-%% 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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