +% and all files in that bundle must be distributed together. +% +% ----------------------------------------------------------------------- +% +% The development version of the bundle can be found at +% +% https://github.com/latex3/latex3 +% +% for those people who are interested. +% +%<*driver> +\documentclass[full,kernel]{l3doc} +\begin{document} + \DocInput{\jobname.dtx} +\end{document} +%</driver> +% \fi +% +% \title{^^A +% The \pkg{l3fp-aux} module\\ Support for floating points^^A +% } +% +% \author{^^A +% The \LaTeX{} Project\thanks +% {^^A +% E-mail: +% \href{mailto:latex-team@latex-project.org} +% {latex-team@latex-project.org}^^A +% }^^A +% } +% +% \date{Released 2024-04-11} +% +% \maketitle +% +% \begin{documentation} +% +% \end{documentation} +% +% \begin{implementation} +% +% \section{\pkg{l3fp-aux} implementation} +% +% \begin{macrocode} +%<*package> +% \end{macrocode} +% +% \begin{macrocode} +%<@@=fp> +% \end{macrocode} +% +% ^^A todo: make sanitize and pack more homogeneous between modules. +% +% \subsection{Access to primitives} +% +% \begin{macro}{\@@_int_eval:w, \@@_int_eval_end:, \@@_int_to_roman:w} +% Largely for performance reasons, we need to directly access primitives +% rather than use \cs{int_eval:n}. This happens \emph{a lot}, so we +% use private names. The same is true for \tn{romannumeral}, although it +% is used much less widely. +% \begin{macrocode} +\cs_new_eq:NN \@@_int_eval:w \tex_numexpr:D +\cs_new_eq:NN \@@_int_eval_end: \scan_stop: +\cs_new_eq:NN \@@_int_to_roman:w \tex_romannumeral:D +% \end{macrocode} +% \end{macro} +% +% \subsection{Internal representation} +% +% Internally, a floating point number \meta{X} is a +% token list containing +% \begin{quote} +% \cs{s_@@} \cs{@@_chk:w} \meta{case} \meta{sign} \meta{body} |;| +% \end{quote} +% Let us explain each piece separately. +% +% Internal floating point numbers are used in expressions, +% and in this context are subject to \texttt{f}-expansion. They must +% leave a recognizable mark after \texttt{f}-expansion, to prevent the +% floating point number from being re-parsed. Thus, \cs{s_@@} +% is simply another name for \tn{relax}. +% +% When used directly without an accessor function, floating points +% should produce an error: this is the role of \cs{@@_chk:w}. We could +% make floating point variables be protected to prevent them from +% expanding under \texttt{e}/\texttt{x}-expansion, but it seems more +% convenient to treat them as a subcase of token list variables. +% +% The (decimal part of the) IEEE-754-2008 standard requires the format +% to be able to represent special floating point numbers besides the +% usual positive and negative cases. We distinguish the various +% possibilities by their \meta{case}, which is a single digit: +% \begin{itemize} +% \item[0] zeros: |+0| and |-0|, +% \item[1] \enquote{normal} numbers (positive and negative), +% \item[2] infinities: |+inf| and |-inf|, +% \item[3] quiet and signalling \texttt{nan}. +% \end{itemize} +% The \meta{sign} is |0| (positive) or |2| (negative), +% except in the case of \texttt{nan}, which have $\meta{sign} = 1$. +% This ensures that changing the \meta{sign} digit to $2-\meta{sign}$ +% is exactly equivalent to changing the sign of the number. +% +% Special floating point numbers have the form +% \begin{quote} +% \cs{s_@@} \cs{@@_chk:w} \meta{case} \meta{sign} \cs[no-index]{s_@@_\ldots} |;| +% \end{quote} +% where \cs[no-index]{s_@@_\ldots} is a scan mark carrying information about how the +% number was formed (useful for debugging). +% +% Normal floating point numbers ($\meta{case} = 1$) have the form +% \begin{quote} +% \cs{s_@@} \cs{@@_chk:w} 1 \meta{sign} \Arg{exponent} +% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} |;| +% \end{quote} +% Here, the \meta{exponent} is an integer, between +% $-\ExplSyntaxOn\int_use:N\c__fp_minus_min_exponent_int$ and +% $\ExplSyntaxOn\int_use:N\c__fp_max_exponent_int$. The body consists +% in four blocks of exactly $4$ digits, +% $0000 \leq \meta{X_i} \leq 9999$, and the floating point is +% \[ +% (-1)^{\meta{sign}/2} \meta{X_1}\meta{X_2}\meta{X_3}\meta{X_4}\cdot 10^{\meta{exponent}-16} +% \] +% where we have concatenated the $16$ digits. Currently, floating point numbers are normalized such that +% the \meta{exponent} is minimal, in other words, $1000 \leq \meta{X_1} \leq 9999$. +% +% \begin{table}\centering +% \caption{Internal representation of floating point numbers.} +% \label{tab:fp-convert-special} +% \begin{tabular}{ll} +% \toprule +% \multicolumn{1}{c}{Representation} & Meaning \\ +% \midrule +% 0 0 \cs[no-index]{s_@@_\ldots} \texttt{;} & Positive zero. \\ +% 0 2 \cs[no-index]{s_@@_\ldots} \texttt{;} & Negative zero. \\ +% 1 0 \Arg{exponent} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \texttt{;} +% & Positive floating point. \\ +% 1 2 \Arg{exponent} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \texttt{;} +% & Negative floating point. \\ +% 2 0 \cs[no-index]{s_@@_\ldots} \texttt{;} & Positive infinity. \\ +% 2 2 \cs[no-index]{s_@@_\ldots} \texttt{;} & Negative infinity. \\ +% 3 1 \cs[no-index]{s_@@_\ldots} \texttt{;} & Quiet \texttt{nan}. \\ +% 3 1 \cs[no-index]{s_@@_\ldots} \texttt{;} & Signalling \texttt{nan}. \\ +% \bottomrule +% \end{tabular} +% \end{table} +% +% Calculations are done in base $10000$, \emph{i.e.} one myriad. +% +% \subsection{Using arguments and semicolons} +% +% \begin{macro}[EXP]{\@@_use_none_stop_f:n} +% This function removes an argument (typically a digit) and replaces +% it by \cs{exp_stop_f:}, a marker which stops \texttt{f}-type +% expansion. +% \begin{macrocode} +\cs_new:Npn \@@_use_none_stop_f:n #1 { \exp_stop_f: } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_use_s:n, \@@_use_s:nn} +% Those functions place a semicolon after one or two arguments +% (typically digits). +% \begin{macrocode} +\cs_new:Npn \@@_use_s:n #1 { #1; } +\cs_new:Npn \@@_use_s:nn #1#2 { #1#2; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP] +% {\@@_use_none_until_s:w, \@@_use_i_until_s:nw, \@@_use_ii_until_s:nnw} +% Those functions select specific arguments among a set of arguments +% delimited by a semicolon. +% \begin{macrocode} +\cs_new:Npn \@@_use_none_until_s:w #1; { } +\cs_new:Npn \@@_use_i_until_s:nw #1#2; {#1} +\cs_new:Npn \@@_use_ii_until_s:nnw #1#2#3; {#2} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_reverse_args:Nww} +% Many internal functions take arguments delimited by semicolons, and +% it is occasionally useful to swap two such arguments. +% \begin{macrocode} +\cs_new:Npn \@@_reverse_args:Nww #1 #2; #3; { #1 #3; #2; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_rrot:www} +% Rotate three arguments delimited by semicolons. This is the inverse +% (or the square) of the Forth primitive |ROT|, hence the name. +% \begin{macrocode} +\cs_new:Npn \@@_rrot:www #1; #2; #3; { #2; #3; #1; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_use_i:ww, \@@_use_i:www} +% Many internal functions take arguments delimited by semicolons, and +% it is occasionally useful to remove one or two such arguments. +% \begin{macrocode} +\cs_new:Npn \@@_use_i:ww #1; #2; { #1; } +\cs_new:Npn \@@_use_i:www #1; #2; #3; { #1; } +% \end{macrocode} +% \end{macro} +% +% \subsection{Constants, and structure of floating points} +% +% \begin{macro}{\@@_misused:n} +% This receives a floating point object (floating point number or +% tuple) and generates an error stating that it was misused. This is +% called when for instance an |fp| variable is left in the input +% stream and its contents reach \TeX{}'s stomach. +% \begin{macrocode} +\cs_new_protected:Npn \@@_misused:n #1 + { \msg_error:nne { fp } { misused } { \fp_to_tl:n {#1} } } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}{\s_@@, \@@_chk:w} +% Floating points numbers all start with \cs{s_@@} \cs{@@_chk:w}, +% where \cs{s_@@} is equal to the \TeX{} primitive \tn{relax}, and +% \cs{@@_chk:w} is protected. The rest of the floating point number +% is made of characters (or \tn{relax}). This ensures that nothing +% expands under \texttt{f}-expansion, nor under +% \texttt{e}/\texttt{x}-expansion. +% However, when typeset, \cs{s_@@} does nothing, and \cs{@@_chk:w} is +% expanded. We define \cs{@@_chk:w} to produce an error. +% \begin{macrocode} +\scan_new:N \s_@@ +\cs_new_protected:Npn \@@_chk:w #1 ; + { \@@_misused:n { \s_@@ \@@_chk:w #1 ; } } +% \end{macrocode} +% \end{macro} +% +% \begin{variable}{\s_@@_expr_mark, \s_@@_expr_stop} +% Aliases of \cs{tex_relax:D}, used to terminate expressions. +% \begin{macrocode} +\scan_new:N \s_@@_expr_mark +\scan_new:N \s_@@_expr_stop +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\s_@@_mark, \s_@@_stop} +% Generic scan marks used throughout the module. +% \begin{macrocode} +\scan_new:N \s_@@_mark +\scan_new:N \s_@@_stop +% \end{macrocode} +% \end{variable} +% +% \begin{macro}[EXP]{\@@_use_i_delimit_by_s_stop:nw} +% Functions to gobble up to a scan mark. +% \begin{macrocode} +\cs_new:Npn \@@_use_i_delimit_by_s_stop:nw #1 #2 \s_@@_stop {#1} +% \end{macrocode} +% \end{macro} +% +% \begin{macro} +% { +% \s_@@_invalid, \s_@@_underflow, \s_@@_overflow, +% \s_@@_division, \s_@@_exact +% } +% A couple of scan marks used to indicate where special floating point +% numbers come from. +% \begin{macrocode} +\scan_new:N \s_@@_invalid +\scan_new:N \s_@@_underflow +\scan_new:N \s_@@_overflow +\scan_new:N \s_@@_division +\scan_new:N \s_@@_exact +% \end{macrocode} +% \end{macro} +% +% \begin{variable} +% {\c_zero_fp, \c_minus_zero_fp, \c_inf_fp, \c_minus_inf_fp, \c_nan_fp} +% The special floating points. We define the floating points here as \enquote{exact}. +% \begin{macrocode} +\tl_const:Nn \c_zero_fp { \s_@@ \@@_chk:w 0 0 \s_@@_exact ; } +\tl_const:Nn \c_minus_zero_fp { \s_@@ \@@_chk:w 0 2 \s_@@_exact ; } +\tl_const:Nn \c_inf_fp { \s_@@ \@@_chk:w 2 0 \s_@@_exact ; } +\tl_const:Nn \c_minus_inf_fp { \s_@@ \@@_chk:w 2 2 \s_@@_exact ; } +\tl_const:Nn \c_nan_fp { \s_@@ \@@_chk:w 3 1 \s_@@_exact ; } +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\c_@@_prec_int, \c_@@_half_prec_int, \c_@@_block_int} +% The number of digits of floating points. +% \begin{macrocode} +\int_const:Nn \c_@@_prec_int { 16 } +\int_const:Nn \c_@@_half_prec_int { 8 } +\int_const:Nn \c_@@_block_int { 4 } +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\c_@@_myriad_int} +% Blocks have $4$~digits so this integer is useful. +% \begin{macrocode} +\int_const:Nn \c_@@_myriad_int { 10000 } +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\c_@@_minus_min_exponent_int, \c_@@_max_exponent_int} +% Normal floating point numbers have an exponent between $-$ +% \texttt{minus_min_exponent} and \texttt{max_exponent} inclusive. +% Larger numbers are rounded to $\pm\infty$. Smaller numbers are +% rounded to $\pm 0$. It would be more natural to define a +% \texttt{min_exponent} with the opposite sign but that would waste +% one \TeX{} count. +% \begin{macrocode} +\int_const:Nn \c_@@_minus_min_exponent_int { 10000 } +\int_const:Nn \c_@@_max_exponent_int { 10000 } +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\c_@@_max_exp_exponent_int} +% If a number's exponent is larger than that, its exponential +% overflows/underflows. +% \begin{macrocode} +\int_const:Nn \c_@@_max_exp_exponent_int { 5 } +% \end{macrocode} +% \end{variable} +% +% \begin{variable}{\c_@@_overflowing_fp} +% A floating point number that is bigger than all normal floating +% point numbers. This replaces infinities when converting to formats +% that do not support infinities. +% \begin{macrocode} +\tl_const:Ne \c_@@_overflowing_fp + { + \s_@@ \@@_chk:w 1 0 + { \int_eval:n { \c_@@_max_exponent_int + 1 } } + {1000} {0000} {0000} {0000} ; + } +% \end{macrocode} +% \end{variable} +% +% \begin{macro}[EXP]{\@@_zero_fp:N, \@@_inf_fp:N} +% In case of overflow or underflow, we have to output +% a zero or infinity with a given sign. +% \begin{macrocode} +\cs_new:Npn \@@_zero_fp:N #1 + { \s_@@ \@@_chk:w 0 #1 \s_@@_underflow ; } +\cs_new:Npn \@@_inf_fp:N #1 + { \s_@@ \@@_chk:w 2 #1 \s_@@_overflow ; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exponent:w} +% For normal numbers, the function expands to the exponent, otherwise +% to $0$. This is used in \pkg{l3str-format}. +% \begin{macrocode} +\cs_new:Npn \@@_exponent:w \s_@@ \@@_chk:w #1 + { + \if_meaning:w 1 #1 + \exp_after:wN \@@_use_ii_until_s:nnw + \else: + \exp_after:wN \@@_use_i_until_s:nw + \exp_after:wN 0 + \fi: + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_neg_sign:N} +% When appearing in an integer expression or after \cs{int_value:w}, +% this expands to the sign opposite to |#1|, namely $0$ (positive) is +% turned to $2$ (negative), $1$ (\texttt{nan}) to $1$, and $2$ to $0$. +% \begin{macrocode} +\cs_new:Npn \@@_neg_sign:N #1 + { \@@_int_eval:w 2 - #1 \@@_int_eval_end: } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_kind:w} +% Expands to $0$ for zeros, $1$ for normal floating point numbers, $2$ +% for infinities, $3$ for \nan{}, $4$ for tuples. +% \begin{macrocode} +\cs_new:Npn \@@_kind:w #1 + { + \@@_if_type_fp:NTwFw + #1 \@@_use_ii_until_s:nnw + \s_@@ { \@@_use_i_until_s:nw 4 } + \s_@@_stop + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Overflow, underflow, and exact zero} +% +%^^A todo: the sign of exact zeros should depend on the rounding mode. +% +% \begin{macro}[EXP]{\@@_sanitize:Nw, \@@_sanitize:wN} +% \begin{macro}[EXP]{\@@_sanitize_zero:w} +% Expects the sign and the exponent in some order, then the +% significand (which we don't touch). Outputs the corresponding +% floating point number, possibly underflowed to $\pm 0$ or overflowed +% to $\pm\infty$. The functions \cs{@@_underflow:w} and +% \cs{@@_overflow:w} are defined in \pkg{l3fp-traps}. +% \begin{macrocode} +\cs_new:Npn \@@_sanitize:Nw #1 #2; + { + \if_case:w + \if_int_compare:w #2 > \c_@@_max_exponent_int 1 ~ \else: + \if_int_compare:w #2 < - \c_@@_minus_min_exponent_int 2 ~ \else: + \if_meaning:w 1 #1 3 ~ \fi: \fi: \fi: 0 ~ + \or: \exp_after:wN \@@_overflow:w + \or: \exp_after:wN \@@_underflow:w + \or: \exp_after:wN \@@_sanitize_zero:w + \fi: + \s_@@ \@@_chk:w 1 #1 {#2} + } +\cs_new:Npn \@@_sanitize:wN #1; #2 { \@@_sanitize:Nw #2 #1; } +\cs_new:Npn \@@_sanitize_zero:w \s_@@ \@@_chk:w #1 #2 #3; + { \c_zero_fp } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Expanding after a floating point number} +% +% \begin{macro}[EXP]{\@@_exp_after_o:w} +% \begin{macro}[EXP]{\@@_exp_after_f:nw} +% \begin{syntax} +% \cs{@@_exp_after_o:w} \meta{floating point} +% \cs{@@_exp_after_f:nw} \Arg{tokens} \meta{floating point} +% \end{syntax} +% Places \meta{tokens} (empty in the case of \cs{@@_exp_after_o:w}) +% between the \meta{floating point} and the following tokens, then +% hits those tokens with \texttt{o} or \texttt{f}-expansion, and +% leaves the floating point number unchanged. +% +% We first distinguish normal floating points, which have a significand, +% from the much simpler special floating points. +% \begin{macrocode} +\cs_new:Npn \@@_exp_after_o:w \s_@@ \@@_chk:w #1 + { + \if_meaning:w 1 #1 + \exp_after:wN \@@_exp_after_normal:nNNw + \else: + \exp_after:wN \@@_exp_after_special:nNNw + \fi: + { } + #1 + } +\cs_new:Npn \@@_exp_after_f:nw #1 \s_@@ \@@_chk:w #2 + { + \if_meaning:w 1 #2 + \exp_after:wN \@@_exp_after_normal:nNNw + \else: + \exp_after:wN \@@_exp_after_special:nNNw + \fi: + { \exp:w \exp_end_continue_f:w #1 } + #2 + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_after_special:nNNw} +% \begin{syntax} +% \cs{@@_exp_after_special:nNNw} \Arg{after} \meta{case} \meta{sign} \meta{scan mark} |;| +% \end{syntax} +% Special floating point numbers are easy to jump over since they +% contain few tokens. +% \begin{macrocode} +\cs_new:Npn \@@_exp_after_special:nNNw #1#2#3#4; + { + \exp_after:wN \s_@@ + \exp_after:wN \@@_chk:w + \exp_after:wN #2 + \exp_after:wN #3 + \exp_after:wN #4 + \exp_after:wN ; + #1 + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_after_normal:nNNw} +% For normal floating point numbers, life is slightly harder, since we +% have many tokens to jump over. Here it would be slightly better if +% the digits were not braced but instead were delimited arguments (for +% instance delimited by |,|). That may be changed some day. +% \begin{macrocode} +\cs_new:Npn \@@_exp_after_normal:nNNw #1 1 #2 #3 #4#5#6#7; + { + \exp_after:wN \@@_exp_after_normal:Nwwwww + \exp_after:wN #2 + \int_value:w #3 \exp_after:wN ; + \int_value:w 1 #4 \exp_after:wN ; + \int_value:w 1 #5 \exp_after:wN ; + \int_value:w 1 #6 \exp_after:wN ; + \int_value:w 1 #7 \exp_after:wN ; #1 + } +\cs_new:Npn \@@_exp_after_normal:Nwwwww + #1 #2; 1 #3 ; 1 #4 ; 1 #5 ; 1 #6 ; + { \s_@@ \@@_chk:w 1 #1 {#2} {#3} {#4} {#5} {#6} ; } +% \end{macrocode} +% \end{macro} +% +% \subsection{Other floating point types} +% +% \begin{macro}{\s_@@_tuple, \@@_tuple_chk:w} +% \begin{variable}{\c_@@_empty_tuple_fp} +% Floating point tuples take the form \cs{s_@@_tuple} +% \cs{@@_tuple_chk:w} |{| \meta{fp 1} \meta{fp 2} \dots |}| |;| where +% each \meta{fp} is a floating point number or tuple, hence ends with +% |;| itself. When a tuple is typeset, \cs{@@_tuple_chk:w} produces +% an error, just like usual floating point numbers. +% Tuples may have zero or one element. +% \begin{macrocode} +\scan_new:N \s_@@_tuple +\cs_new_protected:Npn \@@_tuple_chk:w #1 ; + { \@@_misused:n { \s_@@_tuple \@@_tuple_chk:w #1 ; } } +\tl_const:Nn \c_@@_empty_tuple_fp + { \s_@@_tuple \@@_tuple_chk:w { } ; } +% \end{macrocode} +% \end{variable} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_tuple_count:w, \@@_array_count:n} +% \begin{macro}[EXP]{\@@_tuple_count_loop:Nw} +% Count the number of items in a tuple of floating points by counting +% semicolons. The technique is very similar to \cs{tl_count:n}, but +% with the loop built-in. Checking for the end of the loop is done +% with the |\use_none:n #1| construction. +% \begin{macrocode} +\cs_new:Npn \@@_array_count:n #1 + { \@@_tuple_count:w \s_@@_tuple \@@_tuple_chk:w {#1} ; } +\cs_new:Npn \@@_tuple_count:w \s_@@_tuple \@@_tuple_chk:w #1 ; + { + \int_value:w \@@_int_eval:w 0 + \@@_tuple_count_loop:Nw #1 { ? \prg_break: } ; + \prg_break_point: + \@@_int_eval_end: + } +\cs_new:Npn \@@_tuple_count_loop:Nw #1#2; + { \use_none:n #1 + 1 \@@_tuple_count_loop:Nw } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_if_type_fp:NTwFw} +% Used as \cs{@@_if_type_fp:NTwFw} \meta{marker} \Arg{true code} +% \cs{s_@@} \Arg{false code} \cs{s_@@_stop}, this test whether the +% \meta{marker} is \cs{s_@@} or not and runs the appropriate +% \meta{code}. The very unusual syntax is for optimization purposes +% as that function is used for all floating point operations. +% \begin{macrocode} +\cs_new:Npn \@@_if_type_fp:NTwFw #1 \s_@@ #2 #3 \s_@@_stop {#2} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_array_if_all_fp:nTF, \@@_array_if_all_fp_loop:w} +% True if all items are floating point numbers. Used for |min|. +% \begin{macrocode} +\cs_new:Npn \@@_array_if_all_fp:nTF #1 + { + \@@_array_if_all_fp_loop:w #1 { \s_@@ \prg_break: } ; + \prg_break_point: \use_i:nn + } +\cs_new:Npn \@@_array_if_all_fp_loop:w #1#2 ; + { + \@@_if_type_fp:NTwFw + #1 \@@_array_if_all_fp_loop:w + \s_@@ { \prg_break:n \use_iii:nnn } + \s_@@_stop + } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP] +% {\@@_type_from_scan:N, \@@_type_from_scan_other:N, \@@_type_from_scan:w} +% Used as \cs{@@_type_from_scan:N} \meta{token}. +% Grabs the pieces of the stringified \meta{token} which lies after +% the first |s__fp|. If the \meta{token} does not contain that +% string, the result is |_?|. +% \begin{macrocode} +\cs_new:Npn \@@_type_from_scan:N #1 + { + \@@_if_type_fp:NTwFw + #1 { } + \s_@@ { \@@_type_from_scan_other:N #1 } + \s_@@_stop + } +\cs_new:Npe \@@_type_from_scan_other:N #1 + { + \exp_not:N \exp_after:wN \exp_not:N \@@_type_from_scan:w + \exp_not:N \token_to_str:N #1 \s_@@_mark + \tl_to_str:n { s_@@ _? } \s_@@_mark \s_@@_stop + } +\exp_last_unbraced:NNNNo + \cs_new:Npn \@@_type_from_scan:w #1 + { \tl_to_str:n { s_@@ } } #2 \s_@@_mark #3 \s_@@_stop {#2} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_change_func_type:NNN} +% \begin{macro}[EXP]{\@@_change_func_type_aux:w, \@@_change_func_type_chk:NNN} +% Arguments are \meta{type marker} \meta{function} \meta{recovery}. +% This gives the function obtained by placing the type after |@@|. If +% the function is not defined then \meta{recovery} \meta{function} is +% used instead; however that test is not run when the \meta{type +% marker} is \cs{s_@@}. +% \begin{macrocode} +\cs_new:Npn \@@_change_func_type:NNN #1#2#3 + { + \@@_if_type_fp:NTwFw + #1 #2 + \s_@@ + { + \exp_after:wN \@@_change_func_type_chk:NNN + \cs:w + @@ \@@_type_from_scan_other:N #1 + \exp_after:wN \@@_change_func_type_aux:w \token_to_str:N #2 + \cs_end: + #2 #3 + } + \s_@@_stop + } +\exp_last_unbraced:NNNNo + \cs_new:Npn \@@_change_func_type_aux:w #1 { \tl_to_str:n { @@ } } { } +\cs_new:Npn \@@_change_func_type_chk:NNN #1#2#3 + { + \if_meaning:w \scan_stop: #1 + \exp_after:wN #3 \exp_after:wN #2 + \else: + \exp_after:wN #1 + \fi: + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_after_any_f:Nnw, \@@_exp_after_any_f:nw} +% \begin{macro}[EXP]{\@@_exp_after_expr_stop_f:nw} +% The |Nnw| function simply dispatches to the appropriate +% \cs[no-index]{@@_exp_after\ldots{}_f:nw} with \enquote{\ldots{}} +% (either empty or |_|\meta{type}) extracted from |#1|, which should +% start with |\s__fp|. If it doesn't start with |\s__fp| the function +% \cs{@@_exp_after_?_f:nw} defined in \pkg{l3fp-parse} gives an error; +% another special \meta{type} is |stop|, useful for loops, see below. +% The |nw| function has an important optimization for floating points +% numbers; it also fetches its type marker |#2| from the floating +% point. +% \begin{macrocode} +\cs_new:Npn \@@_exp_after_any_f:Nnw #1 + { \cs:w @@_exp_after \@@_type_from_scan_other:N #1 _f:nw \cs_end: } +\cs_new:Npn \@@_exp_after_any_f:nw #1#2 + { + \@@_if_type_fp:NTwFw + #2 \@@_exp_after_f:nw + \s_@@ { \@@_exp_after_any_f:Nnw #2 } + \s_@@_stop + {#1} #2 + } +\cs_new_eq:NN \@@_exp_after_expr_stop_f:nw \use_none:nn +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_exp_after_tuple_o:w} +% \begin{macro}[EXP]{\@@_exp_after_tuple_f:nw, \@@_exp_after_array_f:w} +% The loop works by using the |n| argument of +% \cs{@@_exp_after_any_f:nw} to place the loop macro after the next +% item in the tuple and expand it. +% \begin{quote} +% \cs{@@_exp_after_array_f:w}\\ +% \meta{fp_1} |;|\\ +% \ldots{}\\ +% \meta{fp_n} |;|\\ +% \cs{s_@@_expr_stop} +% \end{quote} +% \begin{macrocode} +\cs_new:Npn \@@_exp_after_tuple_o:w + { \@@_exp_after_tuple_f:nw { \exp_after:wN \exp_stop_f: } } +\cs_new:Npn \@@_exp_after_tuple_f:nw + #1 \s_@@_tuple \@@_tuple_chk:w #2 ; + { + \exp_after:wN \s_@@_tuple + \exp_after:wN \@@_tuple_chk:w + \exp_after:wN { + \exp:w \exp_end_continue_f:w + \@@_exp_after_array_f:w #2 \s_@@_expr_stop + \exp_after:wN } + \exp_after:wN ; + \exp:w \exp_end_continue_f:w #1 + } +\cs_new:Npn \@@_exp_after_array_f:w + { \@@_exp_after_any_f:nw { \@@_exp_after_array_f:w } } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Packing digits} +% +% When a positive integer |#1| is known to be less than $10^8$, the +% following trick splits it into two blocks of $4$ digits, padding +% with zeros on the left. +% \begin{verbatim} +% \cs_new:Npn \pack:NNNNNw #1 #2#3#4#5 #6; { {#2#3#4#5} {#6} } +% \exp_after:wN \pack:NNNNNw +% \__fp_int_value:w \__fp_int_eval:w 1 0000 0000 + #1 ; +% \end{verbatim} +% The idea is that adding $10^8$ to the number ensures that it has +% exactly $9$ digits, and can then easily find which digits correspond +% to what position in the number. Of course, this can be modified +% for any number of digits less or equal to~$9$ (we are limited by +% \TeX{}'s integers). This method is very heavily relied upon in +% \texttt{l3fp-basics}. +% +% More specifically, the auxiliary inserts |+ #1#2#3#4#5 ; {#6}|, which +% allows us to compute several blocks of $4$ digits in a nested manner, +% performing carries on the fly. Say we want to compute $1\,2345 \times +% 6677\,8899$. With simplified names, we would do +% \begin{verbatim} +% \exp_after:wN \post_processing:w +% \__fp_int_value:w \__fp_int_eval:w - 5 0000 +% \exp_after:wN \pack:NNNNNw +% \__fp_int_value:w \__fp_int_eval:w 4 9995 0000 +% + 12345 * 6677 +% \exp_after:wN \pack:NNNNNw +% \__fp_int_value:w \__fp_int_eval:w 5 0000 0000 +% + 12345 * 8899 ; +% \end{verbatim} +% The \cs{exp_after:wN} triggers \cs{int_value:w} \cs{@@_int_eval:w}, which +% starts a first computation, whose initial value is $- 5\,0000$ (the +% \enquote{leading shift}). In that computation appears an +% \cs{exp_after:wN}, which triggers the nested computation +% \cs{int_value:w} \cs{@@_int_eval:w} with starting value $4\,9995\,0000$ (the +% \enquote{middle shift}). That, in turn, expands \cs{exp_after:wN} +% which triggers the third computation. The third computation's value +% is $5\,0000\,0000 + 12345 \times 8899$, which has $9$ digits. Adding +% $5\cdot 10^{8}$ to the product allowed us to know how many digits to +% expect as long as the numbers to multiply are not too big; it +% also works to some extent with negative results. The \texttt{pack} +% function puts the last $4$ of those $9$ digits into a brace group, +% moves the semi-colon delimiter, and inserts a |+|, which combines the +% carry with the previous computation. The shifts nicely combine into +% $5\,0000\,0000 / 10^{4} + 4\,9995\,0000 = 5\,0000\,0000$. As long as +% the operands are in some range, the result of this second computation +% has $9$ digits. The corresponding \texttt{pack} function, +% expanded after the result is computed, braces the last $4$ digits, and +% leaves |+| \meta{5 digits} for the initial computation. The +% \enquote{leading shift} cancels the combination of the other shifts, +% and the |\post_processing:w| takes care of packing the last few +% digits. +% +% Admittedly, this is quite intricate. It is probably the key in making +% \pkg{l3fp} as fast as other pure \TeX{} floating point units despite +% its increased precision. In fact, this is used so much that we +% provide different sets of packing functions and shifts, depending on +% ranges of input. +% +% \begin{macro}[EXP]{\@@_pack:NNNNNw} +% \begin{variable} +% { +% \c_@@_trailing_shift_int , +% \c_@@_middle_shift_int , +% \c_@@_leading_shift_int , +% } +% This set of shifts allows for computations involving results in the +% range $[-4\cdot 10^{8}, 5\cdot 10^{8}-1]$. Shifted values all have +% exactly $9$ digits. +% \begin{macrocode} +\int_const:Nn \c_@@_leading_shift_int { - 5 0000 } +\int_const:Nn \c_@@_middle_shift_int { 5 0000 * 9999 } +\int_const:Nn \c_@@_trailing_shift_int { 5 0000 * 10000 } +\cs_new:Npn \@@_pack:NNNNNw #1 #2#3#4#5 #6; { + #1#2#3#4#5 ; {#6} } +% \end{macrocode} +% \end{variable} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_pack_big:NNNNNNw} +% \begin{variable} +% { +% \c_@@_big_trailing_shift_int , +% \c_@@_big_middle_shift_int , +% \c_@@_big_leading_shift_int , +% } +% This set of shifts allows for computations involving results in the +% range $[-5\cdot 10^{8}, 6\cdot 10^{8}-1]$ (actually a bit more). +% Shifted values all have exactly $10$ digits. Note that the upper +% bound is due to \TeX{}'s limit of $2^{31}-1$ on integers. The +% shifts are chosen to be roughly the mid-point of $10^{9}$ and +% $2^{31}$, the two bounds on $10$-digit integers in \TeX{}. +% \begin{macrocode} +\int_const:Nn \c_@@_big_leading_shift_int { - 15 2374 } +\int_const:Nn \c_@@_big_middle_shift_int { 15 2374 * 9999 } +\int_const:Nn \c_@@_big_trailing_shift_int { 15 2374 * 10000 } +\cs_new:Npn \@@_pack_big:NNNNNNw #1#2 #3#4#5#6 #7; + { + #1#2#3#4#5#6 ; {#7} } +% \end{macrocode} +% \end{variable} +% \end{macro} +% +% ^^A \@@_pack_Bigg:NNNNNNw = \@@_pack_big:NNNNNNw ? +% \begin{macro}[EXP]{\@@_pack_Bigg:NNNNNNw} +% \begin{variable} +% { +% \c_@@_Bigg_trailing_shift_int , +% \c_@@_Bigg_middle_shift_int , +% \c_@@_Bigg_leading_shift_int , +% } +% This set of shifts allows for computations with results in the +% range $[-1\cdot 10^{9}, 147483647]$; the end-point is $2^{31} - 1 - +% 2\cdot 10^{9} \simeq 1.47\cdot 10^{8}$. Shifted values all have +% exactly $10$ digits. +% \begin{macrocode} +\int_const:Nn \c_@@_Bigg_leading_shift_int { - 20 0000 } +\int_const:Nn \c_@@_Bigg_middle_shift_int { 20 0000 * 9999 } +\int_const:Nn \c_@@_Bigg_trailing_shift_int { 20 0000 * 10000 } +\cs_new:Npn \@@_pack_Bigg:NNNNNNw #1#2 #3#4#5#6 #7; + { + #1#2#3#4#5#6 ; {#7} } +% \end{macrocode} +% \end{variable} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_pack_twice_four:wNNNNNNNN} +% \begin{syntax} +% \cs{@@_pack_twice_four:wNNNNNNNN} \meta{tokens} |;| \meta{$\geq 8$ digits} +% \end{syntax} +% Grabs two sets of $4$ digits and places them before the semi-colon +% delimiter. Putting several copies of this function before a +% semicolon packs more digits since each takes the digits +% packed by the others in its first argument. +% \begin{macrocode} +\cs_new:Npn \@@_pack_twice_four:wNNNNNNNN #1; #2#3#4#5 #6#7#8#9 + { #1 {#2#3#4#5} {#6#7#8#9} ; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_pack_eight:wNNNNNNNN} +% \begin{syntax} +% \cs{@@_pack_eight:wNNNNNNNN} \meta{tokens} |;| \meta{$\geq 8$ digits} +% \end{syntax} +% Grabs one set of $8$ digits and places them before the semi-colon +% delimiter as a single group. Putting several copies of this +% function before a semicolon packs more digits since each +% takes the digits packed by the others in its first argument. +% \begin{macrocode} +\cs_new:Npn \@@_pack_eight:wNNNNNNNN #1; #2#3#4#5 #6#7#8#9 + { #1 {#2#3#4#5#6#7#8#9} ; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP] +% { +% \@@_basics_pack_low:NNNNNw, +% \@@_basics_pack_high:NNNNNw, +% \@@_basics_pack_high_carry:w +% } +% Addition and multiplication of significands are done in two steps: +% first compute a (more or less) exact result, then round and pack +% digits in the final (braced) form. These functions take care of the +% packing, with special attention given to the case where rounding has +% caused a carry. Since rounding can only shift the final digit by +% $1$, a carry always produces an exact power of $10$. Thus, +% \cs{@@_basics_pack_high_carry:w} is always followed by four times +% |{0000}|. +% +% This is used in \pkg{l3fp-basics} and \pkg{l3fp-extended}. +% \begin{macrocode} +\cs_new:Npn \@@_basics_pack_low:NNNNNw #1 #2#3#4#5 #6; + { + #1 - 1 ; {#2#3#4#5} {#6} ; } +\cs_new:Npn \@@_basics_pack_high:NNNNNw #1 #2#3#4#5 #6; + { + \if_meaning:w 2 #1 + \@@_basics_pack_high_carry:w + \fi: + ; {#2#3#4#5} {#6} + } +\cs_new:Npn \@@_basics_pack_high_carry:w \fi: ; #1 + { \fi: + 1 ; {1000} } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP] +% { +% \@@_basics_pack_weird_low:NNNNw, +% \@@_basics_pack_weird_high:NNNNNNNNw +% } +% This is used in \pkg{l3fp-basics} for additions and +% divisions. Their syntax is confusing, hence the name. +% \begin{macrocode} +\cs_new:Npn \@@_basics_pack_weird_low:NNNNw #1 #2#3#4 #5; + { + \if_meaning:w 2 #1 + + 1 + \fi: + \@@_int_eval_end: + #2#3#4; {#5} ; + } +\cs_new:Npn \@@_basics_pack_weird_high:NNNNNNNNw + 1 #1#2#3#4 #5#6#7#8 #9; { ; {#1#2#3#4} {#5#6#7#8} {#9} } +% \end{macrocode} +% \end{macro} +% +% \subsection{Decimate (dividing by a power of 10)} +% +% ^^A begin[todo] +% \begin{macro}[EXP]{\@@_decimate:nNnnnn} +% \begin{syntax} +% \cs{@@_decimate:nNnnnn} \Arg{shift} \meta{f_1} +% ~~\Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} +% \end{syntax} +% Each \meta{X_i} consists in $4$ digits exactly, +% and $1000\leq\meta{X_1}<9999$. The first argument determines +% by how much we shift the digits. \meta{f_1} is called as follows: +% \begin{syntax} +% \meta{f_1} \meta{rounding} \Arg{X'_1} \Arg{X'_2} \meta{extra-digits} |;| +% \end{syntax} +% where $0\leq\meta{X'_i}<10^{8}-1$ are $8$ digit integers, +% forming the truncation of our number. In other words, +% \[ +% \left( +% \sum_{i=1}^{4} \meta{X_i} \cdot 10^{-4i} \cdot 10^{-\meta{shift}} +% \right) +% - \bigl( \meta{X'_1} \cdot 10^{-8} + \meta{X'_2} \cdot 10^{-16} \bigr) +% = 0.\meta{extra-digits} \cdot 10^{-16} +% \in [0,10^{-16}). +% \] +% To round properly later, we need to remember some information +% about the difference. The \meta{rounding} digit is $0$ if and +% only if the difference is exactly $0$, and $5$ if and only if +% the difference is exactly $0.5\cdot 10^{-16}$. Otherwise, it +% is the (non-$0$, non-$5$) digit closest to $10^{17}$ times the +% difference. In particular, if the shift is $17$ or more, all +% the digits are dropped, \meta{rounding} is $1$ (not $0$), and +% \meta{X'_1} and \meta{X'_2} are both zero. +% +% If the shift is $1$, the \meta{rounding} digit is simply the +% only digit that was pushed out of the brace groups (this is +% important for subtraction). It would be more natural for the +% \meta{rounding} digit to be placed after the \meta{X'_i}, +% but the choice we make involves less reshuffling. +% +% Note that this function treats negative \meta{shift} as $0$. +% \begin{macrocode} +\cs_new:Npn \@@_decimate:nNnnnn #1 + { + \cs:w + @@_decimate_ + \if_int_compare:w \@@_int_eval:w #1 > \c_@@_prec_int + tiny + \else: + \@@_int_to_roman:w \@@_int_eval:w #1 + \fi: + :Nnnnn + \cs_end: + } +% \end{macrocode} +% Each of the auxiliaries see the function \meta{f_1}, +% followed by $4$ blocks of $4$ digits. +% \end{macro} +% +% \begin{macro}[EXP]{\@@_decimate_:Nnnnn, \@@_decimate_tiny:Nnnnn} +% If the \meta{shift} is zero, or too big, life is very easy. +% \begin{macrocode} +\cs_new:Npn \@@_decimate_:Nnnnn #1 #2#3#4#5 + { #1 0 {#2#3} {#4#5} ; } +\cs_new:Npn \@@_decimate_tiny:Nnnnn #1 #2#3#4#5 + { #1 1 { 0000 0000 } { 0000 0000 } 0 #2#3#4#5 ; } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP] +% { +% \@@_decimate_auxi:Nnnnn, \@@_decimate_auxii:Nnnnn, +% \@@_decimate_auxiii:Nnnnn, \@@_decimate_auxiv:Nnnnn, +% \@@_decimate_auxv:Nnnnn, \@@_decimate_auxvi:Nnnnn, +% \@@_decimate_auxvii:Nnnnn, \@@_decimate_auxviii:Nnnnn, +% \@@_decimate_auxix:Nnnnn, \@@_decimate_auxx:Nnnnn, +% \@@_decimate_auxxi:Nnnnn, \@@_decimate_auxxii:Nnnnn, +% \@@_decimate_auxxiii:Nnnnn, \@@_decimate_auxxiv:Nnnnn, +% \@@_decimate_auxxv:Nnnnn, \@@_decimate_auxxvi:Nnnnn +% } +% \begin{syntax} +% \cs{@@_decimate_auxi:Nnnnn} \meta{f_1} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} +% \end{syntax} +% Shifting happens in two steps: compute the \meta{rounding} digit, +% and repack digits into two blocks of $8$. The sixteen functions +% are very similar, and defined through \cs{@@_tmp:w}. +% The arguments are as follows: |#1| indicates which function is +% being defined; after one step of expansion, |#2| yields the +% \enquote{extra digits} which are then converted by +% \cs{@@_round_digit:Nw} to the \meta{rounding} digit (note the |+| +% separating blocks of digits to avoid overflowing \TeX{}'s integers). +% This triggers the \texttt{f}-expansion of +% \cs{@@_decimate_pack:nnnnnnnnnnw},\footnote{No, the argument +% spec is not a mistake: the function calls an auxiliary to +% do half of the job.} responsible for building two blocks of +% $8$ digits, and removing the rest. For this to work, |#3| +% alternates between braced and unbraced blocks of $4$ digits, +% in such a way that the $5$ first and $5$ next token groups +% yield the correct blocks of $8$ digits. +% \begin{macrocode} +\cs_new:Npn \@@_tmp:w #1 #2 #3 + { + \cs_new:cpn { @@_decimate_ #1 :Nnnnn } ##1 ##2##3##4##5 + { + \exp_after:wN ##1 + \int_value:w + \exp_after:wN \@@_round_digit:Nw #2 ; + \@@_decimate_pack:nnnnnnnnnnw #3 ; + } + } +\@@_tmp:w {i} {\use_none:nnn #50}{ 0{#2}#3{#4}#5 } +\@@_tmp:w {ii} {\use_none:nn #5 }{ 00{#2}#3{#4}#5 } +\@@_tmp:w {iii} {\use_none:n #5 }{ 000{#2}#3{#4}#5 } +\@@_tmp:w {iv} { #5 }{ {0000}#2{#3}#4 #5 } +\@@_tmp:w {v} {\use_none:nnn #4#5 }{ 0{0000}#2{#3}#4 #5 } +\@@_tmp:w {vi} {\use_none:nn #4#5 }{ 00{0000}#2{#3}#4 #5 } +\@@_tmp:w {vii} {\use_none:n #4#5 }{ 000{0000}#2{#3}#4 #5 } +\@@_tmp:w {viii}{ #4#5 }{ {0000}0000{#2}#3 #4 #5 } +\@@_tmp:w {ix} {\use_none:nnn #3#4+#5}{ 0{0000}0000{#2}#3 #4 #5 } +\@@_tmp:w {x} {\use_none:nn #3#4+#5}{ 00{0000}0000{#2}#3 #4 #5 } +\@@_tmp:w {xi} {\use_none:n #3#4+#5}{ 000{0000}0000{#2}#3 #4 #5 } +\@@_tmp:w {xii} { #3#4+#5}{ {0000}0000{0000}#2 #3 #4 #5 } +\@@_tmp:w {xiii}{\use_none:nnn#2#3+#4#5}{ 0{0000}0000{0000}#2 #3 #4 #5 } +\@@_tmp:w {xiv} {\use_none:nn #2#3+#4#5}{ 00{0000}0000{0000}#2 #3 #4 #5 } +\@@_tmp:w {xv} {\use_none:n #2#3+#4#5}{ 000{0000}0000{0000}#2 #3 #4 #5 } +\@@_tmp:w {xvi} { #2#3+#4#5}{{0000}0000{0000}0000 #2 #3 #4 #5} +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_decimate_pack:nnnnnnnnnnw} +% The computation of the \meta{rounding} digit leaves an unfinished +% \cs{int_value:w}, which expands the following functions. This +% allows us to repack nicely the digits we keep. Those digits come +% as an alternation of unbraced and braced blocks of $4$ digits, +% such that the first $5$ groups of token consist in $4$ single digits, +% and one brace group (in some order), and the next $5$ have the same +% structure. This is followed by some digits and a semicolon. +% \begin{macrocode} +\cs_new:Npn \@@_decimate_pack:nnnnnnnnnnw #1#2#3#4#5 + { \@@_decimate_pack:nnnnnnw { #1#2#3#4#5 } } +\cs_new:Npn \@@_decimate_pack:nnnnnnw #1 #2#3#4#5#6 + { {#1} {#2#3#4#5#6} } +% \end{macrocode} +% \end{macro} +% ^^A end[todo] +% +% \subsection{Functions for use within primitive conditional branches} +% +% The functions described in this section are not pretty and can easily +% be misused. When correctly used, each of them removes one \cs{fi:} as +% part of its parameter text, and puts one back as part of its +% replacement text. +% +% Many computation functions in \pkg{l3fp} must perform tests on the +% type of floating points that they receive. This is often done in an +% \cs{if_case:w} statement or another conditional statement, and only a +% few cases lead to actual computations: most of the special cases are +% treated using a few standard functions which we define now. A typical +% use context for those functions would be +% \begin{syntax} +% \cs{if_case:w} \meta{integer} \cs{exp_stop_f:} +% | |\cs{@@_case_return_o:Nw} \meta{fp var} +% \cs{or:} \cs{@@_case_use:nw} \Arg{some computation} +% \cs{or:} \cs{@@_case_return_same_o:w} +% \cs{or:} \cs{@@_case_return:nw} \Arg{something} +% \cs{fi:} +% \meta{junk} +% \meta{floating point} +% \end{syntax} +% In this example, the case $0$ returns the floating point +% \meta{fp~var}, expanding once after that floating point. Case $1$ +% does \meta{some computation} using the \meta{floating point} +% (presumably compute the operation requested by the user in that +% non-trivial case). Case $2$ returns the \meta{floating point} +% without modifying it, removing the \meta{junk} and expanding once +% after. Case $3$ closes the conditional, removes the \meta{junk} +% and the \meta{floating point}, and expands \meta{something} next. In +% other cases, the \enquote{\meta{junk}} is expanded, performing some +% other operation on the \meta{floating point}. We provide similar +% functions with two trailing \meta{floating points}. +% +% \begin{macro}[EXP]{\@@_case_use:nw} +% This function ends a \TeX{} conditional, removes junk until the next +% floating point, and places its first argument before that floating +% point, to perform some operation on the floating point. +% \begin{macrocode} +\cs_new:Npn \@@_case_use:nw #1#2 \fi: #3 \s_@@ { \fi: #1 \s_@@ } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_case_return:nw} +% This function ends a \TeX{} conditional, removes junk and a floating +% point, and places its first argument in the input stream. A quirk +% is that we don't define this function requiring a floating point to +% follow, simply anything ending in a semicolon. This, in turn, means +% that the \meta{junk} may not contain semicolons. +% \begin{macrocode} +\cs_new:Npn \@@_case_return:nw #1#2 \fi: #3 ; { \fi: #1 } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_case_return_o:Nw} +% This function ends a \TeX{} conditional, removes junk and a floating +% point, and returns its first argument (an \meta{fp~var}) then expands +% once after it. +% \begin{macrocode} +\cs_new:Npn \@@_case_return_o:Nw #1#2 \fi: #3 \s_@@ #4 ; + { \fi: \exp_after:wN #1 } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_case_return_same_o:w} +% This function ends a \TeX{} conditional, removes junk, and returns +% the following floating point, expanding once after it. +% \begin{macrocode} +\cs_new:Npn \@@_case_return_same_o:w #1 \fi: #2 \s_@@ + { \fi: \@@_exp_after_o:w \s_@@ } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_case_return_o:Nww} +% Same as \cs{@@_case_return_o:Nw} but with two trailing floating +% points. +% \begin{macrocode} +\cs_new:Npn \@@_case_return_o:Nww #1#2 \fi: #3 \s_@@ #4 ; #5 ; + { \fi: \exp_after:wN #1 } +% \end{macrocode} +% \end{macro} +% +% \begin{macro}[EXP]{\@@_case_return_i_o:ww, \@@_case_return_ii_o:ww} +% Similar to \cs{@@_case_return_same_o:w}, but this returns the first +% or second of two trailing floating point numbers, expanding once +% after the result. +% \begin{macrocode} +\cs_new:Npn \@@_case_return_i_o:ww #1 \fi: #2 \s_@@ #3 ; \s_@@ #4 ; + { \fi: \@@_exp_after_o:w \s_@@ #3 ; } +\cs_new:Npn \@@_case_return_ii_o:ww #1 \fi: #2 \s_@@ #3 ; + { \fi: \@@_exp_after_o:w } +% \end{macrocode} +% \end{macro} +% +% \subsection{Integer floating points} +% +% \begin{macro}[EXP, pTF]{\@@_int:w} +% Tests if the floating point argument is an integer. For normal +% floating point numbers, this holds if the rounding digit resulting +% from \cs{@@_decimate:nNnnnn} is~$0$. +% \begin{macrocode} +\prg_new_conditional:Npnn \@@_int:w \s_@@ \@@_chk:w #1 #2 #3 #4; + { TF , T , F , p } + { + \if_case:w #1 \exp_stop_f: + \prg_return_true: + \or: + \if_charcode:w 0 + \@@_decimate:nNnnnn { \c_@@_prec_int - #3 } + \@@_use_i_until_s:nw #4 + \prg_return_true: + \else: + \prg_return_false: + \fi: + \else: \prg_return_false: + \fi: + } +% \end{macrocode} +% \end{macro} +% +% \subsection{Small integer floating points} +% +% \begin{macro}[EXP]{\@@_small_int:wTF} +% \begin{macro}[EXP] +% { +% \@@_small_int_true:wTF, +% \@@_small_int_normal:NnwTF, +% \@@_small_int_test:NnnwNTF +% } +% Tests if the floating point argument is an integer or $\pm\infty$. +% If so, it is clipped to an integer in the range $[-10^{8},10^{8}]$ +% and fed as a braced argument to the \meta{true code}. +% Otherwise, the \meta{false code} is performed. +% +% First filter special cases: zeros and infinities are integers, +% \texttt{nan} is not. For normal numbers, decimate. If the rounding +% digit is not $0$ run the \meta{false code}. If it is, then the +% integer is |#2| |#3|; use |#3| if |#2| vanishes and otherwise +% $10^{8}$. +% \begin{macrocode} +\cs_new:Npn \@@_small_int:wTF \s_@@ \@@_chk:w #1#2 + { + \if_case:w #1 \exp_stop_f: + \@@_case_return:nw { \@@_small_int_true:wTF 0 ; } + \or: \exp_after:wN \@@_small_int_normal:NnwTF + \or: + \@@_case_return:nw + { + \exp_after:wN \@@_small_int_true:wTF \int_value:w + \if_meaning:w 2 #2 - \fi: 1 0000 0000 ; + } + \else: \@@_case_return:nw \use_ii:nn + \fi: + #2 + } +\cs_new:Npn \@@_small_int_true:wTF #1; #2#3 { #2 {#1} } +\cs_new:Npn \@@_small_int_normal:NnwTF #1#2#3; + { + \@@_decimate:nNnnnn { \c_@@_prec_int - #2 } + \@@_small_int_test:NnnwNw + #3 #1 + } +\cs_new:Npn \@@_small_int_test:NnnwNw #1#2#3#4; #5 + { + \if_meaning:w 0 #1 + \exp_after:wN \@@_small_int_true:wTF + \int_value:w \if_meaning:w 2 #5 - \fi: + \if_int_compare:w #2 > \c_zero_int + 1 0000 0000 + \else: + #3 + \fi: + \exp_after:wN ; + \else: + \exp_after:wN \use_ii:nn + \fi: + } +% \end{macrocode} +% \end{macro} +% \end{macro} +% +% \subsection{Fast string comparison} +% +% \begin{macro}{\@@_str_if_eq:nn} +% A private version of the low-level string comparison function. +% \begin{macrocode} +\cs_new_eq:NN \@@_str_if_eq:nn \tex_strcmp:D +% \end{macrocode} +% \end{macro} +% +% \subsection{Name of a function from its \pkg{l3fp-parse} name} +% +% \begin{macro}[EXP]{\@@_func_to_name:N, \@@_func_to_name_aux:w} +% The goal is to convert for instance \cs{@@_sin_o:w} to |sin|. +% This is used in error messages hence does not need to be fast. +% \begin{macrocode} +\cs_new:Npn \@@_func_to_name:N #1 + { + \exp_last_unbraced:Nf + \@@_func_to_name_aux:w { \cs_to_str:N #1 } X + } +\cs_set_protected:Npn \@@_tmp:w #1 #2 + { \cs_new:Npn \@@_func_to_name_aux:w ##1 #1 ##2 #2 ##3 X {##2} } +\exp_args:Nff \@@_tmp:w { \tl_to_str:n { @@_ } } + { \tl_to_str:n { _o: } } +% \end{macrocode} +% \end{macro} +% +% \subsection{Messages} +% +% Using a floating point directly is an error. +% \begin{macrocode} +\msg_new:nnnn { fp } { misused } + { A~floating~point~with~value~'#1'~was~misused. } + { + To~obtain~the~value~of~a~floating~point~variable,~use~ + '\token_to_str:N \fp_to_decimal:N',~ + '\token_to_str:N \fp_to_tl:N',~or~other~ + conversion~functions. + } +\prop_gput:Nnn \g_msg_module_name_prop { fp } { LaTeX } +\prop_gput:Nnn \g_msg_module_type_prop { fp } { } +% \end{macrocode} +% +% \begin{macrocode} +%</package> +% \end{macrocode} +% +% \end{implementation} +% +% \PrintChanges +% +% \PrintIndex |