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|
%% poormanlog.tex v0.04, 2019/02/17
%%
%% Copyright (C) 2019, Jean-Francois Burnol
%%
%% This Work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License version 1.3c.
%%
%% This Work has the LPPL maintenance status `author-maintained`.
%%
%% The Author of this Work is Jean-Francois Burnol (jfbu AT free DOT fr)
%%
%% This Work consists of files poormanlog.tex, poormanlog.sty and the
%% README
%%
%% ************ BARELY TESTED ****************
%% >>>>>>> WARNING FOR LACK OF TIME I HAVE NOT YET VALIDATED FULLY <<<<<
%% >>>>>>> NOR HAVE I MADE PRECISE ESTIMATES. USE AT YOUR OWN RISK <<<<<
%% (bug reports welcome)
%% ************ BARELY TESTED ****************
%
% Package macro private prefix: \PML@
%
% SEE README FOR USER DOCUMENTATION
%
\ifx\numexpr\JFBUundefined
\message{**** poormanlog requires e-TeX **** LOADING ABORTED ****}%
\endinput\fi
%
% CATCODES (AND AUXILIARIES FOR PLAIN TEX)
%
\expandafter\edef\csname @tempa\endcsname
{%
\catcode0 \the\catcode0 %
\catcode`\noexpand\_ \the\catcode`\_ %
\catcode`\noexpand\^ \the\catcode`\^ %
\catcode`\noexpand\: \the\catcode`\: %
\catcode`\noexpand\* \the\catcode`\* %
\catcode`\noexpand\@ \the\catcode`\@ %
}%
\catcode`\@ 11 %
\if1\ifx\@namedef\JFBUundefined1\else\ifx\@namedef\relax1\else0\fi\fi
\def\@namedef#1{\expandafter\def\csname #1\endcsname}%
\def\@nameuse#1{\csname #1\endcsname}%
\fi
\long\def\@gobble#1{}%
%
% TABLES
%
% The algorithm for \PMLogZ is the reverse engineering of the one I did
% for \PMPowTen. I had an other approach for log10 initially but it was
% less accurate than what testings of \PMPowTen showed. The same accuracy, even
% slightly better, was then obtain with \PMLogZ by imitating the latter
% algorithm (the original was faster but achieved only about 6ulp max error,
% although most of the time the result was correct up to +2 or -2,
% the present one appears to achieve not much worse than 1ulp maximal error).
%
% method of CORDIX type combined with usage of \numexpr "scaling" operations
%
\@namedef{PML@1@10}{}%
\@namedef{PML@1@9}{*1349157701/1071674055}%
\@namedef{PML@1@8}{*946017331/596896583}%
\@namedef{PML@1@7}{*495834591/248505967}%
\@namedef{PML@1@6}{*870020383/346361353}%
\@namedef{PML@1@5}{*1499219281/474094764}%
\@namedef{PML@1@4}{*1978893117/497075477}%
\@namedef{PML@1@3}{*368683859/73562101}%
\@namedef{PML@1@2}{*1295603161/205339263}%
\@namedef{PML@1@1}{*1268797901/159732192}%
\@namedef{PML@1@0}{*10}%
%
\@namedef{PML@P1@10}{}%
\@namedef{PML@P1@9}{*1071674055/1349157701}%
\@namedef{PML@P1@8}{*596896583/946017331}%
\@namedef{PML@P1@7}{*248505967/495834591}%
\@namedef{PML@P1@6}{*346361353/870020383}%
\@namedef{PML@P1@5}{*474094764/1499219281}%
\@namedef{PML@P1@4}{*497075477/1978893117}%
\@namedef{PML@P1@3}{*73562101/368683859}%
\@namedef{PML@P1@2}{*205339263/1295603161}%
\@namedef{PML@P1@1}{*159732192/1268797901}%
\@namedef{PML@P1@0}{/10}%
%
\@namedef{PML@2@12}{*1759219355/1983514284}%
\@namedef{PML@2@11}{*1038555297/1159316503}%
\@namedef{PML@2@10}{*1446048050/1598130251}%
\@namedef{PML@2@9}{*200795995/219705814}%
\@namedef{PML@2@8}{*19701438/21342313}%
\@namedef{PML@2@7}{*699035545/749721341}%
\@namedef{PML@2@6}{*342419462/363593499}%
\@namedef{PML@2@5}{*262151601/275592401}%
\@namedef{PML@2@4}{*643237001/669488001}%
\@namedef{PML@2@3}{*909207215/936896697}%
\@namedef{PML@2@2}{*14850599/15150601}%
\@namedef{PML@2@1}{*119401199/120601201}%
\@namedef{PML@2@0}{}%
\@namedef{PML@2@-1}{*120601201/119401199}%
\@namedef{PML@2@-2}{*15150601/14850599}%
\@namedef{PML@2@-3}{*936896697/909207215}%
\@namedef{PML@2@-4}{*669488001/643237001}%
\@namedef{PML@2@-5}{*275592401/262151601}%
\@namedef{PML@2@-6}{*363593499/342419462}%
\@namedef{PML@2@-7}{*749721341/699035545}%
\@namedef{PML@2@-8}{*21342313/19701438}%
\@namedef{PML@2@-9}{*219705814/200795995}%
\@namedef{PML@2@-10}{*1598130251/1446048050}%
\@namedef{PML@2@-11}{*1159316503/1038555297}%
\@namedef{PML@2@-12}{*1983514284/1759219355}%
%
%\@namedef{PML@3@9}{*166909427/168418392}%
%\@namedef{PML@3@8}{*233438999/235314001}%
%\@namedef{PML@3@7}{*921380630/927852921}%
\@namedef{PML@3@6}{*92333611/92889278}%
\@namedef{PML@3@5}{*957602399/962402401}%
\@namedef{PML@3@4}{*1871252999/1878753001}%
\@namedef{PML@3@3}{*739112555/741333222}%
\@namedef{PML@3@2}{*2997001/3003001}%
\@namedef{PML@3@1}{*11994001/12006001}%
\@namedef{PML@3@0}{}%
\@namedef{PML@3@-1}{*12006001/11994001}%
\@namedef{PML@3@-2}{*3003001/2997001}%
\@namedef{PML@3@-3}{*741333222/739112555}%
\@namedef{PML@3@-4}{*1878753001/1871252999}%
\@namedef{PML@3@-5}{*962402401/957602399}%
\@namedef{PML@3@-6}{*92889278/92333611}%
%\@namedef{PML@3@-7}{*927852921/921380630}%
%\@namedef{PML@3@-8}{*235314001/233438999}%
%\@namedef{PML@3@-9}{*168418392/166909427}%
%
\@namedef{PML@4@5}{*47988001/48012001}%
\@namedef{PML@4@4}{*74985001/75015001}%
\@namedef{PML@4@3}{*399940003/400060003}%
\@namedef{PML@4@2}{*299970001/300030001}%
\@namedef{PML@4@1}{*1199940001/1200060001}%
\@namedef{PML@4@0}{}%
\@namedef{PML@4@-1}{*1200060001/1199940001}%
\@namedef{PML@4@-2}{*300030001/299970001}%
\@namedef{PML@4@-3}{*400060003/399940003}%
\@namedef{PML@4@-4}{*75015001/74985001}%
\@namedef{PML@4@-5}{*48012001/47988001}%
%%
%%
%% LOG IN BASE 10 : \the\numexpr\PML@ ddddddddd.\relax
%%
%% Exactly 9 digits representing d.dddddddd, first one at least 1.
%% Goal is to compute log10(d.dddddddd) with 9 digits
%%
\def\PML@#1.{\expandafter\PML@a\the\numexpr#1/100000.#1.}%
\def\PML@a#1.{\expandafter\PML@ai
\the\numexpr
\ifnum#1>2817 %
\ifnum#1>5622 %
\ifnum#1>7078 \ifnum#1>8912 10\else 9\fi
\else 8%
\fi
\else
\ifnum#1>3547 \ifnum#1>4466 7\else 6\fi
\else 5%
\fi
\fi
\else
\ifnum#1>1412 %
\ifnum#1>1777 \ifnum#1>2238 4\else 3\fi
\else 2%
\fi
\else
\ifnum#1>1121 1\else 0\fi
\fi
\fi
.%
}%
\def\PML@ai #1.#2.%
{\expandafter\PML@b\the\numexpr#2\@nameuse{PML@1@#1}.%
)*774923109/1784326399+#100000000}%
%
\def\PML@b#1.{\expandafter\PML@bi\the\numexpr#1/1000000.#1.}%
\def\PML@bi#1.{\expandafter\PML@bii
\the\numexpr
\ifnum#1>994 %
\ifnum#1>1056 %
\ifnum#1>1088 %
\ifnum#1>1110 \ifnum#1>1121 12\else 11\fi
\else \ifnum#1>1099 10\else 9\fi
\fi
\else
\ifnum#1>1066 \ifnum#1>1077 8\else 7\fi
\else 6%
\fi
\fi
\else
\ifnum#1>1024 %
\ifnum#1>1035 \ifnum#1>1045 5\else 4\fi
\else 3%
\fi
\else
\ifnum#1>1004 \ifnum#1>1014 2\else 1\fi
\else 0%
\fi
\fi
\fi
\else
\ifnum#1>936 %
\ifnum#1>965 %
\ifnum#1>974 \ifnum#1>984 -1\else -2\fi
\else -3%
\fi
\else
\ifnum#1>945 \ifnum#1>955 -4\else -5\fi
\else -6%
\fi
\fi
\else
\ifnum#1>908 %
\ifnum#1>918 \ifnum#1>927 -7\else -8\fi
\else -9%
\fi
\else
\ifnum#1>890 \ifnum#1>899 -10\else -11\fi
\else -12%
\fi
\fi
\fi
\fi
.%
}%
\def\PML@bii#1.#2.%
{\expandafter\PML@c\the\numexpr#2\@nameuse{PML@2@#1}.+#10000000}%
%
\def\PML@c#1.%
{\expandafter\PML@ci\the\numexpr#1/1000000-1000.#1.}%
\def\PML@ci#1.#2.%
{\expandafter\PML@d\the\numexpr#2\@nameuse{PML@3@#1}.+#1000000}%
%
\def\PML@d#1.%
{\expandafter\PML@di\the\numexpr#1/100000-10000.#1.}%
\def\PML@di#1.#2.%
{\expandafter\PML@e\the\numexpr#2\@nameuse{PML@4@#1}-1000000000.+#100000}%
% we have reached
% x = 1 + t/10^9 represented by t
% log(x) represented by 10^9 times (t/10^9 (2.10^9 - t)/2.10^9)
\def\PML@e #1.{(#1*(2000000000-#1)/2000000000+}%
%%
%%
%% POWER OF TEN : \the\numexpr\PML@Pa ddddddddd.\relax
%%
%% Exactly 9 digits in input representing number 0.ddddddddd
%% Goal is to compute 10^0.ddddddddd with 9 digits.
%%
\def\PML@Pa#1#2{\expandafter\PML@Pai\the\numexpr#1#2/10.#1#2}%
\def\PML@Pai#1.#2.{%
\expandafter\PML@Pb\the\numexpr(#2-#100000000)*2079839159/90326267.%
\@nameuse{PML@P1@#1}%
}%
% problem with minus sign interfering with token count so simply fetch all
% the -#1 is to re-use constants already defined for log
\def\PML@Pb#1.{\expandafter\PML@Pbi\the\numexpr-#1/100000000.#1.}%
\def\PML@Pbi#1.#2.%
{\expandafter\PML@Pc\the\numexpr#2+#100000000.\@nameuse{PML@2@#1}}%
% \pm 0.00d, d at most 5, 8 digits integer N for N/10^10
\def\PML@Pc#1.{\expandafter\PML@Pci\the\numexpr-#1/10000000.#1.}%
\def\PML@Pci#1.#2.%
{\expandafter\PML@Pd\the\numexpr#2+#10000000.\@nameuse{PML@3@#1}}%
% \pm 0.000d, d at most 5, 7 digits
\def\PML@Pd#1.{\expandafter\PML@Pdi\the\numexpr-#1/1000000.#1.}%
\def\PML@Pdi#1.#2.%
{\expandafter\PML@Pe\the\numexpr#2+#1000000.\@nameuse{PML@4@#1}}%
% \pm 0.0000d, d at most 5, 6 digits integer N for N/10^10
% exp of that must fit in 10 digits, and there will be leading 1, hence
% only 9 digits available for fitting x + x^2/2, x = Ne-10
% so we must compute via numexpr
% 10^9*x*(1+x/2) = 10^9*N/10^10*(2.10^10+ N)/2.10^10
% with N in absolute value at most 500000.
% N*(2.10^9+N/10)/2.10^10 which is about N/10
\def\PML@Pe#1.{(1000000000+#1*(200000000+#1/100)/2000000000)}%
%%
%%
%% BASIC USER INTERFACE : \PMLogZ, \PMPowTen
%%
% Another way of inserting leading zeros is found in xint source code,
% maybe faster. "Z" is for reminding that output has leading zeros.
\def\PMLogZ#1%
{%
\romannumeral-`0\expandafter\@gobble
% \PML@ never gives something negative, hopefully...
\the\numexpr1000000000+\expandafter\PML@\romannumeral-`0#1.\relax
}%
\def\PMPowTen#1{\the\numexpr\expandafter\PML@Pa\romannumeral-`0#1.\relax}%
%%
%%
%% ADD-ONS TO XINTFRAC : \PoorManLogBaseTen{#1}, \PoorManPowerOfTen{#1}
%% and \PoorManPower{#1}{#2}
%% Does not matter if xintfrac loaded before or after.
%%
\def\PoorManLogBaseTen{\romannumeral0\poormanlogbaseten}%
\def\poormanlogbaseten #1%
{\expandafter\PML@logbaseten\romannumeral0\XINTinfloat[9]{#1}}%
\def\PML@logbaseten#1[#2]%
{%
\xintiiadd{\xintDSx{-9}{\the\numexpr#2+8\relax}}{\the\numexpr\PML@#1.}%
[-9]%
}%
%
\def\PoorManPowerOfTen{\the\numexpr\poormanpoweroften}%
\def\poormanpoweroften #1%
{\expandafter\PML@powoften\romannumeral0\xintraw{#1}}%
\catcode`\_ 11 %
\def\PML@powoften#1{%
\xint_UDzerominusfork
#1-\PML@powoften@zero
0#1\PML@powoften@neg
0-\PML@powoften@pos
\krof #1%
}%
\catcode`\_ 8 %
\def\PML@powoften@zero 0{1\relax}%/1[0]
\def\PML@powoften@pos#1[#2]{%
\expandafter\PML@powoften@pos@a\romannumeral0\xintround{9}{#1[#2]}.%
}%
\def\PML@powoften@pos@a#1.#2.{\PML@Pa#2.\expandafter[\the\numexpr-8+#1]}%
\def\PML@powoften@neg#1[#2]{%
\expandafter\PML@powoften@neg@a\romannumeral0\xintround{9}{#1[#2]}.%
}%
\def\PML@powoften@neg@a#1.#2.{\ifnum#2=0 \xint_afterfi{1\relax/1[#1]}\else
\expandafter\expandafter\expandafter
\PML@Pa\expandafter\@gobble\the\numexpr2000000000-#2.%
\expandafter[\the\numexpr-9+#1\expandafter]\fi
}%
%
\def\PoorManPower#1#2% #1**#2
{%
\PoorManPowerOfTen{\xintMul{#2}{\PoorManLogBaseTen{#1}}}%
}%
%%
%% ADD-ONS TO XINTEXPR :
%%
%% log10(), pow10()
%%
%% Does not matter if xintexpr loaded before or after.
%%
\catcode`\_ 11 \catcode`\: 11 %
\unless\ifcsname XINT_expr_func_log10\endcsname
\@namedef{XINT_expr_func_log10}#1#2#3%
{%
\expandafter #1\expandafter #2\csname.=%
\XINT:NEhook:one\PoorManLogBaseTen{\XINT_expr_unlock #3}\endcsname
}%
\fi
\unless\ifcsname XINT_flexpr_func_log10\endcsname
\expandafter\let\csname XINT_flexpr_func_log10\expandafter\endcsname
\csname XINT_expr_func_log10\endcsname
\fi
%
\unless\ifcsname XINT_expr_func_pow10\endcsname
\@namedef{XINT_expr_func_pow10}#1#2#3%
{%
\expandafter #1\expandafter #2\csname.=%
\XINT:NEhook:one\PoorManPowerOfTen{\XINT_expr_unlock #3}\endcsname
}%
\fi
\unless\ifcsname XINT_flexpr_func_pow10\endcsname
\expandafter\let\csname XINT_flexpr_func_pow10\expandafter\endcsname
\csname XINT_expr_func_pow10\endcsname
\fi
%%
%% On-demand modification of ** or ^ operators in \xintexpr and \xintfloatexpr
%%
\catcode`\* 11 \catcode0 12 %
%
\def\poormanloghack**{%
\def\XINT_expr_op_** ##1%
{%
\expandafter \XINT_expr_until_**_a
\expandafter ##1\romannumeral`^^@\expandafter\XINT_expr_getnext
}%
\def\XINT_expr_until_**_a ##1{%
\def\XINT_expr_until_**_a ####1####2%
{%
\xint_UDsignfork
####2{\expandafter \XINT_expr_until_**_a \expandafter ##1%
\romannumeral`^^@##1}%
-{\XINT_expr_until_**_b ####1####2}%
\krof
}}\expandafter\XINT_expr_until_**_a\csname XINT_expr_op_-ix\endcsname
\def\XINT_expr_until_**_b ##1##2##3##4%
{%
\ifnum ##2>\XINT_expr_precedence_**
\xint_afterfi
{\expandafter \XINT_expr_until_**_a \expandafter ##1%
\romannumeral`^^@\csname XINT_expr_op_##3\endcsname {##4}}%
\else
\xint_afterfi
{\expandafter ##2\expandafter ##3%
\csname .=\XINT:NEhook:two
\PoorManPower{\XINT_expr_unlock ##1}{\XINT_expr_unlock ##4}\endcsname
}%
\fi
}%
\let\XINT_flexpr_op_** \XINT_expr_op_**
\let\XINT_flexpr_until_**_a\XINT_expr_until_**_a
\let\XINT_flexpr_until_**_b\XINT_expr_until_**_b
}%
\catcode`\^ 11 \catcode`\* 7 %
\def\poormanloghack^{%
\def\XINT_expr_until_^_b ##1##2##3##4%
{%
\ifnum ##2>\XINT_expr_precedence_^
\xint_afterfi
{\expandafter \XINT_expr_until_^_a \expandafter ##1%
\romannumeral`**@\csname XINT_expr_op_##3\endcsname {##4}}%
\else
\xint_afterfi
{\expandafter ##2\expandafter ##3%
\csname .=\XINT:NEhook:two
\PoorManPower{\XINT_expr_unlock ##1}{\XINT_expr_unlock ##4}\endcsname
}%
\fi
}%
\let\XINT_flexpr_until_^_b\XINT_expr_until_^_b
}%
\@tempa
\def\poormanloghack#1{\csname poormanloghack#1\endcsname}%
\endinput
|
