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diff --git a/macros/latex/contrib/l3kernel/l3fp-basics.dtx b/macros/latex/contrib/l3kernel/l3fp-basics.dtx deleted file mode 100644 index a7401407cd..0000000000 --- a/macros/latex/contrib/l3kernel/l3fp-basics.dtx +++ /dev/null @@ -1,2181 +0,0 @@ -% \iffalse meta-comment -% -%% File: l3fp-basics.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-basics} module\\ -% Floating point arithmetic^^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 2024-03-14} -% -% \maketitle -% -% \begin{documentation} -% -% \end{documentation} -% -% \begin{implementation} -% -% \section{\pkg{l3fp-basics} implementation} -% -% \begin{macrocode} -%<*package> -% \end{macrocode} -% -% \begin{macrocode} -%<@@=fp> -% \end{macrocode} -% -% The \pkg{l3fp-basics} module implements addition, subtraction, -% multiplication, and division of two floating points, and the absolute -% value and sign-changing operations on one floating point. -% All operations implemented in this module yield the outcome of -% rounding the infinitely precise result of the operation to the -% nearest floating point. -% -% Some algorithms used below end up being quite similar to some -% described in \enquote{What Every Computer Scientist Should Know About -% Floating Point Arithmetic}, by David Goldberg, which can be found at -% \texttt{http://cr.yp.to/2005-590/goldberg.pdf}. -% -% \begin{macro}[EXP] -% { -% \@@_parse_word_abs:N , -% \@@_parse_word_logb:N , -% \@@_parse_word_sign:N , -% \@@_parse_word_sqrt:N , -% } -% Unary functions. -% \begin{macrocode} -\cs_new:Npn \@@_parse_word_abs:N - { \@@_parse_unary_function:NNN \@@_set_sign_o:w 0 } -\cs_new:Npn \@@_parse_word_logb:N - { \@@_parse_unary_function:NNN \@@_logb_o:w ? } -\cs_new:Npn \@@_parse_word_sign:N - { \@@_parse_unary_function:NNN \@@_sign_o:w ? } -\cs_new:Npn \@@_parse_word_sqrt:N - { \@@_parse_unary_function:NNN \@@_sqrt_o:w ? } -% \end{macrocode} -% \end{macro} -% -% \subsection{Addition and subtraction} -% -% We define here two functions, \cs{@@_-_o:ww} and \cs{@@_+_o:ww}, which -% perform the subtraction and addition of their two floating point -% operands, and expand the tokens following the result once. -% -% A more obscure function, \cs{@@_add_big_i_o:wNww}, is used in -% \pkg{l3fp-expo}. -% -% The logic goes as follows: -% \begin{itemize} -% \item \cs{@@_-_o:ww} calls \cs{@@_+_o:ww} to do the work, with the -% sign of the second operand flipped; -% \item \cs{@@_+_o:ww} dispatches depending on the type of floating -% point, calling specialized auxiliaries; -% \item in all cases except summing two normal floating point numbers, -% we return one or the other operands depending on the signs, or -% detect an invalid operation in the case of $\infty - \infty$; -% \item for normal floating point numbers, compare the signs; -% \item to add two floating point numbers of the same sign or of -% opposite signs, shift the significand of the smaller one to match the -% bigger one, perform the addition or subtraction of significands, -% check for a carry, round, and pack using the -% \cs[no-index]{@@_basics_pack_\ldots{}} functions. -% \end{itemize} -% The trickiest part is to round correctly when adding or subtracting -% normal floating point numbers. -% -% \subsubsection{Sign, exponent, and special numbers} -% -% \begin{macro}[EXP]{\@@_-_o:ww} -% The \cs{@@_+_o:ww} auxiliary has a hook: it takes one argument -% between the first \cs{s_@@} and \cs{@@_chk:w}, which is applied to -% the sign of the second operand. Positioning the hook there means -% that \cs{@@_+_o:ww} can still perform the sanity check that it was -% followed by \cs{s_@@}. -% \begin{macrocode} -\cs_new:cpe { @@_-_o:ww } \s_@@ - { - \exp_not:c { @@_+_o:ww } - \exp_not:n { \s_@@ \@@_neg_sign:N } - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_+_o:ww} -% This function is either called directly with an empty |#1| to -% compute an addition, or it is called by \cs{@@_-_o:ww} with -% \cs{@@_neg_sign:N} as |#1| to compute a subtraction, in which case -% the second operand's sign should be changed. If the -% \meta{types} |#2| and |#4| are the same, dispatch to case |#2| ($0$, -% $1$, $2$, or $3$), where we call specialized functions: thanks to -% \cs{int_value:w}, those receive the tweaked \meta{sign_2} -% (expansion of |#1#5|) as an argument. If the \meta{types} are -% distinct, the result is simply the floating point number with the -% highest \meta{type}. Since case $3$ (used for two \texttt{nan}) -% also picks the first operand, we can also use it when \meta{type_1} -% is greater than \meta{type_2}. Also note that we don't need to -% worry about \meta{sign_2} in that case since the second operand is -% discarded. -% \begin{macrocode} -\cs_new:cpn { @@_+_o:ww } - \s_@@ #1 \@@_chk:w #2 #3 ; \s_@@ \@@_chk:w #4 #5 - { - \if_case:w - \if_meaning:w #2 #4 - #2 - \else: - \if_int_compare:w #2 > #4 \exp_stop_f: - 3 - \else: - 4 - \fi: - \fi: - \exp_stop_f: - \exp_after:wN \@@_add_zeros_o:Nww \int_value:w - \or: \exp_after:wN \@@_add_normal_o:Nww \int_value:w - \or: \exp_after:wN \@@_add_inf_o:Nww \int_value:w - \or: \@@_case_return_i_o:ww - \else: \exp_after:wN \@@_add_return_ii_o:Nww \int_value:w - \fi: - #1 #5 - \s_@@ \@@_chk:w #2 #3 ; - \s_@@ \@@_chk:w #4 #5 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_add_return_ii_o:Nww} -% Ignore the first operand, and return the second, but using the sign -% |#1| rather than |#4|. As usual, expand after the floating point. -% \begin{macrocode} -\cs_new:Npn \@@_add_return_ii_o:Nww #1 #2 ; \s_@@ \@@_chk:w #3 #4 - { \@@_exp_after_o:w \s_@@ \@@_chk:w #3 #1 } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_add_zeros_o:Nww} -% Adding two zeros yields \cs{c_zero_fp}, except if both zeros were -% $-0$. -% \begin{macrocode} -\cs_new:Npn \@@_add_zeros_o:Nww #1 \s_@@ \@@_chk:w 0 #2 - { - \if_int_compare:w #2 #1 = 20 \exp_stop_f: - \exp_after:wN \@@_add_return_ii_o:Nww - \else: - \@@_case_return_i_o:ww - \fi: - #1 - \s_@@ \@@_chk:w 0 #2 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_add_inf_o:Nww} -% If both infinities have the same sign, just return that infinity, -% otherwise, it is an invalid operation. We find out if that invalid -% operation is an addition or a subtraction by testing whether the -% tweaked \meta{sign_2} (|#1|) and the \meta{sign_2} (|#4|) are -% identical. -% \begin{macrocode} -\cs_new:Npn \@@_add_inf_o:Nww - #1 \s_@@ \@@_chk:w 2 #2 #3; \s_@@ \@@_chk:w 2 #4 - { - \if_meaning:w #1 #2 - \@@_case_return_i_o:ww - \else: - \@@_case_use:nw - { - \exp_last_unbraced:Nf \@@_invalid_operation_o:Nww - { \token_if_eq_meaning:NNTF #1 #4 + - } - } - \fi: - \s_@@ \@@_chk:w 2 #2 #3; - \s_@@ \@@_chk:w 2 #4 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_add_normal_o:Nww} -% \begin{quote} -% \cs{@@_add_normal_o:Nww} \meta{sign_2} -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{sign_1} -% \meta{exp_1} \meta{body_1} |;| -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{initial sign_2} -% \meta{exp_2} \meta{body_2} |;| -% \end{quote} -% We now have two normal numbers to add, and we have to check signs -% and exponents more carefully before performing the addition. -% \begin{macrocode} -\cs_new:Npn \@@_add_normal_o:Nww #1 \s_@@ \@@_chk:w 1 #2 - { - \if_meaning:w #1#2 - \exp_after:wN \@@_add_npos_o:NnwNnw - \else: - \exp_after:wN \@@_sub_npos_o:NnwNnw - \fi: - #2 - } -% \end{macrocode} -% \end{macro} -% -% \subsubsection{Absolute addition} -% -% In this subsection, we perform the addition of two positive normal -% numbers. -% -% \begin{macro}[EXP]{\@@_add_npos_o:NnwNnw} -% \begin{quote} -% \cs{@@_add_npos_o:NnwNnw} \meta{sign_1} \meta{exp_1} \meta{body_1} -% |;| \cs{s_@@} \cs{@@_chk:w} |1| \meta{initial sign_2} \meta{exp_2} -% \meta{body_2} |;| -% \end{quote} -% Since we are doing an addition, the final sign is \meta{sign_1}. -% Start an \cs{@@_int_eval:w}, responsible for computing the exponent: -% the result, and the \meta{final sign} are then given to -% \cs{@@_sanitize:Nw} which checks for overflow. The exponent is -% computed as the largest exponent |#2| or |#5|, incremented if there -% is a carry. To add the significands, we decimate the smaller number by -% the difference between the exponents. This is done by -% \cs{@@_add_big_i:wNww} or \cs{@@_add_big_ii:wNww}. We need to bring -% the final sign with us in the midst of the calculation to round -% properly at the end. -% \begin{macrocode} -\cs_new:Npn \@@_add_npos_o:NnwNnw #1#2#3 ; \s_@@ \@@_chk:w 1 #4 #5 - { - \exp_after:wN \@@_sanitize:Nw - \exp_after:wN #1 - \int_value:w \@@_int_eval:w - \if_int_compare:w #2 > #5 \exp_stop_f: - #2 - \exp_after:wN \@@_add_big_i_o:wNww \int_value:w - - \else: - #5 - \exp_after:wN \@@_add_big_ii_o:wNww \int_value:w - \fi: - \@@_int_eval:w #5 - #2 ; #1 #3; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_add_big_i_o:wNww} -% \begin{macro}[rEXP]{\@@_add_big_ii_o:wNww} -% \begin{quote} -% \cs{@@_add_big_i_o:wNww} \meta{shift} |;| \meta{final sign} -% \meta{body_1} |;| \meta{body_2} |;| -% \end{quote} -% Used in \pkg{l3fp-expo}. -% Shift the significand of the small number, then add with -% \cs{@@_add_significand_o:NnnwnnnnN}. -% \begin{macrocode} -\cs_new:Npn \@@_add_big_i_o:wNww #1; #2 #3; #4; - { - \@@_decimate:nNnnnn {#1} - \@@_add_significand_o:NnnwnnnnN - #4 - #3 - #2 - } -\cs_new:Npn \@@_add_big_ii_o:wNww #1; #2 #3; #4; - { - \@@_decimate:nNnnnn {#1} - \@@_add_significand_o:NnnwnnnnN - #3 - #4 - #2 - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_add_significand_o:NnnwnnnnN} -% \begin{macro}[rEXP] -% {\@@_add_significand_pack:NNNNNNN, \@@_add_significand_test_o:N} -% \begin{quote}\raggedright -% \cs{@@_add_significand_o:NnnwnnnnN} -% \meta{rounding digit} -% \Arg{Y'_1} \Arg{Y'_2} \meta{extra-digits} |;| -% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} -% \meta{final sign} -% \end{quote} -% To round properly, we must know at which digit the rounding -% should occur. This requires to know whether the addition -% produces an overall carry or not. Thus, we do the computation -% now and check for a carry, then go back and do the rounding. -% The rounding may cause a carry in very rare cases such as -% $0.99\cdots 95 \to 1.00\cdots 0$, but this situation always -% give an exact power of $10$, for which it is easy to correct -% the result at the end. -% \begin{macrocode} -\cs_new:Npn \@@_add_significand_o:NnnwnnnnN #1 #2#3 #4; #5#6#7#8 - { - \exp_after:wN \@@_add_significand_test_o:N - \int_value:w \@@_int_eval:w 1#5#6 + #2 - \exp_after:wN \@@_add_significand_pack:NNNNNNN - \int_value:w \@@_int_eval:w 1#7#8 + #3 ; #1 - } -\cs_new:Npn \@@_add_significand_pack:NNNNNNN #1 #2#3#4#5#6#7 - { - \if_meaning:w 2 #1 - + 1 - \fi: - ; #2 #3 #4 #5 #6 #7 ; - } -\cs_new:Npn \@@_add_significand_test_o:N #1 - { - \if_meaning:w 2 #1 - \exp_after:wN \@@_add_significand_carry_o:wwwNN - \else: - \exp_after:wN \@@_add_significand_no_carry_o:wwwNN - \fi: - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_add_significand_no_carry_o:wwwNN} -% \begin{quote} -% \cs{@@_add_significand_no_carry_o:wwwNN} -% \meta{8d} |;| \meta{6d} |;| \meta{2d} |;| -% \meta{rounding digit} \meta{sign} -% \end{quote} -% If there's no carry, grab all the digits again and round. The -% packing function \cs{@@_basics_pack_high:NNNNNw} takes care of the -% case where rounding brings a carry. -% \begin{macrocode} -\cs_new:Npn \@@_add_significand_no_carry_o:wwwNN - #1; #2; #3#4 ; #5#6 - { - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1 #1 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 1 #2 #3#4 - + \@@_round:NNN #6 #4 #5 - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_add_significand_carry_o:wwwNN} -% \begin{quote} -% \cs{@@_add_significand_carry_o:wwwNN} -% \meta{8d} |;| \meta{6d} |;| \meta{2d} |;| -% \meta{rounding digit} \meta{sign} -% \end{quote} -% The case where there is a carry is very similar. Rounding can even -% raise the first digit from $1$ to $2$, but we don't care. -% \begin{macrocode} -\cs_new:Npn \@@_add_significand_carry_o:wwwNN - #1; #2; #3#4; #5#6 - { - + 1 - \exp_after:wN \@@_basics_pack_weird_high:NNNNNNNNw - \int_value:w \@@_int_eval:w 1 1 #1 - \exp_after:wN \@@_basics_pack_weird_low:NNNNw - \int_value:w \@@_int_eval:w 1 #2#3 + - \exp_after:wN \@@_round:NNN - \exp_after:wN #6 - \exp_after:wN #3 - \int_value:w \@@_round_digit:Nw #4 #5 ; - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \subsubsection{Absolute subtraction} -% -% \begin{macro}[EXP]{\@@_sub_npos_o:NnwNnw} -% \begin{macro}[EXP]{\@@_sub_eq_o:Nnwnw, \@@_sub_npos_ii_o:Nnwnw} -% \begin{quote} -% \cs{@@_sub_npos_o:NnwNnw} -% \meta{sign_1} \meta{exp_1} \meta{body_1} |;| -% \cs{s_@@} \cs{@@_chk:w} |1| -% \meta{initial sign_2} \meta{exp_2} \meta{body_2} |;| -% \end{quote} -% Rounding properly in some modes requires to know what the sign of -% the result will be. Thus, we start by comparing the exponents and -% significands. If the numbers coincide, return zero. If the second -% number is larger, swap the numbers and call -% \cs{@@_sub_npos_i_o:Nnwnw} with the opposite of \meta{sign_1}. -% \begin{macrocode} -\cs_new:Npn \@@_sub_npos_o:NnwNnw #1#2#3; \s_@@ \@@_chk:w 1 #4#5#6; - { - \if_case:w \@@_compare_npos:nwnw {#2} #3; {#5} #6; \exp_stop_f: - \exp_after:wN \@@_sub_eq_o:Nnwnw - \or: - \exp_after:wN \@@_sub_npos_i_o:Nnwnw - \else: - \exp_after:wN \@@_sub_npos_ii_o:Nnwnw - \fi: - #1 {#2} #3; {#5} #6; - } -\cs_new:Npn \@@_sub_eq_o:Nnwnw #1#2; #3; { \exp_after:wN \c_zero_fp } -\cs_new:Npn \@@_sub_npos_ii_o:Nnwnw #1 #2; #3; - { - \exp_after:wN \@@_sub_npos_i_o:Nnwnw - \int_value:w \@@_neg_sign:N #1 - #3; #2; - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_sub_npos_i_o:Nnwnw} -% After the computation is done, \cs{@@_sanitize:Nw} checks for -% overflow/underflow. It expects the \meta{final sign} and the -% \meta{exponent} (delimited by |;|). Start an integer expression for -% the exponent, which starts with the exponent of the largest number, -% and may be decreased if the two numbers are very close. If the two -% numbers have the same exponent, call the \texttt{near} auxiliary. -% Otherwise, decimate $y$, then call the \texttt{far} auxiliary to -% evaluate the difference between the two significands. Note that we -% decimate by $1$ less than one could expect. -% \begin{macrocode} -\cs_new:Npn \@@_sub_npos_i_o:Nnwnw #1 #2#3; #4#5; - { - \exp_after:wN \@@_sanitize:Nw - \exp_after:wN #1 - \int_value:w \@@_int_eval:w - #2 - \if_int_compare:w #2 = #4 \exp_stop_f: - \exp_after:wN \@@_sub_back_near_o:nnnnnnnnN - \else: - \exp_after:wN \@@_decimate:nNnnnn \exp_after:wN - { \int_value:w \@@_int_eval:w #2 - #4 - 1 \exp_after:wN } - \exp_after:wN \@@_sub_back_far_o:NnnwnnnnN - \fi: - #5 - #3 - #1 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sub_back_near_o:nnnnnnnnN} -% \begin{macro}[rEXP] -% {\@@_sub_back_near_pack:NNNNNNw, \@@_sub_back_near_after:wNNNNw} -% \begin{quote} -% \cs{@@_sub_back_near_o:nnnnnnnnN} -% \Arg{Y_1} \Arg{Y_2} \Arg{Y_3} \Arg{Y_4} -% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} -% \meta{final sign} -% \end{quote} -% In this case, the subtraction is exact, so we discard the -% \meta{final sign} |#9|. The very large shifts of $10^{9}$ and -% $1.1\cdot10^{9}$ are unnecessary here, but allow the auxiliaries to -% be reused later. Each integer expression produces a $10$ digit -% result. If the resulting $16$ digits start with a $0$, then we need -% to shift the group, padding with trailing zeros. -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_near_o:nnnnnnnnN #1#2#3#4 #5#6#7#8 #9 - { - \exp_after:wN \@@_sub_back_near_after:wNNNNw - \int_value:w \@@_int_eval:w 10#5#6 - #1#2 - 11 - \exp_after:wN \@@_sub_back_near_pack:NNNNNNw - \int_value:w \@@_int_eval:w 11#7#8 - #3#4 \exp_after:wN ; - } -\cs_new:Npn \@@_sub_back_near_pack:NNNNNNw #1#2#3#4#5#6#7 ; - { + #1#2 ; {#3#4#5#6} {#7} ; } -\cs_new:Npn \@@_sub_back_near_after:wNNNNw 10 #1#2#3#4 #5 ; - { - \if_meaning:w 0 #1 - \exp_after:wN \@@_sub_back_shift:wnnnn - \fi: - ; {#1#2#3#4} {#5} - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sub_back_shift:wnnnn} -% \begin{macro}[rEXP] -% { -% \@@_sub_back_shift_ii:ww, -% \@@_sub_back_shift_iii:NNNNNNNNw, -% \@@_sub_back_shift_iv:nnnnw -% } -% \begin{quote} -% \cs{@@_sub_back_shift:wnnnn} |;| -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} |;| -% \end{quote} -% This function is called with $\meta{Z_1}\leq 999$. Act with -% \tn{number} to trim leading zeros from \meta{Z_1} \meta{Z_2} (we -% don't do all four blocks at once, since non-zero blocks would then -% overflow \TeX{}'s integers). If the first two blocks are zero, the -% auxiliary receives an empty |#1| and trims |#2#30| from leading -% zeros, yielding a total shift between $7$ and~$16$ to the exponent. -% Otherwise we get the shift from |#1| alone, yielding a result -% between $1$ and~$6$. Once the exponent is taken care of, trim -% leading zeros from |#1#2#3| (when |#1| is empty, the space before -% |#2#3| is ignored), get four blocks of $4$~digits and finally clean -% up. Trailing zeros are added so that digits can be grabbed safely. -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_shift:wnnnn ; #1#2 - { - \exp_after:wN \@@_sub_back_shift_ii:ww - \int_value:w #1 #2 0 ; - } -\cs_new:Npn \@@_sub_back_shift_ii:ww #1 0 ; #2#3 ; - { - \if_meaning:w @ #1 @ - - 7 - - \exp_after:wN \use_i:nnn - \exp_after:wN \@@_sub_back_shift_iii:NNNNNNNNw - \int_value:w #2#3 0 ~ 123456789; - \else: - - \@@_sub_back_shift_iii:NNNNNNNNw #1 123456789; - \fi: - \exp_after:wN \@@_pack_twice_four:wNNNNNNNN - \exp_after:wN \@@_pack_twice_four:wNNNNNNNN - \exp_after:wN \@@_sub_back_shift_iv:nnnnw - \exp_after:wN ; - \int_value:w - #1 ~ #2#3 0 ~ 0000 0000 0000 000 ; - } -\cs_new:Npn \@@_sub_back_shift_iii:NNNNNNNNw #1#2#3#4#5#6#7#8#9; {#8} -\cs_new:Npn \@@_sub_back_shift_iv:nnnnw #1 ; #2 ; { ; #1 ; } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sub_back_far_o:NnnwnnnnN} -% \begin{quote}\raggedright -% \cs{@@_sub_back_far_o:NnnwnnnnN} -% \meta{rounding} \Arg{Y'_1} \Arg{Y'_2} \meta{extra-digits} |;| -% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} -% \meta{final sign} -% \end{quote} -% If the difference is greater than $10^{\meta{expo_x}}$, call the -% \texttt{very_far} auxiliary. If the result is less than -% $10^{\meta{expo_x}}$, call the \texttt{not_far} auxiliary. If it is -% too close a call to know yet, namely if $1 \meta{Y'_1} \meta{Y'_2} = -% \meta{X_1} \meta{X_2} \meta{X_3} \meta{X_4} 0$, then call the -% \texttt{quite_far} auxiliary. We use the odd combination of space -% and semi-colon delimiters to allow the \texttt{not_far} auxiliary to -% grab each piece individually, the \texttt{very_far} auxiliary to use -% \cs{@@_pack_eight:wNNNNNNNN}, and the \texttt{quite_far} to ignore -% the significands easily (using the |;| delimiter). -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_far_o:NnnwnnnnN #1 #2#3 #4; #5#6#7#8 - { - \if_case:w - \if_int_compare:w 1 #2 = #5#6 \use_i:nnnn #7 \exp_stop_f: - \if_int_compare:w #3 = \use_none:n #7#8 0 \exp_stop_f: - 0 - \else: - \if_int_compare:w #3 > \use_none:n #7#8 0 - \fi: 1 - \fi: - \else: - \if_int_compare:w 1 #2 > #5#6 \use_i:nnnn #7 - \fi: 1 - \fi: - \exp_stop_f: - \exp_after:wN \@@_sub_back_quite_far_o:wwNN - \or: \exp_after:wN \@@_sub_back_very_far_o:wwwwNN - \else: \exp_after:wN \@@_sub_back_not_far_o:wwwwNN - \fi: - #2 ~ #3 ; #5 #6 ~ #7 #8 ; #1 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_sub_back_quite_far_o:wwNN} -% \begin{macro}[EXP]{\@@_sub_back_quite_far_ii:NN} -% The easiest case is when $x-y$ is extremely close to a power of -% $10$, namely the first digit of $x$ is $1$, and all others vanish -% when subtracting $y$. Then the \meta{rounding} |#3| and the -% \meta{final sign} |#4| control whether we get $1$ or $0.9999 9999 -% 9999 9999$. In the usual round-to-nearest mode, we get $1$ -% whenever the \meta{rounding} digit is less than or equal to $5$ -% (remember that the \meta{rounding} digit is only equal to $5$ if -% there was no further non-zero digit). -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_quite_far_o:wwNN #1; #2; #3#4 - { - \exp_after:wN \@@_sub_back_quite_far_ii:NN - \exp_after:wN #3 - \exp_after:wN #4 - } -\cs_new:Npn \@@_sub_back_quite_far_ii:NN #1#2 - { - \if_case:w \@@_round_neg:NNN #2 0 #1 - \exp_after:wN \use_i:nn - \else: - \exp_after:wN \use_ii:nn - \fi: - { ; {1000} {0000} {0000} {0000} ; } - { - 1 ; {9999} {9999} {9999} {9999} ; } - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sub_back_not_far_o:wwwwNN} -% In the present case, $x$ and $y$ have different exponents, but -% $y$~is large enough that $x-y$ has a smaller exponent than~$x$. -% Decrement the exponent (with |-1|). Then proceed in a way -% similar to the \texttt{near} auxiliaries seen earlier, but -% multiplying $x$ by~$10$ (|#30| and |#40| below), and with the added -% quirk that the \meta{rounding} digit has to be taken into account. -% Namely, we may have to decrease the result by one unit if -% \cs{@@_round_neg:NNN} returns~$1$. This function expects the -% \meta{final sign}~|#6|, the last digit of |1100000000+#40-#2|, and -% the \meta{rounding} digit. Instead of redoing the computation for -% the second argument, we note that \cs{@@_round_neg:NNN} only cares -% about its parity, which is identical to that of the last digit -% of~|#2|. -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_not_far_o:wwwwNN #1 ~ #2; #3 ~ #4; #5#6 - { - - 1 - \exp_after:wN \@@_sub_back_near_after:wNNNNw - \int_value:w \@@_int_eval:w 1#30 - #1 - 11 - \exp_after:wN \@@_sub_back_near_pack:NNNNNNw - \int_value:w \@@_int_eval:w 11 0000 0000 + #40 - #2 - - \exp_after:wN \@@_round_neg:NNN - \exp_after:wN #6 - \use_none:nnnnnnn #2 #5 - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_sub_back_very_far_o:wwwwNN} -% \begin{macro}[EXP]{\@@_sub_back_very_far_ii_o:nnNwwNN} -% The case where $x-y$ and $x$ have the same exponent is a bit more -% tricky, mostly because it cannot reuse the same auxiliaries. Shift -% the $y$~significand by adding a leading~$0$. Then the logic is similar -% to the \texttt{not_far} functions above. Rounding is a bit more -% complicated: we have two \meta{rounding} digits |#3| and |#6| (from -% the decimation, and from the new shift) to take into account, and -% getting the parity of the main result requires a computation. The -% first \cs{int_value:w} triggers the second one because the number -% is unfinished; we can thus not use $0$ in place of $2$ there. -% \begin{macrocode} -\cs_new:Npn \@@_sub_back_very_far_o:wwwwNN #1#2#3#4#5#6#7 - { - \@@_pack_eight:wNNNNNNNN - \@@_sub_back_very_far_ii_o:nnNwwNN - { 0 #1#2#3 #4#5#6#7 } - ; - } -\cs_new:Npn \@@_sub_back_very_far_ii_o:nnNwwNN #1#2 ; #3 ; #4 ~ #5; #6#7 - { - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1#4 - #1 - 1 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 2#5 - #2 - - \exp_after:wN \@@_round_neg:NNN - \exp_after:wN #7 - \int_value:w - \if_int_odd:w \@@_int_eval:w #5 - #2 \@@_int_eval_end: - 1 \else: 2 \fi: - \int_value:w \@@_round_digit:Nw #3 #6 ; - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \subsection{Multiplication} -% -% \subsubsection{Signs, and special numbers} -% -% \begin{macro}[EXP]{\@@_*_o:ww} -% We go through an auxiliary, which is common with \cs{@@_/_o:ww}. -% The first argument is the operation, used for the invalid operation -% exception. The second is inserted in a formula to dispatch cases -% slightly differently between multiplication and division. The third -% is the operation for normal floating points. The fourth is there -% for extra cases needed in \cs{@@_/_o:ww}. -% \begin{macrocode} -\cs_new:cpn { @@_*_o:ww } - { - \@@_mul_cases_o:NnNnww - * - { - 2 + } - \@@_mul_npos_o:Nww - { } - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_mul_cases_o:nNnnww} -% Split into $10$ cases ($12$ for division). -% If both numbers are normal, go to case $0$ -% (same sign) or case $1$ (opposite signs): in both cases, call -% \cs{@@_mul_npos_o:Nww} to do the work. If the first operand is -% \texttt{nan}, go to case $2$, in which the second operand is -% discarded; if the second operand is \texttt{nan}, go to case $3$, in -% which the first operand is discarded (note the weird interaction -% with the final test on signs). Then we separate the case where the -% first number is normal and the second is zero: this goes to cases -% $4$ and $5$ for multiplication, $10$ and $11$ for division. -% Otherwise, we do a computation which -% dispatches the products $0\times 0 = 0\times 1 = 1\times 0 = 0$ to -% case $4$ or $5$ depending on the combined sign, the products -% $0\times\infty$ and $\infty\times0$ to case $6$ or $7$ (invalid -% operation), and the products $1\times\infty = \infty\times1 = -% \infty\times\infty = \infty$ to cases $8$ and $9$. Note that the -% code for these two cases (which return $\pm\infty$) is inserted as -% argument |#4|, because it differs in the case of divisions. -% \begin{macrocode} -\cs_new:Npn \@@_mul_cases_o:NnNnww - #1#2#3#4 \s_@@ \@@_chk:w #5#6#7; \s_@@ \@@_chk:w #8#9 - { - \if_case:w \@@_int_eval:w - \if_int_compare:w #5 #8 = 11 ~ - 1 - \else: - \if_meaning:w 3 #8 - 3 - \else: - \if_meaning:w 3 #5 - 2 - \else: - \if_int_compare:w #5 #8 = 10 ~ - 9 #2 - 2 - \else: - (#5 #2 #8) / 2 * 2 + 7 - \fi: - \fi: - \fi: - \fi: - \if_meaning:w #6 #9 - 1 \fi: - \@@_int_eval_end: - \@@_case_use:nw { #3 0 } - \or: \@@_case_use:nw { #3 2 } - \or: \@@_case_return_i_o:ww - \or: \@@_case_return_ii_o:ww - \or: \@@_case_return_o:Nww \c_zero_fp - \or: \@@_case_return_o:Nww \c_minus_zero_fp - \or: \@@_case_use:nw { \@@_invalid_operation_o:Nww #1 } - \or: \@@_case_use:nw { \@@_invalid_operation_o:Nww #1 } - \or: \@@_case_return_o:Nww \c_inf_fp - \or: \@@_case_return_o:Nww \c_minus_inf_fp - #4 - \fi: - \s_@@ \@@_chk:w #5 #6 #7; - \s_@@ \@@_chk:w #8 #9 - } -% \end{macrocode} -% \end{macro} -% -% \subsubsection{Absolute multiplication} -% -% In this subsection, we perform the multiplication -% of two positive normal numbers. -% -% \begin{macro}[EXP]{\@@_mul_npos_o:Nww} -% \begin{quote} -% \cs{@@_mul_npos_o:Nww} \meta{final sign} -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{sign_1} \Arg{exp_1} \meta{body_1} |;| -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{sign_2} \Arg{exp_2} \meta{body_2} |;| -% \end{quote} -% After the computation, \cs{@@_sanitize:Nw} checks for overflow or -% underflow. As we did for addition, \cs{@@_int_eval:w} computes the -% exponent, catching any shift coming from the computation in the -% significand. The \meta{final sign} is needed to do the rounding -% properly in the significand computation. We setup the post-expansion -% here, triggered by \cs{@@_mul_significand_o:nnnnNnnnn}. -% -% This is also used in \pkg{l3fp-convert}. -% \begin{macrocode} -\cs_new:Npn \@@_mul_npos_o:Nww - #1 \s_@@ \@@_chk:w #2 #3 #4 #5 ; \s_@@ \@@_chk:w #6 #7 #8 #9 ; - { - \exp_after:wN \@@_sanitize:Nw - \exp_after:wN #1 - \int_value:w \@@_int_eval:w - #4 + #8 - \@@_mul_significand_o:nnnnNnnnn #5 #1 #9 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_mul_significand_o:nnnnNnnnn} -% \begin{macro}[EXP] -% {\@@_mul_significand_drop:NNNNNw, \@@_mul_significand_keep:NNNNNw} -% \begin{quote} -% \cs{@@_mul_significand_o:nnnnNnnnn} -% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \meta{sign} -% \Arg{Y_1} \Arg{Y_2} \Arg{Y_3} \Arg{Y_4} -% \end{quote} -% Note the three semicolons at the end of the definition. One is for -% the last \cs{@@_mul_significand_drop:NNNNNw}; one is for -% \cs{@@_round_digit:Nw} later on; and one, preceded by -% \cs{exp_after:wN}, which is correctly expanded (within an -% \cs{@@_int_eval:w}), is used by \cs{@@_basics_pack_low:NNNNNw}. -% -% The product of two $16$ digit integers has $31$ or $32$ digits, -% but it is impossible to know which one before computing. The place -% where we round depends on that number of digits, and may depend -% on all digits until the last in some rare cases. The approach is -% thus to compute the $5$ first blocks of $4$ digits (the first one -% is between $100$ and $9999$ inclusive), and a compact version of -% the remaining $3$ blocks. Afterwards, the number of digits is -% known, and we can do the rounding within yet another set of -% \cs{@@_int_eval:w}. -% \begin{macrocode} -\cs_new:Npn \@@_mul_significand_o:nnnnNnnnn #1#2#3#4 #5 #6#7#8#9 - { - \exp_after:wN \@@_mul_significand_test_f:NNN - \exp_after:wN #5 - \int_value:w \@@_int_eval:w 99990000 + #1*#6 + - \exp_after:wN \@@_mul_significand_keep:NNNNNw - \int_value:w \@@_int_eval:w 99990000 + #1*#7 + #2*#6 + - \exp_after:wN \@@_mul_significand_keep:NNNNNw - \int_value:w \@@_int_eval:w 99990000 + #1*#8 + #2*#7 + #3*#6 + - \exp_after:wN \@@_mul_significand_drop:NNNNNw - \int_value:w \@@_int_eval:w 99990000 + #1*#9 + #2*#8 + - #3*#7 + #4*#6 + - \exp_after:wN \@@_mul_significand_drop:NNNNNw - \int_value:w \@@_int_eval:w 99990000 + #2*#9 + #3*#8 + - #4*#7 + - \exp_after:wN \@@_mul_significand_drop:NNNNNw - \int_value:w \@@_int_eval:w 99990000 + #3*#9 + #4*#8 + - \exp_after:wN \@@_mul_significand_drop:NNNNNw - \int_value:w \@@_int_eval:w 100000000 + #4*#9 ; - ; \exp_after:wN ; - } -\cs_new:Npn \@@_mul_significand_drop:NNNNNw #1#2#3#4#5 #6; - { #1#2#3#4#5 ; + #6 } -\cs_new:Npn \@@_mul_significand_keep:NNNNNw #1#2#3#4#5 #6; - { #1#2#3#4#5 ; #6 ; } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_mul_significand_test_f:NNN} -% \begin{quote} -% \cs{@@_mul_significand_test_f:NNN} \meta{sign} |1| -% \meta{digits 1--8} |;| \meta{digits 9--12} |;| \meta{digits 13--16} |;| -% |+| \meta{digits 17--20} |+| \meta{digits 21--24} -% |+| \meta{digits 25--28} |+| \meta{digits 29--32} |;| -% \cs{exp_after:wN} |;| -% \end{quote} -% If the \meta{digit 1} is non-zero, then for rounding we only care -% about the digits $16$ and $17$, and whether further digits are zero -% or not (check for exact ties). On the other hand, if \meta{digit 1} -% is zero, we care about digits $17$ and $18$, and whether further -% digits are zero. -% \begin{macrocode} -\cs_new:Npn \@@_mul_significand_test_f:NNN #1 #2 #3 - { - \if_meaning:w 0 #3 - \exp_after:wN \@@_mul_significand_small_f:NNwwwN - \else: - \exp_after:wN \@@_mul_significand_large_f:NwwNNNN - \fi: - #1 #3 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_mul_significand_large_f:NwwNNNN} -% In this branch, \meta{digit 1} is non-zero. The result is thus -% \meta{digits 1--16}, plus some rounding which depends on the digits -% $16$, $17$, and whether all subsequent digits are zero or not. -% Here, \cs{@@_round_digit:Nw} takes digits $17$ and further (as an -% integer expression), and replaces it by a \meta{rounding digit}, -% suitable for \cs{@@_round:NNN}. -% \begin{macrocode} -\cs_new:Npn \@@_mul_significand_large_f:NwwNNNN #1 #2; #3; #4#5#6#7; + - { - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1#2 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 1#3#4#5#6#7 - + \exp_after:wN \@@_round:NNN - \exp_after:wN #1 - \exp_after:wN #7 - \int_value:w \@@_round_digit:Nw - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_mul_significand_small_f:NNwwwN} -% In this branch, \meta{digit 1} is zero. Our result is thus -% \meta{digits 2--17}, plus some rounding which depends on the digits -% $17$, $18$, and whether all subsequent digits are zero or not. -% The $8$ digits |1#3| are followed, after expansion of the -% \texttt{small_pack} auxiliary, by the next digit, to form a $9$ -% digit number. -% \begin{macrocode} -\cs_new:Npn \@@_mul_significand_small_f:NNwwwN #1 #2#3; #4#5; #6; + #7 - { - - 1 - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1#3#4 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 1#5#6#7 - + \exp_after:wN \@@_round:NNN - \exp_after:wN #1 - \exp_after:wN #7 - \int_value:w \@@_round_digit:Nw - } -% \end{macrocode} -% \end{macro} -% -% \subsection{Division} -% -% \subsubsection{Signs, and special numbers} -% -% Time is now ripe to tackle the hardest of the four elementary -% operations: division. -% -% \begin{macro}[EXP]{\@@_/_o:ww} -% Filtering special floating point is very similar to what we did for -% multiplications, with a few variations. Invalid operation -% exceptions display |/| rather than |*|. In the formula for -% dispatch, we replace |- 2 +| by |-|. The case of normal -% numbers is treated using \cs{@@_div_npos_o:Nww} rather than -% \cs{@@_mul_npos_o:Nww}. There are two additional cases: if the -% first operand is normal and the second is a zero, then the division -% by zero exception is raised: cases $10$ and $11$ of the -% \cs{if_case:w} construction in \cs{@@_mul_cases_o:NnNnww} are -% provided as the fourth argument here. -% \begin{macrocode} -\cs_new:cpn { @@_/_o:ww } - { - \@@_mul_cases_o:NnNnww - / - { - } - \@@_div_npos_o:Nww - { - \or: - \@@_case_use:nw - { \@@_division_by_zero_o:NNww \c_inf_fp / } - \or: - \@@_case_use:nw - { \@@_division_by_zero_o:NNww \c_minus_inf_fp / } - } - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_div_npos_o:Nww} -% \begin{quote} -% \cs{@@_div_npos_o:Nww} \meta{final sign} -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{sign_A} \Arg{exp A} -% \Arg{A_1} \Arg{A_2} \Arg{A_3} \Arg{A_4} |;| -% \cs{s_@@} \cs{@@_chk:w} |1| \meta{sign_Z} \Arg{exp Z} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} |;| -% \end{quote} -% We want to compute $A/Z$. As for multiplication, -% \cs{@@_sanitize:Nw} checks for overflow or underflow; we provide it -% with the \meta{final sign}, and an integer expression in which we -% compute the exponent. We set up the arguments of -% \cs{@@_div_significand_i_o:wnnw}, namely an integer \meta{y} obtained -% by adding $1$ to the first $5$ digits of $Z$ (explanation given soon -% below), then the four \Arg{A_{i}}, then the four \Arg{Z_{i}}, a -% semi-colon, and the \meta{final sign}, used for rounding at the end. -% \begin{macrocode} -\cs_new:Npn \@@_div_npos_o:Nww - #1 \s_@@ \@@_chk:w 1 #2 #3 #4 ; \s_@@ \@@_chk:w 1 #5 #6 #7#8#9; - { - \exp_after:wN \@@_sanitize:Nw - \exp_after:wN #1 - \int_value:w \@@_int_eval:w - #3 - #6 - \exp_after:wN \@@_div_significand_i_o:wnnw - \int_value:w \@@_int_eval:w #7 \use_i:nnnn #8 + 1 ; - #4 - {#7}{#8}#9 ; - #1 - } -% \end{macrocode} -% \end{macro} -% -% \subsubsection{Work plan} -% -% In this subsection, we explain how to avoid overflowing \TeX{}'s -% integers when performing the division of two positive normal numbers. -% -% We are given two numbers, $A=0.A_{1}A_{2}A_{3}A_{4}$ and -% $Z=0.Z_{1}Z_{2}Z_{3}Z_{4}$, in blocks of $4$ digits, and we know that -% the first digits of $A_{1}$ and of $Z_{1}$ are non-zero. To compute -% $A/Z$, we proceed as follows. -% \begin{itemize} -% \item Find an integer $Q_{A} \simeq 10^{4} A / Z$. -% \item Replace $A$ by $B = 10^{4} A - Q_{A} Z$. -% \item Find an integer $Q_{B} \simeq 10^{4} B / Z$. -% \item Replace $B$ by $C = 10^{4} B - Q_{B} Z$. -% \item Find an integer $Q_{C} \simeq 10^{4} C / Z$. -% \item Replace $C$ by $D = 10^{4} C - Q_{C} Z$. -% \item Find an integer $Q_{D} \simeq 10^{4} D / Z$. -% \item Consider $E = 10^{4} D - Q_{D} Z$, and ensure -% correct rounding. -% \end{itemize} -% The result is then $Q = 10^{-4} Q_{A} + 10^{-8} Q_{B} + 10^{-12} Q_{C} -% + 10^{-16} Q_{D} + \text{rounding}$. Since the $Q_{i}$ are integers, -% $B$, $C$, $D$, and~$E$ are all exact multiples of $10^{-16}$, in other -% words, computing with $16$ digits after the decimal separator yields -% exact results. The problem is the risk of overflow: in general $B$, $C$, -% $D$, and $E$ may be greater than $1$. -% -% Unfortunately, things are not as easy as they seem. In particular, we -% want all intermediate steps to be positive, since negative results -% would require extra calculations at the end. This requires that -% $Q_{A} \leq 10^{4} A / Z$ \emph{etc.} A reasonable attempt would be -% to define $Q_{A}$ as -% \begin{equation*} -% \cs{int_eval:n} \left\{ -% \frac{ A_{1} A_{2} }{ Z_{1} + 1 } - 1 \right\} -% \leq 10^{4} \frac{A}{Z} -% \end{equation*} -% Subtracting $1$ at the end takes care of the fact that \eTeX{}'s -% \cs{@@_int_eval:w} rounds divisions instead of truncating (really, -% $1/2$ would be sufficient, but we work with integers). We add $1$ to -% $Z_{1}$ because $Z_{1} \leq 10^{4}Z < Z_{1}+1$ and we need $Q_{A}$ to -% be an underestimate. However, we are now underestimating $Q_{A}$ too -% much: it can be wrong by up to $100$, for instance when $Z = 0.1$ and -% $A \simeq 1$. Then $B$ could take values up to $10$ (maybe more), and -% a few steps down the line, we would run into arithmetic overflow, -% since \TeX{} can only handle integers less than roughly $2\cdot -% 10^{9}$. -% -% A better formula is to take -% \begin{equation*} -% Q_{A} = \cs{int_eval:n} \left\{ -% \frac{ 10 \cdot A_{1} A_{2} } -% { \left\lfloor 10^{-3} \cdot Z_{1} Z_{2} \right\rfloor + 1 } -% - 1 \right\}. -% \end{equation*} -% This is always less than $10^{9} A / (10^{5} Z)$, as we wanted. In -% words, we take the $5$ first digits of $Z$ into account, and the $8$ -% first digits of $A$, using $0$ as a $9$-th digit rather than the true -% digit for efficiency reasons. We shall prove that using this formula -% to define all the $Q_{i}$ avoids any overflow. For convenience, let -% us denote -% \begin{equation*} -% y = \left\lfloor 10^{-3} \cdot Z_{1} Z_{2} \right\rfloor + 1, -% \end{equation*} -% so that, taking into account the fact that \eTeX{} rounds ties away -% from zero, -% \begin{align*} -% Q_{A} -% &= \left\lfloor \frac{A_{1}A_{2}0}{y} - \frac{1}{2} \right\rfloor -% \\ -% &>\frac{A_{1}A_{2}0}{y} - \frac{3}{2}. -% \end{align*} -% Note that $10^{4}<y\leq 10^{5}$, and $999 \leq Q_{A} \leq 99989$. -% Also note that this formula does not cause an overflow as long as $A < -% (2^{31}-1) / 10^{9} \simeq 2.147\cdots$, since the numerator involves an -% integer slightly smaller than $10^{9} A$. -% -% Let us bound $B$: -% \begin{align*} -% 10^{5} B -% &= -% A_{1}A_{2}0 + 10 \cdot 0.A_{3}A_{4} -% - 10 \cdot Z_{1}.Z_{2}Z_{3}Z_{4} \cdot Q_{A} -% \\ -% &< -% A_{1}A_{2}0 -% \cdot \left( 1 - 10 \cdot \frac{Z_{1}.Z_{2}Z_{3}Z_{4}}{y} \right) -% + \frac{3}{2} \cdot 10 \cdot Z_{1}.Z_{2}Z_{3}Z_{4} + 10 -% \\ -% &\leq -% \frac{A_{1}A_{2}0 \cdot (y - 10 \cdot Z_{1}.Z_{2}Z_{3}Z_{4})}{y} -% + \frac{3}{2} y + 10 -% \\ -% &\leq -% \frac{A_{1}A_{2}0\cdot 1}{y} + \frac{3}{2} y + 10 -% \leq -% \frac{10^{9} A}{y} + 1.6\cdot y. -% \end{align*} -% At the last step, we hide $10$ into the second term for later -% convenience. The same reasoning yields -% \begin{align*} -% 10^{5} B &< 10^{9} A/y + 1.6 y, \\ -% 10^{5} C &< 10^{9} B/y + 1.6 y, \\ -% 10^{5} D &< 10^{9} C/y + 1.6 y, \\ -% 10^{5} E &< 10^{9} D/y + 1.6 y. \\ -% \end{align*} -% The goal is now to prove that none of $B$, $C$, $D$, and $E$ can go -% beyond $(2^{31}-1) / 10^{9} = 2.147\cdots$. -% -% Combining the various inequalities together with $A<1$, we get -% \begin{align*} -% 10^{5} B &< 10^{9}/y + 1.6 y, \\ -% 10^{5} C &< 10^{13}/y^{2} + 1.6 (y + 10^{4}), \\ -% 10^{5} D &< 10^{17}/y^{3} + 1.6 (y + 10^{4} + 10^{8}/y), \\ -% 10^{5} E &< 10^{21}/y^{4} + 1.6 (y + 10^{4} + 10^{8}/y + 10^{12}/y^{2}). \\ -% \end{align*} -% All of those bounds are convex functions of $y$ (since every power of -% $y$ involved is convex, and the coefficients are positive), and thus -% maximal at one of the end-points of the allowed range $10^{4} < y \leq -% 10^{5}$. Thus, -% \begin{align*} -% 10^{5} B &< \mathrm{max} ( 1.16\cdot 10^{5}, 1.7 \cdot 10^{5}), \\ -% 10^{5} C &< \mathrm{max} ( 1.32\cdot 10^{5}, 1.77 \cdot 10^{5}), \\ -% 10^{5} D &< \mathrm{max} ( 1.48\cdot 10^{5}, 1.777 \cdot 10^{5}), \\ -% 10^{5} E &< \mathrm{max} ( 1.64\cdot 10^{5}, 1.7777 \cdot 10^{5}). \\ -% \end{align*} -% All of those bounds are less than $2.147\cdot 10^{5}$, and we are thus -% within \TeX{}'s bounds in all cases! -% -% We later need to have a bound on the $Q_{i}$. Their definitions -% imply that $Q_{A} < 10^{9} A/y - 1/2 < 10^{5} A$ and similarly for the -% other $Q_{i}$. Thus, all of them are less than $177770$. -% -% The last step is to ensure correct rounding. We have -% \begin{equation*} -% A/Z = \sum_{i=1}^{4} \left(10^{-4i} Q_{i}\right) + 10^{-16} E/Z -% \end{equation*} -% exactly. Furthermore, we know that the result is in $[0.1,10)$, -% hence will be rounded to a multiple of $10^{-16}$ or of $10^{-15}$, so -% we only need to know the integer part of $E/Z$, and a -% \enquote{rounding} digit encoding the rest. Equivalently, we need to -% find the integer part of $2E/Z$, and determine whether it was an -% exact integer or not (this serves to detect ties). Since -% \begin{equation*} -% \frac{2E}{Z} = 2\frac{10^{5} E}{10^{5} Z} -% \leq 2\frac{10^{5} E}{10^{4}} < 36, -% \end{equation*} -% this integer part is between $0$ and $35$ inclusive. We let \eTeX{} -% round -% \begin{equation*} -% P = \cs{int_eval:n} \left\{ -% \frac{2\cdot E_{1}E_{2}}{Z_{1}Z_{2}} \right\}, -% \end{equation*} -% which differs from $2E/Z$ by at most -% \begin{equation*} -% \frac{1}{2} -% + 2 \left\lvert \frac{E}{Z} - \frac{E}{10^{-8} Z_{1}Z_{2}}\right\rvert -% + 2 \left\lvert \frac{10^{8} E - E_{1}E_{2}}{Z_{1}Z_{2}}\right\rvert -% < 1, -% \end{equation*} -% ($1/2$ comes from \eTeX{}'s rounding) because each absolute value is -% less than $10^{-7}$. Thus $P$ is either the correct integer part, or -% is off by $1$; furthermore, if $2 E / Z$ is an integer, $P = 2 E / Z$. -% We will check the sign of $2 E - P Z$. If it is negative, then $E / Z -% \in \big((P - 1) / 2, P / 2\big)$. If it is zero, then $E / Z = P / -% 2$. If it is positive, then $E / Z \in \big(P / 2, (P - 1) / 2\big)$. -% In each case, we know how to round to an integer, depending on the -% parity of $P$, and the rounding mode. -% -% \subsubsection{Implementing the significand division} -% -% \begin{macro}[rEXP]{\@@_div_significand_i_o:wnnw} -% \begin{quote} -% \cs{@@_div_significand_i_o:wnnw} \meta{y} |;| -% \Arg{A_1} \Arg{A_2} \Arg{A_3} \Arg{A_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} |;| \meta{sign} -% \end{quote} -% Compute $10^{6} + Q_{A}$ (a $7$~digit number thanks to the shift), -% unbrace \meta{A_1} and \meta{A_2}, and prepare the -% \meta{continuation} arguments for $4$ consecutive calls to -% \cs{@@_div_significand_calc:wwnnnnnnn}. Each of these calls needs -% \meta{y} (|#1|), and it turns out that we need post-expansion there, -% hence the \cs{int_value:w}. Here, |#4| is six brace groups, which -% give the six first |n|-type arguments of the \texttt{calc} function. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_i_o:wnnw #1 ; #2#3 #4 ; - { - \exp_after:wN \@@_div_significand_test_o:w - \int_value:w \@@_int_eval:w - \exp_after:wN \@@_div_significand_calc:wwnnnnnnn - \int_value:w \@@_int_eval:w 999999 + #2 #3 0 / #1 ; - #2 #3 ; - #4 - { \exp_after:wN \@@_div_significand_ii:wwn \int_value:w #1 } - { \exp_after:wN \@@_div_significand_ii:wwn \int_value:w #1 } - { \exp_after:wN \@@_div_significand_ii:wwn \int_value:w #1 } - { \exp_after:wN \@@_div_significand_iii:wwnnnnn \int_value:w #1 } - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_div_significand_calc:wwnnnnnnn} -% \begin{macro}[rEXP] -% { -% \@@_div_significand_calc_i:wwnnnnnnn, -% \@@_div_significand_calc_ii:wwnnnnnnn, -% } -% \begin{quote} -% \cs{@@_div_significand_calc:wwnnnnnnn} \meta{$10^{6}+{}$Q_{A}} |;| -% \meta{A_1} \meta{A_2} |;| \Arg{A_3} \Arg{A_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} -% \Arg{continuation} -% \end{quote} -% expands to -% \begin{quote} -% \meta{$10^{6}+{}$Q_{A}} \meta{continuation} |;| -% \meta{B_1} \meta{B_2} |;| \Arg{B_3} \Arg{B_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} -% \end{quote} -% where $B = 10^{4} A - Q_{A} \cdot Z$. This function is also used to -% compute $C$, $D$, $E$ (with the input shifted accordingly), and is -% used in \pkg{l3fp-expo}. -% -% We know that $0<Q_{A}<1.8\cdot 10^{5}$, so the product of $Q_{A}$ -% with each $Z_{i}$ is within \TeX{}'s bounds. However, it is a -% little bit too large for our purposes: we would not be able to use -% the usual trick of adding a large power of $10$ to ensure that the -% number of digits is fixed. -% -% The bound on $Q_{A}$, implies that $10^{6}+Q_{A}$ starts with the -% digit $1$, followed by $0$ or $1$. We test, and call different -% auxiliaries for the two cases. An earlier implementation did the -% tests within the computation, but since we added a -% \meta{continuation}, this is not possible because the macro has $9$ -% parameters. -% -% The result we want is then (the overall power of $10$ is arbitrary): -% \begin{align*} -% &10^{-4} ( \#2 - \#1 \cdot \#5 - 10 \cdot \meta{i} \cdot \#5\#6 ) -% + 10^{-8} ( \#3 - \#1 \cdot \#6 - 10 \cdot \meta{i} \cdot \#7 ) \\ -% &+ 10^{-12}( \#4 - \#1 \cdot \#7 - 10 \cdot \meta{i} \cdot \#8 ) -% + 10^{-16}( - \#1 \cdot \#8 ), -% \end{align*} -% where \meta{i} stands for the $10^{5}$ digit of $Q_{A}$, which is -% $0$ or~$1$, and $\#1$, $\#2$, \emph{etc.\@} are the parameters of -% either auxiliary. The factors of $10$ come from the fact that -% $Q_{A} = 10\cdot 10^{4} \cdot \meta{i} + \#1$. As usual, to combine -% all the terms, we need to choose some shifts which must ensure that -% the number of digits of the second, third, and fourth terms are each -% fixed. Here, the positive contributions are at most $10^{8}$ and -% the negative contributions can go up to $10^{9}$. Indeed, for the -% auxiliary with $\meta{i}=1$, |#1| is at most $80000$, leading to -% contributions of at worse $-8\cdot 10^{8}4$, while the other -% negative term is very small $<10^{6}$ (except in the first -% expression, where we don't care about the number of digits); for the -% auxiliary with $\meta{i}=0$, |#1| can go up to $99999$, but there is -% no other negative term. Hence, a good choice is $2\cdot 10^{9}$, -% which produces totals in the range $[10^{9}, 2.1\cdot 10^{9}]$. We -% are flirting with \TeX{}'s limits once more. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_calc:wwnnnnnnn 1#1 - { - \if_meaning:w 1 #1 - \exp_after:wN \@@_div_significand_calc_i:wwnnnnnnn - \else: - \exp_after:wN \@@_div_significand_calc_ii:wwnnnnnnn - \fi: - } -\cs_new:Npn \@@_div_significand_calc_i:wwnnnnnnn - #1; #2;#3#4 #5#6#7#8 #9 - { - 1 1 #1 - #9 \exp_after:wN ; - \int_value:w \@@_int_eval:w \c_@@_Bigg_leading_shift_int - + #2 - #1 * #5 - #5#60 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_middle_shift_int - + #3 - #1 * #6 - #70 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_middle_shift_int - + #4 - #1 * #7 - #80 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_trailing_shift_int - - #1 * #8 ; - {#5}{#6}{#7}{#8} - } -\cs_new:Npn \@@_div_significand_calc_ii:wwnnnnnnn - #1; #2;#3#4 #5#6#7#8 #9 - { - 1 0 #1 - #9 \exp_after:wN ; - \int_value:w \@@_int_eval:w \c_@@_Bigg_leading_shift_int - + #2 - #1 * #5 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_middle_shift_int - + #3 - #1 * #6 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_middle_shift_int - + #4 - #1 * #7 - \exp_after:wN \@@_pack_Bigg:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_Bigg_trailing_shift_int - - #1 * #8 ; - {#5}{#6}{#7}{#8} - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_div_significand_ii:wwn} -% \begin{quote} -% \cs{@@_div_significand_ii:wwn} \meta{y} |;| -% \meta{B_1} |;| \Arg{B_2} \Arg{B_3} \Arg{B_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} -% \meta{continuations} \meta{sign} -% \end{quote} -% Compute $Q_{B}$ by evaluating $\meta{B_1}\meta{B_2}0 / y - 1$. The -% result is output to the left, in an \cs{@@_int_eval:w} which we -% start now. Once that is evaluated (and the other $Q_{i}$ also, -% since later expansions are triggered by this one), a packing -% auxiliary takes care of placing the digits of $Q_{B}$ in an -% appropriate way for the final addition to obtain $Q$. This -% auxiliary is also used to compute $Q_{C}$ and $Q_{D}$ with the -% inputs $C$ and $D$ instead of $B$. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_ii:wwn #1; #2;#3 - { - \exp_after:wN \@@_div_significand_pack:NNN - \int_value:w \@@_int_eval:w - \exp_after:wN \@@_div_significand_calc:wwnnnnnnn - \int_value:w \@@_int_eval:w 999999 + #2 #3 0 / #1 ; #2 #3 ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_div_significand_iii:wwnnnnn} -% \begin{quote} -% \cs{@@_div_significand_iii:wwnnnnn} \meta{y} |;| -% \meta{E_1} |;| \Arg{E_2} \Arg{E_3} \Arg{E_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} \meta{sign} -% \end{quote} -% We compute $P \simeq 2E/Z$ by rounding $2 E_{1} E_{2}/Z_{1}Z_{2}$. -% Note the first $0$, which multiplies $Q_{D}$ by $10$: we later -% add (roughly) $5\cdot P$, which amounts to adding $P/2 \simeq E/Z$ -% to $Q_{D}$, the appropriate correction from a hypothetical $Q_{E}$. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_iii:wwnnnnn #1; #2;#3#4#5 #6#7 - { - 0 - \exp_after:wN \@@_div_significand_iv:wwnnnnnnn - \int_value:w \@@_int_eval:w ( 2 * #2 #3) / #6 #7 ; % <- P - #2 ; {#3} {#4} {#5} - {#6} {#7} - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP] -% { -% \@@_div_significand_iv:wwnnnnnnn, -% \@@_div_significand_v:NNw, -% \@@_div_significand_vi:Nw -% } -% \begin{quote} -% \cs{@@_div_significand_iv:wwnnnnnnn} \meta{P} |;| -% \meta{E_1} |;| \Arg{E_2} \Arg{E_3} \Arg{E_4} -% \Arg{Z_1} \Arg{Z_2} \Arg{Z_3} \Arg{Z_4} \meta{sign} -% \end{quote} -% This adds to the current expression ($10^{7} + 10\cdot Q_{D}$) a -% contribution of $5 \cdot P + \operatorname{sign}(T)$ with $T = 2 E - -% P Z$. This amounts to adding $P / 2$ to $Q_{D}$, with an extra -% \meta{rounding} digit. This \meta{rounding} digit is $0$ or $5$ if -% $T$ does not contribute, \emph{i.e.,} if $0 = T = 2 E - P Z$, in -% other words if $10^{16} A / Z$ is an integer or half-integer. -% Otherwise it is in the appropriate range, $[1,4]$ or $[6,9]$. This -% is precise enough for rounding purposes (in any mode). -% -% It seems an overkill to compute $T$ exactly as I do here, but I see -% no faster way right now. -% -% Once more, we need to be careful and show that the calculation -% $\#1\cdot\#6\#7$ below does not cause an overflow: naively, $P$ can -% be up to $35$, and $\#6\#7$ up to $10^{8}$, but both cannot happen -% simultaneously. To show that things are fine, we split in two -% (non-disjoint) cases. -% \begin{itemize} -% \item For $P < 10$, the product obeys $P\cdot\#6\#7 < 10^{8} \cdot P -% < 10^{9} $. -% \item For large $P\geq 3$, the rounding error on $P$, which is at -% most $1$, is less than a factor of $2$, hence $P\leq 4E/Z$. Also, -% $\#6\#7 \leq 10^{8} \cdot Z$, hence $P\cdot \#6\#7 \leq 4E\cdot -% 10^{8} < 10^{9}$. -% \end{itemize} -% Both inequalities could be made tighter if needed. -% -% Note however that $P\cdot \#8\#9$ may overflow, since the two -% factors are now independent, and the result may reach $3.5\cdot -% 10^{9}$. Thus we compute the two lower levels separately. The rest -% is standard, except that we use |+| as a separator (ending integer -% expressions explicitly). $T$ is negative if the first character is -% |-|, it is positive if the first character is neither |0| nor |-|. -% It is also positive if the first character is |0| and second -% argument of \cs{@@_div_significand_vi:Nw}, a sum of several terms, is -% also zero. Otherwise, there was an exact agreement: $T = 0$. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_iv:wwnnnnnnn #1; #2;#3#4#5 #6#7#8#9 - { - + 5 * #1 - \exp_after:wN \@@_div_significand_vi:Nw - \int_value:w \@@_int_eval:w -50 + 2*#2#3 - #1*#6#7 + - \exp_after:wN \@@_div_significand_v:NN - \int_value:w \@@_int_eval:w 499950 + 2*#4 - #1*#8 + - \exp_after:wN \@@_div_significand_v:NN - \int_value:w \@@_int_eval:w 500000 + 2*#5 - #1*#9 ; - } -\cs_new:Npn \@@_div_significand_v:NN #1#2 { #1#2 \@@_int_eval_end: + } -\cs_new:Npn \@@_div_significand_vi:Nw #1#2; - { - \if_meaning:w 0 #1 - \if_int_compare:w \@@_int_eval:w #2 > 0 + 1 \fi: - \else: - \if_meaning:w - #1 - \else: + \fi: 1 - \fi: - ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_div_significand_pack:NNN} -% At this stage, we are in the following situation: \TeX{} is in the -% process of expanding several integer expressions, thus functions at -% the bottom expand before those above. -% \begin{quote} -% \cs{@@_div_significand_test_o:w} $10^{6} + Q_{A}$ -% \cs{@@_div_significand_pack:NNN} $10^{6} + Q_{B}$ -% \cs{@@_div_significand_pack:NNN} $10^{6} + Q_{C}$ -% \cs{@@_div_significand_pack:NNN} -% $10^{7} + 10\cdot Q_{D} + 5 \cdot P + \varepsilon$ |;| \meta{sign} -% \end{quote} -% Here, $\varepsilon = \operatorname{sign}(T)$ is $0$ in case $2E=PZ$, -% $1$ in case $2E>PZ$, which means that $P$ was the correct value, but -% not with an exact quotient, and $-1$ if $2E<PZ$, \emph{i.e.}, $P$ -% was an overestimate. The packing function we define now does -% nothing special: it removes the $10^{6}$ and carries two digits (for -% the $10^{5}$'s and the $10^{4}$'s). -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_pack:NNN 1 #1 #2 { + #1 #2 ; } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_div_significand_test_o:w} -% \begin{quote} -% \cs{@@_div_significand_test_o:w} |1| |0| \meta{5d} |;| -% ~~\meta{4d} |;| \meta{4d} |;| \meta{5d} |;| \meta{sign} -% \end{quote} -% The reason we know that the first two digits are |1| and |0| is that -% the final result is known to be between $0.1$ (inclusive) and $10$, -% hence $\widetilde{Q_{A}}$ (the tilde denoting the contribution from -% the other $Q_{i}$) is at most $99999$, and $10^{6}+\widetilde{Q_{A}} -% = 10\cdots$. -% -% It is now time to round. This depends on how many digits the final -% result will have. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_test_o:w 10 #1 - { - \if_meaning:w 0 #1 - \exp_after:wN \@@_div_significand_small_o:wwwNNNNwN - \else: - \exp_after:wN \@@_div_significand_large_o:wwwNNNNwN - \fi: - #1 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_div_significand_small_o:wwwNNNNwN} -% \begin{quote} -% \cs{@@_div_significand_small_o:wwwNNNNwN} |0| \meta{4d} |;| -% ~~\meta{4d} |;| \meta{4d} |;| \meta{5d} |;| \meta{final sign} -% \end{quote} -% Standard use of the functions \cs{@@_basics_pack_low:NNNNNw} and -% \cs{@@_basics_pack_high:NNNNNw}. We finally get to use the -% \meta{final sign} which has been sitting there for a while. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_small_o:wwwNNNNwN - 0 #1; #2; #3; #4#5#6#7#8; #9 - { - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1 #1#2 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 1 #3#4#5#6#7 - + \@@_round:NNN #9 #7 #8 - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_div_significand_large_o:wwwNNNNwN} -% \begin{quote} -% \cs{@@_div_significand_large_o:wwwNNNNwN} \meta{5d} |;| -% ~~\meta{4d} |;| \meta{4d} |;| \meta{5d} |;| \meta{sign} -% \end{quote} -% We know that the final result cannot reach $10$, hence |1#1#2|, -% together with contributions from the level below, cannot reach -% $2\cdot 10^{9}$. For rounding, we build the \meta{rounding digit} -% from the last two of our $18$ digits. -% \begin{macrocode} -\cs_new:Npn \@@_div_significand_large_o:wwwNNNNwN - #1; #2; #3; #4#5#6#7#8; #9 - { - + 1 - \exp_after:wN \@@_basics_pack_weird_high:NNNNNNNNw - \int_value:w \@@_int_eval:w 1 #1 #2 - \exp_after:wN \@@_basics_pack_weird_low:NNNNw - \int_value:w \@@_int_eval:w 1 #3 #4 #5 #6 + - \exp_after:wN \@@_round:NNN - \exp_after:wN #9 - \exp_after:wN #6 - \int_value:w \@@_round_digit:Nw #7 #8 ; - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \subsection{Square root} -% -% \begin{macro}[EXP]{\@@_sqrt_o:w} -% Zeros are unchanged: $\sqrt{-0} = -0$ and $\sqrt{+0} = +0$. -% Negative numbers (other than $-0$) have no real square root. -% Positive infinity, and \texttt{nan}, are unchanged. Finally, for -% normal positive numbers, there is some work to do. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_o:w #1 \s_@@ \@@_chk:w #2#3#4; @ - { - \if_meaning:w 0 #2 \@@_case_return_same_o:w \fi: - \if_meaning:w 2 #3 - \@@_case_use:nw { \@@_invalid_operation_o:nw { sqrt } } - \fi: - \if_meaning:w 1 #2 \else: \@@_case_return_same_o:w \fi: - \@@_sqrt_npos_o:w - \s_@@ \@@_chk:w #2 #3 #4; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_sqrt_npos_o:w} -% \begin{macro}[rEXP] -% {\@@_sqrt_npos_auxi_o:wwnnN, \@@_sqrt_npos_auxii_o:wNNNNNNNN} -% Prepare \cs{@@_sanitize:Nw} to receive the final sign~|0| (the -% result is always positive) and the exponent, equal to half of the -% exponent~|#1| of the argument. If the exponent~|#1| is even, find a -% first approximation of the square root of the significand $10^{8} -% a_1 + a_2 = 10^{8} |#2#3| + |#4#5|$ through Newton's method, -% starting at $x = 57234133 \simeq 10^{7.75}$. Otherwise, first shift -% the significand of the argument by one digit, getting -% $a_1'\in[10^{6}, 10^{7})$ instead of $[10^{7}, 10^{8})$, then use -% Newton's method starting at $17782794 \simeq 10^{7.25}$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_npos_o:w \s_@@ \@@_chk:w 1 0 #1#2#3#4#5; - { - \exp_after:wN \@@_sanitize:Nw - \exp_after:wN 0 - \int_value:w \@@_int_eval:w - \if_int_odd:w #1 \exp_stop_f: - \exp_after:wN \@@_sqrt_npos_auxi_o:wwnnN - \fi: - #1 / 2 - \@@_sqrt_Newton_o:wwn 56234133; 0; {#2#3} {#4#5} 0 - } -\cs_new:Npn \@@_sqrt_npos_auxi_o:wwnnN #1 / 2 #2; 0; #3#4#5 - { - ( #1 + 1 ) / 2 - \@@_pack_eight:wNNNNNNNN - \@@_sqrt_npos_auxii_o:wNNNNNNNN - ; - 0 #3 #4 - } -\cs_new:Npn \@@_sqrt_npos_auxii_o:wNNNNNNNN #1; #2#3#4#5#6#7#8#9 - { \@@_sqrt_Newton_o:wwn 17782794; 0; {#1} {#2#3#4#5#6#7#8#9} } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sqrt_Newton_o:wwn} -% Newton's method maps $x\mapsto\bigl[(x + [10^{8} a_1 / x])/2\bigr]$ -% in each iteration, where $[b/c]$ denotes \eTeX{}'s division. This -% division rounds the real number $b/c$ to the closest integer, -% rounding ties away from zero, hence when $c$~is even, -% $b/c - 1/2 + 1/c \leq [b/c] \leq b/c + 1/2$ -% and when $c$~is odd, -% $b/c - 1/2 + 1/(2c) \leq [b/c] \leq b/c + 1/2 - 1/(2c)$. -% For all~$c$, $b/c - 1/2 + 1/(2c) \leq [b/c] \leq b/c + 1/2$. -% -% Let us prove that the method converges when implemented with \eTeX{} -% integer division, for any $10^{6} \leq a_1 < 10^{8}$ and starting -% value $10^{6} \leq x < 10^{8}$. Using the inequalities above and -% the arithmetic--geometric inequality $(x+t)/2 \geq \sqrt{xt}$ for $t -% = 10^{8} a_1 / x$, we find -% \[ -% x' -% = \left[\frac{x + [10^{8} a_1 / x]}{2}\right] -% \geq \frac{x + 10^{8} a_1 / x - 1/2 + 1/(2x)}{2} -% \geq \sqrt{10^{8} a_1} - \frac{1}{4} + \frac{1}{4x} \,. -% \] -% After any step of iteration, we thus have $\delta = x - \sqrt{10^{8} -% a_1} \geq -0.25 + 0.25 \cdot 10^{-8}$. The new difference -% $\delta' = x' - \sqrt{10^{8} a_1}$ after one step is bounded above -% as -% \[ -% x' - \sqrt{10^{8} a_1} -% \leq \frac{x + 10^{8} a_1 / x + 1/2}{2} + \frac{1}{2} -% - \sqrt{10^{8} a_1} -% \leq \frac{\delta}{2} \frac{\delta}{\sqrt{10^{8} a_1} + \delta} -% + \frac{3}{4} \,. -% \] -% For $\delta > 3/2$, this last expression is -% $\leq\delta/2+3/4<\delta$, hence $\delta$~decreases at each step: -% since all~$x$ are integers, $\delta$~must reach a value -% $-1/4<\delta\leq 3/2$. In this range of values, we get $\delta' -% \leq \frac{3}{4} \frac{3}{2\sqrt{10^{8} a_1}} + \frac{3}{4} \leq -% 0.75 + 1.125 \cdot 10^{-7}$. We deduce that the difference $\delta -% = x - \sqrt{10^{8} a_1}$ eventually reaches a value in the interval -% $[-0.25 + 0.25\cdot 10^{-8}, 0.75 + 11.25 \cdot 10^{-8}]$, whose -% width is $1 + 11 \cdot 10^{-8}$. The corresponding interval for~$x$ -% may contain two integers, hence $x$~might oscillate between those -% two values. -% -% However, the fact that $x\mapsto x-1$ and $x-1 \mapsto x$ puts -% stronger constraints, which are not compatible: the first implies -% \[ -% x + [10^{8} a_1 / x] \leq 2x - 2 -% \] -% hence $10^{8} a_1 / x \leq x - 3/2$, while the second implies -% \[ -% x - 1 + [10^{8} a_1 / (x - 1)] \geq 2x - 1 -% \] -% hence $10^{8} a_1 / (x - 1) \geq x - 1/2$. Combining the two -% inequalities yields $x^2 - 3x/2 \geq 10^{8} a_1 \geq x - 3x/2 + -% 1/2$, which cannot hold. Therefore, the iteration always converges -% to a single integer~$x$. To stop the iteration when two consecutive -% results are equal, the function \cs{@@_sqrt_Newton_o:wwn} receives -% the newly computed result as~|#1|, the previous result as~|#2|, and -% $a_1$ as~|#3|. Note that \eTeX{} combines the computation of a -% multiplication and a following division, thus avoiding overflow in -% |#3 * 100000000 / #1|. In any case, the result is within $[10^{7}, -% 10^{8}]$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_Newton_o:wwn #1; #2; #3 - { - \if_int_compare:w #1 = #2 \exp_stop_f: - \exp_after:wN \@@_sqrt_auxi_o:NNNNwnnN - \int_value:w \@@_int_eval:w 9999 9999 + - \exp_after:wN \@@_use_none_until_s:w - \fi: - \exp_after:wN \@@_sqrt_Newton_o:wwn - \int_value:w \@@_int_eval:w (#1 + #3 * 1 0000 0000 / #1) / 2 ; - #1; {#3} - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sqrt_auxi_o:NNNNwnnN} -% This function is followed by $10^{8}+x-1$, which has~$9$ digits -% starting with~$1$, then |;| \Arg{a_1} \Arg{a_2} \meta{a'}. Here, $x -% \simeq \sqrt{10^{8} a_1}$ and we want to estimate the square root of -% $a = 10^{-8} a_1 + 10^{-16} a_2 + 10^{-17} a'$. We set up an -% initial underestimate -% \[ -% y = (x - 1) 10^{-8} + 0.2499998875 \cdot 10^{-8} \lesssim \sqrt{a}\,. -% \] -% From the inequalities shown earlier, we know that $y \leq -% \sqrt{10^{-8} a_1} \leq \sqrt{a}$ and that $\sqrt{10^{-8} a_1} \leq -% y + 10^{-8} + 11\cdot 10^{-16}$ hence (using $0.1\leq y\leq -% \sqrt{a}\leq 1$) -% \[ -% a - y^2 \leq 10^{-8} a_1 + 10^{-8} - y^2 -% \leq (y + 10^{-8} + 11\cdot 10^{-16})^2 - y^2 + 10^{-8} -% < 3.2 \cdot 10^{-8} \,, -% \] -% and $\sqrt{a} - y = (a - y^2)/(\sqrt{a} + y) \leq 16 \cdot 10^{-8}$. -% Next, \cs{@@_sqrt_auxii_o:NnnnnnnnN} is called several times to -% get closer and closer underestimates of~$\sqrt{a}$. By -% construction, the underestimates~$y$ are always increasing, $a - y^2 -% < 3.2 \cdot 10^{-8}$ for all. Also, $y<1$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxi_o:NNNNwnnN 1 #1#2#3#4#5; - { - \@@_sqrt_auxii_o:NnnnnnnnN - \@@_sqrt_auxiii_o:wnnnnnnnn - {#1#2#3#4} {#5} {2499} {9988} {7500} - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sqrt_auxii_o:NnnnnnnnN} -% This receives a continuation function~|#1|, then five blocks of~$4$ -% digits for~$y$, then two $8$-digit blocks and a single digit -% for~$a$. A common estimate of $\sqrt{a} - y = (a - y^2) / (\sqrt{a} -% + y)$ is $(a - y^2)/(2y)$, which leads to alternating overestimates -% and underestimates. We tweak this, to only work with underestimates -% (no need then to worry about signs in the computation). Each step -% finds the largest integer $j\leq 6$ such that $10^{4j}(a-y^2) < -% 2\cdot 10^{8}$, then computes the integer (with \eTeX{}'s rounding -% division) -% \[ -% 10^{4j} z = -% \Bigl[\bigl(\lfloor 10^{4j}(a-y^2)\rfloor - 257\bigr) -% \cdot (0.5\cdot 10^{8}) -% \Bigm/ \lfloor 10^{8} y + 1\rfloor\Bigr] \,. -% \] -% The choice of~$j$ ensures that $10^{4j} z < 2\cdot 10^{8} \cdot -% 0.5\cdot 10^{8} / 10^{7} = 10^{9}$, thus $10^{9} + 10^{4j} z$ has -% exactly $10$~digits, does not overflow \TeX{}'s integer range, and -% starts with~$1$. Incidentally, since all $a - y^2 \leq 3.2\cdot -% 10^{-8}$, we know that $j\geq 3$. -% -% Let us show that $z$ is an underestimate of $\sqrt{a} - y$. On the -% one hand, $\sqrt{a} - y \leq 16\cdot 10^{-8}$ because this holds for -% the initial~$y$ and values of~$y$ can only increase. On the other -% hand, the choice of~$j$ implies that $\sqrt{a} - y \leq -% 5(\sqrt{a}+y)(\sqrt{a}-y) = 5(a - y^2) < 10^{9-4j}$. For $j=3$, the -% first bound is better, while for larger~$j$, the second bound is -% better. For all $j\in[3,6]$, we find $\sqrt{a}-y < 16\cdot -% 10^{-2j}$. From this, we deduce that -% \[ -% 10^{4j} (\sqrt{a}-y) -% = \frac{10^{4j}\bigl(a-y^2-(\sqrt{a}-y)^2\bigr)}{2y} -% \geq \frac{\bigl\lfloor 10^{4j}(a-y^2)\bigr\rfloor-257} -% {2\cdot 10^{-8} \lfloor 10^{8}y+1\rfloor} -% + \frac{1}{2} -% \] -% where we have replaced the bound $10^{4j}(16\cdot 10^{-2j}) = 256$ -% by~$257$ and extracted the corresponding term $1/\bigl(2\cdot -% 10^{-8} \lfloor 10^{8}y+1\rfloor\bigr) \geq 1/2$. Given that -% \eTeX{}'s integer division obeys $[b/c] \leq b/c + 1/2$, we deduce -% that $10^{4j} z \leq 10^{4j} (\sqrt{a}-y)$, hence $y+z\leq\sqrt{a}$ -% is an underestimate of~$\sqrt{a}$, as claimed. One implementation -% detail: because the computation involves |-#4*#4| |-| |2*#3*#5| |-| -% |2*#2*#6| which may be as low as $-5\cdot 10^{8}$, we need to use -% the \texttt{pack_big} functions, and the \texttt{big} shifts. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxii_o:NnnnnnnnN #1 #2#3#4#5#6 #7#8#9 - { - \exp_after:wN #1 - \int_value:w \@@_int_eval:w \c_@@_big_leading_shift_int - + #7 - #2 * #2 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - - 2 * #2 * #3 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - + #8 - #3 * #3 - 2 * #2 * #4 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - - 2 * #3 * #4 - 2 * #2 * #5 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - + #9 000 0000 - #4 * #4 - 2 * #3 * #5 - 2 * #2 * #6 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - - 2 * #4 * #5 - 2 * #3 * #6 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w \c_@@_big_middle_shift_int - - #5 * #5 - 2 * #4 * #6 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w - \c_@@_big_middle_shift_int - - 2 * #5 * #6 - \exp_after:wN \@@_pack_big:NNNNNNw - \int_value:w \@@_int_eval:w - \c_@@_big_trailing_shift_int - - #6 * #6 ; - % ( - - 257 ) * 5000 0000 / (#2#3 + 1) + 10 0000 0000 ; - {#2}{#3}{#4}{#5}{#6} {#7}{#8}#9 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP] -% { -% \@@_sqrt_auxiii_o:wnnnnnnnn, -% \@@_sqrt_auxiv_o:NNNNNw, -% \@@_sqrt_auxv_o:NNNNNw, -% \@@_sqrt_auxvi_o:NNNNNw, -% \@@_sqrt_auxvii_o:NNNNNw -% } -% We receive here the difference $a-y^2=d=\sum_i d_i \cdot 10^{-4i}$, -% as \meta{d_2} |;| \Arg{d_3} \ldots{} \Arg{d_{10}}, where each block -% has~$4$ digits, except \meta{d_2}. This function finds the largest -% $j\leq 6$ such that $10^{4j}(a-y^2) < 2\cdot 10^{8}$, then leaves an -% open parenthesis and the integer -% $\bigl\lfloor 10^{4j}(a-y^2)\bigr\rfloor$ in an integer -% expression. The closing parenthesis is provided by the caller -% \cs{@@_sqrt_auxii_o:NnnnnnnnN}, which completes the expression -% \[ -% 10^{4j} z = -% \Bigl[\bigl(\lfloor 10^{4j}(a-y^2)\rfloor - 257\bigr) -% \cdot (0.5\cdot 10^{8}) -% \Bigm/ \lfloor 10^{8} y + 1\rfloor\Bigr] -% \] -% for an estimate of $10^{4j} (\sqrt{a} - y)$. If $d_2\geq 2$, $j=3$ -% and the \texttt{auxiv} auxiliary receives $10^{12} z$. If $d_2\leq -% 1$ but $10^{4} d_2 + d_3 \geq 2$, $j=4$ and the \texttt{auxv} -% auxiliary is called, and receives $10^{16} z$, and so on. In all -% those cases, the \texttt{auxviii} auxiliary is set up to add~$z$ -% to~$y$, then go back to the \texttt{auxii} step with continuation -% \texttt{auxiii} (the function we are currently describing). The -% maximum value of $j$ is~$6$, regardless of whether $10^{12} d_2 + -% 10^{8} d_3 + 10^{4} d_4 + d_5 \geq 1$. In this last case, we detect -% when $10^{24} z < 10^{7}$, which essentially means $\sqrt{a} - y -% \lesssim 10^{-17}$: once this threshold is reached, there is enough -% information to find the correctly rounded~$\sqrt{a}$ with only one -% more call to \cs{@@_sqrt_auxii_o:NnnnnnnnN}. Note that the -% iteration cannot be stuck before reaching $j=6$, because for $j<6$, -% one has $2\cdot 10^{8}\leq 10^{4(j+1)}(a-y^2)$, hence -% \[ -% 10^{4j} z -% \geq \frac{(20000-257)(0.5\cdot 10^{8})}{\lfloor 10^{8} y + 1\rfloor} -% \geq (20000-257)\cdot 0.5 > 0 \,. -% \] -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxiii_o:wnnnnnnnn - #1; #2#3#4#5#6#7#8#9 - { - \if_int_compare:w #1 > \c_one_int - \exp_after:wN \@@_sqrt_auxiv_o:NNNNNw - \int_value:w \@@_int_eval:w (#1#2 %) - \else: - \if_int_compare:w #1#2 > \c_one_int - \exp_after:wN \@@_sqrt_auxv_o:NNNNNw - \int_value:w \@@_int_eval:w (#1#2#3 %) - \else: - \if_int_compare:w #1#2#3 > \c_one_int - \exp_after:wN \@@_sqrt_auxvi_o:NNNNNw - \int_value:w \@@_int_eval:w (#1#2#3#4 %) - \else: - \exp_after:wN \@@_sqrt_auxvii_o:NNNNNw - \int_value:w \@@_int_eval:w (#1#2#3#4#5 %) - \fi: - \fi: - \fi: - } -\cs_new:Npn \@@_sqrt_auxiv_o:NNNNNw 1#1#2#3#4#5#6; - { \@@_sqrt_auxviii_o:nnnnnnn {#1#2#3#4#5#6} {00000000} } -\cs_new:Npn \@@_sqrt_auxv_o:NNNNNw 1#1#2#3#4#5#6; - { \@@_sqrt_auxviii_o:nnnnnnn {000#1#2#3#4#5} {#60000} } -\cs_new:Npn \@@_sqrt_auxvi_o:NNNNNw 1#1#2#3#4#5#6; - { \@@_sqrt_auxviii_o:nnnnnnn {0000000#1} {#2#3#4#5#6} } -\cs_new:Npn \@@_sqrt_auxvii_o:NNNNNw 1#1#2#3#4#5#6; - { - \if_int_compare:w #1#2 = \c_zero_int - \exp_after:wN \@@_sqrt_auxx_o:Nnnnnnnn - \fi: - \@@_sqrt_auxviii_o:nnnnnnn {00000000} {000#1#2#3#4#5} - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP] -% {\@@_sqrt_auxviii_o:nnnnnnn, \@@_sqrt_auxix_o:wnwnw} -% Simply add the two $8$-digit blocks of~$z$, aligned to the last four -% of the five $4$-digit blocks of~$y$, then call the \texttt{auxii} -% auxiliary to evaluate $y'^{2} = (y+z)^{2}$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxviii_o:nnnnnnn #1#2 #3#4#5#6#7 - { - \exp_after:wN \@@_sqrt_auxix_o:wnwnw - \int_value:w \@@_int_eval:w #3 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w #1 + 1#4#5 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w #2 + 1#6#7 ; - } -\cs_new:Npn \@@_sqrt_auxix_o:wnwnw #1; #2#3; #4#5; - { - \@@_sqrt_auxii_o:NnnnnnnnN - \@@_sqrt_auxiii_o:wnnnnnnnn {#1}{#2}{#3}{#4}{#5} - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP] -% {\@@_sqrt_auxx_o:Nnnnnnnn, \@@_sqrt_auxxi_o:wwnnN} -% At this stage, $j=6$ and $10^{24} z < 10^{7}$, hence -% \[ -% 10^{7} + 1/2 > 10^{24} z + 1/2 \geq -% \bigl(10^{24}(a-y^2) - 258\bigr) \cdot (0.5\cdot 10^{8}) -% \Bigm/ (10^{8} y + 1) \,, -% \] -% then $10^{24}(a-y^2) - 258 < 2 (10^{7} + 1/2) (y + 10^{-8})$, and -% \[ -% 10^{24}(a-y^2) -% < (10^{7} + 1290.5) (1 + 10^{-8}/y) (2y) -% < (10^{7} + 1290.5) (1 + 10^{-7}) (y + \sqrt{a}) \,, -% \] -% which finally implies $0\leq\sqrt{a}-y < 0.2\cdot 10^{-16}$. In -% particular, $y$~is an underestimate of~$\sqrt{a}$ and $y+0.5\cdot -% 10^{-16}$ is a (strict) overestimate. There is at exactly one -% multiple $m$~of $0.5\cdot 10^{-16}$ in the interval $[y, y+0.5\cdot -% 10^{-16})$. If $m^2>a$, then the square root is inexact and is -% obtained by rounding $m-\epsilon$ to a multiple of $10^{-16}$ (the -% precise shift $0<\epsilon<0.5\cdot 10^{-16}$ is irrelevant for -% rounding). If $m^2=a$ then the square root is exactly~$m$, and -% there is no rounding. If $m^2<a$ then we round $m+\epsilon$. For -% now, discard a few irrelevant arguments |#1|, |#2|, |#3|, and find -% the multiple of $0.5\cdot 10^{-16}$ within $[y, y+0.5\cdot -% 10^{-16})$; rather, only the last $4$~digits |#8| of~$y$ are -% considered, and we do not perform any carry yet. The \texttt{auxxi} -% auxiliary sets up \texttt{auxii} with a continuation function -% \texttt{auxxii} instead of \texttt{auxiii} as before. To prevent -% \texttt{auxii} from giving a negative results $a-m^2$, we compute -% $a+10^{-16}-m^2$ instead, always positive since $m<\sqrt{a}+0.5\cdot -% 10^{-16}$ and $a\leq 1-10^{-16}$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxx_o:Nnnnnnnn #1#2#3 #4#5#6#7#8 - { - \exp_after:wN \@@_sqrt_auxxi_o:wwnnN - \int_value:w \@@_int_eval:w - (#8 + 2499) / 5000 * 5000 ; - {#4} {#5} {#6} {#7} ; - } -\cs_new:Npn \@@_sqrt_auxxi_o:wwnnN #1; #2; #3#4#5 - { - \@@_sqrt_auxii_o:NnnnnnnnN - \@@_sqrt_auxxii_o:nnnnnnnnw - #2 {#1} - {#3} { #4 + 1 } #5 - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP] -% {\@@_sqrt_auxxii_o:nnnnnnnnw, \@@_sqrt_auxxiii_o:w} -% The difference $0\leq a+10^{-16}-m^2\leq -% 10^{-16}+(\sqrt{a}-m)(\sqrt{a}+m)\leq 2\cdot 10^{-16}$ was just -% computed: its first $8$~digits vanish, as do the next four,~|#1|, -% and most of the following four,~|#2|. The guess~$m$ is an -% overestimate if $a+10^{-16}-m^2 < 10^{-16}$, that is, |#1#2| -% vanishes. Otherwise it is an underestimate, unless -% $a+10^{-16}-m^2=10^{-16}$ exactly. For an underestimate, call the -% \texttt{auxxiv} function with argument~$9998$. For an exact result -% call it with~$9999$, and for an overestimate call it with~$10000$. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxxii_o:nnnnnnnnw 0; #1#2#3#4#5#6#7#8 #9; - { - \if_int_compare:w #1#2 > \c_zero_int - \if_int_compare:w #1#2 = \c_one_int - \if_int_compare:w #3#4 = \c_zero_int - \if_int_compare:w #5#6 = \c_zero_int - \if_int_compare:w #7#8 = \c_zero_int - \@@_sqrt_auxxiii_o:w - \fi: - \fi: - \fi: - \fi: - \exp_after:wN \@@_sqrt_auxxiv_o:wnnnnnnnN - \int_value:w 9998 - \else: - \exp_after:wN \@@_sqrt_auxxiv_o:wnnnnnnnN - \int_value:w 10000 - \fi: - ; - } -\cs_new:Npn \@@_sqrt_auxxiii_o:w \fi: \fi: \fi: \fi: #1 \fi: ; - { - \fi: \fi: \fi: \fi: \fi: - \@@_sqrt_auxxiv_o:wnnnnnnnN 9999 ; - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[rEXP]{\@@_sqrt_auxxiv_o:wnnnnnnnN} -% This receives $9998$, $9999$ or $10000$ as~|#1| when $m$~is an -% underestimate, exact, or an overestimate, respectively. Then -% comes~$m$ as five blocks of~$4$ digits, but where the last -% block~|#6| may be $0$, $5000$, or~$10000$. In the latter case, we -% need to add a carry, unless $m$~is an overestimate (|#1|~is then -% $10000$). Then comes~$a$ as three arguments. Rounding is done by -% \cs{@@_round:NNN}, whose first argument is the final sign~$0$ -% (square roots are positive). We fake its second argument. It -% should be the last digit kept, but this is only used when ties are -% \enquote{rounded to even}, and only when the result is exactly -% half-way between two representable numbers rational square roots of -% numbers with $16$~significant digits have: this situation never -% arises for the square root, as any exact square root of a $16$~digit -% number has at most $8$~significant digits. Finally, the last -% argument is the next digit, possibly shifted by~$1$ when there are -% further nonzero digits. This is achieved by \cs{@@_round_digit:Nw}, -% which receives (after removal of the $10000$'s digit) one of $0000$, -% $0001$, $4999$, $5000$, $5001$, or~$9999$, which it converts to $0$, -% $1$, $4$, $5$, $6$, and~$9$, respectively. -% \begin{macrocode} -\cs_new:Npn \@@_sqrt_auxxiv_o:wnnnnnnnN #1; #2#3#4#5#6 #7#8#9 - { - \exp_after:wN \@@_basics_pack_high:NNNNNw - \int_value:w \@@_int_eval:w 1 0000 0000 + #2#3 - \exp_after:wN \@@_basics_pack_low:NNNNNw - \int_value:w \@@_int_eval:w 1 0000 0000 - + #4#5 - \if_int_compare:w #6 > #1 \exp_stop_f: + 1 \fi: - + \exp_after:wN \@@_round:NNN - \exp_after:wN 0 - \exp_after:wN 0 - \int_value:w - \exp_after:wN \use_i:nn - \exp_after:wN \@@_round_digit:Nw - \int_value:w \@@_int_eval:w #6 + 19999 - #1 ; - \exp_after:wN ; - } -% \end{macrocode} -% \end{macro} -% -% \subsection{About the sign and exponent} -% -% \begin{macro}[EXP]{\@@_logb_o:w, \@@_logb_aux_o:w} -% The exponent of a normal number is its \meta{exponent} minus one. -% \begin{macrocode} -\cs_new:Npn \@@_logb_o:w ? \s_@@ \@@_chk:w #1#2; @ - { - \if_case:w #1 \exp_stop_f: - \@@_case_use:nw - { \@@_division_by_zero_o:Nnw \c_minus_inf_fp { logb } } - \or: \exp_after:wN \@@_logb_aux_o:w - \or: \@@_case_return_o:Nw \c_inf_fp - \else: \@@_case_return_same_o:w - \fi: - \s_@@ \@@_chk:w #1 #2; - } -\cs_new:Npn \@@_logb_aux_o:w \s_@@ \@@_chk:w #1 #2 #3 #4 ; - { - \exp_after:wN \@@_parse:n \exp_after:wN - { \int_value:w \int_eval:w #3 - 1 \exp_after:wN } - } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_sign_o:w} -% \begin{macro}[EXP]{\@@_sign_aux_o:w} -% Find the sign of the floating point: \texttt{nan}, |+0|, |-0|, |+1| or |-1|. -% \begin{macrocode} -\cs_new:Npn \@@_sign_o:w ? \s_@@ \@@_chk:w #1#2; @ - { - \if_case:w #1 \exp_stop_f: - \@@_case_return_same_o:w - \or: \exp_after:wN \@@_sign_aux_o:w - \or: \exp_after:wN \@@_sign_aux_o:w - \else: \@@_case_return_same_o:w - \fi: - \s_@@ \@@_chk:w #1 #2; - } -\cs_new:Npn \@@_sign_aux_o:w \s_@@ \@@_chk:w #1 #2 #3 ; - { \exp_after:wN \@@_set_sign_o:w \exp_after:wN #2 \c_one_fp @ } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_set_sign_o:w} -% This function is used for the unary minus and for \texttt{abs}. It -% leaves the sign of \texttt{nan} invariant, turns negative numbers -% (sign~$2$) to positive numbers (sign~$0$) and positive numbers -% (sign~$0$) to positive or negative numbers depending on~|#1|. It -% also expands after itself in the input stream, just like -% \cs{@@_+_o:ww}. -% \begin{macrocode} -\cs_new:Npn \@@_set_sign_o:w #1 \s_@@ \@@_chk:w #2#3#4; @ - { - \exp_after:wN \@@_exp_after_o:w - \exp_after:wN \s_@@ - \exp_after:wN \@@_chk:w - \exp_after:wN #2 - \int_value:w - \if_case:w #3 \exp_stop_f: #1 \or: 1 \or: 0 \fi: \exp_stop_f: - #4; - } -% \end{macrocode} -% \end{macro} -% -% \subsection{Operations on tuples} -% -% \begin{macro}[EXP]{\@@_tuple_set_sign_o:w} -% \begin{macro}[EXP]{\@@_tuple_set_sign_aux_o:Nnw, \@@_tuple_set_sign_aux_o:w} -% Two cases: |abs(|\meta{tuple}|)| for which |#1| is $0$ (invalid for -% tuples) and |-|\meta{tuple} for which |#1| is $2$. In that case, -% map over all items in the tuple an auxiliary that dispatches to the -% type-appropriate sign-flipping function. -% \begin{macrocode} -\cs_new:Npn \@@_tuple_set_sign_o:w #1#2 @ - { - \if_meaning:w 2 #1 - \exp_after:wN \@@_tuple_set_sign_aux_o:Nnw - \fi: - \@@_invalid_operation_o:nw { abs } - #2 - } -\cs_new:Npn \@@_tuple_set_sign_aux_o:Nnw #1#2 - { \@@_tuple_map_o:nw \@@_tuple_set_sign_aux_o:w } -\cs_new:Npn \@@_tuple_set_sign_aux_o:w #1#2 ; - { - \@@_change_func_type:NNN #1 \@@_set_sign_o:w - \@@_parse_apply_unary_error:NNw - 2 #1 #2 ; @ - } -% \end{macrocode} -% \end{macro} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_*_tuple_o:ww, \@@_tuple_*_o:ww, \@@_tuple_/_o:ww} -% For \meta{number}|*|\meta{tuple} and \meta{tuple}|*|\meta{number} -% and \meta{tuple}|/|\meta{number}, loop through the \meta{tuple} some -% code that multiplies or divides by the appropriate \meta{number}. -% Importantly we need to dispatch according to the type, and we make -% sure to apply the operator in the correct order. -% \begin{macrocode} -\cs_new:cpn { @@_*_tuple_o:ww } #1 ; - { \@@_tuple_map_o:nw { \@@_binary_type_o:Nww * #1 ; } } -\cs_new:cpn { @@_tuple_*_o:ww } #1 ; #2 ; - { \@@_tuple_map_o:nw { \@@_binary_rev_type_o:Nww * #2 ; } #1 ; } -\cs_new:cpn { @@_tuple_/_o:ww } #1 ; #2 ; - { \@@_tuple_map_o:nw { \@@_binary_rev_type_o:Nww / #2 ; } #1 ; } -% \end{macrocode} -% \end{macro} -% -% \begin{macro}[EXP]{\@@_tuple_+_tuple_o:ww, \@@_tuple_-_tuple_o:ww} -% Check the two tuples have the same number of items and map through -% these a helper that dispatches appropriately depending on the types. -% This means |(1,2)+((1,1),2)| gives |(nan,4)|. -% \begin{macrocode} -\cs_set_protected:Npn \@@_tmp:w #1 - { - \cs_new:cpn { @@_tuple_#1_tuple_o:ww } - \s_@@_tuple \@@_tuple_chk:w ##1 ; - \s_@@_tuple \@@_tuple_chk:w ##2 ; - { - \int_compare:nNnTF - { \@@_array_count:n {##1} } = { \@@_array_count:n {##2} } - { \@@_tuple_mapthread_o:nww { \@@_binary_type_o:Nww #1 } } - { \@@_invalid_operation_o:nww #1 } - \s_@@_tuple \@@_tuple_chk:w {##1} ; - \s_@@_tuple \@@_tuple_chk:w {##2} ; - } - } -\@@_tmp:w + -\@@_tmp:w - -% \end{macrocode} -% \end{macro} -% -% \begin{macrocode} -%</package> -% \end{macrocode} -% -% \end{implementation} -% -% \PrintChanges -% -% \PrintIndex |