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