| 1 | package Math::BigInt::Calc; |
| 2 | |
| 3 | use 5.005; |
| 4 | use strict; |
| 5 | # use warnings; # dont use warnings for older Perls |
| 6 | |
| 7 | require Exporter; |
| 8 | use vars qw/@ISA $VERSION/; |
| 9 | @ISA = qw(Exporter); |
| 10 | |
| 11 | $VERSION = '0.30'; |
| 12 | |
| 13 | # Package to store unsigned big integers in decimal and do math with them |
| 14 | |
| 15 | # Internally the numbers are stored in an array with at least 1 element, no |
| 16 | # leading zero parts (except the first) and in base 1eX where X is determined |
| 17 | # automatically at loading time to be the maximum possible value |
| 18 | |
| 19 | # todo: |
| 20 | # - fully remove funky $# stuff (maybe) |
| 21 | |
| 22 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
| 23 | # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms |
| 24 | # BS2000, some Crays need USE_DIV instead. |
| 25 | # The BEGIN block is used to determine which of the two variants gives the |
| 26 | # correct result. |
| 27 | |
| 28 | ############################################################################## |
| 29 | # global constants, flags and accessory |
| 30 | |
| 31 | # constants for easier life |
| 32 | my $nan = 'NaN'; |
| 33 | my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL); |
| 34 | my ($AND_BITS,$XOR_BITS,$OR_BITS); |
| 35 | my ($AND_MASK,$XOR_MASK,$OR_MASK); |
| 36 | my ($LEN_CONVERT); |
| 37 | |
| 38 | sub _base_len |
| 39 | { |
| 40 | # set/get the BASE_LEN and assorted other, connected values |
| 41 | # used only be the testsuite, set is used only by the BEGIN block below |
| 42 | shift; |
| 43 | |
| 44 | my $b = shift; |
| 45 | if (defined $b) |
| 46 | { |
| 47 | # find whether we can use mul or div or none in mul()/div() |
| 48 | # (in last case reduce BASE_LEN_SMALL) |
| 49 | $BASE_LEN_SMALL = $b+1; |
| 50 | my $caught = 0; |
| 51 | while (--$BASE_LEN_SMALL > 5) |
| 52 | { |
| 53 | $MBASE = int("1e".$BASE_LEN_SMALL); |
| 54 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
| 55 | $caught = 0; |
| 56 | $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1 |
| 57 | $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1 |
| 58 | last if $caught != 3; |
| 59 | } |
| 60 | # BASE_LEN is used for anything else than mul()/div() |
| 61 | $BASE_LEN = $BASE_LEN_SMALL; |
| 62 | $BASE_LEN = shift if (defined $_[0]); # one more arg? |
| 63 | $BASE = int("1e".$BASE_LEN); |
| 64 | |
| 65 | $BASE_LEN2 = int($BASE_LEN_SMALL / 2); # for mul shortcut |
| 66 | $MBASE = int("1e".$BASE_LEN_SMALL); |
| 67 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
| 68 | $MAX_VAL = $MBASE-1; |
| 69 | $LEN_CONVERT = 0; |
| 70 | $LEN_CONVERT = 1 if $BASE_LEN_SMALL != $BASE_LEN; |
| 71 | |
| 72 | #print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL BASE: $BASE RBASE: $RBASE "; |
| 73 | #print "BASE_LEN_SMALL: $BASE_LEN_SMALL MBASE: $MBASE\n"; |
| 74 | |
| 75 | undef &_mul; |
| 76 | undef &_div; |
| 77 | |
| 78 | if ($caught & 1 != 0) |
| 79 | { |
| 80 | # must USE_MUL |
| 81 | *{_mul} = \&_mul_use_mul; |
| 82 | *{_div} = \&_div_use_mul; |
| 83 | } |
| 84 | else # $caught must be 2, since it can't be 1 nor 3 |
| 85 | { |
| 86 | # can USE_DIV instead |
| 87 | *{_mul} = \&_mul_use_div; |
| 88 | *{_div} = \&_div_use_div; |
| 89 | } |
| 90 | } |
| 91 | return $BASE_LEN unless wantarray; |
| 92 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL); |
| 93 | } |
| 94 | |
| 95 | BEGIN |
| 96 | { |
| 97 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
| 98 | # multipliable with itself plus carry |
| 99 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
| 100 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
| 101 | do |
| 102 | { |
| 103 | $num = ('9' x ++$e) + 0; |
| 104 | $num *= $num + 1.0; |
| 105 | } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern |
| 106 | $e--; # last test failed, so retract one step |
| 107 | # the limits below brush the problems with the test above under the rug: |
| 108 | # the test should be able to find the proper $e automatically |
| 109 | $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment |
| 110 | $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work |
| 111 | # there, but we play safe) |
| 112 | $e = 5 if $] < 5.006; # cap, for older Perls |
| 113 | $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems |
| 114 | # 8 fails inside random testsuite, so take 7 |
| 115 | |
| 116 | # determine how many digits fit into an integer and can be safely added |
| 117 | # together plus carry w/o causing an overflow |
| 118 | |
| 119 | # this below detects 15 on a 64 bit system, because after that it becomes |
| 120 | # 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of |
| 121 | # test failures. Ugh! (Tomake detect 18: uncomment lines marked with *) |
| 122 | use integer; |
| 123 | my $bi = 5; # approx. 16 bit |
| 124 | $num = int('9' x $bi); |
| 125 | # $num = 99999; # * |
| 126 | # while ( ($num+$num+1) eq '1' . '9' x $bi) # * |
| 127 | while ( int($num+$num+1) eq '1' . '9' x $bi) |
| 128 | { |
| 129 | $bi++; $num = int('9' x $bi); |
| 130 | # $bi++; $num *= 10; $num += 9; # * |
| 131 | } |
| 132 | $bi--; # back off one step |
| 133 | # by setting them equal, we ignore the findings and use the default |
| 134 | # one-size-fits-all approach from former versions |
| 135 | $bi = $e; # XXX, this should work always |
| 136 | |
| 137 | __PACKAGE__->_base_len($e,$bi); # set and store |
| 138 | |
| 139 | # find out how many bits _and, _or and _xor can take (old default = 16) |
| 140 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
| 141 | local $^W = 0; # don't warn about 'nonportable number' |
| 142 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
| 143 | |
| 144 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
| 145 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
| 146 | my $max = 16; |
| 147 | while (2 ** $max < $BASE) { $max++; } |
| 148 | { |
| 149 | no integer; |
| 150 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
| 151 | } |
| 152 | my ($x,$y,$z); |
| 153 | do { |
| 154 | $AND_BITS++; |
| 155 | $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
| 156 | $z = (2 ** $AND_BITS) - 1; |
| 157 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
| 158 | $AND_BITS --; # retreat one step |
| 159 | do { |
| 160 | $XOR_BITS++; |
| 161 | $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
| 162 | $z = (2 ** $XOR_BITS) - 1; |
| 163 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
| 164 | $XOR_BITS --; # retreat one step |
| 165 | do { |
| 166 | $OR_BITS++; |
| 167 | $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
| 168 | $z = (2 ** $OR_BITS) - 1; |
| 169 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
| 170 | $OR_BITS --; # retreat one step |
| 171 | |
| 172 | } |
| 173 | |
| 174 | ############################################################################## |
| 175 | # convert between the "small" and the "large" representation |
| 176 | |
| 177 | sub _to_large |
| 178 | { |
| 179 | # take an array in base $BASE_LEN_SMALL and convert it in-place to $BASE_LEN |
| 180 | my ($c,$x) = @_; |
| 181 | |
| 182 | # print "_to_large $BASE_LEN_SMALL => $BASE_LEN\n"; |
| 183 | |
| 184 | return $x if $LEN_CONVERT == 0 || # nothing to converconvertor |
| 185 | @$x == 1; # only one element => early out |
| 186 | |
| 187 | # 12345 67890 12345 67890 contents |
| 188 | # to 3 2 1 0 index |
| 189 | # 123456 7890123 4567890 contents |
| 190 | |
| 191 | # # faster variant |
| 192 | # my @d; my $str = ''; |
| 193 | # my $z = '0' x $BASE_LEN_SMALL; |
| 194 | # foreach (@$x) |
| 195 | # { |
| 196 | # # ... . 04321 . 000321 |
| 197 | # $str = substr($z.$_,-$BASE_LEN_SMALL,$BASE_LEN_SMALL) . $str; |
| 198 | # if (length($str) > $BASE_LEN) |
| 199 | # { |
| 200 | # push @d, substr($str,-$BASE_LEN,$BASE_LEN); # extract one piece |
| 201 | # substr($str,-$BASE_LEN,$BASE_LEN) = ''; # remove it |
| 202 | # } |
| 203 | # } |
| 204 | # push @d, $str if $str !~ /^0*$/; # extract last piece |
| 205 | # @$x = @d; |
| 206 | # $x->[-1] = int($x->[-1]); # strip leading zero |
| 207 | # $x; |
| 208 | |
| 209 | my $ret = ""; |
| 210 | my $l = scalar @$x; # number of parts |
| 211 | $l --; $ret .= int($x->[$l]); $l--; |
| 212 | my $z = '0' x ($BASE_LEN_SMALL-1); |
| 213 | while ($l >= 0) |
| 214 | { |
| 215 | $ret .= substr($z.$x->[$l],-$BASE_LEN_SMALL); |
| 216 | $l--; |
| 217 | } |
| 218 | my $str = _new($c,\$ret); # make array |
| 219 | @$x = @$str; # clobber contents of $x |
| 220 | $x->[-1] = int($x->[-1]); # strip leading zero |
| 221 | } |
| 222 | |
| 223 | sub _to_small |
| 224 | { |
| 225 | # take an array in base $BASE_LEN and convert it in-place to $BASE_LEN_SMALL |
| 226 | my ($c,$x) = @_; |
| 227 | |
| 228 | return $x if $LEN_CONVERT == 0; # nothing to do |
| 229 | return $x if @$x == 1 && length(int($x->[0])) <= $BASE_LEN_SMALL; |
| 230 | |
| 231 | my $d = _str($c,$x); |
| 232 | my $il = length($$d)-1; |
| 233 | ## this leaves '00000' instead of int 0 and will be corrected after any op |
| 234 | # clobber contents of $x |
| 235 | @$x = reverse(unpack("a" . ($il % $BASE_LEN_SMALL+1) |
| 236 | . ("a$BASE_LEN_SMALL" x ($il / $BASE_LEN_SMALL)), $$d)); |
| 237 | |
| 238 | $x->[-1] = int($x->[-1]); # strip leading zero |
| 239 | } |
| 240 | |
| 241 | ############################################################################### |
| 242 | |
| 243 | sub _new |
| 244 | { |
| 245 | # (ref to string) return ref to num_array |
| 246 | # Convert a number from string format (without sign) to internal base |
| 247 | # 1ex format. Assumes normalized value as input. |
| 248 | my $d = $_[1]; |
| 249 | my $il = length($$d)-1; |
| 250 | # this leaves '00000' instead of int 0 and will be corrected after any op |
| 251 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
| 252 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ]; |
| 253 | } |
| 254 | |
| 255 | BEGIN |
| 256 | { |
| 257 | $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS )); |
| 258 | $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS )); |
| 259 | $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS )); |
| 260 | } |
| 261 | |
| 262 | sub _zero |
| 263 | { |
| 264 | # create a zero |
| 265 | [ 0 ]; |
| 266 | } |
| 267 | |
| 268 | sub _one |
| 269 | { |
| 270 | # create a one |
| 271 | [ 1 ]; |
| 272 | } |
| 273 | |
| 274 | sub _two |
| 275 | { |
| 276 | # create a two (used internally for shifting) |
| 277 | [ 2 ]; |
| 278 | } |
| 279 | |
| 280 | sub _copy |
| 281 | { |
| 282 | [ @{$_[1]} ]; |
| 283 | } |
| 284 | |
| 285 | # catch and throw away |
| 286 | sub import { } |
| 287 | |
| 288 | ############################################################################## |
| 289 | # convert back to string and number |
| 290 | |
| 291 | sub _str |
| 292 | { |
| 293 | # (ref to BINT) return num_str |
| 294 | # Convert number from internal base 100000 format to string format. |
| 295 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
| 296 | my $ar = $_[1]; |
| 297 | my $ret = ""; |
| 298 | |
| 299 | my $l = scalar @$ar; # number of parts |
| 300 | return $nan if $l < 1; # should not happen |
| 301 | |
| 302 | # handle first one different to strip leading zeros from it (there are no |
| 303 | # leading zero parts in internal representation) |
| 304 | $l --; $ret .= int($ar->[$l]); $l--; |
| 305 | # Interestingly, the pre-padd method uses more time |
| 306 | # the old grep variant takes longer (14 to 10 sec) |
| 307 | my $z = '0' x ($BASE_LEN-1); |
| 308 | while ($l >= 0) |
| 309 | { |
| 310 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
| 311 | $l--; |
| 312 | } |
| 313 | \$ret; |
| 314 | } |
| 315 | |
| 316 | sub _num |
| 317 | { |
| 318 | # Make a number (scalar int/float) from a BigInt object |
| 319 | my $x = $_[1]; |
| 320 | return $x->[0] if scalar @$x == 1; # below $BASE |
| 321 | my $fac = 1; |
| 322 | my $num = 0; |
| 323 | foreach (@$x) |
| 324 | { |
| 325 | $num += $fac*$_; $fac *= $BASE; |
| 326 | } |
| 327 | $num; |
| 328 | } |
| 329 | |
| 330 | ############################################################################## |
| 331 | # actual math code |
| 332 | |
| 333 | sub _add |
| 334 | { |
| 335 | # (ref to int_num_array, ref to int_num_array) |
| 336 | # routine to add two base 1eX numbers |
| 337 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
| 338 | # there are separate routines to add and sub as per Knuth pg 233 |
| 339 | # This routine clobbers up array x, but not y. |
| 340 | |
| 341 | my ($c,$x,$y) = @_; |
| 342 | |
| 343 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
| 344 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
| 345 | { |
| 346 | # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :( |
| 347 | @$x = @$y; return $x; |
| 348 | } |
| 349 | |
| 350 | # for each in Y, add Y to X and carry. If after that, something is left in |
| 351 | # X, foreach in X add carry to X and then return X, carry |
| 352 | # Trades one "$j++" for having to shift arrays, $j could be made integer |
| 353 | # but this would impose a limit to number-length of 2**32. |
| 354 | my $i; my $car = 0; my $j = 0; |
| 355 | for $i (@$y) |
| 356 | { |
| 357 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
| 358 | $j++; |
| 359 | } |
| 360 | while ($car != 0) |
| 361 | { |
| 362 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
| 363 | } |
| 364 | $x; |
| 365 | } |
| 366 | |
| 367 | sub _inc |
| 368 | { |
| 369 | # (ref to int_num_array, ref to int_num_array) |
| 370 | # routine to add 1 to a base 1eX numbers |
| 371 | # This routine clobbers up array x, but not y. |
| 372 | my ($c,$x) = @_; |
| 373 | |
| 374 | for my $i (@$x) |
| 375 | { |
| 376 | return $x if (($i += 1) < $BASE); # early out |
| 377 | $i = 0; # overflow, next |
| 378 | } |
| 379 | push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend |
| 380 | $x; |
| 381 | } |
| 382 | |
| 383 | sub _dec |
| 384 | { |
| 385 | # (ref to int_num_array, ref to int_num_array) |
| 386 | # routine to add 1 to a base 1eX numbers |
| 387 | # This routine clobbers up array x, but not y. |
| 388 | my ($c,$x) = @_; |
| 389 | |
| 390 | my $MAX = $BASE-1; # since MAX_VAL based on MBASE |
| 391 | for my $i (@$x) |
| 392 | { |
| 393 | last if (($i -= 1) >= 0); # early out |
| 394 | $i = $MAX; # overflow, next |
| 395 | } |
| 396 | pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0) |
| 397 | $x; |
| 398 | } |
| 399 | |
| 400 | sub _sub |
| 401 | { |
| 402 | # (ref to int_num_array, ref to int_num_array, swap) |
| 403 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
| 404 | # subtract Y from X by modifying x in place |
| 405 | my ($c,$sx,$sy,$s) = @_; |
| 406 | |
| 407 | my $car = 0; my $i; my $j = 0; |
| 408 | if (!$s) |
| 409 | { |
| 410 | #print "case 2\n"; |
| 411 | for $i (@$sx) |
| 412 | { |
| 413 | last unless defined $sy->[$j] || $car; |
| 414 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
| 415 | } |
| 416 | # might leave leading zeros, so fix that |
| 417 | return __strip_zeros($sx); |
| 418 | } |
| 419 | #print "case 1 (swap)\n"; |
| 420 | for $i (@$sx) |
| 421 | { |
| 422 | # we can't do an early out if $x is < than $y, since we |
| 423 | # need to copy the high chunks from $y. Found by Bob Mathews. |
| 424 | #last unless defined $sy->[$j] || $car; |
| 425 | $sy->[$j] += $BASE |
| 426 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
| 427 | $j++; |
| 428 | } |
| 429 | # might leave leading zeros, so fix that |
| 430 | __strip_zeros($sy); |
| 431 | } |
| 432 | |
| 433 | sub _square_use_mul |
| 434 | { |
| 435 | # compute $x ** 2 or $x * $x in-place and return $x |
| 436 | my ($c,$x) = @_; |
| 437 | |
| 438 | # From: Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and |
| 439 | # S. Vanstone., Chapter 14 |
| 440 | |
| 441 | #14.16 Algorithm Multiple-precision squaring |
| 442 | #INPUT: positive integer x = (xt 1 xt 2 ... x1 x0)b. |
| 443 | #OUTPUT: x * x = x ** 2 in radix b representation. |
| 444 | #1. For i from 0 to (2t - 1) do: wi <- 0. |
| 445 | #2. For i from 0 to (t - 1) do the following: |
| 446 | # 2.1 (uv)b w2i + xi * xi, w2i v, c u. |
| 447 | # 2.2 For j from (i + 1)to (t - 1) do the following: |
| 448 | # (uv)b <- wi+j + 2*xj * xi + c, wi+j <- v, c <- u. |
| 449 | # 2.3 wi+t <- u. |
| 450 | #3. Return((w2t-1 w2t-2 ... w1 w0)b). |
| 451 | |
| 452 | # # Note: That description is crap. Half of the symbols are not explained or |
| 453 | # # used with out beeing set. |
| 454 | # my $t = scalar @$x; # count |
| 455 | # my ($c,$i,$j); |
| 456 | # for ($i = 0; $i < $t; $i++) |
| 457 | # { |
| 458 | # $x->[$i] = $x->[$i*2] + $x[$i]*$x[$i]; |
| 459 | # $x->[$i*2] = $x[$i]; $c = $x[$i]; |
| 460 | # for ($j = $i+1; $j < $t; $j++) |
| 461 | # { |
| 462 | # $x->[$i] = $x->[$i+$j] + 2 * $x->[$i] * $x->[$j]; |
| 463 | # $x->[$i+$j] = $x[$j]; $c = $x[$i]; |
| 464 | # } |
| 465 | # $x->[$i+$t] = $x[$i]; |
| 466 | # } |
| 467 | $x; |
| 468 | } |
| 469 | |
| 470 | sub _mul_use_mul |
| 471 | { |
| 472 | # (ref to int_num_array, ref to int_num_array) |
| 473 | # multiply two numbers in internal representation |
| 474 | # modifies first arg, second need not be different from first |
| 475 | my ($c,$xv,$yv) = @_; |
| 476 | |
| 477 | # shortcut for two very short numbers (improved by Nathan Zook) |
| 478 | # works also if xv and yv are the same reference |
| 479 | if ((@$xv == 1) && (@$yv == 1)) |
| 480 | { |
| 481 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
| 482 | { |
| 483 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE; |
| 484 | }; |
| 485 | return $xv; |
| 486 | } |
| 487 | # shortcut for result == 0 |
| 488 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
| 489 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
| 490 | { |
| 491 | @$xv = (0); |
| 492 | return $xv; |
| 493 | } |
| 494 | |
| 495 | # since multiplying $x with $x fails, make copy in this case |
| 496 | $yv = [@$xv] if $xv == $yv; # same references? |
| 497 | # $yv = [@$xv] if "$xv" eq "$yv"; # same references? |
| 498 | |
| 499 | # since multiplying $x with $x would fail here, use the faster squaring |
| 500 | # return _square($c,$xv) if $xv == $yv; # same reference? |
| 501 | |
| 502 | if ($LEN_CONVERT != 0) |
| 503 | { |
| 504 | $c->_to_small($xv); $c->_to_small($yv); |
| 505 | } |
| 506 | |
| 507 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
| 508 | |
| 509 | for $xi (@$xv) |
| 510 | { |
| 511 | $car = 0; $cty = 0; |
| 512 | |
| 513 | # slow variant |
| 514 | # for $yi (@$yv) |
| 515 | # { |
| 516 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
| 517 | # $prod[$cty++] = |
| 518 | # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL |
| 519 | # } |
| 520 | # $prod[$cty] += $car if $car; # need really to check for 0? |
| 521 | # $xi = shift @prod; |
| 522 | |
| 523 | # faster variant |
| 524 | # looping through this if $xi == 0 is silly - so optimize it away! |
| 525 | $xi = (shift @prod || 0), next if $xi == 0; |
| 526 | for $yi (@$yv) |
| 527 | { |
| 528 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
| 529 | ## this is actually a tad slower |
| 530 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
| 531 | $prod[$cty++] = |
| 532 | $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL |
| 533 | } |
| 534 | $prod[$cty] += $car if $car; # need really to check for 0? |
| 535 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
| 536 | } |
| 537 | push @$xv, @prod; |
| 538 | if ($LEN_CONVERT != 0) |
| 539 | { |
| 540 | $c->_to_large($yv); |
| 541 | $c->_to_large($xv); |
| 542 | } |
| 543 | else |
| 544 | { |
| 545 | __strip_zeros($xv); |
| 546 | } |
| 547 | $xv; |
| 548 | } |
| 549 | |
| 550 | sub _mul_use_div |
| 551 | { |
| 552 | # (ref to int_num_array, ref to int_num_array) |
| 553 | # multiply two numbers in internal representation |
| 554 | # modifies first arg, second need not be different from first |
| 555 | my ($c,$xv,$yv) = @_; |
| 556 | |
| 557 | # shortcut for two very short numbers (improved by Nathan Zook) |
| 558 | # works also if xv and yv are the same reference |
| 559 | if ((@$xv == 1) && (@$yv == 1)) |
| 560 | { |
| 561 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
| 562 | { |
| 563 | $xv->[0] = |
| 564 | $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE; |
| 565 | }; |
| 566 | return $xv; |
| 567 | } |
| 568 | # shortcut for result == 0 |
| 569 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
| 570 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
| 571 | { |
| 572 | @$xv = (0); |
| 573 | return $xv; |
| 574 | } |
| 575 | |
| 576 | |
| 577 | # since multiplying $x with $x fails, make copy in this case |
| 578 | $yv = [@$xv] if $xv == $yv; # same references? |
| 579 | # $yv = [@$xv] if "$xv" eq "$yv"; # same references? |
| 580 | # since multiplying $x with $x would fail here, use the faster squaring |
| 581 | # return _square($c,$xv) if $xv == $yv; # same reference? |
| 582 | |
| 583 | if ($LEN_CONVERT != 0) |
| 584 | { |
| 585 | $c->_to_small($xv); $c->_to_small($yv); |
| 586 | } |
| 587 | |
| 588 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
| 589 | for $xi (@$xv) |
| 590 | { |
| 591 | $car = 0; $cty = 0; |
| 592 | # looping through this if $xi == 0 is silly - so optimize it away! |
| 593 | $xi = (shift @prod || 0), next if $xi == 0; |
| 594 | for $yi (@$yv) |
| 595 | { |
| 596 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
| 597 | $prod[$cty++] = |
| 598 | $prod - ($car = int($prod / $MBASE)) * $MBASE; |
| 599 | } |
| 600 | $prod[$cty] += $car if $car; # need really to check for 0? |
| 601 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
| 602 | } |
| 603 | push @$xv, @prod; |
| 604 | if ($LEN_CONVERT != 0) |
| 605 | { |
| 606 | $c->_to_large($yv); |
| 607 | $c->_to_large($xv); |
| 608 | } |
| 609 | else |
| 610 | { |
| 611 | __strip_zeros($xv); |
| 612 | } |
| 613 | $xv; |
| 614 | } |
| 615 | |
| 616 | sub _div_use_mul |
| 617 | { |
| 618 | # ref to array, ref to array, modify first array and return remainder if |
| 619 | # in list context |
| 620 | my ($c,$x,$yorg) = @_; |
| 621 | |
| 622 | if (@$x == 1 && @$yorg == 1) |
| 623 | { |
| 624 | # shortcut, $yorg and $x are two small numbers |
| 625 | if (wantarray) |
| 626 | { |
| 627 | my $r = [ $x->[0] % $yorg->[0] ]; |
| 628 | $x->[0] = int($x->[0] / $yorg->[0]); |
| 629 | return ($x,$r); |
| 630 | } |
| 631 | else |
| 632 | { |
| 633 | $x->[0] = int($x->[0] / $yorg->[0]); |
| 634 | return $x; |
| 635 | } |
| 636 | } |
| 637 | if (@$yorg == 1) |
| 638 | { |
| 639 | my $rem; |
| 640 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
| 641 | |
| 642 | # shortcut, $y is < $BASE |
| 643 | my $j = scalar @$x; my $r = 0; |
| 644 | my $y = $yorg->[0]; my $b; |
| 645 | while ($j-- > 0) |
| 646 | { |
| 647 | $b = $r * $MBASE + $x->[$j]; |
| 648 | $x->[$j] = int($b/$y); |
| 649 | $r = $b % $y; |
| 650 | } |
| 651 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
| 652 | return ($x,$rem) if wantarray; |
| 653 | return $x; |
| 654 | } |
| 655 | |
| 656 | my $y = [ @$yorg ]; # always make copy to preserve |
| 657 | if ($LEN_CONVERT != 0) |
| 658 | { |
| 659 | $c->_to_small($x); $c->_to_small($y); |
| 660 | } |
| 661 | |
| 662 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
| 663 | |
| 664 | $car = $bar = $prd = 0; |
| 665 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
| 666 | { |
| 667 | for $xi (@$x) |
| 668 | { |
| 669 | $xi = $xi * $dd + $car; |
| 670 | $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL |
| 671 | } |
| 672 | push(@$x, $car); $car = 0; |
| 673 | for $yi (@$y) |
| 674 | { |
| 675 | $yi = $yi * $dd + $car; |
| 676 | $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL |
| 677 | } |
| 678 | } |
| 679 | else |
| 680 | { |
| 681 | push(@$x, 0); |
| 682 | } |
| 683 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
| 684 | $v2 = 0 unless $v2; |
| 685 | while ($#$x > $#$y) |
| 686 | { |
| 687 | ($u2,$u1,$u0) = @$x[-3..-1]; |
| 688 | $u2 = 0 unless $u2; |
| 689 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
| 690 | # if $v1 == 0; |
| 691 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
| 692 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
| 693 | if ($q) |
| 694 | { |
| 695 | ($car, $bar) = (0,0); |
| 696 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
| 697 | { |
| 698 | $prd = $q * $y->[$yi] + $car; |
| 699 | $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL |
| 700 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
| 701 | } |
| 702 | if ($x->[-1] < $car + $bar) |
| 703 | { |
| 704 | $car = 0; --$q; |
| 705 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
| 706 | { |
| 707 | $x->[$xi] -= $MBASE |
| 708 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
| 709 | } |
| 710 | } |
| 711 | } |
| 712 | pop(@$x); unshift(@q, $q); |
| 713 | } |
| 714 | if (wantarray) |
| 715 | { |
| 716 | @d = (); |
| 717 | if ($dd != 1) |
| 718 | { |
| 719 | $car = 0; |
| 720 | for $xi (reverse @$x) |
| 721 | { |
| 722 | $prd = $car * $MBASE + $xi; |
| 723 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
| 724 | unshift(@d, $tmp); |
| 725 | } |
| 726 | } |
| 727 | else |
| 728 | { |
| 729 | @d = @$x; |
| 730 | } |
| 731 | @$x = @q; |
| 732 | my $d = \@d; |
| 733 | if ($LEN_CONVERT != 0) |
| 734 | { |
| 735 | $c->_to_large($x); $c->_to_large($d); |
| 736 | } |
| 737 | else |
| 738 | { |
| 739 | __strip_zeros($x); |
| 740 | __strip_zeros($d); |
| 741 | } |
| 742 | return ($x,$d); |
| 743 | } |
| 744 | @$x = @q; |
| 745 | if ($LEN_CONVERT != 0) |
| 746 | { |
| 747 | $c->_to_large($x); |
| 748 | } |
| 749 | else |
| 750 | { |
| 751 | __strip_zeros($x); |
| 752 | } |
| 753 | $x; |
| 754 | } |
| 755 | |
| 756 | sub _div_use_div |
| 757 | { |
| 758 | # ref to array, ref to array, modify first array and return remainder if |
| 759 | # in list context |
| 760 | my ($c,$x,$yorg) = @_; |
| 761 | |
| 762 | if (@$x == 1 && @$yorg == 1) |
| 763 | { |
| 764 | # shortcut, $yorg and $x are two small numbers |
| 765 | if (wantarray) |
| 766 | { |
| 767 | my $r = [ $x->[0] % $yorg->[0] ]; |
| 768 | $x->[0] = int($x->[0] / $yorg->[0]); |
| 769 | return ($x,$r); |
| 770 | } |
| 771 | else |
| 772 | { |
| 773 | $x->[0] = int($x->[0] / $yorg->[0]); |
| 774 | return $x; |
| 775 | } |
| 776 | } |
| 777 | if (@$yorg == 1) |
| 778 | { |
| 779 | my $rem; |
| 780 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
| 781 | |
| 782 | # shortcut, $y is < $BASE |
| 783 | my $j = scalar @$x; my $r = 0; |
| 784 | my $y = $yorg->[0]; my $b; |
| 785 | while ($j-- > 0) |
| 786 | { |
| 787 | $b = $r * $MBASE + $x->[$j]; |
| 788 | $x->[$j] = int($b/$y); |
| 789 | $r = $b % $y; |
| 790 | } |
| 791 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
| 792 | return ($x,$rem) if wantarray; |
| 793 | return $x; |
| 794 | } |
| 795 | |
| 796 | my $y = [ @$yorg ]; # always make copy to preserve |
| 797 | if ($LEN_CONVERT != 0) |
| 798 | { |
| 799 | $c->_to_small($x); $c->_to_small($y); |
| 800 | } |
| 801 | |
| 802 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
| 803 | |
| 804 | $car = $bar = $prd = 0; |
| 805 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
| 806 | { |
| 807 | for $xi (@$x) |
| 808 | { |
| 809 | $xi = $xi * $dd + $car; |
| 810 | $xi -= ($car = int($xi / $MBASE)) * $MBASE; |
| 811 | } |
| 812 | push(@$x, $car); $car = 0; |
| 813 | for $yi (@$y) |
| 814 | { |
| 815 | $yi = $yi * $dd + $car; |
| 816 | $yi -= ($car = int($yi / $MBASE)) * $MBASE; |
| 817 | } |
| 818 | } |
| 819 | else |
| 820 | { |
| 821 | push(@$x, 0); |
| 822 | } |
| 823 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
| 824 | $v2 = 0 unless $v2; |
| 825 | while ($#$x > $#$y) |
| 826 | { |
| 827 | ($u2,$u1,$u0) = @$x[-3..-1]; |
| 828 | $u2 = 0 unless $u2; |
| 829 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
| 830 | # if $v1 == 0; |
| 831 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
| 832 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
| 833 | if ($q) |
| 834 | { |
| 835 | ($car, $bar) = (0,0); |
| 836 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
| 837 | { |
| 838 | $prd = $q * $y->[$yi] + $car; |
| 839 | $prd -= ($car = int($prd / $MBASE)) * $MBASE; |
| 840 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
| 841 | } |
| 842 | if ($x->[-1] < $car + $bar) |
| 843 | { |
| 844 | $car = 0; --$q; |
| 845 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
| 846 | { |
| 847 | $x->[$xi] -= $MBASE |
| 848 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
| 849 | } |
| 850 | } |
| 851 | } |
| 852 | pop(@$x); unshift(@q, $q); |
| 853 | } |
| 854 | if (wantarray) |
| 855 | { |
| 856 | @d = (); |
| 857 | if ($dd != 1) |
| 858 | { |
| 859 | $car = 0; |
| 860 | for $xi (reverse @$x) |
| 861 | { |
| 862 | $prd = $car * $MBASE + $xi; |
| 863 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
| 864 | unshift(@d, $tmp); |
| 865 | } |
| 866 | } |
| 867 | else |
| 868 | { |
| 869 | @d = @$x; |
| 870 | } |
| 871 | @$x = @q; |
| 872 | my $d = \@d; |
| 873 | if ($LEN_CONVERT != 0) |
| 874 | { |
| 875 | $c->_to_large($x); $c->_to_large($d); |
| 876 | } |
| 877 | else |
| 878 | { |
| 879 | __strip_zeros($x); |
| 880 | __strip_zeros($d); |
| 881 | } |
| 882 | return ($x,$d); |
| 883 | } |
| 884 | @$x = @q; |
| 885 | if ($LEN_CONVERT != 0) |
| 886 | { |
| 887 | $c->_to_large($x); |
| 888 | } |
| 889 | else |
| 890 | { |
| 891 | __strip_zeros($x); |
| 892 | } |
| 893 | $x; |
| 894 | } |
| 895 | |
| 896 | ############################################################################## |
| 897 | # testing |
| 898 | |
| 899 | sub _acmp |
| 900 | { |
| 901 | # internal absolute post-normalized compare (ignore signs) |
| 902 | # ref to array, ref to array, return <0, 0, >0 |
| 903 | # arrays must have at least one entry; this is not checked for |
| 904 | |
| 905 | my ($c,$cx,$cy) = @_; |
| 906 | |
| 907 | # fast comp based on number of array elements (aka pseudo-length) |
| 908 | my $lxy = scalar @$cx - scalar @$cy; |
| 909 | return -1 if $lxy < 0; # already differs, ret |
| 910 | return 1 if $lxy > 0; # ditto |
| 911 | |
| 912 | # now calculate length based on digits, not parts |
| 913 | $lxy = _len($c,$cx) - _len($c,$cy); # difference |
| 914 | return -1 if $lxy < 0; |
| 915 | return 1 if $lxy > 0; |
| 916 | |
| 917 | # hm, same lengths, but same contents? |
| 918 | my $i = 0; my $a; |
| 919 | # first way takes 5.49 sec instead of 4.87, but has the early out advantage |
| 920 | # so grep is slightly faster, but more inflexible. hm. $_ instead of $k |
| 921 | # yields 5.6 instead of 5.5 sec huh? |
| 922 | # manual way (abort if unequal, good for early ne) |
| 923 | my $j = scalar @$cx - 1; |
| 924 | while ($j >= 0) |
| 925 | { |
| 926 | last if ($a = $cx->[$j] - $cy->[$j]); $j--; |
| 927 | } |
| 928 | # my $j = scalar @$cx; |
| 929 | # while (--$j >= 0) |
| 930 | # { |
| 931 | # last if ($a = $cx->[$j] - $cy->[$j]); |
| 932 | # } |
| 933 | return 1 if $a > 0; |
| 934 | return -1 if $a < 0; |
| 935 | 0; # equal |
| 936 | |
| 937 | # while it early aborts, it is even slower than the manual variant |
| 938 | #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx; |
| 939 | # grep way, go trough all (bad for early ne) |
| 940 | #grep { $a = $_ - $cy->[$i++]; } @$cx; |
| 941 | #return $a; |
| 942 | } |
| 943 | |
| 944 | sub _len |
| 945 | { |
| 946 | # compute number of digits in bigint, minus the sign |
| 947 | |
| 948 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
| 949 | # '5' in this place, thus causing length() to report wrong length |
| 950 | my $cx = $_[1]; |
| 951 | |
| 952 | return (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
| 953 | } |
| 954 | |
| 955 | sub _digit |
| 956 | { |
| 957 | # return the nth digit, negative values count backward |
| 958 | # zero is rightmost, so _digit(123,0) will give 3 |
| 959 | my ($c,$x,$n) = @_; |
| 960 | |
| 961 | my $len = _len('',$x); |
| 962 | |
| 963 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
| 964 | $n = abs($n); # if negative was too big |
| 965 | $len--; $n = $len if $n > $len; # n to big? |
| 966 | |
| 967 | my $elem = int($n / $BASE_LEN); # which array element |
| 968 | my $digit = $n % $BASE_LEN; # which digit in this element |
| 969 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
| 970 | return substr($elem,-$digit-1,1); |
| 971 | } |
| 972 | |
| 973 | sub _zeros |
| 974 | { |
| 975 | # return amount of trailing zeros in decimal |
| 976 | # check each array elem in _m for having 0 at end as long as elem == 0 |
| 977 | # Upon finding a elem != 0, stop |
| 978 | my $x = $_[1]; |
| 979 | my $zeros = 0; my $elem; |
| 980 | foreach my $e (@$x) |
| 981 | { |
| 982 | if ($e != 0) |
| 983 | { |
| 984 | $elem = "$e"; # preserve x |
| 985 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
| 986 | $zeros *= $BASE_LEN; # elems * 5 |
| 987 | $zeros += length($elem); # count trailing zeros |
| 988 | last; # early out |
| 989 | } |
| 990 | $zeros ++; # real else branch: 50% slower! |
| 991 | } |
| 992 | $zeros; |
| 993 | } |
| 994 | |
| 995 | ############################################################################## |
| 996 | # _is_* routines |
| 997 | |
| 998 | sub _is_zero |
| 999 | { |
| 1000 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
| 1001 | my $x = $_[1]; |
| 1002 | |
| 1003 | (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0; |
| 1004 | } |
| 1005 | |
| 1006 | sub _is_even |
| 1007 | { |
| 1008 | # return true if arg (BINT or num_str) is even |
| 1009 | my $x = $_[1]; |
| 1010 | (!($x->[0] & 1)) <=> 0; |
| 1011 | } |
| 1012 | |
| 1013 | sub _is_odd |
| 1014 | { |
| 1015 | # return true if arg (BINT or num_str) is even |
| 1016 | my $x = $_[1]; |
| 1017 | |
| 1018 | (($x->[0] & 1)) <=> 0; |
| 1019 | } |
| 1020 | |
| 1021 | sub _is_one |
| 1022 | { |
| 1023 | # return true if arg (BINT or num_str) is one (array '+', '1') |
| 1024 | my $x = $_[1]; |
| 1025 | |
| 1026 | (scalar @$x == 1) && ($x->[0] == 1) <=> 0; |
| 1027 | } |
| 1028 | |
| 1029 | sub __strip_zeros |
| 1030 | { |
| 1031 | # internal normalization function that strips leading zeros from the array |
| 1032 | # args: ref to array |
| 1033 | my $s = shift; |
| 1034 | |
| 1035 | my $cnt = scalar @$s; # get count of parts |
| 1036 | my $i = $cnt-1; |
| 1037 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
| 1038 | |
| 1039 | return $s if @$s == 1; # early out |
| 1040 | |
| 1041 | #print "strip: cnt $cnt i $i\n"; |
| 1042 | # '0', '3', '4', '0', '0', |
| 1043 | # 0 1 2 3 4 |
| 1044 | # cnt = 5, i = 4 |
| 1045 | # i = 4 |
| 1046 | # i = 3 |
| 1047 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
| 1048 | # >= 1: skip first part (this can be zero) |
| 1049 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
| 1050 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
| 1051 | $s; |
| 1052 | } |
| 1053 | |
| 1054 | ############################################################################### |
| 1055 | # check routine to test internal state of corruptions |
| 1056 | |
| 1057 | sub _check |
| 1058 | { |
| 1059 | # used by the test suite |
| 1060 | my $x = $_[1]; |
| 1061 | |
| 1062 | return "$x is not a reference" if !ref($x); |
| 1063 | |
| 1064 | # are all parts are valid? |
| 1065 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
| 1066 | while ($i < $j) |
| 1067 | { |
| 1068 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
| 1069 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
| 1070 | last if $e !~ /^[+]?[0-9]+$/; |
| 1071 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
| 1072 | last if "$e" !~ /^[+]?[0-9]+$/; |
| 1073 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
| 1074 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
| 1075 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
| 1076 | last if $e <0 || $e >= $BASE; |
| 1077 | # this test is disabled, since new/bnorm and certain ops (like early out |
| 1078 | # in add/sub) are allowed/expected to leave '00000' in some elements |
| 1079 | #$try = '=~ /^00+/; '."($x, $e)"; |
| 1080 | #last if $e =~ /^00+/; |
| 1081 | $i++; |
| 1082 | } |
| 1083 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
| 1084 | return 0; |
| 1085 | } |
| 1086 | |
| 1087 | |
| 1088 | ############################################################################### |
| 1089 | ############################################################################### |
| 1090 | # some optional routines to make BigInt faster |
| 1091 | |
| 1092 | sub _mod |
| 1093 | { |
| 1094 | # if possible, use mod shortcut |
| 1095 | my ($c,$x,$yo) = @_; |
| 1096 | |
| 1097 | # slow way since $y to big |
| 1098 | if (scalar @$yo > 1) |
| 1099 | { |
| 1100 | my ($xo,$rem) = _div($c,$x,$yo); |
| 1101 | return $rem; |
| 1102 | } |
| 1103 | my $y = $yo->[0]; |
| 1104 | # both are single element arrays |
| 1105 | if (scalar @$x == 1) |
| 1106 | { |
| 1107 | $x->[0] %= $y; |
| 1108 | return $x; |
| 1109 | } |
| 1110 | |
| 1111 | # @y is single element, but @x has more than one |
| 1112 | my $b = $BASE % $y; |
| 1113 | if ($b == 0) |
| 1114 | { |
| 1115 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
| 1116 | # (B * BASE) % $y + A % Y => A % Y |
| 1117 | # so need to consider only last element: O(1) |
| 1118 | $x->[0] %= $y; |
| 1119 | } |
| 1120 | elsif ($b == 1) |
| 1121 | { |
| 1122 | # else need to go trough all elements: O(N), but loop is a bit simplified |
| 1123 | my $r = 0; |
| 1124 | foreach (@$x) |
| 1125 | { |
| 1126 | $r = ($r + $_) % $y; # not much faster, but heh... |
| 1127 | #$r += $_ % $y; $r %= $y; |
| 1128 | } |
| 1129 | $r = 0 if $r == $y; |
| 1130 | $x->[0] = $r; |
| 1131 | } |
| 1132 | else |
| 1133 | { |
| 1134 | # else need to go trough all elements: O(N) |
| 1135 | my $r = 0; my $bm = 1; |
| 1136 | foreach (@$x) |
| 1137 | { |
| 1138 | $r = ($_ * $bm + $r) % $y; |
| 1139 | $bm = ($bm * $b) % $y; |
| 1140 | |
| 1141 | #$r += ($_ % $y) * $bm; |
| 1142 | #$bm *= $b; |
| 1143 | #$bm %= $y; |
| 1144 | #$r %= $y; |
| 1145 | } |
| 1146 | $r = 0 if $r == $y; |
| 1147 | $x->[0] = $r; |
| 1148 | } |
| 1149 | splice (@$x,1); |
| 1150 | $x; |
| 1151 | } |
| 1152 | |
| 1153 | ############################################################################## |
| 1154 | # shifts |
| 1155 | |
| 1156 | sub _rsft |
| 1157 | { |
| 1158 | my ($c,$x,$y,$n) = @_; |
| 1159 | |
| 1160 | if ($n != 10) |
| 1161 | { |
| 1162 | $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y)); |
| 1163 | } |
| 1164 | |
| 1165 | # shortcut (faster) for shifting by 10) |
| 1166 | # multiples of $BASE_LEN |
| 1167 | my $dst = 0; # destination |
| 1168 | my $src = _num($c,$y); # as normal int |
| 1169 | my $rem = $src % $BASE_LEN; # remainder to shift |
| 1170 | $src = int($src / $BASE_LEN); # source |
| 1171 | if ($rem == 0) |
| 1172 | { |
| 1173 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
| 1174 | } |
| 1175 | else |
| 1176 | { |
| 1177 | my $len = scalar @$x - $src; # elems to go |
| 1178 | my $vd; my $z = '0'x $BASE_LEN; |
| 1179 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
| 1180 | while ($dst < $len) |
| 1181 | { |
| 1182 | $vd = $z.$x->[$src]; |
| 1183 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
| 1184 | $src++; |
| 1185 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
| 1186 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
| 1187 | $x->[$dst] = int($vd); |
| 1188 | $dst++; |
| 1189 | } |
| 1190 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
| 1191 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
| 1192 | } # else rem == 0 |
| 1193 | $x; |
| 1194 | } |
| 1195 | |
| 1196 | sub _lsft |
| 1197 | { |
| 1198 | my ($c,$x,$y,$n) = @_; |
| 1199 | |
| 1200 | if ($n != 10) |
| 1201 | { |
| 1202 | $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y)); |
| 1203 | } |
| 1204 | |
| 1205 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
| 1206 | # multiples of $BASE_LEN: |
| 1207 | my $src = scalar @$x; # source |
| 1208 | my $len = _num($c,$y); # shift-len as normal int |
| 1209 | my $rem = $len % $BASE_LEN; # remainder to shift |
| 1210 | my $dst = $src + int($len/$BASE_LEN); # destination |
| 1211 | my $vd; # further speedup |
| 1212 | $x->[$src] = 0; # avoid first ||0 for speed |
| 1213 | my $z = '0' x $BASE_LEN; |
| 1214 | while ($src >= 0) |
| 1215 | { |
| 1216 | $vd = $x->[$src]; $vd = $z.$vd; |
| 1217 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
| 1218 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
| 1219 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
| 1220 | $x->[$dst] = int($vd); |
| 1221 | $dst--; $src--; |
| 1222 | } |
| 1223 | # set lowest parts to 0 |
| 1224 | while ($dst >= 0) { $x->[$dst--] = 0; } |
| 1225 | # fix spurios last zero element |
| 1226 | splice @$x,-1 if $x->[-1] == 0; |
| 1227 | $x; |
| 1228 | } |
| 1229 | |
| 1230 | sub _pow |
| 1231 | { |
| 1232 | # power of $x to $y |
| 1233 | # ref to array, ref to array, return ref to array |
| 1234 | my ($c,$cx,$cy) = @_; |
| 1235 | |
| 1236 | my $pow2 = _one(); |
| 1237 | |
| 1238 | my $y_bin = ${_as_bin($c,$cy)}; $y_bin =~ s/^0b//; |
| 1239 | my $len = length($y_bin); |
| 1240 | while (--$len > 0) |
| 1241 | { |
| 1242 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
| 1243 | _mul($c,$cx,$cx); |
| 1244 | } |
| 1245 | |
| 1246 | _mul($c,$cx,$pow2); |
| 1247 | $cx; |
| 1248 | } |
| 1249 | |
| 1250 | sub _fac |
| 1251 | { |
| 1252 | # factorial of $x |
| 1253 | # ref to array, return ref to array |
| 1254 | my ($c,$cx) = @_; |
| 1255 | |
| 1256 | if ((@$cx == 1) && ($cx->[0] <= 2)) |
| 1257 | { |
| 1258 | $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2 |
| 1259 | return $cx; |
| 1260 | } |
| 1261 | |
| 1262 | # go forward until $base is exceeded |
| 1263 | # limit is either $x or $base (x == 100 means as result too high) |
| 1264 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
| 1265 | my $r = 2; my $cf = 3; my $step = 1; my $last = $r; |
| 1266 | while ($r < $BASE && $step < $steps) |
| 1267 | { |
| 1268 | $last = $r; $r *= $cf++; $step++; |
| 1269 | } |
| 1270 | if ((@$cx == 1) && ($step == $cx->[0])) |
| 1271 | { |
| 1272 | # completely done |
| 1273 | $cx = [$last]; |
| 1274 | return $cx; |
| 1275 | } |
| 1276 | my $n = _copy($c,$cx); |
| 1277 | $cx = [$last]; |
| 1278 | |
| 1279 | #$cx = _one(); |
| 1280 | while (!(@$n == 1 && $n->[0] == $step)) |
| 1281 | { |
| 1282 | _mul($c,$cx,$n); _dec($c,$n); |
| 1283 | } |
| 1284 | $cx; |
| 1285 | } |
| 1286 | |
| 1287 | use constant DEBUG => 0; |
| 1288 | |
| 1289 | my $steps = 0; |
| 1290 | |
| 1291 | sub steps { $steps }; |
| 1292 | |
| 1293 | sub _sqrt |
| 1294 | { |
| 1295 | # square-root of $x |
| 1296 | # ref to array, return ref to array |
| 1297 | my ($c,$x) = @_; |
| 1298 | |
| 1299 | if (scalar @$x == 1) |
| 1300 | { |
| 1301 | # fit's into one Perl scalar |
| 1302 | $x->[0] = int(sqrt($x->[0])); |
| 1303 | return $x; |
| 1304 | } |
| 1305 | my $y = _copy($c,$x); |
| 1306 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
| 1307 | # since our guess will "grow" |
| 1308 | my $l = int((_len($c,$x)-1) / 2); |
| 1309 | |
| 1310 | my $lastelem = $x->[-1]; # for guess |
| 1311 | my $elems = scalar @$x - 1; |
| 1312 | # not enough digits, but could have more? |
| 1313 | if ((length($lastelem) <= 3) && ($elems > 1)) |
| 1314 | { |
| 1315 | # right-align with zero pad |
| 1316 | my $len = length($lastelem) & 1; |
| 1317 | print "$lastelem => " if DEBUG; |
| 1318 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
| 1319 | # former odd => make odd again, or former even to even again |
| 1320 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
| 1321 | print "$lastelem\n" if DEBUG; |
| 1322 | } |
| 1323 | |
| 1324 | # construct $x (instead of _lsft($c,$x,$l,10) |
| 1325 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
| 1326 | $l = int($l / $BASE_LEN); |
| 1327 | print "l = $l " if DEBUG; |
| 1328 | |
| 1329 | splice @$x,$l; # keep ref($x), but modify it |
| 1330 | |
| 1331 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
| 1332 | # that gives us: |
| 1333 | # 14400 00000 => sqrt(14400) => 120 |
| 1334 | # 144000 000000 => sqrt(144000) => 379 |
| 1335 | |
| 1336 | # $x->[$l--] = int('1' . '0' x $r); # old way of guessing |
| 1337 | print "$lastelem (elems $elems) => " if DEBUG; |
| 1338 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
| 1339 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
| 1340 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
| 1341 | |
| 1342 | # padd with zeros if result is too short |
| 1343 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
| 1344 | print "now ",$x->[-1] if DEBUG; |
| 1345 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
| 1346 | |
| 1347 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
| 1348 | # an even better guess. Not implemented yet. |
| 1349 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
| 1350 | |
| 1351 | print "start x= ",${_str($c,$x)},"\n" if DEBUG; |
| 1352 | my $two = _two(); |
| 1353 | my $last = _zero(); |
| 1354 | my $lastlast = _zero(); |
| 1355 | $steps = 0 if DEBUG; |
| 1356 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
| 1357 | { |
| 1358 | $steps++ if DEBUG; |
| 1359 | $lastlast = _copy($c,$last); |
| 1360 | $last = _copy($c,$x); |
| 1361 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
| 1362 | _div($c,$x, $two ); |
| 1363 | print " x= ",${_str($c,$x)},"\n" if DEBUG; |
| 1364 | } |
| 1365 | print "\nsteps in sqrt: $steps, " if DEBUG; |
| 1366 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
| 1367 | print " final ",$x->[-1],"\n" if DEBUG; |
| 1368 | $x; |
| 1369 | } |
| 1370 | |
| 1371 | ############################################################################## |
| 1372 | # binary stuff |
| 1373 | |
| 1374 | sub _and |
| 1375 | { |
| 1376 | my ($c,$x,$y) = @_; |
| 1377 | |
| 1378 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
| 1379 | # very small performance drop for small numbers (e.g. something with less |
| 1380 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
| 1381 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
| 1382 | |
| 1383 | my $m = _one(); my ($xr,$yr); |
| 1384 | my $mask = $AND_MASK; |
| 1385 | |
| 1386 | my $x1 = $x; |
| 1387 | my $y1 = _copy($c,$y); # make copy |
| 1388 | $x = _zero(); |
| 1389 | my ($b,$xrr,$yrr); |
| 1390 | use integer; |
| 1391 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
| 1392 | { |
| 1393 | ($x1, $xr) = _div($c,$x1,$mask); |
| 1394 | ($y1, $yr) = _div($c,$y1,$mask); |
| 1395 | |
| 1396 | # make ints() from $xr, $yr |
| 1397 | # this is when the AND_BITS are greater tahn $BASE and is slower for |
| 1398 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
| 1399 | # due to KISS principle |
| 1400 | |
| 1401 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
| 1402 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
| 1403 | # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) ); |
| 1404 | |
| 1405 | # 0+ due to '&' doesn't work in strings |
| 1406 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
| 1407 | _mul($c,$m,$mask); |
| 1408 | } |
| 1409 | $x; |
| 1410 | } |
| 1411 | |
| 1412 | sub _xor |
| 1413 | { |
| 1414 | my ($c,$x,$y) = @_; |
| 1415 | |
| 1416 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
| 1417 | |
| 1418 | my $m = _one(); my ($xr,$yr); |
| 1419 | my $mask = $XOR_MASK; |
| 1420 | |
| 1421 | my $x1 = $x; |
| 1422 | my $y1 = _copy($c,$y); # make copy |
| 1423 | $x = _zero(); |
| 1424 | my ($b,$xrr,$yrr); |
| 1425 | use integer; |
| 1426 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
| 1427 | { |
| 1428 | ($x1, $xr) = _div($c,$x1,$mask); |
| 1429 | ($y1, $yr) = _div($c,$y1,$mask); |
| 1430 | # make ints() from $xr, $yr (see _and()) |
| 1431 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
| 1432 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
| 1433 | #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) ); |
| 1434 | |
| 1435 | # 0+ due to '^' doesn't work in strings |
| 1436 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
| 1437 | _mul($c,$m,$mask); |
| 1438 | } |
| 1439 | # the loop stops when the shorter of the two numbers is exhausted |
| 1440 | # the remainder of the longer one will survive bit-by-bit, so we simple |
| 1441 | # multiply-add it in |
| 1442 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
| 1443 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
| 1444 | |
| 1445 | $x; |
| 1446 | } |
| 1447 | |
| 1448 | sub _or |
| 1449 | { |
| 1450 | my ($c,$x,$y) = @_; |
| 1451 | |
| 1452 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
| 1453 | |
| 1454 | my $m = _one(); my ($xr,$yr); |
| 1455 | my $mask = $OR_MASK; |
| 1456 | |
| 1457 | my $x1 = $x; |
| 1458 | my $y1 = _copy($c,$y); # make copy |
| 1459 | $x = _zero(); |
| 1460 | my ($b,$xrr,$yrr); |
| 1461 | use integer; |
| 1462 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
| 1463 | { |
| 1464 | ($x1, $xr) = _div($c,$x1,$mask); |
| 1465 | ($y1, $yr) = _div($c,$y1,$mask); |
| 1466 | # make ints() from $xr, $yr (see _and()) |
| 1467 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
| 1468 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
| 1469 | # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) ); |
| 1470 | |
| 1471 | # 0+ due to '|' doesn't work in strings |
| 1472 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
| 1473 | _mul($c,$m,$mask); |
| 1474 | } |
| 1475 | # the loop stops when the shorter of the two numbers is exhausted |
| 1476 | # the remainder of the longer one will survive bit-by-bit, so we simple |
| 1477 | # multiply-add it in |
| 1478 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
| 1479 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
| 1480 | |
| 1481 | $x; |
| 1482 | } |
| 1483 | |
| 1484 | sub _as_hex |
| 1485 | { |
| 1486 | # convert a decimal number to hex (ref to array, return ref to string) |
| 1487 | my ($c,$x) = @_; |
| 1488 | |
| 1489 | my $x1 = _copy($c,$x); |
| 1490 | |
| 1491 | my $es = ''; |
| 1492 | my ($xr, $h, $x10000); |
| 1493 | if ($] >= 5.006) |
| 1494 | { |
| 1495 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
| 1496 | } |
| 1497 | else |
| 1498 | { |
| 1499 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
| 1500 | } |
| 1501 | while (! _is_zero($c,$x1)) |
| 1502 | { |
| 1503 | ($x1, $xr) = _div($c,$x1,$x10000); |
| 1504 | $es .= unpack($h,pack('v',$xr->[0])); |
| 1505 | } |
| 1506 | $es = reverse $es; |
| 1507 | $es =~ s/^[0]+//; # strip leading zeros |
| 1508 | $es = '0x' . $es; |
| 1509 | \$es; |
| 1510 | } |
| 1511 | |
| 1512 | sub _as_bin |
| 1513 | { |
| 1514 | # convert a decimal number to bin (ref to array, return ref to string) |
| 1515 | my ($c,$x) = @_; |
| 1516 | |
| 1517 | my $x1 = _copy($c,$x); |
| 1518 | |
| 1519 | my $es = ''; |
| 1520 | my ($xr, $b, $x10000); |
| 1521 | if ($] >= 5.006) |
| 1522 | { |
| 1523 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
| 1524 | } |
| 1525 | else |
| 1526 | { |
| 1527 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
| 1528 | } |
| 1529 | while (! _is_zero($c,$x1)) |
| 1530 | { |
| 1531 | ($x1, $xr) = _div($c,$x1,$x10000); |
| 1532 | $es .= unpack($b,pack('v',$xr->[0])); |
| 1533 | } |
| 1534 | $es = reverse $es; |
| 1535 | $es =~ s/^[0]+//; # strip leading zeros |
| 1536 | $es = '0b' . $es; |
| 1537 | \$es; |
| 1538 | } |
| 1539 | |
| 1540 | sub _from_hex |
| 1541 | { |
| 1542 | # convert a hex number to decimal (ref to string, return ref to array) |
| 1543 | my ($c,$hs) = @_; |
| 1544 | |
| 1545 | my $mul = _one(); |
| 1546 | my $m = [ 0x10000 ]; # 16 bit at a time |
| 1547 | my $x = _zero(); |
| 1548 | |
| 1549 | my $len = length($$hs)-2; |
| 1550 | $len = int($len/4); # 4-digit parts, w/o '0x' |
| 1551 | my $val; my $i = -4; |
| 1552 | while ($len >= 0) |
| 1553 | { |
| 1554 | $val = substr($$hs,$i,4); |
| 1555 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because |
| 1556 | $val = hex($val); # hex does not like wrong chars |
| 1557 | $i -= 4; $len --; |
| 1558 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
| 1559 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
| 1560 | } |
| 1561 | $x; |
| 1562 | } |
| 1563 | |
| 1564 | sub _from_bin |
| 1565 | { |
| 1566 | # convert a hex number to decimal (ref to string, return ref to array) |
| 1567 | my ($c,$bs) = @_; |
| 1568 | |
| 1569 | # instead of converting 8 bit at a time, it is faster to convert the |
| 1570 | # number to hex, and then call _from_hex. |
| 1571 | |
| 1572 | my $hs = $$bs; |
| 1573 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
| 1574 | my $l = length($hs); # bits |
| 1575 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
| 1576 | my $h = unpack('H*', pack ('B*', $hs)); # repack as hex |
| 1577 | return $c->_from_hex(\('0x'.$h)); |
| 1578 | |
| 1579 | my $mul = _one(); |
| 1580 | my $m = [ 0x100 ]; # 8 bit at a time |
| 1581 | my $x = _zero(); |
| 1582 | |
| 1583 | my $len = length($$bs)-2; |
| 1584 | $len = int($len/8); # 4-digit parts, w/o '0x' |
| 1585 | my $val; my $i = -8; |
| 1586 | while ($len >= 0) |
| 1587 | { |
| 1588 | $val = substr($$bs,$i,8); |
| 1589 | $val =~ s/^[+-]?0b// if $len == 0; # for last part only |
| 1590 | |
| 1591 | $val = ord(pack('B8',substr('00000000'.$val,-8,8))); |
| 1592 | |
| 1593 | $i -= 8; $len --; |
| 1594 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
| 1595 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
| 1596 | } |
| 1597 | $x; |
| 1598 | } |
| 1599 | |
| 1600 | ############################################################################## |
| 1601 | # special modulus functions |
| 1602 | |
| 1603 | # not ready yet, since it would need to deal with unsigned numbers |
| 1604 | sub _modinv1 |
| 1605 | { |
| 1606 | # inverse modulus |
| 1607 | my ($c,$num,$mod) = @_; |
| 1608 | |
| 1609 | my $u = _zero(); my $u1 = _one(); |
| 1610 | my $a = _copy($c,$mod); my $b = _copy($c,$num); |
| 1611 | |
| 1612 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
| 1613 | # result ($u) at the same time |
| 1614 | while (!_is_zero($c,$b)) |
| 1615 | { |
| 1616 | # print ${_str($c,$a)}, " ", ${_str($c,$b)}, " ", ${_str($c,$u)}, " ", |
| 1617 | # ${_str($c,$u1)}, "\n"; |
| 1618 | ($a, my $q, $b) = ($b, _div($c,$a,$b)); |
| 1619 | # print ${_str($c,$a)}, " ", ${_str($c,$q)}, " ", ${_str($c,$b)}, "\n"; |
| 1620 | # original: ($u,$u1) = ($u1, $u - $u1 * $q); |
| 1621 | my $t = _copy($c,$u); |
| 1622 | $u = _copy($c,$u1); |
| 1623 | _mul($c,$u1,$q); |
| 1624 | $u1 = _sub($t,$u1); |
| 1625 | # print ${_str($c,$a)}, " ", ${_str($c,$b)}, " ", ${_str($c,$u)}, " ", |
| 1626 | # ${_str($c,$u1)}, "\n"; |
| 1627 | } |
| 1628 | |
| 1629 | # if the gcd is not 1, then return NaN |
| 1630 | return undef unless _is_one($c,$a); |
| 1631 | |
| 1632 | $num = _mod($c,$u,$mod); |
| 1633 | # print ${_str($c,$num)},"\n"; |
| 1634 | $num; |
| 1635 | } |
| 1636 | |
| 1637 | sub _modpow |
| 1638 | { |
| 1639 | # modulus of power ($x ** $y) % $z |
| 1640 | my ($c,$num,$exp,$mod) = @_; |
| 1641 | |
| 1642 | # in the trivial case, |
| 1643 | if (_is_one($c,$mod)) |
| 1644 | { |
| 1645 | splice @$num,0,1; $num->[0] = 0; |
| 1646 | return $num; |
| 1647 | } |
| 1648 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
| 1649 | { |
| 1650 | $num->[0] = 1; |
| 1651 | return $num; |
| 1652 | } |
| 1653 | |
| 1654 | # $num = _mod($c,$num,$mod); # this does not make it faster |
| 1655 | |
| 1656 | my $acc = _copy($c,$num); my $t = _one(); |
| 1657 | |
| 1658 | my $expbin = ${_as_bin($c,$exp)}; $expbin =~ s/^0b//; |
| 1659 | my $len = length($expbin); |
| 1660 | while (--$len >= 0) |
| 1661 | { |
| 1662 | if ( substr($expbin,$len,1) eq '1') # is_odd |
| 1663 | { |
| 1664 | _mul($c,$t,$acc); |
| 1665 | $t = _mod($c,$t,$mod); |
| 1666 | } |
| 1667 | _mul($c,$acc,$acc); |
| 1668 | $acc = _mod($c,$acc,$mod); |
| 1669 | } |
| 1670 | @$num = @$t; |
| 1671 | $num; |
| 1672 | } |
| 1673 | |
| 1674 | ############################################################################## |
| 1675 | ############################################################################## |
| 1676 | |
| 1677 | 1; |
| 1678 | __END__ |
| 1679 | |
| 1680 | =head1 NAME |
| 1681 | |
| 1682 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
| 1683 | |
| 1684 | =head1 SYNOPSIS |
| 1685 | |
| 1686 | Provides support for big integer calculations. Not intended to be used by other |
| 1687 | modules (except Math::BigInt::Cached). Other modules which sport the same |
| 1688 | functions can also be used to support Math::Bigint, like Math::BigInt::Pari. |
| 1689 | |
| 1690 | =head1 DESCRIPTION |
| 1691 | |
| 1692 | In order to allow for multiple big integer libraries, Math::BigInt was |
| 1693 | rewritten to use library modules for core math routines. Any module which |
| 1694 | follows the same API as this can be used instead by using the following: |
| 1695 | |
| 1696 | use Math::BigInt lib => 'libname'; |
| 1697 | |
| 1698 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
| 1699 | version like 'Pari'. |
| 1700 | |
| 1701 | =head1 EXPORT |
| 1702 | |
| 1703 | The following functions MUST be defined in order to support the use by |
| 1704 | Math::BigInt: |
| 1705 | |
| 1706 | _new(string) return ref to new object from ref to decimal string |
| 1707 | _zero() return a new object with value 0 |
| 1708 | _one() return a new object with value 1 |
| 1709 | |
| 1710 | _str(obj) return ref to a string representing the object |
| 1711 | _num(obj) returns a Perl integer/floating point number |
| 1712 | NOTE: because of Perl numeric notation defaults, |
| 1713 | the _num'ified obj may lose accuracy due to |
| 1714 | machine-dependend floating point size limitations |
| 1715 | |
| 1716 | _add(obj,obj) Simple addition of two objects |
| 1717 | _mul(obj,obj) Multiplication of two objects |
| 1718 | _div(obj,obj) Division of the 1st object by the 2nd |
| 1719 | In list context, returns (result,remainder). |
| 1720 | NOTE: this is integer math, so no |
| 1721 | fractional part will be returned. |
| 1722 | _sub(obj,obj) Simple subtraction of 1 object from another |
| 1723 | a third, optional parameter indicates that the params |
| 1724 | are swapped. In this case, the first param needs to |
| 1725 | be preserved, while you can destroy the second. |
| 1726 | sub (x,y,1) => return x - y and keep x intact! |
| 1727 | _dec(obj) decrement object by one (input is garant. to be > 0) |
| 1728 | _inc(obj) increment object by one |
| 1729 | |
| 1730 | |
| 1731 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
| 1732 | |
| 1733 | _len(obj) returns count of the decimal digits of the object |
| 1734 | _digit(obj,n) returns the n'th decimal digit of object |
| 1735 | |
| 1736 | _is_one(obj) return true if argument is +1 |
| 1737 | _is_zero(obj) return true if argument is 0 |
| 1738 | _is_even(obj) return true if argument is even (0,2,4,6..) |
| 1739 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
| 1740 | |
| 1741 | _copy return a ref to a true copy of the object |
| 1742 | |
| 1743 | _check(obj) check whether internal representation is still intact |
| 1744 | return 0 for ok, otherwise error message as string |
| 1745 | |
| 1746 | The following functions are optional, and can be defined if the underlying lib |
| 1747 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
| 1748 | slow) fallback routines to emulate these: |
| 1749 | |
| 1750 | _from_hex(str) return ref to new object from ref to hexadecimal string |
| 1751 | _from_bin(str) return ref to new object from ref to binary string |
| 1752 | |
| 1753 | _as_hex(str) return ref to scalar string containing the value as |
| 1754 | unsigned hex string, with the '0x' prepended. |
| 1755 | Leading zeros must be stripped. |
| 1756 | _as_bin(str) Like as_hex, only as binary string containing only |
| 1757 | zeros and ones. Leading zeros must be stripped and a |
| 1758 | '0b' must be prepended. |
| 1759 | |
| 1760 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
| 1761 | For unsupported bases B, return undef to signal failure |
| 1762 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
| 1763 | For unsupported bases B, return undef to signal failure |
| 1764 | |
| 1765 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
| 1766 | Note: XOR, AND and OR pad with zeros if size mismatches |
| 1767 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
| 1768 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
| 1769 | |
| 1770 | _mod(obj,obj) Return remainder of div of the 1st by the 2nd object |
| 1771 | _sqrt(obj) return the square root of object (truncate to int) |
| 1772 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
| 1773 | _pow(obj,obj) return object 1 to the power of object 2 |
| 1774 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
| 1775 | |
| 1776 | _zeros(obj) return number of trailing decimal zeros |
| 1777 | _modinv return inverse modulus |
| 1778 | _modpow return modulus of power ($x ** $y) % $z |
| 1779 | |
| 1780 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
| 1781 | or '0b1101'). |
| 1782 | |
| 1783 | Testing of input parameter validity is done by the caller, so you need not |
| 1784 | worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by |
| 1785 | zero or similar cases. |
| 1786 | |
| 1787 | The first parameter can be modified, that includes the possibility that you |
| 1788 | return a reference to a completely different object instead. Although keeping |
| 1789 | the reference and just changing it's contents is prefered over creating and |
| 1790 | returning a different reference. |
| 1791 | |
| 1792 | Return values are always references to objects or strings. Exceptions are |
| 1793 | C<_lsft()> and C<_rsft()>, which return undef if they can not shift the |
| 1794 | argument. This is used to delegate shifting of bases different than the one |
| 1795 | you can support back to Math::BigInt, which will use some generic code to |
| 1796 | calculate the result. |
| 1797 | |
| 1798 | =head1 WRAP YOUR OWN |
| 1799 | |
| 1800 | If you want to port your own favourite c-lib for big numbers to the |
| 1801 | Math::BigInt interface, you can take any of the already existing modules as |
| 1802 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
| 1803 | testsuites with your module, and replace in them any of the following: |
| 1804 | |
| 1805 | use Math::BigInt; |
| 1806 | |
| 1807 | by this: |
| 1808 | |
| 1809 | use Math::BigInt lib => 'yourlib'; |
| 1810 | |
| 1811 | This way you ensure that your library really works 100% within Math::BigInt. |
| 1812 | |
| 1813 | =head1 LICENSE |
| 1814 | |
| 1815 | This program is free software; you may redistribute it and/or modify it under |
| 1816 | the same terms as Perl itself. |
| 1817 | |
| 1818 | =head1 AUTHORS |
| 1819 | |
| 1820 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
| 1821 | in late 2000, 2001. |
| 1822 | Seperated from BigInt and shaped API with the help of John Peacock. |
| 1823 | |
| 1824 | =head1 SEE ALSO |
| 1825 | |
| 1826 | L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>, |
| 1827 | L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>. |
| 1828 | |
| 1829 | =cut |