| 1 | /* |
| 2 | * Matt McCutchen's Big Integer Library |
| 3 | */ |
| 4 | |
| 5 | #include "BigInteger.hh" |
| 6 | |
| 7 | // MANAGEMENT |
| 8 | |
| 9 | // Assignment operator |
| 10 | void BigInteger::operator =(const BigInteger &x) { |
| 11 | // Calls like a = a have no effect |
| 12 | if (this == &x) |
| 13 | return; |
| 14 | // Copy sign |
| 15 | sign = x.sign; |
| 16 | // Copy the rest |
| 17 | BigUnsigned::operator =(x); |
| 18 | } |
| 19 | |
| 20 | // Constructor from an array of blocks and a sign |
| 21 | BigInteger::BigInteger(const Blk *b, Index l, Sign s) : BigUnsigned(b, l) { |
| 22 | switch (s) { |
| 23 | case zero: |
| 24 | case positive: |
| 25 | case negative: |
| 26 | sign = (len == 0) ? zero : s; |
| 27 | break; |
| 28 | default: |
| 29 | throw "BigInteger::BigInteger(Blk *, Index, Sign): Invalid sign"; |
| 30 | } |
| 31 | } |
| 32 | |
| 33 | // Constructor from a BigUnsigned and a sign |
| 34 | BigInteger::BigInteger(const BigUnsigned &x, Sign s) : BigUnsigned(x) { |
| 35 | switch (s) { |
| 36 | case zero: |
| 37 | case positive: |
| 38 | case negative: |
| 39 | sign = (len == 0) ? zero : s; |
| 40 | break; |
| 41 | default: |
| 42 | throw "BigInteger::BigInteger(Blk *, Index, Sign): Invalid sign"; |
| 43 | } |
| 44 | } |
| 45 | |
| 46 | /* |
| 47 | * The steps for construction of a BigInteger |
| 48 | * from an integral value x are as follows: |
| 49 | * 1. If x is zero, create an empty BigInteger and stop. |
| 50 | * 2. Allocate a one-block number array. |
| 51 | * 3. If x is positive (or of an unsigned type), set the |
| 52 | * sign of the BigInteger to positive. |
| 53 | * 4. If x is of a signed type and is negative, set the |
| 54 | * sign of the BigInteger to negative. |
| 55 | * 5. If x is of a signed type, convert x (or -x if x < 0) |
| 56 | * to the unsigned type of the same length. |
| 57 | * 6. Expand x (or the result of step 5) to a Blk, |
| 58 | * and store it in the number array. |
| 59 | * |
| 60 | * See remarks in `BigUnsigned.cc' and `NumberlikeArray.hh' |
| 61 | * about new handling of zero-length arrays. |
| 62 | */ |
| 63 | |
| 64 | BigInteger::BigInteger(unsigned long x) { |
| 65 | if (x == 0) |
| 66 | sign = zero; // NumberlikeArray did the rest |
| 67 | else { |
| 68 | cap = 1; |
| 69 | blk = new Blk[1]; |
| 70 | sign = positive; |
| 71 | len = 1; |
| 72 | blk[0] = Blk(x); |
| 73 | } |
| 74 | } |
| 75 | |
| 76 | BigInteger::BigInteger(long x) { |
| 77 | if (x > 0) { |
| 78 | cap = 1; |
| 79 | blk = new Blk[1]; |
| 80 | sign = positive; |
| 81 | len = 1; |
| 82 | blk[0] = Blk(x); |
| 83 | } else if (x < 0) { |
| 84 | cap = 1; |
| 85 | blk = new Blk[1]; |
| 86 | sign = negative; |
| 87 | len = 1; |
| 88 | blk[0] = Blk(-x); |
| 89 | } else |
| 90 | sign = zero; |
| 91 | } |
| 92 | |
| 93 | BigInteger::BigInteger(unsigned int x) { |
| 94 | if (x == 0) |
| 95 | sign = zero; |
| 96 | else { |
| 97 | cap = 1; |
| 98 | blk = new Blk[1]; |
| 99 | sign = positive; |
| 100 | len = 1; |
| 101 | blk[0] = Blk(x); |
| 102 | } |
| 103 | } |
| 104 | |
| 105 | BigInteger::BigInteger(int x) { |
| 106 | if (x > 0) { |
| 107 | cap = 1; |
| 108 | blk = new Blk[1]; |
| 109 | sign = positive; |
| 110 | len = 1; |
| 111 | blk[0] = Blk(x); |
| 112 | } else if (x < 0) { |
| 113 | cap = 1; |
| 114 | blk = new Blk[1]; |
| 115 | sign = negative; |
| 116 | len = 1; |
| 117 | blk[0] = Blk(-x); |
| 118 | } else |
| 119 | sign = zero; |
| 120 | } |
| 121 | |
| 122 | BigInteger::BigInteger(unsigned short x) { |
| 123 | if (x == 0) |
| 124 | sign = zero; |
| 125 | else { |
| 126 | cap = 1; |
| 127 | blk = new Blk[1]; |
| 128 | sign = positive; |
| 129 | len = 1; |
| 130 | blk[0] = Blk(x); |
| 131 | } |
| 132 | } |
| 133 | |
| 134 | BigInteger::BigInteger(short x) { |
| 135 | if (x > 0) { |
| 136 | cap = 1; |
| 137 | blk = new Blk[1]; |
| 138 | sign = positive; |
| 139 | len = 1; |
| 140 | blk[0] = Blk(x); |
| 141 | } else if (x < 0) { |
| 142 | cap = 1; |
| 143 | blk = new Blk[1]; |
| 144 | sign = negative; |
| 145 | len = 1; |
| 146 | blk[0] = Blk(-x); |
| 147 | } else |
| 148 | sign = zero; |
| 149 | } |
| 150 | |
| 151 | // CONVERTERS |
| 152 | /* |
| 153 | * The steps for conversion of a BigInteger to an |
| 154 | * integral type are as follows: |
| 155 | * 1. If the BigInteger is zero, return zero. |
| 156 | * 2. If the BigInteger is positive: |
| 157 | * 3. If it is more than one block long or its lowest |
| 158 | * block has bits set out of the range of the target |
| 159 | * type, throw an exception. |
| 160 | * 4. Otherwise, convert the lowest block to the |
| 161 | * target type and return it. |
| 162 | * 5. If the BigInteger is negative: |
| 163 | * 6. If the target type is unsigned, throw an exception. |
| 164 | * 7. If it is more than one block long or its lowest |
| 165 | * block has bits set out of the range of the target |
| 166 | * type, throw an exception. |
| 167 | * 8. Otherwise, convert the lowest block to the |
| 168 | * target type, negate it, and return it. |
| 169 | */ |
| 170 | |
| 171 | namespace { |
| 172 | // These masks are used to test whether a Blk has bits |
| 173 | // set out of the range of a smaller integral type. Note |
| 174 | // that this range is not considered to include the sign bit. |
| 175 | const BigUnsigned::Blk lMask = ~0 >> 1; |
| 176 | const BigUnsigned::Blk uiMask = (unsigned int)(~0); |
| 177 | const BigUnsigned::Blk iMask = uiMask >> 1; |
| 178 | const BigUnsigned::Blk usMask = (unsigned short)(~0); |
| 179 | const BigUnsigned::Blk sMask = usMask >> 1; |
| 180 | } |
| 181 | |
| 182 | BigInteger::operator unsigned long() const { |
| 183 | switch (sign) { |
| 184 | case zero: |
| 185 | return 0; |
| 186 | case positive: |
| 187 | if (len == 1) |
| 188 | return blk[0]; |
| 189 | else |
| 190 | throw "BigInteger operator unsigned long() const: Value is too big for an unsigned long"; |
| 191 | case negative: |
| 192 | throw "BigInteger operator unsigned long() const: Cannot convert a negative integer to an unsigned type"; |
| 193 | default: |
| 194 | throw "BigInteger: Internal error"; |
| 195 | } |
| 196 | } |
| 197 | |
| 198 | BigInteger::operator long() const { |
| 199 | switch (sign) { |
| 200 | case zero: |
| 201 | return 0; |
| 202 | case positive: |
| 203 | if (len == 1 && (blk[0] & ~lMask) == 0) |
| 204 | return long(blk[0]); |
| 205 | else |
| 206 | throw "BigInteger operator long() const: Value is too big for a long"; |
| 207 | case negative: |
| 208 | if (len == 1 && (blk[0] & ~lMask) == 0) |
| 209 | return -long(blk[0]); |
| 210 | else |
| 211 | throw "BigInteger operator long() const: Value is too big for a long"; |
| 212 | default: |
| 213 | throw "BigInteger: Internal error"; |
| 214 | } |
| 215 | } |
| 216 | |
| 217 | BigInteger::operator unsigned int() const { |
| 218 | switch (sign) { |
| 219 | case zero: |
| 220 | return 0; |
| 221 | case positive: |
| 222 | if (len == 1 && (blk[0] & ~uiMask) == 0) |
| 223 | return (unsigned int)(blk[0]); |
| 224 | else |
| 225 | throw "BigInteger operator unsigned int() const: Value is too big for an unsigned int"; |
| 226 | case negative: |
| 227 | throw "BigInteger operator unsigned int() const: Cannot convert a negative integer to an unsigned type"; |
| 228 | default: |
| 229 | throw "BigInteger: Internal error"; |
| 230 | } |
| 231 | } |
| 232 | |
| 233 | BigInteger::operator int() const { |
| 234 | switch (sign) { |
| 235 | case zero: |
| 236 | return 0; |
| 237 | case positive: |
| 238 | if (len == 1 && (blk[0] & ~iMask) == 0) |
| 239 | return int(blk[0]); |
| 240 | else |
| 241 | throw "BigInteger operator int() const: Value is too big for an int"; |
| 242 | case negative: |
| 243 | if (len == 1 && (blk[0] & ~iMask) == 0) |
| 244 | return -int(blk[0]); |
| 245 | else |
| 246 | throw "BigInteger operator int() const: Value is too big for an int"; |
| 247 | default: |
| 248 | throw "BigInteger: Internal error"; |
| 249 | } |
| 250 | } |
| 251 | |
| 252 | BigInteger::operator unsigned short() const { |
| 253 | switch (sign) { |
| 254 | case zero: |
| 255 | return 0; |
| 256 | case positive: |
| 257 | if (len == 1 && (blk[0] & ~usMask) == 0) |
| 258 | return (unsigned short)(blk[0]); |
| 259 | else |
| 260 | throw "BigInteger operator unsigned short() const: Value is too big for an unsigned short"; |
| 261 | case negative: |
| 262 | throw "BigInteger operator unsigned short() const: Cannot convert a negative integer to an unsigned type"; |
| 263 | default: |
| 264 | throw "BigInteger: Internal error"; |
| 265 | } |
| 266 | } |
| 267 | |
| 268 | BigInteger::operator short() const { |
| 269 | switch (sign) { |
| 270 | case zero: |
| 271 | return 0; |
| 272 | case positive: |
| 273 | if (len == 1 && (blk[0] & ~sMask) == 0) |
| 274 | return short(blk[0]); |
| 275 | else |
| 276 | throw "BigInteger operator short() const: Value is too big for a short"; |
| 277 | case negative: |
| 278 | if (len == 1 && (blk[0] & ~sMask) == 0) |
| 279 | return -short(blk[0]); |
| 280 | else |
| 281 | throw "BigInteger operator short() const: Value is too big for a short"; |
| 282 | default: |
| 283 | throw "BigInteger: Internal error"; |
| 284 | } |
| 285 | } |
| 286 | |
| 287 | // COMPARISON |
| 288 | BigInteger::CmpRes BigInteger::compareTo(const BigInteger &x) const { |
| 289 | // A greater sign implies a greater number |
| 290 | if (sign < x.sign) |
| 291 | return less; |
| 292 | else if (sign > x.sign) |
| 293 | return greater; |
| 294 | else switch (sign) { |
| 295 | // If the signs are the same... |
| 296 | case zero: |
| 297 | return equal; // Two zeros are equal |
| 298 | case positive: |
| 299 | // Compare the magnitudes |
| 300 | return BigUnsigned::compareTo(x); |
| 301 | case negative: |
| 302 | // Compare the magnitudes, but return the opposite result |
| 303 | return CmpRes(-BigUnsigned::compareTo(x)); |
| 304 | default: |
| 305 | throw "BigInteger: Internal error"; |
| 306 | } |
| 307 | } |
| 308 | |
| 309 | // PUT-HERE OPERATIONS |
| 310 | // These do some messing around to determine the sign of the result, |
| 311 | // then call one of BigUnsigned's put-heres. |
| 312 | |
| 313 | // See remarks about aliased calls in BigUnsigned.cc . |
| 314 | #define DOTR_ALIASED(cond, op) \ |
| 315 | if (cond) { \ |
| 316 | BigInteger tmpThis; \ |
| 317 | tmpThis.op; \ |
| 318 | *this = tmpThis; \ |
| 319 | return; \ |
| 320 | } |
| 321 | |
| 322 | // Addition |
| 323 | void BigInteger::add(const BigInteger &a, const BigInteger &b) { |
| 324 | DOTR_ALIASED(this == &a || this == &b, add(a, b)); |
| 325 | // If one argument is zero, copy the other. |
| 326 | if (a.sign == zero) |
| 327 | operator =(b); |
| 328 | else if (b.sign == zero) |
| 329 | operator =(a); |
| 330 | // If the arguments have the same sign, take the |
| 331 | // common sign and add their magnitudes. |
| 332 | else if (a.sign == b.sign) { |
| 333 | sign = a.sign; |
| 334 | BigUnsigned::add(a, b); |
| 335 | } else { |
| 336 | // Otherwise, their magnitudes must be compared. |
| 337 | switch (a.BigUnsigned::compareTo(b)) { |
| 338 | // If their magnitudes are the same, copy zero. |
| 339 | case equal: |
| 340 | len = 0; |
| 341 | sign = zero; |
| 342 | break; |
| 343 | // Otherwise, take the sign of the greater, and subtract |
| 344 | // the lesser magnitude from the greater magnitude. |
| 345 | case greater: |
| 346 | sign = a.sign; |
| 347 | BigUnsigned::subtract(a, b); |
| 348 | break; |
| 349 | case less: |
| 350 | sign = b.sign; |
| 351 | BigUnsigned::subtract(b, a); |
| 352 | break; |
| 353 | } |
| 354 | } |
| 355 | } |
| 356 | |
| 357 | // Subtraction |
| 358 | void BigInteger::subtract(const BigInteger &a, const BigInteger &b) { |
| 359 | // Notice that this routine is identical to BigInteger::add, |
| 360 | // if one replaces b.sign by its opposite. |
| 361 | DOTR_ALIASED(this == &a || this == &b, subtract(a, b)); |
| 362 | // If a is zero, copy b and flip its sign. If b is zero, copy a. |
| 363 | if (a.sign == zero) { |
| 364 | BigUnsigned::operator =(b); |
| 365 | // Take the negative of _b_'s, sign, not ours. |
| 366 | // Bug pointed out by Sam Larkin on 2005.03.30. |
| 367 | sign = Sign(-b.sign); |
| 368 | } else if (b.sign == zero) |
| 369 | operator =(a); |
| 370 | // If their signs differ, take a.sign and add the magnitudes. |
| 371 | else if (a.sign != b.sign) { |
| 372 | sign = a.sign; |
| 373 | BigUnsigned::add(a, b); |
| 374 | } else { |
| 375 | // Otherwise, their magnitudes must be compared. |
| 376 | switch (a.BigUnsigned::compareTo(b)) { |
| 377 | // If their magnitudes are the same, copy zero. |
| 378 | case equal: |
| 379 | len = 0; |
| 380 | sign = zero; |
| 381 | break; |
| 382 | // If a's magnitude is greater, take a.sign and |
| 383 | // subtract a from b. |
| 384 | case greater: |
| 385 | sign = a.sign; |
| 386 | BigUnsigned::subtract(a, b); |
| 387 | break; |
| 388 | // If b's magnitude is greater, take the opposite |
| 389 | // of b.sign and subtract b from a. |
| 390 | case less: |
| 391 | sign = Sign(-b.sign); |
| 392 | BigUnsigned::subtract(b, a); |
| 393 | break; |
| 394 | } |
| 395 | } |
| 396 | } |
| 397 | |
| 398 | // Multiplication |
| 399 | void BigInteger::multiply(const BigInteger &a, const BigInteger &b) { |
| 400 | DOTR_ALIASED(this == &a || this == &b, multiply(a, b)); |
| 401 | // If one object is zero, copy zero and return. |
| 402 | if (a.sign == zero || b.sign == zero) { |
| 403 | sign = zero; |
| 404 | len = 0; |
| 405 | return; |
| 406 | } |
| 407 | // If the signs of the arguments are the same, the result |
| 408 | // is positive, otherwise it is negative. |
| 409 | sign = (a.sign == b.sign) ? positive : negative; |
| 410 | // Multiply the magnitudes. |
| 411 | BigUnsigned::multiply(a, b); |
| 412 | } |
| 413 | |
| 414 | /* |
| 415 | * DIVISION WITH REMAINDER |
| 416 | * Please read the comments before the definition of |
| 417 | * `BigUnsigned::divideWithRemainder' in `BigUnsigned.cc' for lots of |
| 418 | * information you should know before reading this function. |
| 419 | * |
| 420 | * Following Knuth, I decree that x / y is to be |
| 421 | * 0 if y==0 and floor(real-number x / y) if y!=0. |
| 422 | * Then x % y shall be x - y*(integer x / y). |
| 423 | * |
| 424 | * Note that x = y * (x / y) + (x % y) always holds. |
| 425 | * In addition, (x % y) is from 0 to y - 1 if y > 0, |
| 426 | * and from -(|y| - 1) to 0 if y < 0. (x % y) = x if y = 0. |
| 427 | * |
| 428 | * Examples: (q = a / b, r = a % b) |
| 429 | * a b q r |
| 430 | * === === === === |
| 431 | * 4 3 1 1 |
| 432 | * -4 3 -2 2 |
| 433 | * 4 -3 -2 -2 |
| 434 | * -4 -3 1 -1 |
| 435 | */ |
| 436 | void BigInteger::divideWithRemainder(const BigInteger &b, BigInteger &q) { |
| 437 | // Defend against aliased calls; |
| 438 | // same idea as in BigUnsigned::divideWithRemainder . |
| 439 | if (this == &q) |
| 440 | throw "BigInteger::divideWithRemainder: Cannot write quotient and remainder into the same variable"; |
| 441 | if (this == &b || &q == &b) { |
| 442 | BigInteger tmpB(b); |
| 443 | divideWithRemainder(tmpB, q); |
| 444 | return; |
| 445 | } |
| 446 | |
| 447 | // Division by zero gives quotient 0 and remainder *this |
| 448 | if (b.sign == zero) { |
| 449 | q.len = 0; |
| 450 | q.sign = zero; |
| 451 | return; |
| 452 | } |
| 453 | // 0 / b gives quotient 0 and remainder 0 |
| 454 | if (sign == zero) { |
| 455 | q.len = 0; |
| 456 | q.sign = zero; |
| 457 | return; |
| 458 | } |
| 459 | |
| 460 | // Here *this != 0, b != 0. |
| 461 | |
| 462 | // Do the operands have the same sign? |
| 463 | if (sign == b.sign) { |
| 464 | // Yes: easy case. Quotient is zero or positive. |
| 465 | q.sign = positive; |
| 466 | } else { |
| 467 | // No: harder case. Quotient is negative. |
| 468 | q.sign = negative; |
| 469 | // Decrease the magnitude of the dividend by one. |
| 470 | BigUnsigned::operator --(); |
| 471 | /* |
| 472 | * We tinker with the dividend before and with the |
| 473 | * quotient and remainder after so that the result |
| 474 | * comes out right. To see why it works, consider the following |
| 475 | * list of examples, where A is the magnitude-decreased |
| 476 | * a, Q and R are the results of BigUnsigned division |
| 477 | * with remainder on A and |b|, and q and r are the |
| 478 | * final results we want: |
| 479 | * |
| 480 | * a A b Q R q r |
| 481 | * -3 -2 3 0 2 -1 0 |
| 482 | * -4 -3 3 1 0 -2 2 |
| 483 | * -5 -4 3 1 1 -2 1 |
| 484 | * -6 -5 3 1 2 -2 0 |
| 485 | * |
| 486 | * It appears that we need a total of 3 corrections: |
| 487 | * Decrease the magnitude of a to get A. Increase the |
| 488 | * magnitude of Q to get q (and make it negative). |
| 489 | * Find r = (b - 1) - R and give it the desired sign. |
| 490 | */ |
| 491 | } |
| 492 | |
| 493 | // Divide the magnitudes. |
| 494 | BigUnsigned::divideWithRemainder(b, q); |
| 495 | |
| 496 | if (sign != b.sign) { |
| 497 | // More for the harder case (as described): |
| 498 | // Increase the magnitude of the quotient by one. |
| 499 | q.BigUnsigned::operator ++(); |
| 500 | // Modify the remainder. |
| 501 | BigUnsigned temp(*this); |
| 502 | BigUnsigned::subtract(b, temp); |
| 503 | BigUnsigned::operator --(); |
| 504 | } |
| 505 | |
| 506 | // Sign of the remainder is always the sign of the divisor b. |
| 507 | sign = b.sign; |
| 508 | |
| 509 | // Set signs to zero as necessary. (Thanks David Allen!) |
| 510 | if (len == 0) |
| 511 | sign = zero; |
| 512 | if (q.len == 0) |
| 513 | q.sign = zero; |
| 514 | |
| 515 | // WHEW!!! |
| 516 | } |
| 517 | |
| 518 | // Negation |
| 519 | void BigInteger::negate(const BigInteger &a) { |
| 520 | DOTR_ALIASED(this == &a, negate(a)); |
| 521 | // Copy a's magnitude |
| 522 | BigUnsigned::operator =(a); |
| 523 | // Copy the opposite of a.sign |
| 524 | sign = Sign(-a.sign); |
| 525 | } |
| 526 | |
| 527 | // INCREMENT/DECREMENT OPERATORS |
| 528 | |
| 529 | // Prefix increment |
| 530 | void BigInteger::operator ++() { |
| 531 | switch (sign) { |
| 532 | case zero: |
| 533 | allocate(1); |
| 534 | sign = positive; |
| 535 | len = 1; |
| 536 | blk[0] = 1; |
| 537 | break; |
| 538 | case positive: |
| 539 | BigUnsigned::operator ++(); |
| 540 | break; |
| 541 | case negative: |
| 542 | BigUnsigned::operator --(); |
| 543 | if (len == 0) |
| 544 | sign = zero; |
| 545 | break; |
| 546 | } |
| 547 | } |
| 548 | |
| 549 | // Postfix increment: same as prefix |
| 550 | void BigInteger::operator ++(int) { |
| 551 | operator ++(); |
| 552 | } |
| 553 | |
| 554 | // Prefix decrement |
| 555 | void BigInteger::operator --() { |
| 556 | switch (sign) { |
| 557 | case zero: |
| 558 | allocate(1); |
| 559 | sign = negative; |
| 560 | len = 1; |
| 561 | blk[0] = 1; |
| 562 | break; |
| 563 | case negative: |
| 564 | BigUnsigned::operator ++(); |
| 565 | break; |
| 566 | case positive: |
| 567 | BigUnsigned::operator --(); |
| 568 | if (len == 0) |
| 569 | sign = zero; |
| 570 | break; |
| 571 | } |
| 572 | } |
| 573 | |
| 574 | // Postfix decrement: same as prefix |
| 575 | void BigInteger::operator --(int) { |
| 576 | operator --(); |
| 577 | } |
| 578 | |