package Math::BigInt; use overload '+' => sub {new Math::BigInt &badd}, '-' => sub {new Math::BigInt \$_[2]? bsub(\$_[1],\${\$_[0]}) : bsub(\${\$_[0]},\$_[1])}, '<=>' => sub {new Math::BigInt \$_[2]? bcmp(\$_[1],\${\$_[0]}) : bcmp(\${\$_[0]},\$_[1])}, 'cmp' => sub {new Math::BigInt \$_[2]? (\$_[1] cmp \${\$_[0]}) : (\${\$_[0]} cmp \$_[1])}, '*' => sub {new Math::BigInt &bmul}, '/' => sub {new Math::BigInt \$_[2]? scalar bdiv(\$_[1],\${\$_[0]}) : scalar bdiv(\${\$_[0]},\$_[1])}, '%' => sub {new Math::BigInt \$_[2]? bmod(\$_[1],\${\$_[0]}) : bmod(\${\$_[0]},\$_[1])}, '**' => sub {new Math::BigInt \$_[2]? bpow(\$_[1],\${\$_[0]}) : bpow(\${\$_[0]},\$_[1])}, 'neg' => sub {new Math::BigInt &bneg}, 'abs' => sub {new Math::BigInt &babs}, qw( "" stringify 0+ numify) # Order of arguments unsignificant ; \$NaNOK=1; sub new { my(\$class) = shift; my(\$foo) = bnorm(shift); die "Not a number initialized to Math::BigInt" if !\$NaNOK && \$foo eq "NaN"; bless \\$foo, \$class; } sub stringify { "\${\$_[0]}" } sub numify { 0 + "\${\$_[0]}" } # Not needed, additional overhead # comparing to direct compilation based on # stringify sub import { shift; return unless @_; die "unknown import: @_" unless @_ == 1 and \$_[0] eq ':constant'; overload::constant integer => sub {Math::BigInt->new(shift)}; } \$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 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 bneg { #(num_str) return num_str local(\$_) = &bnorm(@_); vec(\$_,0,8) ^= ord('+') ^ ord('-') unless \$_ eq '+0'; s/^H/N/; \$_; } # Returns the absolute value of the input. sub 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 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 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 bsub { #(num_str, num_str) return num_str &badd(\$_[\$[],&bneg(\$_[\$[+1])); } # GCD -- Euclids algorithm Knuth Vol 2 pg 296 sub 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 += (@y ? shift(@y) : 0) + \$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 @sy || \$bar; \$sx += 1e5 if \$bar = ((\$sx -= (@sy ? shift(@sy) : 0) + \$bar) < 0); } @sx; } # multiply two numbers -- stolen from Knuth Vol 2 pg 233 sub 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); &external(&mul(*x,*y)); } } # multiply two numbers in internal representation # destroys the arguments, supposes that two arguments are different sub mul { #(*int_num_array, *int_num_array) return int_num_array local(*x, *y) = (shift, shift); local(\$signr) = (shift @x ne shift @y) ? '-' : '+'; @prod = (); for \$x (@x) { (\$car, \$cty) = (0, \$[); for \$y (@y) { \$prod = \$x * \$y + (\$prod[\$cty] || 0) + \$car; \$prod[\$cty++] = \$prod - (\$car = int(\$prod * 1e-5)) * 1e5; } \$prod[\$cty] += \$car if \$car; \$x = shift @prod; } (\$signr, @x, @prod); } # modulus sub bmod { #(num_str, num_str) return num_str (&bdiv(@_))[\$[+1]; } sub 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); } } # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 sub bpow { #(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'; } elsif (\$x eq '+1') { '+1'; } elsif (\$x eq '-1') { &bmod(\$x,2) ? '-1': '+1'; } elsif (\$y =~ /^-/) { 'NaN'; } elsif (\$x eq '+0' && \$y eq '+0') { 'NaN'; } else { @x = &internal(\$x); local(@pow2)=@x; local(@pow)=&internal("+1"); local(\$y1,\$res,@tmp1,@tmp2)=(1); # need tmp to send to mul while (\$y ne '+0') { (\$y,\$res)=&bdiv(\$y,2); if (\$res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} if (\$y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} } &external(@pow); } } 1; __END__ =head1 NAME Math::BigInt - Arbitrary size integer math package =head1 SYNOPSIS use Math::BigInt; \$i = Math::BigInt->new(\$string); \$i->bneg return BINT negation \$i->babs return BINT absolute value \$i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0) \$i->badd(BINT) return BINT addition \$i->bsub(BINT) return BINT subtraction \$i->bmul(BINT) return BINT multiplication \$i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar \$i->bmod(BINT) return BINT modulus \$i->bgcd(BINT) return BINT greatest common divisor \$i->bnorm return BINT normalization =head1 DESCRIPTION All basic math operations are overloaded if you declare your big integers as \$i = new Math::BigInt '123 456 789 123 456 789'; =over 2 =item Canonical notation Big integer value are strings of the form C with leading zeros suppressed. =item Input Input values to these routines may be strings of the form C. =item Output Output values always always in canonical form =back 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. =head1 EXAMPLES '+0' canonical zero value ' -123 123 123' canonical value '-123123123' '1 23 456 7890' canonical value '+1234567890' =head1 Autocreating constants After C all the integer decimal constants in the given scope are converted to C. This convertion happens at compile time. In particular perl -MMath::BigInt=:constant -e 'print 2**100' print the integer value of C<2**100>. Note that without convertion of constants the expression 2**100 will be calculatted as floating point number. =head1 BUGS The current version of this module is a preliminary version of the real thing that is currently (as of perl5.002) under development. =head1 AUTHOR Mark Biggar, overloaded interface by Ilya Zakharevich. =cut