--- /dev/null
+%% realcalc.tex
+%%
+%% (C) Frank Buchholz, Jan. 1993
+%% e-mail: buchholz@lusty.informatik.uni-dortmund.de
+% User documentation: All lines starting with two %
+% Try grep to extract these lines.
+%%
+%% Real arithmetic with big values and high precision.
+%% Calculations are done with 9 decimal digits.
+%% Usable with TeX and LaTeX.
+%%
+%% Parameter:
+%%
+%% <macro> Macro, witch catches the result.
+%% <value> String or macro witch expands to a value in the range
+%% from -2147483647.999999999 to 2147483647.999999999
+%% <precision> Number from 0 to 9
+%%
+%% Functions:
+%%
+
+% Don't read this file twice
+\ifx\Radd\undefined \else \endinput \fi
+
+% Make "@" be a letters
+\chardef\catamp=\the\catcode`\@
+\catcode`@=11
+
+\def\@Rversion{Version 1.0, Jan. 1993}
+\message{`Real arithmetic', \@Rversion (C) Frank Buchholz}
+
+% Allocation of temporary registers
+\newcount\@ta
+\newcount\@tb
+%
+\newcount\@xs % Sign of 1st value
+\newcount\@xi % Integer part of 1st value
+\newcount\@xf % Fractional part of 1st value
+%
+\newcount\@ys % Sign of 2nd value
+\newcount\@yi % Integer part of 2nd value
+\newcount\@yf % Fractional part of 2nd value
+%
+\newcount\@mil \@mil=1000000000 % 10^9
+%
+\countdef\@xa240
+\countdef\@xb241
+\countdef\@xc242
+\countdef\@xd243
+\countdef\@xe244
+\countdef\@ya245
+\countdef\@yb246
+\countdef\@yc247
+\countdef\@yd248
+\countdef\@ye249
+\let\@f=\@tb
+\let\@product=\@ta
+
+\chardef\@zero=0
+
+%% \Radd<macro><value><value> Addition
+\def\Radd#1#2#3{\@callc\@Radd#1{#2}{#3}}
+
+%% \Rsub<macro><value><value> Subtraction
+\def\Rsub#1#2#3{\@callc\@Radd#1{#2}{-#3}}
+
+\def\@Radd#1#2.#3.#4\relax#5.#6.#7\relax{%
+ % #1 Macro, witch gets the result
+ % #2 integer part of 1st value
+ % #3 fractional part of 1st value
+ % #4 dummy to swallow everthing after the 2nd '.'
+ % #5 integer part of 2nd value
+ % #6 fractional part of 2nd value
+ % #7 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#2}{#3}%
+ \@readvalue\@ys\@yi\@yf{#5}{#6}%
+ %
+ % Add integer parts
+ \@ta=\@xi \advance\@ta\@yi%
+ %
+ % Add fractional parts if they are not zero
+ \ifnum\@xf=\@zero%
+ \ifnum\@yf=\@zero%
+ \@tb=\@zero%
+ \else%
+ \@longadd%
+ \fi%
+ \else%
+ \@longadd%
+ \fi%
+ %
+ % Store result
+ \@store#1\@ta\@tb%
+}
+
+\def\@longadd{%
+ % Propagate sign to fractional parts
+ \multiply\@xf\@xs \multiply\@yf\@ys%
+ %
+ % Add fractional parts
+ \@tb=\@xf \advance\@tb\@yf%
+ %
+ % Calculate carry
+ \ifnum\@tb<\@zero%
+ \ifnum\@tb<-\@mil \advance\@ta -\@ne \advance\@tb\@mil \fi%
+ \else%
+ \ifnum\@tb<\@mil \else \advance\@ta \@ne \advance\@tb-\@mil \fi%
+ \fi%
+}
+
+%% \Rmul<macro><value><value> Multiplication
+\def\Rmul#1#2#3{\@callc\@Rmul#1{#2}{#3}}
+
+\def\@Rmul#1#2.#3.#4\relax#5.#6.#7\relax{%
+ % #1 Macro, witch gets the result
+ % #2 integer part of 1st value
+ % #3 fractional part of 1st value
+ % #4 dummy to swallow everthing after the 2nd '.'
+ % #5 integer part of 2nd value
+ % #6 fractional part of 2nd value
+ % #7 dummy to swallow everthing after the 2nd '.'
+ %
+ % How to do real multiplications:
+ % Split values in various parts
+ % x y = 1234 567890 . 123 456 789
+ % -> xa xb xc xd xe
+ % -> ya yb yc yd ye
+ % Do 21 (!) integer multiplications if there are any fractional parts
+ % r = x * Y
+ % -> r = xi*yi
+ % + (xa*yc + xa*yc )*10^3
+ % + (xa*yd + xd*ya )*10^-0
+ % + (xa*ye + xe*ya + xb*yc + xc*yb)*10^-3
+ % + (xb*yd + xd*yb + xc*yc )*10^-6
+ % + (xb*ye + xe*yb + xc*yd + xd*yc)*10^-9
+ % + (xc*ye + xe*yc + xd*yd )*10^-12
+ % + (xd*ye + xe*yd )*10^-15
+ % + (xe*ye )*10^-18
+ %
+ % -> r = ((((( xe*ye / 1000
+ % + xd*ye + xe*yd ) / 1000
+ % + xc*ye + xe*yc + xd*yd ) / 1000
+ % + xb*ye + xe*yb + xc*yd + xd*yc ) / 1000
+ % + xb*yd + xd*yb + xc*yc ) / 1000
+ % + xa*ye + ye*xa + xb*yc + xc*yb ) / 1000
+ % + xa*yd + xd*ya + (xa*yc + ya*xc)*1000 + xi*yi
+ % The last three digigs are saved just before the divisions.
+ %
+ % Init
+ %
+ \@readvalue\@xs\@xi\@xf{#2}{#3}%
+ \@readvalue\@ys\@yi\@yf{#5}{#6}%
+ %
+ % Sign
+ \multiply\@xi\@xs%
+ \multiply\@yi\@ys%
+ \multiply\@xs\@ys%
+ %
+ \@product=\@zero%
+ %
+ \ifnum\@xf=\@zero%
+ \ifnum\@yf=\@zero%
+ % No fractional parts
+ \edef\@frac{000000000} % digits of fractional part
+ \else%
+ \@longmul % Long multiplication
+ \fi%
+ \else%
+ \@longmul % Long multiplication
+ \fi%
+ %
+ % Multiply integer parts
+ \count@=\@xi \multiply\count@\@yi%
+ \advance\@product\count@%
+ %
+ % Sign
+ \multiply\@product\@xs%
+ %
+ % Store result
+ \edef#1{\number\@product.\@frac}%
+}
+
+% Do long multiplication
+\def\@longmul{%
+ % Split values in pieces
+ \@f=1000000%
+ \count@=\@xi%
+ \divide\count@\@f \@xa=\count@%
+ \multiply\count@-\@f \advance\count@\@xi \@xb=\count@%
+ \count@=\@xf%
+ \divide\count@\@f \@xc=\count@%
+ \multiply\count@-\@f \advance\count@\@xf%
+ \@f=1000%
+ \@xe=\count@%
+ \divide\count@\@f \@xd=\count@%
+ \multiply\count@-\@f \advance\count@\@xe \@xe=\count@%
+ %
+ \@f=1000000%
+ \count@=\@yi%
+ \divide\count@\@f \@ya=\count@%
+ \multiply\count@-\@f \advance\count@\@yi \@yb=\count@%
+ \count@=\@yf%
+ \divide\count@\@f \@yc=\count@%
+ \multiply\count@-\@f \advance\count@\@yf%
+ \@f=1000%
+ \@ye=\count@%
+ \divide\count@\@f \@yd=\count@%
+ \multiply\count@-\@f \advance\count@\@ye \@ye=\count@%
+ %
+ \edef\@frac{} % digits of fractional part
+ %
+ \@mul ee \relax\@shift % 10^-18
+ \@mul de ed \relax\@shift % 10^-15
+ \@mul ce ec dd \relax\@shift % 10^-12
+ \@mul be eb cd dc \relax\@saveshift % 10^-9
+ \@mul bd db cc \relax\@saveshift % 10^-6
+ \@mul ae ea bc cb \relax\@saveshift % 10^-3
+ \multiply\@xa\@f \multiply\@xc\@f % 10^3
+ \@mul ad da ac ca \relax%
+}
+
+\def\@mul#1#2#3{%
+ \count@=\csname @x#1\endcsname%
+ \multiply\count@\csname @y#2\endcsname%
+ \advance\@product\count@%
+ %
+ \ifx#3\relax%
+ \let\next=\relax%
+ \else%
+ \let\next=\@mul%
+ \fi%
+ \next#3%
+}
+
+\def\@shift{%
+ \divide\@product\@f%
+}
+
+\def\@saveshift{%
+ % Save rightmost digits
+ \count@=\@product%
+ \divide\@product\@f%
+ \multiply\@product\@f%
+ \advance\count@-\@product%
+ \advance\count@\@f%
+ \edef\@frac{\expandafter\@ignorenext\number\count@\@frac}%
+ %
+ \divide\@product\@f%
+}
+
+%% \Rdiv<macro><value><value> Division
+\def\Rdiv#1#2#3{\@callc\@Radd#1{#2}{#3}}
+
+\def\@Rdiv#1#2.#3.#4\relax#5.#6.#7\relax{%
+ % #1 Macro, witch gets the result
+ % #2 integer part of 1st value
+ % #3 fractional part of 1st value
+ % #4 dummy to swallow everthing after the 2nd '.'
+ % #5 integer part of 2nd value
+ % #6 fractional part of 2nd value
+ % #7 dummy to swallow everthing after the 2nd '.'
+ %
+ \message{Error: Rdiv not ready}
+ %
+ \@readvalue\@xs\@xi\@xf{#2}{#3}%
+ \@readvalue\@ys\@yi\@yf{#5}{#6}%
+ %
+ \ifnum\@yi=\@zero
+ \@xi=\@zero
+ \else
+ \divide\@xi\@yi
+ \fi
+ %
+ \@store#1\@xi\@xf%
+}
+
+%% \Rneg<macro><value> Negation
+\def\Rneg#1#2{%
+ % Expand the value and split it into the integer and the fractional part
+ \edef\next{\noexpand\@Rneg\noexpand#1#2..\noexpand\relax}%
+ \next%
+}
+
+\def\@Rneg#1#2.#3.#4\relax{
+ % #1 Macro, witch gets the result
+ % #2 Integer part of value
+ % #3 Fractional part of value
+ % #4 Remaining tokens
+ %
+ \@readvalue\@xs\@xi\@xf{#2}{#3}%
+ %
+ % Change sign
+ \multiply\@xi-\@ne%
+ \multiply\@xf-\@xs%
+ %
+ % Store result
+ \@store#1\@xi\@xf%
+}
+
+%%
+%% \Rtrunc<macro><precision><value> Truncate value to specified precision
+\def\Rtrunc#1#2#3{%
+ % Expand parameter and split the value into the integer and fractional part
+ \edef\next{\noexpand\@Rtrunc\noexpand#1#2#3..\noexpand\relax}%
+ \next%
+}
+
+\def\@Rtrunc#1#2#3.#4.#5\relax{%
+ % #1 Macro, witch gets the result
+ % #2 Number of decimal places
+ % #3 Integer part of value
+ % #4 Fractional part of value
+ % #5 dummy to swallow everthing after the 2nd '.'
+ %
+ \count@=#2%
+ \edef\next{\empty}%
+ \ifnum\count@>\@zero%
+ \expandafter\@@Rtrunc#4000000000\relax%
+ \fi%
+ \edef#1{#3\next}%
+}
+
+\def\@@Rtrunc#1#2#3#4#5#6#7#8#9{%
+ % #1...#9 fractional part, padded with trailing zeros
+ \edef\next{.#1%
+ \ifnum\count@>1 #2\fi%
+ \ifnum\count@>2 #3\fi%
+ \ifnum\count@>3 #4\fi%
+ \ifnum\count@>4 #5\fi%
+ \ifnum\count@>5 #6\fi%
+ \ifnum\count@>6 #7\fi%
+ \ifnum\count@>7 #8\fi%
+ \ifnum\count@>8 #9\fi%
+ }%
+ \@swallow%
+}
+
+%%
+%% \Rifle<value><value> Test v1 < v2
+\def\Rifle#1#2{\@callb\@Rifle{#1}{#2}}
+
+\def\@Rifle#1.#2.#3\relax#4.#5.#6\relax{%
+ % #1 integer part of 1st value
+ % #2 fractional part of 1st value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ % #4 integer part of 2nd value
+ % #5 fractional part of 2nd value
+ % #6 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ \@readvalue\@ys\@yi\@yf{#4}{#5}%
+ %
+ \ifnum\@xs<\@ys%
+ \Rtesttrue%
+ \else%
+ \ifnum\@xs>\@ys%
+ \Rtestfalse%
+ \else%
+ \Rtestfalse%
+ \ifnum\@xi<\@yi \Rtesttrue \fi%
+ \ifnum\@xi=\@yi%
+ \multiply\@xf\@xs \multiply\@yf\@ys%
+ \ifnum\@xf<\@yf \Rtesttrue \fi%
+ \fi%
+ \fi%
+ \fi%
+ %
+ \ifRtest%
+}
+
+%% \Rifeq<value><value> Test v1 = v2
+\def\Rifeq#1#2{\@callb\@Rifeq{#1}{#2}}
+
+\def\@Rifeq#1.#2.#3\relax#4.#5.#6\relax{%
+ % #1 integer part of 1st value
+ % #2 fractional part of 1st value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ % #4 integer part of 2nd value
+ % #5 fractional part of 2nd value
+ % #6 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ \@readvalue\@ys\@yi\@yf{#4}{#5}%
+ %
+ \Rtestfalse%
+ \ifnum\@xi=\@yi\ifnum\@xf=\@yf \Rtesttrue \fi\fi%
+ \ifRtest%
+}
+
+%% \Rifgt<value><value> Test v1 > v2
+\def\Rifgt#1#2{\@callb\@Rifgt{#1}{#2}}
+
+\def\@Rifgt#1.#2.#3\relax#4.#5.#6\relax{%
+ % #1 integer part of 1st value
+ % #2 fractional part of 1st value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ % #4 integer part of 2nd value
+ % #5 fractional part of 2nd value
+ % #6 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ \@readvalue\@ys\@yi\@yf{#4}{#5}%
+ %
+ \ifnum\@xs>\@ys%
+ \Rtesttrue%
+ \else%
+ \ifnum\@xs<\@ys%
+ \Rtestfalse%
+ \else%
+ \Rtestfalse%
+ \ifnum\@xi>\@yi \Rtesttrue \fi%
+ \ifnum\@xi=\@yi%
+ \multiply\@xf\@xs \multiply\@yf\@ys%
+ \ifnum\@xf>\@yf \Rtesttrue \fi%
+ \fi%
+ \fi%
+ \fi%
+ %
+ \ifRtest%
+}
+
+%% \Rifneg<value> Test v < 0
+\def\Rifneg#1{\@calla\@Rifneg{#1}}
+
+\def\@Rifneg#1.#2.#3\relax{%
+ % #1 Integer part of value
+ % #2 Fractional part of value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ %
+ \ifnum\@xs<0 \Rtesttrue \else \Rtestfalse \fi%
+ \ifRtest%
+ }
+
+%% \Rifzero<value> Test v = 0
+\def\Rifzero#1{\@calla\@Rifzero{#1}}
+
+\def\@Rifzero#1.#2.#3\relax{%
+ % #1 Integer part of value
+ % #2 Fractional part of value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ %
+ \Rtestfalse%
+ \ifnum\@xi=\@zero \ifnum\@xf=\@zero \Rtesttrue \fi \fi%
+ \ifRtest%
+}
+
+%% \Rifpos<value> Test v >= 0
+%% Take care to have a corresponding \fi
+\def\Rifpos#1{\@calla\@Rifpos{#1}}
+
+\def\@Rifpos#1.#2.#3\relax{%
+ % #1 Integer part of value
+ % #2 Fractional part of value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ %
+ \ifnum\@xs<0 \Rtestfalse \else \Rtesttrue \fi%
+ \ifRtest%
+}
+
+%% \Rifint<value> Test v is an integer value
+%% Take care to have a corresponding \fi
+\def\Rifint#1{\@calla\@Rifint{#1}}
+
+\def\@Rifint#1.#2.#3\relax{%
+ % #1 Integer part of value
+ % #2 Fractional part of value
+ % #3 dummy to swallow everthing after the 2nd '.'
+ %
+ \@readvalue\@xs\@xi\@xf{#1}{#2}%
+ %
+ \ifnum\@xf=0 \Rtesttrue \else \Rtestfalse \fi%
+ \ifRtest%
+}
+
+% Utility macros
+
+\def\@calla#1#2{%
+ % #1 Macro to call
+ % #2 Value
+ % Expand the value and split them into the integer and the fractional part
+ \edef\next{\noexpand#1#2..\noexpand\relax}%
+ \next%
+}
+
+\def\@callb#1#2#3{%
+ % #1 Macro to call
+ % #2 1st value
+ % #3 2nd value
+ % Expand the values and split them into the integer and the fractional parts
+ \edef\next{\noexpand#1#2..\noexpand\relax#3..\noexpand\relax}%
+ \next%
+}
+
+\def\@callc#1#2#3#4{%
+ % #1 Macro to call
+ % #2 Macro, witch gets the result
+ % #3 1st value
+ % #4 2nd value
+ % Expand the values and split them into the integer and the fractional parts
+ \edef\next{\noexpand#1\noexpand#2#3..\noexpand\relax#4..\noexpand\relax}%
+ \next%
+}
+
+%% \ifRtest ... \else ... \fi Re-execute the last test
+\newif\ifRtest
+
+%
+% Read value
+%
+\def\@readvalue#1#2#3#4#5{%
+ % #1 macro to catch the sign ( -1 or 1 )
+ % #2 macro to catch the integer part (maybe negative)
+ % #3 macro to catch the fractional part (positive)
+ % #4.#5 value
+ %
+ % Regular expression [-...d]d...["."[d...]]
+ % Attention: There has to be at least one digit between "-" and "."!
+ %
+ % Integer part
+ \if !#4! #2=\@zero \else #2=#4 \fi%
+ %
+ % Sign
+ \ifnum#2<\@zero #1=-\@ne \else #1=\@ne \fi%
+ \ifnum#2=\@zero \ifnum #4#5<\@zero #1=-\@ne \fi \fi%
+ %
+ % Fractional part
+ \if !#5!%
+ #3=\@zero%
+ \else%
+ \@@setcount@#5000000000\relax%
+ #3=\count@%
+ \fi%
+}
+
+% Set count@ to fractional part with trailing zeros
+\def\@@setcount@#1#2#3#4#5#6#7#8#9{%
+ % #1...#9 fractional part, padded with '0'
+ \count@=#1#2#3#4#5#6#7#8#9%
+ \@swallow%
+}
+
+% Swallow everything up to next \relax
+\def\@swallow#1\relax{}
+
+%
+% Store result
+%
+\def\@store#1#2#3{%
+ % #1 marcro to catch the result
+ % #2 counter with integer part (maybe negative)
+ % #3 counter with fractional part (maybe negative}
+ %
+ % Sign
+ \edef#1{}%
+ \count@=#3%
+ \ifnum\count@<\@zero%
+ \count@=-\count@%
+ \ifnum#2=\@zero%
+ \edef#1{-}%
+ \fi%
+ \fi%
+ %
+ % Pad fractional part with leading zeros
+ \advance\count@\@mil%
+ %
+ % Store result
+ \edef#1{#1\number#2.\expandafter\@ignorenext\number\count@}%
+}
+
+% Ignore next token
+\def\@ignorenext#1{}
+
+% Restore catcode of "@"
+\catcode`\@=\catamp
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