%% poormanlog.tex 0.07, 2022/05/25
%%   (macros not modified since 0.05, 2019/04/22)
%% Copyright (C) 2019-2022, 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
%% 
%
% 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)
%
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{%
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\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}%
\@tempa
\endinput