package bigint;
# arbitrary size integer math package
#
# by Mark Biggar
#
# Canonical Big integer value are strings of the form
# /^[+-]\d+$/ with leading zeros suppressed
# Input values to these routines may be strings of the form
# /^\s*[+-]?[\d\s]+$/.
# Examples:
# '+0' canonical zero value
# ' -123 123 123' canonical value '-123123123'
# '1 23 456 7890' canonical value '+1234567890'
# Output values always always in canonical form
#
# Actual math is done in an internal format consisting of an array
# whose first element is the sign (/^[+-]$/) and whose remaining
# elements are base 100000 digits with the least significant digit first.
# The string 'NaN' is used to represent the result when input arguments
# are not numbers, as well as the result of dividing by zero
#
# routines provided are:
#
# bneg(BINT) return BINT negation
# babs(BINT) return BINT absolute value
# bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0)
# badd(BINT,BINT) return BINT addition
# bsub(BINT,BINT) return BINT subtraction
# bmul(BINT,BINT) return BINT multiplication
# bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar
# bmod(BINT,BINT) return BINT modulus
# bgcd(BINT,BINT) return BINT greatest common divisor
# bnorm(BINT) return BINT normalization
#
$zero = 0;
# normalize string form of number. Strip leading zeros. Strip any
# white space and add a sign, if missing.
# Strings that are not numbers result the value 'NaN'.
sub main'bnorm { #(num_str) return num_str
local($_) = @_;
s/\s+//g; # strip white space
if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number
substr($_,$[,0) = '+' unless $1; # Add missing sign
s/^-0/+0/;
$_;
} else {
'NaN';
}
}
# Convert a number from string format to internal base 100000 format.
# Assumes normalized value as input.
sub internal { #(num_str) return int_num_array
local($d) = @_;
($is,$il) = (substr($d,$[,1),length($d)-2);
substr($d,$[,1) = '';
($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
}
# Convert a number from internal base 100000 format to string format.
# This routine scribbles all over input array.
sub external { #(int_num_array) return num_str
$es = shift;
grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad
&'bnorm(join('', $es, reverse(@_))); # reverse concat and normalize
}
# Negate input value.
sub main'bneg { #(num_str) return num_str
local($_) = &'bnorm(@_);
vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
s/^./N/ unless /^[-+]/; # works both in ASCII and EBCDIC
$_;
}
# Returns the absolute value of the input.
sub main'babs { #(num_str) return num_str
&abs(&'bnorm(@_));
}
sub abs { # post-normalized abs for internal use
local($_) = @_;
s/^-/+/;
$_;
}
# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
sub main'bcmp { #(num_str, num_str) return cond_code
local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
if ($x eq 'NaN') {
undef;
} elsif ($y eq 'NaN') {
undef;
} else {
&cmp($x,$y);
}
}
sub cmp { # post-normalized compare for internal use
local($cx, $cy) = @_;
return 0 if ($cx eq $cy);
local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1));
local($ld);
if ($sx eq '+') {
return 1 if ($sy eq '-' || $cy eq '+0');
$ld = length($cx) - length($cy);
return $ld if ($ld);
return $cx cmp $cy;
} else { # $sx eq '-'
return -1 if ($sy eq '+');
$ld = length($cy) - length($cx);
return $ld if ($ld);
return $cy cmp $cx;
}
}
sub main'badd { #(num_str, num_str) return num_str
local(*x, *y); ($x, $y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
if ($x eq 'NaN') {
'NaN';
} elsif ($y eq 'NaN') {
'NaN';
} else {
@x = &internal($x); # convert to internal form
@y = &internal($y);
local($sx, $sy) = (shift @x, shift @y); # get signs
if ($sx eq $sy) {
&external($sx, &add(*x, *y)); # if same sign add
} else {
($x, $y) = (&abs($x),&abs($y)); # make abs
if (&cmp($y,$x) > 0) {
&external($sy, &sub(*y, *x));
} else {
&external($sx, &sub(*x, *y));
}
}
}
}
sub main'bsub { #(num_str, num_str) return num_str
&'badd($_[$[],&'bneg($_[$[+1]));
}
# GCD -- Euclids algorithm Knuth Vol 2 pg 296
sub main'bgcd { #(num_str, num_str) return num_str
local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
if ($x eq 'NaN' || $y eq 'NaN') {
'NaN';
} else {
($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0';
$x;
}
}
# routine to add two base 1e5 numbers
# stolen from Knuth Vol 2 Algorithm A pg 231
# there are separate routines to add and sub as per Kunth pg 233
sub add { #(int_num_array, int_num_array) return int_num_array
local(*x, *y) = @_;
$car = 0;
for $x (@x) {
last unless @y || $car;
$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5) ? 1 : 0;
}
for $y (@y) {
last unless $car;
$y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0;
}
(@x, @y, $car);
}
# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
sub sub { #(int_num_array, int_num_array) return int_num_array
local(*sx, *sy) = @_;
$bar = 0;
for $sx (@sx) {
last unless @y || $bar;
$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
}
@sx;
}
# multiply two numbers -- stolen from Knuth Vol 2 pg 233
sub main'bmul { #(num_str, num_str) return num_str
local(*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
if ($x eq 'NaN') {
'NaN';
} elsif ($y eq 'NaN') {
'NaN';
} else {
@x = &internal($x);
@y = &internal($y);
local($signr) = (shift @x ne shift @y) ? '-' : '+';
@prod = ();
for $x (@x) {
($car, $cty) = (0, $[);
for $y (@y) {
$prod = $x * $y + $prod[$cty] + $car;
$prod[$cty++] =
$prod - ($car = int($prod * 1e-5)) * 1e5;
}
$prod[$cty] += $car if $car;
$x = shift @prod;
}
&external($signr, @x, @prod);
}
}
# modulus
sub main'bmod { #(num_str, num_str) return num_str
(&'bdiv(@_))[$[+1];
}
sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str
local (*x, *y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
return wantarray ? ('NaN','NaN') : 'NaN'
if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0');
return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
@x = &internal($x); @y = &internal($y);
$srem = $y[$[];
$sr = (shift @x ne shift @y) ? '-' : '+';
$car = $bar = $prd = 0;
if (($dd = int(1e5/($y[$#y]+1))) != 1) {
for $x (@x) {
$x = $x * $dd + $car;
$x -= ($car = int($x * 1e-5)) * 1e5;
}
push(@x, $car); $car = 0;
for $y (@y) {
$y = $y * $dd + $car;
$y -= ($car = int($y * 1e-5)) * 1e5;
}
}
else {
push(@x, 0);
}
@q = (); ($v2,$v1) = @y[-2,-1];
while ($#x > $#y) {
($u2,$u1,$u0) = @x[-3..-1];
$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
--$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2);
if ($q) {
($car, $bar) = (0,0);
for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
$prd = $q * $y[$y] + $car;
$prd -= ($car = int($prd * 1e-5)) * 1e5;
$x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
}
if ($x[$#x] < $car + $bar) {
$car = 0; --$q;
for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
$x[$x] -= 1e5
if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
}
}
}
pop(@x); unshift(@q, $q);
}
if (wantarray) {
@d = ();
if ($dd != 1) {
$car = 0;
for $x (reverse @x) {
$prd = $car * 1e5 + $x;
$car = $prd - ($tmp = int($prd / $dd)) * $dd;
unshift(@d, $tmp);
}
}
else {
@d = @x;
}
(&external($sr, @q), &external($srem, @d, $zero));
} else {
&external($sr, @q);
}
}
1;
