package bigfloat; require "bigint.pl"; # Arbitrary length float math package # # by Mark Biggar # # number format # canonical strings have the form /[+-]\d+E[+-]\d+/ # Input values can have inbedded whitespace # Error returns # 'NaN' An input parameter was "Not a Number" or # divide by zero or sqrt of negative number # Division is computed to # max($div_scale,length(dividend)+length(divisor)) # digits by default. # Also used for default sqrt scale $div_scale = 40; # Rounding modes one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'. $rnd_mode = 'even'; # bigfloat routines # # fadd(NSTR, NSTR) return NSTR addition # fsub(NSTR, NSTR) return NSTR subtraction # fmul(NSTR, NSTR) return NSTR multiplication # fdiv(NSTR, NSTR[,SCALE]) returns NSTR division to SCALE places # fneg(NSTR) return NSTR negation # fabs(NSTR) return NSTR absolute value # fcmp(NSTR,NSTR) return CODE compare undef,<0,=0,>0 # fround(NSTR, SCALE) return NSTR round to SCALE digits # ffround(NSTR, SCALE) return NSTR round at SCALEth place # fnorm(NSTR) return (NSTR) normalize # fsqrt(NSTR[, SCALE]) return NSTR sqrt to SCALE places # Convert a number to canonical string form. # Takes something that looks like a number and converts it to # the form /^[+-]\d+E[+-]\d+$/. sub main'fnorm { #(string) return fnum_str local($_) = @_; s/\s+//g; # strip white space if (/^([+-]?)(\d*)(\.(\d*))?([Ee]([+-]?\d+))?$/ && ($2 ne '' || defined($4))) { my $x = defined($4) ? $4 : ''; &norm(($1 ? "$1$2$x" : "+$2$x"), (($x ne '') ? $6-length($x) : $6)); } else { 'NaN'; } } # normalize number -- for internal use sub norm { #(mantissa, exponent) return fnum_str local($_, $exp) = @_; if ($_ eq 'NaN') { 'NaN'; } else { s/^([+-])0+/$1/; # strip leading zeros if (length($_) == 1) { '+0E+0'; } else { $exp += length($1) if (s/(0+)$//); # strip trailing zeros sprintf("%sE%+ld", $_, $exp); } } } # negation sub main'fneg { #(fnum_str) return fnum_str local($_) = &'fnorm($_[$[]); vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0E+0'; # flip sign s/^H/N/; $_; } # absolute value sub main'fabs { #(fnum_str) return fnum_str local($_) = &'fnorm($_[$[]); s/^-/+/; # mash sign $_; } # multiplication sub main'fmul { #(fnum_str, fnum_str) return fnum_str local($x,$y) = (&'fnorm($_[$[]),&'fnorm($_[$[+1])); if ($x eq 'NaN' || $y eq 'NaN') { 'NaN'; } else { local($xm,$xe) = split('E',$x); local($ym,$ye) = split('E',$y); &norm(&'bmul($xm,$ym),$xe+$ye); } } # addition sub main'fadd { #(fnum_str, fnum_str) return fnum_str local($x,$y) = (&'fnorm($_[$[]),&'fnorm($_[$[+1])); if ($x eq 'NaN' || $y eq 'NaN') { 'NaN'; } else { local($xm,$xe) = split('E',$x); local($ym,$ye) = split('E',$y); ($xm,$xe,$ym,$ye) = ($ym,$ye,$xm,$xe) if ($xe < $ye); &norm(&'badd($ym,$xm.('0' x ($xe-$ye))),$ye); } } # subtraction sub main'fsub { #(fnum_str, fnum_str) return fnum_str &'fadd($_[$[],&'fneg($_[$[+1])); } # division # args are dividend, divisor, scale (optional) # result has at most max(scale, length(dividend), length(divisor)) digits sub main'fdiv #(fnum_str, fnum_str[,scale]) return fnum_str { local($x,$y,$scale) = (&'fnorm($_[$[]),&'fnorm($_[$[+1]),$_[$[+2]); if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0E+0') { 'NaN'; } else { local($xm,$xe) = split('E',$x); local($ym,$ye) = split('E',$y); $scale = $div_scale if (!$scale); $scale = length($xm)-1 if (length($xm)-1 > $scale); $scale = length($ym)-1 if (length($ym)-1 > $scale); $scale = $scale + length($ym) - length($xm); &norm(&round(&'bdiv($xm.('0' x $scale),$ym),$ym), $xe-$ye-$scale); } } # round int $q based on fraction $r/$base using $rnd_mode sub round { #(int_str, int_str, int_str) return int_str local($q,$r,$base) = @_; if ($q eq 'NaN' || $r eq 'NaN') { 'NaN'; } elsif ($rnd_mode eq 'trunc') { $q; # just truncate } else { local($cmp) = &'bcmp(&'bmul($r,'+2'),$base); if ( $cmp < 0 || ($cmp == 0 && ( $rnd_mode eq 'zero' || ($rnd_mode eq '-inf' && (substr($q,$[,1) eq '+')) || ($rnd_mode eq '+inf' && (substr($q,$[,1) eq '-')) || ($rnd_mode eq 'even' && $q =~ /[24680]$/) || ($rnd_mode eq 'odd' && $q =~ /[13579]$/) )) ) { $q; # round down } else { &'badd($q, ((substr($q,$[,1) eq '-') ? '-1' : '+1')); # round up } } } # round the mantissa of $x to $scale digits sub main'fround { #(fnum_str, scale) return fnum_str local($x,$scale) = (&'fnorm($_[$[]),$_[$[+1]); if ($x eq 'NaN' || $scale <= 0) { $x; } else { local($xm,$xe) = split('E',$x); if (length($xm)-1 <= $scale) { $x; } else { &norm(&round(substr($xm,$[,$scale+1), "+0".substr($xm,$[+$scale+1,1),"+10"), $xe+length($xm)-$scale-1); } } } # round $x at the 10 to the $scale digit place sub main'ffround { #(fnum_str, scale) return fnum_str local($x,$scale) = (&'fnorm($_[$[]),$_[$[+1]); if ($x eq 'NaN') { 'NaN'; } else { local($xm,$xe) = split('E',$x); if ($xe >= $scale) { $x; } else { $xe = length($xm)+$xe-$scale; if ($xe < 1) { '+0E+0'; } elsif ($xe == 1) { &norm(&round('+0',"+0".substr($xm,$[+1,1),"+10"), $scale); } else { &norm(&round(substr($xm,$[,$xe), "+0".substr($xm,$[+$xe,1),"+10"), $scale); } } } } # compare 2 values returns one of undef, <0, =0, >0 # returns undef if either or both input value are not numbers sub main'fcmp #(fnum_str, fnum_str) return cond_code { local($x, $y) = (&'fnorm($_[$[]),&'fnorm($_[$[+1])); if ($x eq "NaN" || $y eq "NaN") { undef; } else { ord($y) <=> ord($x) || ( local($xm,$xe,$ym,$ye) = split('E', $x."E$y"), (($xe <=> $ye) * (substr($x,$[,1).'1') || &bigint'cmp($xm,$ym)) ); } } # square root by Newtons method. sub main'fsqrt { #(fnum_str[, scale]) return fnum_str local($x, $scale) = (&'fnorm($_[$[]), $_[$[+1]); if ($x eq 'NaN' || $x =~ /^-/) { 'NaN'; } elsif ($x eq '+0E+0') { '+0E+0'; } else { local($xm, $xe) = split('E',$x); $scale = $div_scale if (!$scale); $scale = length($xm)-1 if ($scale < length($xm)-1); local($gs, $guess) = (1, sprintf("1E%+d", (length($xm)+$xe-1)/2)); while ($gs < 2*$scale) { $guess = &'fmul(&'fadd($guess,&'fdiv($x,$guess,$gs*2)),".5"); $gs *= 2; } &'fround($guess, $scale); } } 1;