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+% \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