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1 | package Math::BigInt; |
| 2 | |
| 3 | %OVERLOAD = ( |
| 4 | # Anonymous subroutines: |
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5 | '+' => sub {new Math::BigInt &badd}, |
| 6 | '-' => sub {new Math::BigInt |
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7 | $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])}, |
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8 | '<=>' => sub {new Math::BigInt |
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9 | $_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])}, |
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10 | 'cmp' => sub {new Math::BigInt |
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11 | $_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])}, |
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12 | '*' => sub {new Math::BigInt &bmul}, |
| 13 | '/' => sub {new Math::BigInt |
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14 | $_[2]? scalar bdiv($_[1],${$_[0]}) : |
| 15 | scalar bdiv(${$_[0]},$_[1])}, |
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16 | '%' => sub {new Math::BigInt |
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17 | $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])}, |
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18 | '**' => sub {new Math::BigInt |
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19 | $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])}, |
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20 | 'neg' => sub {new Math::BigInt &bneg}, |
| 21 | 'abs' => sub {new Math::BigInt &babs}, |
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22 | |
| 23 | qw( |
| 24 | "" stringify |
| 25 | 0+ numify) # Order of arguments unsignificant |
| 26 | ); |
| 27 | |
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28 | $NaNOK=1; |
| 29 | |
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30 | sub new { |
| 31 | my $foo = bnorm($_[1]); |
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32 | die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN"; |
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33 | bless \$foo; |
| 34 | } |
| 35 | sub stringify { "${$_[0]}" } |
| 36 | sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead |
| 37 | # comparing to direct compilation based on |
| 38 | # stringify |
| 39 | |
| 40 | # arbitrary size integer math package |
| 41 | # |
| 42 | # by Mark Biggar |
| 43 | # |
| 44 | # Canonical Big integer value are strings of the form |
| 45 | # /^[+-]\d+$/ with leading zeros suppressed |
| 46 | # Input values to these routines may be strings of the form |
| 47 | # /^\s*[+-]?[\d\s]+$/. |
| 48 | # Examples: |
| 49 | # '+0' canonical zero value |
| 50 | # ' -123 123 123' canonical value '-123123123' |
| 51 | # '1 23 456 7890' canonical value '+1234567890' |
| 52 | # Output values always always in canonical form |
| 53 | # |
| 54 | # Actual math is done in an internal format consisting of an array |
| 55 | # whose first element is the sign (/^[+-]$/) and whose remaining |
| 56 | # elements are base 100000 digits with the least significant digit first. |
| 57 | # The string 'NaN' is used to represent the result when input arguments |
| 58 | # are not numbers, as well as the result of dividing by zero |
| 59 | # |
| 60 | # routines provided are: |
| 61 | # |
| 62 | # bneg(BINT) return BINT negation |
| 63 | # babs(BINT) return BINT absolute value |
| 64 | # bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0) |
| 65 | # badd(BINT,BINT) return BINT addition |
| 66 | # bsub(BINT,BINT) return BINT subtraction |
| 67 | # bmul(BINT,BINT) return BINT multiplication |
| 68 | # bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar |
| 69 | # bmod(BINT,BINT) return BINT modulus |
| 70 | # bgcd(BINT,BINT) return BINT greatest common divisor |
| 71 | # bnorm(BINT) return BINT normalization |
| 72 | # |
| 73 | |
| 74 | $zero = 0; |
| 75 | |
| 76 | \f |
| 77 | # normalize string form of number. Strip leading zeros. Strip any |
| 78 | # white space and add a sign, if missing. |
| 79 | # Strings that are not numbers result the value 'NaN'. |
| 80 | |
| 81 | sub bnorm { #(num_str) return num_str |
| 82 | local($_) = @_; |
| 83 | s/\s+//g; # strip white space |
| 84 | if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number |
| 85 | substr($_,$[,0) = '+' unless $1; # Add missing sign |
| 86 | s/^-0/+0/; |
| 87 | $_; |
| 88 | } else { |
| 89 | 'NaN'; |
| 90 | } |
| 91 | } |
| 92 | |
| 93 | # Convert a number from string format to internal base 100000 format. |
| 94 | # Assumes normalized value as input. |
| 95 | sub internal { #(num_str) return int_num_array |
| 96 | local($d) = @_; |
| 97 | ($is,$il) = (substr($d,$[,1),length($d)-2); |
| 98 | substr($d,$[,1) = ''; |
| 99 | ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d))); |
| 100 | } |
| 101 | |
| 102 | # Convert a number from internal base 100000 format to string format. |
| 103 | # This routine scribbles all over input array. |
| 104 | sub external { #(int_num_array) return num_str |
| 105 | $es = shift; |
| 106 | grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad |
| 107 | &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize |
| 108 | } |
| 109 | |
| 110 | # Negate input value. |
| 111 | sub bneg { #(num_str) return num_str |
| 112 | local($_) = &bnorm(@_); |
| 113 | vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0'; |
| 114 | s/^H/N/; |
| 115 | $_; |
| 116 | } |
| 117 | |
| 118 | # Returns the absolute value of the input. |
| 119 | sub babs { #(num_str) return num_str |
| 120 | &abs(&bnorm(@_)); |
| 121 | } |
| 122 | |
| 123 | sub abs { # post-normalized abs for internal use |
| 124 | local($_) = @_; |
| 125 | s/^-/+/; |
| 126 | $_; |
| 127 | } |
| 128 | \f |
| 129 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
| 130 | sub bcmp { #(num_str, num_str) return cond_code |
| 131 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
| 132 | if ($x eq 'NaN') { |
| 133 | undef; |
| 134 | } elsif ($y eq 'NaN') { |
| 135 | undef; |
| 136 | } else { |
| 137 | &cmp($x,$y); |
| 138 | } |
| 139 | } |
| 140 | |
| 141 | sub cmp { # post-normalized compare for internal use |
| 142 | local($cx, $cy) = @_; |
| 143 | $cx cmp $cy |
| 144 | && |
| 145 | ( |
| 146 | ord($cy) <=> ord($cx) |
| 147 | || |
| 148 | ($cx cmp ',') * (length($cy) <=> length($cx) || $cy cmp $cx) |
| 149 | ); |
| 150 | } |
| 151 | |
| 152 | sub badd { #(num_str, num_str) return num_str |
| 153 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
| 154 | if ($x eq 'NaN') { |
| 155 | 'NaN'; |
| 156 | } elsif ($y eq 'NaN') { |
| 157 | 'NaN'; |
| 158 | } else { |
| 159 | @x = &internal($x); # convert to internal form |
| 160 | @y = &internal($y); |
| 161 | local($sx, $sy) = (shift @x, shift @y); # get signs |
| 162 | if ($sx eq $sy) { |
| 163 | &external($sx, &add(*x, *y)); # if same sign add |
| 164 | } else { |
| 165 | ($x, $y) = (&abs($x),&abs($y)); # make abs |
| 166 | if (&cmp($y,$x) > 0) { |
| 167 | &external($sy, &sub(*y, *x)); |
| 168 | } else { |
| 169 | &external($sx, &sub(*x, *y)); |
| 170 | } |
| 171 | } |
| 172 | } |
| 173 | } |
| 174 | |
| 175 | sub bsub { #(num_str, num_str) return num_str |
| 176 | &badd($_[$[],&bneg($_[$[+1])); |
| 177 | } |
| 178 | |
| 179 | # GCD -- Euclids algorithm Knuth Vol 2 pg 296 |
| 180 | sub bgcd { #(num_str, num_str) return num_str |
| 181 | local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1])); |
| 182 | if ($x eq 'NaN' || $y eq 'NaN') { |
| 183 | 'NaN'; |
| 184 | } else { |
| 185 | ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0'; |
| 186 | $x; |
| 187 | } |
| 188 | } |
| 189 | \f |
| 190 | # routine to add two base 1e5 numbers |
| 191 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
| 192 | # there are separate routines to add and sub as per Kunth pg 233 |
| 193 | sub add { #(int_num_array, int_num_array) return int_num_array |
| 194 | local(*x, *y) = @_; |
| 195 | $car = 0; |
| 196 | for $x (@x) { |
| 197 | last unless @y || $car; |
| 198 | $x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5); |
| 199 | } |
| 200 | for $y (@y) { |
| 201 | last unless $car; |
| 202 | $y -= 1e5 if $car = (($y += $car) >= 1e5); |
| 203 | } |
| 204 | (@x, @y, $car); |
| 205 | } |
| 206 | |
| 207 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
| 208 | sub sub { #(int_num_array, int_num_array) return int_num_array |
| 209 | local(*sx, *sy) = @_; |
| 210 | $bar = 0; |
| 211 | for $sx (@sx) { |
| 212 | last unless @y || $bar; |
| 213 | $sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0); |
| 214 | } |
| 215 | @sx; |
| 216 | } |
| 217 | |
| 218 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
| 219 | sub bmul { #(num_str, num_str) return num_str |
| 220 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
| 221 | if ($x eq 'NaN') { |
| 222 | 'NaN'; |
| 223 | } elsif ($y eq 'NaN') { |
| 224 | 'NaN'; |
| 225 | } else { |
| 226 | @x = &internal($x); |
| 227 | @y = &internal($y); |
| 228 | &external(&mul(*x,*y)); |
| 229 | } |
| 230 | } |
| 231 | |
| 232 | # multiply two numbers in internal representation |
| 233 | # destroys the arguments, supposes that two arguments are different |
| 234 | sub mul { #(*int_num_array, *int_num_array) return int_num_array |
| 235 | local(*x, *y) = (shift, shift); |
| 236 | local($signr) = (shift @x ne shift @y) ? '-' : '+'; |
| 237 | @prod = (); |
| 238 | for $x (@x) { |
| 239 | ($car, $cty) = (0, $[); |
| 240 | for $y (@y) { |
| 241 | $prod = $x * $y + $prod[$cty] + $car; |
| 242 | $prod[$cty++] = |
| 243 | $prod - ($car = int($prod * 1e-5)) * 1e5; |
| 244 | } |
| 245 | $prod[$cty] += $car if $car; |
| 246 | $x = shift @prod; |
| 247 | } |
| 248 | ($signr, @x, @prod); |
| 249 | } |
| 250 | |
| 251 | # modulus |
| 252 | sub bmod { #(num_str, num_str) return num_str |
| 253 | (&bdiv(@_))[$[+1]; |
| 254 | } |
| 255 | \f |
| 256 | sub bdiv { #(dividend: num_str, divisor: num_str) return num_str |
| 257 | local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
| 258 | return wantarray ? ('NaN','NaN') : 'NaN' |
| 259 | if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0'); |
| 260 | return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0); |
| 261 | @x = &internal($x); @y = &internal($y); |
| 262 | $srem = $y[$[]; |
| 263 | $sr = (shift @x ne shift @y) ? '-' : '+'; |
| 264 | $car = $bar = $prd = 0; |
| 265 | if (($dd = int(1e5/($y[$#y]+1))) != 1) { |
| 266 | for $x (@x) { |
| 267 | $x = $x * $dd + $car; |
| 268 | $x -= ($car = int($x * 1e-5)) * 1e5; |
| 269 | } |
| 270 | push(@x, $car); $car = 0; |
| 271 | for $y (@y) { |
| 272 | $y = $y * $dd + $car; |
| 273 | $y -= ($car = int($y * 1e-5)) * 1e5; |
| 274 | } |
| 275 | } |
| 276 | else { |
| 277 | push(@x, 0); |
| 278 | } |
| 279 | @q = (); ($v2,$v1) = @y[-2,-1]; |
| 280 | while ($#x > $#y) { |
| 281 | ($u2,$u1,$u0) = @x[-3..-1]; |
| 282 | $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1)); |
| 283 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2); |
| 284 | if ($q) { |
| 285 | ($car, $bar) = (0,0); |
| 286 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
| 287 | $prd = $q * $y[$y] + $car; |
| 288 | $prd -= ($car = int($prd * 1e-5)) * 1e5; |
| 289 | $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0)); |
| 290 | } |
| 291 | if ($x[$#x] < $car + $bar) { |
| 292 | $car = 0; --$q; |
| 293 | for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) { |
| 294 | $x[$x] -= 1e5 |
| 295 | if ($car = (($x[$x] += $y[$y] + $car) > 1e5)); |
| 296 | } |
| 297 | } |
| 298 | } |
| 299 | pop(@x); unshift(@q, $q); |
| 300 | } |
| 301 | if (wantarray) { |
| 302 | @d = (); |
| 303 | if ($dd != 1) { |
| 304 | $car = 0; |
| 305 | for $x (reverse @x) { |
| 306 | $prd = $car * 1e5 + $x; |
| 307 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
| 308 | unshift(@d, $tmp); |
| 309 | } |
| 310 | } |
| 311 | else { |
| 312 | @d = @x; |
| 313 | } |
| 314 | (&external($sr, @q), &external($srem, @d, $zero)); |
| 315 | } else { |
| 316 | &external($sr, @q); |
| 317 | } |
| 318 | } |
| 319 | |
| 320 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
| 321 | sub bpow { #(num_str, num_str) return num_str |
| 322 | local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1])); |
| 323 | if ($x eq 'NaN') { |
| 324 | 'NaN'; |
| 325 | } elsif ($y eq 'NaN') { |
| 326 | 'NaN'; |
| 327 | } elsif ($x eq '+1') { |
| 328 | '+1'; |
| 329 | } elsif ($x eq '-1') { |
| 330 | &bmod($x,2) ? '-1': '+1'; |
| 331 | } elsif ($y =~ /^-/) { |
| 332 | 'NaN'; |
| 333 | } elsif ($x eq '+0' && $y eq '+0') { |
| 334 | 'NaN'; |
| 335 | } else { |
| 336 | @x = &internal($x); |
| 337 | local(@pow2)=@x; |
| 338 | local(@pow)=&internal("+1"); |
| 339 | local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul |
| 340 | while ($y ne '+0') { |
| 341 | ($y,$res)=&bdiv($y,2); |
| 342 | if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);} |
| 343 | if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);} |
| 344 | } |
| 345 | &external(@pow); |
| 346 | } |
| 347 | } |
| 348 | |
| 349 | 1; |
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