# arbitrary size integer math package
# Canonical Big integer value are strings of the form
# /^[+-]\d+$/ with leading zeros suppressed
# Input values to these routines may be strings of the form
# '+0' canonical zero value
# ' -123 123 123' canonical value '-123123123'
# '1 23 456 7890' canonical value '+1234567890'
# Output values always always in canonical form
# Actual math is done in an internal format consisting of an array
# whose first element is the sign (/^[+-]$/) 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
# 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
# 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
s/\s+//g; # strip white space
if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number
substr($_,0,0) = '+' unless $1; # Add missing sign
# Convert a number from string format to internal base 100000 format.
# Assumes normalized value as input.
sub internal { #(num_str) return int_num_array
($is,$il) = (substr($d,0,1),length($d)-2);
($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
# Convert a number from internal base 100000 format to string format.
# This routine scribbles all over input array.
sub external { #(int_num_array) return num_str
grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad
&'bnorm
(join('', $es, reverse(@_))); # reverse concat and normalize
sub main
'bneg { #(num_str) return num_str
vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
# Returns the absolute value of the input.
sub main
'babs { #(num_str) return num_str
sub abs { # post-normalized abs for internal use
# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
sub main
'bcmp { #(num_str, num_str) return cond_code
local($x,$y) = (&'bnorm
($_[0]),&'bnorm($_[1]));
sub cmp { # post-normalized compare for internal use
($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx)
sub main'badd
{ #(num_str, num_str) return num_str
local(*x
, *y
); ($x, $y) = (&'bnorm($_[0]),&'bnorm
($_[1]));
@x = &internal
($x); # convert to internal form
local($sx, $sy) = (shift @x, shift @y); # get signs
&external
($sx, &add
(*x
, *y
)); # if same sign add
($x, $y) = (&abs($x),&abs($y)); # make abs
&external
($sy, &sub(*y
, *x
));
&external
($sx, &sub(*x
, *y
));
sub main
'bsub { #(num_str, num_str) return num_str
&'badd
($_[0],&'bneg($_[1]));
# GCD -- Euclids algorithm Knuth Vol 2 pg 296
sub main'bgcd
{ #(num_str, num_str) return num_str
local($x,$y) = (&'bnorm($_[0]),&'bnorm
($_[1]));
if ($x eq 'NaN' || $y eq 'NaN') {
($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0';
# 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
$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5);
$y -= 1e5 if $car = (($y += $car) >= 1e5);
# 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
$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
# multiply two numbers -- stolen from Knuth Vol 2 pg 233
sub main'bmul
{ #(num_str, num_str) return num_str
local(*x
, *y
); ($x, $y) = (&'bnorm($_[0]), &'bnorm
($_[1]));
local($signr) = (shift @x ne shift @y) ?
'-' : '+';
$prod = $x * $y + $prod[$cty] + $car;
$prod - ($car = int($prod * 1e-5)) * 1e5
;
$prod[$cty] += $car if $car;
&external
($signr, @x, @prod);
sub main
'bmod { #(num_str, num_str) return num_str
sub main
'bdiv { #(dividend: num_str, divisor: num_str) return num_str
local (*x, *y); ($x, $y) = (&'bnorm
($_[0]), &'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);
$sr = (shift @x ne shift @y) ? '-' : '+';
if (($dd = int(1e5/($y[$#y]+1))) != 1) {
$x -= ($car = int($x * 1e-5)) * 1e5;
push(@x, $car); $car = 0;
$y -= ($car = int($y * 1e-5)) * 1e5;
@q = (); ($v2,$v1) = @y[$#y-1,$#y];
($u2,$u1,$u0) = @x[($#x-2)..$#x];
$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
--$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2);
for ($y = 0, $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 + $bar) {
for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) {
if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
pop(@x); unshift(@q, $q);
$car = $prd - ($tmp = int($prd / $dd)) * $dd;
(&external($sr, @q), &external($srem, @d, 0));