| 1 | /* Copyright (c) 1995, 1996, 2001 by Arkkra Enterprises */ |
| 2 | /* All rights reserved */ |
| 3 | /* |
| 4 | * rational.c functions to do operations on rational numbers |
| 5 | * |
| 6 | * Contents: |
| 7 | * |
| 8 | * radd(), rsub(), rmul(), rdiv(), rneg(), rinv(), rrai(), rred(), |
| 9 | * ator(), rtoa(); also gtrat(), called by macros GT,GE,LT,LE |
| 10 | * |
| 11 | * ratmsg(), add64_64(), mul32_64(), divmod64(), red64_64() |
| 12 | * |
| 13 | * The first group of functions are for the user. The second |
| 14 | * are for internal use only. |
| 15 | * |
| 16 | * Description: |
| 17 | * The functions in this file do operations on rational numbers. |
| 18 | * The rational arguments to functions that the users can call |
| 19 | * must be in standard form (lowest terms, with positive |
| 20 | * denominator) except for rred(). There are checks for division |
| 21 | * by zero and for overflow of numerators and denominators. |
| 22 | * (The absolute values of each are limited to MAXLONG, defined |
| 23 | * in rational.h.) If there is an error, the external int |
| 24 | * raterrno is set to RATDIV0 or RATOVER, as the case may be, |
| 25 | * and *raterrfuncp is checked. If nonzero, it is assumed to |
| 26 | * point at the user's error handler, and it is called with a |
| 27 | * parameter equal to raterrno. Otherwise, a message is printed |
| 28 | * to stderr. In any case, the answer returned to the user is |
| 29 | * 0/1. If there was no error, raterrno is set to RATNOERR. |
| 30 | * |
| 31 | * In general, the functions assume they are being called with |
| 32 | * valid parameters. If they are not, results are not guaranteed |
| 33 | * to be correct. However, they are defensive enough so that |
| 34 | * invalid parameters will not cause a crash in these routines. |
| 35 | * They will not always detect invalid parameters, but if they |
| 36 | * do, they will use the raterrno/raterrfuncp mechanism described |
| 37 | * above, with the value RATPARM. |
| 38 | * |
| 39 | * These routines depend on a INT32B being a 32-bit number, |
| 40 | * stored in two's complement form, and UINT32B being the same |
| 41 | * for unsigned. See rational.h. Numerators and denominators |
| 42 | * are assumed to be INT32B. Furthermore, the number 0x80000000 |
| 43 | * is not allowed. The routines should work on any machine and |
| 44 | * compiler where these requirements are met. |
| 45 | * |
| 46 | * Internally, when 64-bit numbers are used, they are represented |
| 47 | * by an array of two INT32B. The 0 subscript contains the low |
| 48 | * order bits and the 1 subscript contains the high order bits. |
| 49 | * The numbers are usually used as two's complement signed |
| 50 | * integers, so the high bit of the 1 subscript is a sign bit. |
| 51 | */ |
| 52 | |
| 53 | #ifndef stderr |
| 54 | # include <stdio.h> |
| 55 | #endif |
| 56 | |
| 57 | #ifndef isspace |
| 58 | # include <ctype.h> |
| 59 | #endif |
| 60 | |
| 61 | #ifndef _RATIONAL |
| 62 | # include "rational.h" |
| 63 | #endif |
| 64 | |
| 65 | |
| 66 | |
| 67 | /* |
| 68 | * Define SMALL to be a number that uses less than half as many bits as |
| 69 | * MAXLONG (15 as compared to 31). Define a SMALLRAT as a rational number |
| 70 | * whose numerator (absolute value) and denominator are in that range. |
| 71 | * The denominator is assumed to be positive. |
| 72 | */ |
| 73 | #define SMALL 0x7fff |
| 74 | #define SMALLRAT(q) ((q).n <= SMALL && (q).n >= -SMALL && (q).d <= SMALL) |
| 75 | |
| 76 | |
| 77 | /* |
| 78 | * This macro checks whether a 64-bit integer is actually less than or |
| 79 | * equal to MAXLONG in absolute value, 0x7fffffff. |
| 80 | */ |
| 81 | #define INT32(n) ((n)[1] == 0 && (n)[0] >= 0 || \ |
| 82 | (n)[1] == -1 && (n)[0] < 0 && (n)[0] != 0x80000000) |
| 83 | |
| 84 | |
| 85 | /* |
| 86 | * Return whether the first 64-bit number equals the second. |
| 87 | * To be equal, both words must be equal. |
| 88 | */ |
| 89 | #define EQ64(x, y) ( (x)[1] == (y)[1] && (x)[0] == (y)[0] ) |
| 90 | |
| 91 | |
| 92 | /* |
| 93 | * Return whether the first 64-bit number is greater than the second. |
| 94 | * If the high order words are equal, use the low order words, unsigned; |
| 95 | * otherwise, just use the high order words. |
| 96 | */ |
| 97 | #define GT64(x, y) ( \ |
| 98 | (x)[1] == (y)[1] ? \ |
| 99 | (UINT32B)(x)[0] > (UINT32B)(y)[0] \ |
| 100 | : \ |
| 101 | (x)[1] > (y)[1] \ |
| 102 | ) |
| 103 | |
| 104 | |
| 105 | /* |
| 106 | * Return whether the first 64-bit number is less than the second. |
| 107 | * If the high order words are equal, use the low order words, unsigned; |
| 108 | * otherwise, just use the high order words. |
| 109 | */ |
| 110 | #define LT64(x, y) ( \ |
| 111 | (x)[1] == (y)[1] ? \ |
| 112 | (UINT32B)(x)[0] < (UINT32B)(y)[0] \ |
| 113 | : \ |
| 114 | (x)[1] < (y)[1] \ |
| 115 | ) |
| 116 | |
| 117 | |
| 118 | /* |
| 119 | * Return whether the first *unsigned* 64-bit number is less than or equal to |
| 120 | * the second. If the high order words are equal, use the low order words; |
| 121 | * otherwise, just use the high order words. |
| 122 | */ |
| 123 | #define LEU64(x, y) ( \ |
| 124 | (x)[1] == (y)[1] ? \ |
| 125 | (UINT32B)(x)[0] <= (UINT32B)(y)[0] \ |
| 126 | : \ |
| 127 | (UINT32B)(x)[1] <= (UINT32B)(y)[1] \ |
| 128 | ) |
| 129 | |
| 130 | |
| 131 | /* |
| 132 | * Negate a 64-bit number. |
| 133 | */ |
| 134 | #define NEG64(x) { \ |
| 135 | (x)[1] = ~(x)[1]; /* one's complement */ \ |
| 136 | (x)[0] = -(x)[0]; /* two's complement */ \ |
| 137 | if ((x)[0] == 0) /* if "carry" must inc high word */ \ |
| 138 | (x)[1]++; \ |
| 139 | } |
| 140 | |
| 141 | |
| 142 | /* |
| 143 | * Shift a 64-bit number left one bit as unsigned (not that it matters). |
| 144 | */ |
| 145 | #define SHL1U64(x) { \ |
| 146 | (x)[1] <<= 1; /* shift high word */ \ |
| 147 | if ((x)[0] < 0) /* if high bit of low word is set */ \ |
| 148 | (x)[1]++; /* shift it into the high word */ \ |
| 149 | (x)[0] <<= 1; /* shift low word */ \ |
| 150 | } |
| 151 | |
| 152 | |
| 153 | /* |
| 154 | * Shift a 64-bit number right one bit as unsigned. |
| 155 | */ |
| 156 | #define SHR1U64(x) { \ |
| 157 | (x)[0] = (UINT32B)(x)[0] >> 1; /* shift low word */ \ |
| 158 | if ((x)[1] & 1) /* if low bit of high word set*/\ |
| 159 | (x)[0] |= 0x80000000; /* shift it into low word */ \ |
| 160 | (x)[1] = (UINT32B)(x)[1] >> 1; /* shift low word */ \ |
| 161 | } |
| 162 | |
| 163 | |
| 164 | |
| 165 | /* declare as static the functions that are only used internally */ |
| 166 | #ifdef __STDC__ |
| 167 | static void ratmsg(int code); |
| 168 | static void add64_64(INT32B a[], INT32B x[], INT32B y[]); |
| 169 | static void mul32_64(INT32B a[], INT32B x, INT32B y); |
| 170 | static void divmod64(INT32B x[], INT32B y[], INT32B q[], INT32B r[]); |
| 171 | static void red64_64(INT32B num[], INT32B den[]); |
| 172 | #else |
| 173 | static void ratmsg(), add64_64(), mul32_64(), divmod64(), red64_64(); |
| 174 | #endif |
| 175 | |
| 176 | |
| 177 | int raterrno; /* set to error type upon return to user */ |
| 178 | void (*raterrfuncp)(); /* error handler functions to be called */ |
| 179 | |
| 180 | static RATIONAL zero = {0,1}; |
| 181 | \f |
| 182 | /* |
| 183 | * radd() add two rational numbers |
| 184 | * |
| 185 | * This function adds two rational numbers. They must be in standard |
| 186 | * form. |
| 187 | * |
| 188 | * Parameters: x the first number |
| 189 | * y the second number |
| 190 | * |
| 191 | * Return value: The sum (x + y), if it can be represented as a |
| 192 | * RATIONAL, else 0/1. |
| 193 | * |
| 194 | * Side effects: If radd() succeeds, it sets raterrno to RATNOERR. |
| 195 | * Otherwise, the numerator or denominator must have |
| 196 | * overflowed, so it sets raterrno to RATOVER and |
| 197 | * either prints a message or calls a user-supplied |
| 198 | * error handler. |
| 199 | */ |
| 200 | |
| 201 | RATIONAL |
| 202 | radd(x, y) |
| 203 | |
| 204 | RATIONAL x, y; |
| 205 | |
| 206 | { |
| 207 | RATIONAL a; /* the answer */ |
| 208 | INT32B bign[2]; /* 64-bit numerator */ |
| 209 | INT32B bigd[2]; /* 64-bit denominator */ |
| 210 | INT32B bigt[2]; /* temp storage */ |
| 211 | |
| 212 | |
| 213 | raterrno = RATNOERR; /* no error yet */ |
| 214 | |
| 215 | /* |
| 216 | * If the numbers are small enough, do it the easy way, since there is |
| 217 | * then no danger of overflow. |
| 218 | */ |
| 219 | if (SMALLRAT(x) && SMALLRAT(y)) { |
| 220 | a.n = x.n * y.d + x.d * y.n; |
| 221 | a.d = x.d * y.d; |
| 222 | rred(&a); /* reduce to standard form */ |
| 223 | return(a); |
| 224 | } |
| 225 | |
| 226 | /* |
| 227 | * To avoid overflow during the calculations, use two INT32B to |
| 228 | * hold numbers. |
| 229 | */ |
| 230 | mul32_64(bign, x.n, y.d); /* get first part of numerator */ |
| 231 | mul32_64(bigt, x.d, y.n); /* get second part of numerator */ |
| 232 | add64_64(bign, bign, bigt); /* add to get full numerator */ |
| 233 | mul32_64(bigd, x.d, y.d); /* get denominator */ |
| 234 | red64_64(bign, bigd); /* reduce */ |
| 235 | |
| 236 | /* overflow if the result can't fit in a RATIONAL */ |
| 237 | if ( ! INT32(bign) || ! INT32(bigd) ) { |
| 238 | ratmsg(RATOVER); /* set raterrno, report error */ |
| 239 | return(zero); |
| 240 | } |
| 241 | |
| 242 | a.n = bign[0]; /* set answer */ |
| 243 | a.d = bigd[0]; |
| 244 | |
| 245 | return(a); |
| 246 | } |
| 247 | \f |
| 248 | /* |
| 249 | * rsub() subtract two rational numbers |
| 250 | * |
| 251 | * This function subtracts two rational numbers. They must be in standard |
| 252 | * form. |
| 253 | * |
| 254 | * Parameters: x the first number |
| 255 | * y the second number |
| 256 | * |
| 257 | * Return value: The difference (x - y), if it can be represented as a |
| 258 | * RATIONAL, else 0/1. |
| 259 | * |
| 260 | * Side effects: If rsub() succeeds, it sets raterrno to RATNOERR. |
| 261 | * Otherwise, the numerator or denominator must have |
| 262 | * overflowed, so it sets raterrno to RATOVER and |
| 263 | * either prints a message or calls a user-supplied |
| 264 | * error handler. |
| 265 | */ |
| 266 | |
| 267 | RATIONAL |
| 268 | rsub(x, y) |
| 269 | |
| 270 | RATIONAL x, y; |
| 271 | |
| 272 | { |
| 273 | /* |
| 274 | * Just negate the second operand and add. We could call rneg() to |
| 275 | * negate y, but why waste the time? |
| 276 | */ |
| 277 | y.n = -y.n; |
| 278 | return(radd(x, y)); |
| 279 | } |
| 280 | \f |
| 281 | /* |
| 282 | * rmul() multiply two rational numbers |
| 283 | * |
| 284 | * This function multiplies two rational numbers. They must be in standard |
| 285 | * form. |
| 286 | * |
| 287 | * Parameters: x the first number |
| 288 | * y the second number |
| 289 | * |
| 290 | * Return value: The product (x * y), if it can be represented as a |
| 291 | * RATIONAL, else 0/1. |
| 292 | * |
| 293 | * Side effects: If rsub() succeeds, it sets raterrno to RATNOERR. |
| 294 | * Otherwise, the numerator or denominator must have |
| 295 | * overflowed, so it sets raterrno to RATOVER and |
| 296 | * either prints a message or calls a user-supplied |
| 297 | * error handler. |
| 298 | */ |
| 299 | |
| 300 | RATIONAL |
| 301 | rmul(x, y) |
| 302 | |
| 303 | RATIONAL x, y; |
| 304 | |
| 305 | { |
| 306 | RATIONAL a; /* the answer */ |
| 307 | INT32B bign[2]; /* 64-bit numerator */ |
| 308 | INT32B bigd[2]; /* 64-bit denominator */ |
| 309 | |
| 310 | |
| 311 | raterrno = RATNOERR; /* no error yet */ |
| 312 | |
| 313 | /* |
| 314 | * If the numbers are small enough, do it the easy way, since there is |
| 315 | * then no danger of overflow. |
| 316 | */ |
| 317 | if (SMALLRAT(x) && SMALLRAT(y)) { |
| 318 | a.n = x.n * y.n; |
| 319 | a.d = x.d * y.d; |
| 320 | rred(&a); /* reduce to standard form */ |
| 321 | return(a); |
| 322 | } |
| 323 | |
| 324 | /* |
| 325 | * To avoid overflow during the calculations, use two INT32B to |
| 326 | * hold numbers. |
| 327 | */ |
| 328 | mul32_64(bign, x.n, y.n); /* get numerator */ |
| 329 | mul32_64(bigd, x.d, y.d); /* get denominator */ |
| 330 | red64_64(bign, bigd); /* reduce */ |
| 331 | |
| 332 | /* overflow if the result can't fit in a RATIONAL */ |
| 333 | if ( ! INT32(bign) || ! INT32(bigd) ) { |
| 334 | ratmsg(RATOVER); /* set raterrno, report error */ |
| 335 | return(zero); |
| 336 | } |
| 337 | |
| 338 | a.n = bign[0]; /* set answer */ |
| 339 | a.d = bigd[0]; |
| 340 | |
| 341 | return(a); |
| 342 | } |
| 343 | \f |
| 344 | /* |
| 345 | * rdiv() divide two rational numbers |
| 346 | * |
| 347 | * This function divides two rational numbers. They must be in standard |
| 348 | * form. |
| 349 | * |
| 350 | * Parameters: x the first number |
| 351 | * y the second number |
| 352 | * |
| 353 | * Return value: The quotient (x / y), if it is defined and can be |
| 354 | * represented as a RATIONAL, else 0/1. |
| 355 | * |
| 356 | * Side effects: If rdiv() succeeds, it sets raterrno to RATNOERR. |
| 357 | * Otherwise, either the second number was zero or the |
| 358 | * numerator or denominator overflowed. In this case, |
| 359 | * it sets raterrno to RATDIV0 or RATOVER, respectively, |
| 360 | * and either prints a message or calls a user-supplied |
| 361 | * error handler. |
| 362 | */ |
| 363 | |
| 364 | RATIONAL |
| 365 | rdiv(x, y) |
| 366 | |
| 367 | RATIONAL x, y; |
| 368 | |
| 369 | { |
| 370 | RATIONAL r; /* reciprocal of y */ |
| 371 | |
| 372 | |
| 373 | r = rinv(y); /* first find 1/y */ |
| 374 | |
| 375 | if (raterrno != RATNOERR) /* if y was 0, return failure now */ |
| 376 | return(zero); |
| 377 | |
| 378 | /* |
| 379 | * Return x * r. Whether rmul() succeeds or fails, we still just want |
| 380 | * to leave raterrno the same and return what rmul() returns. |
| 381 | */ |
| 382 | return(rmul(x, r)); |
| 383 | } |
| 384 | \f |
| 385 | /* |
| 386 | * rneg() negate a rational number |
| 387 | * |
| 388 | * This function negates a rational number. It must be in standard form. |
| 389 | * |
| 390 | * Parameters: x the number |
| 391 | * |
| 392 | * Return value: The negative (-x). |
| 393 | * |
| 394 | * Side effects: It sets raterrno to RATNOERR. |
| 395 | */ |
| 396 | |
| 397 | RATIONAL |
| 398 | rneg(x) |
| 399 | |
| 400 | RATIONAL x; |
| 401 | |
| 402 | { |
| 403 | raterrno = RATNOERR; /* no errors are possible */ |
| 404 | |
| 405 | x.n = -x.n; |
| 406 | |
| 407 | /* answer is already in standard form since x was */ |
| 408 | return(x); |
| 409 | } |
| 410 | \f |
| 411 | /* |
| 412 | * rinv() invert a rational number |
| 413 | * |
| 414 | * This function inverts a rational number. It must be in standard form. |
| 415 | * |
| 416 | * Parameters: x the number |
| 417 | * |
| 418 | * Return value: The reciprocal (1 / x), if it is defined, else 0/1. |
| 419 | * |
| 420 | * Side effects: If rinv() succeeds, it sets raterrno to RATNOERR. |
| 421 | * Otherwise, the second number must have been zero, |
| 422 | * so it sets raterrno to RATDIV0 and either prints a |
| 423 | * message or calls a user-supplied error handler. |
| 424 | */ |
| 425 | |
| 426 | RATIONAL |
| 427 | rinv(x) |
| 428 | |
| 429 | RATIONAL x; |
| 430 | |
| 431 | { |
| 432 | RATIONAL a; /* the answer */ |
| 433 | |
| 434 | |
| 435 | /* check for division by 0 */ |
| 436 | if (ZE(x)) { |
| 437 | ratmsg(RATDIV0); /* set raterrno, report error */ |
| 438 | return(zero); |
| 439 | } |
| 440 | |
| 441 | raterrno = RATNOERR; /* no errors from here on */ |
| 442 | |
| 443 | a.n = x.d; /* flip numerator and denominator */ |
| 444 | a.d = x.n; |
| 445 | |
| 446 | if (a.d < 0) { /* if x was negative, reverse signs */ |
| 447 | a.n = -a.n; |
| 448 | a.d = -a.d; |
| 449 | } |
| 450 | |
| 451 | return(a); |
| 452 | } |
| 453 | \f |
| 454 | /* |
| 455 | * rrai() raise a rational number to an integral power |
| 456 | * |
| 457 | * This function raises a rational number to an integral power. The |
| 458 | * rational number must be in standard form. |
| 459 | * |
| 460 | * Parameters: x the rational number |
| 461 | * n the power, an integer |
| 462 | * |
| 463 | * Return value: The result (x to the nth power), if it is defined and |
| 464 | * can be represented as a RATIONAL, else 0/1. |
| 465 | * |
| 466 | * Side effects: If rrai() succeeds, it sets raterrno to RATNOERR. |
| 467 | * Otherwise, either zero is being raised to a non- |
| 468 | * positive power, or the numerator or denominator |
| 469 | * overflowed. In this case, it sets raterrno to |
| 470 | * RATDIV0 or RATOVER, respectively, and either prints |
| 471 | * a message or calls a user-supplied error handler. |
| 472 | */ |
| 473 | |
| 474 | RATIONAL |
| 475 | rrai(x, n) |
| 476 | |
| 477 | RATIONAL x; |
| 478 | register int n; |
| 479 | |
| 480 | { |
| 481 | static RATIONAL one = {1,1}; |
| 482 | |
| 483 | RATIONAL a; /* the answer */ |
| 484 | register int i; /* loop counter */ |
| 485 | |
| 486 | |
| 487 | /* it is undefined to raise zero to a nonpositive power */ |
| 488 | if (ZE(x) && n <= 0) { |
| 489 | ratmsg(RATDIV0); /* set raterrno, report error */ |
| 490 | return(zero); |
| 491 | } |
| 492 | |
| 493 | raterrno = RATNOERR; /* no error yet */ |
| 494 | |
| 495 | a = one; /* init to 1 */ |
| 496 | if (n >= 0) { |
| 497 | for (i = 0; i < n; i++) { |
| 498 | a = rmul(a, x); /* mul again by x */ |
| 499 | if (raterrno != RATNOERR) |
| 500 | return(zero); |
| 501 | } |
| 502 | } else { |
| 503 | for (i = 0; i > n; i--) { |
| 504 | a = rdiv(a, x); /* div again by x */ |
| 505 | if (raterrno != RATNOERR) |
| 506 | return(zero); |
| 507 | } |
| 508 | } |
| 509 | |
| 510 | return(a); |
| 511 | } |
| 512 | \f |
| 513 | /* |
| 514 | * rred() reduce a rational number to standard form |
| 515 | * |
| 516 | * This function puts a rational number into standard form; that is, |
| 517 | * numerator and denominator will be relatively prime and the denominator |
| 518 | * will be positive. On input, they may be any integers whose absolute |
| 519 | * values do not exceed MAXLONG. |
| 520 | * |
| 521 | * Parameters: ap pointer to the rational number |
| 522 | * |
| 523 | * Return value: None. |
| 524 | * |
| 525 | * Side effects: If ap->d is 0, the function sets raterrno to RATDIV0, |
| 526 | * either prints a message or calls a user-supplied |
| 527 | * error handler, and sets *ap to 0/1. Otherwise, it |
| 528 | * sets raterrno to RATNOERR and puts *ap in standard form. |
| 529 | */ |
| 530 | |
| 531 | void |
| 532 | rred(ap) |
| 533 | |
| 534 | register RATIONAL *ap; |
| 535 | |
| 536 | { |
| 537 | register INT32B b, c, r; /* temp variables for Euclidean algorithm */ |
| 538 | register int sign; /* answer is pos (1) or neg (-1) */ |
| 539 | |
| 540 | |
| 541 | /* |
| 542 | * Since the numerator and denominator can be anything <= MAXLONG, |
| 543 | * we must guard against division by 0. |
| 544 | */ |
| 545 | if (ap->d == 0) { |
| 546 | ratmsg(RATDIV0); /* set raterrno, report error */ |
| 547 | *ap = zero; |
| 548 | return; |
| 549 | } |
| 550 | |
| 551 | raterrno = RATNOERR; /* no errors possible from here on */ |
| 552 | |
| 553 | if (ap->n == 0) { /* if so, answer is "0/1" */ |
| 554 | ap->d = 1; |
| 555 | return; |
| 556 | } |
| 557 | |
| 558 | /* now figure out sign of answer, and make n & d positive */ |
| 559 | sign = 1; /* init to positive */ |
| 560 | if (ap->n < 0) { /* reverse if numerator neg */ |
| 561 | sign = -sign; |
| 562 | ap->n = -(ap->n); |
| 563 | } |
| 564 | if (ap->d < 0) { /* reverse if denominator neg */ |
| 565 | sign = -sign; |
| 566 | ap->d = -(ap->d); |
| 567 | } |
| 568 | |
| 569 | /* now check whether numerator or denominator are equal */ |
| 570 | if (ap->n == ap->d) { /* if so, answer is +1 or -1 */ |
| 571 | ap->n = sign; |
| 572 | ap->d = 1; |
| 573 | return; |
| 574 | } |
| 575 | |
| 576 | if (ap->n < ap->d) { /* set so that c > b */ |
| 577 | c = ap->d; |
| 578 | b = ap->n; |
| 579 | } else { |
| 580 | c = ap->n; |
| 581 | b = ap->d; |
| 582 | } |
| 583 | |
| 584 | /* use Euclidean Algorithm to find greatest common divisor of c & b */ |
| 585 | do { |
| 586 | r = c % b; |
| 587 | c = b; |
| 588 | b = r; |
| 589 | } while (r != 0); |
| 590 | |
| 591 | /* now c is the greatest common divisor */ |
| 592 | |
| 593 | ap->n /= c; /* divide out greatest common divisor */ |
| 594 | ap->d /= c; /* divide out greatest common divisor */ |
| 595 | |
| 596 | if (sign < 0) /* put sign in if result should be negative */ |
| 597 | ap->n = -(ap->n); |
| 598 | |
| 599 | return; |
| 600 | } |
| 601 | \f |
| 602 | /* |
| 603 | * ator() convert an ascii string to a rational number |
| 604 | * |
| 605 | * This function takes an ascii string as input and interprets it as |
| 606 | * a rational number. White space may precede the number, but the |
| 607 | * number may not contain white space. The numerator may be preceded |
| 608 | * by a minus sign. The denomintor is optional, but if present, must |
| 609 | * not contain a sign. In short, the number must match one of the |
| 610 | * following lex regular expressions, which starts where s points and |
| 611 | * ends before the first character not matching the pattern: |
| 612 | * [ \t\n]*-?[0-9]+ |
| 613 | * [ \t\n]*-?[0-9]+\/[0-9]+ |
| 614 | * Further restrictions are that the absolute values of numerator and |
| 615 | * denominator cannot exceed MAXLONG, and the denominator cannot be 0. |
| 616 | * If neither pattern is matched, or the further restrictions are |
| 617 | * violated, the function sets *rp to 0/1 and returns NULL. Otherwise, |
| 618 | * it sets *rp to the result in standard form, and returns a pointer to |
| 619 | * the first char after the number found. |
| 620 | * |
| 621 | * Parameters: rp pointer to where the answer goes |
| 622 | * s string containing ascii rational number |
| 623 | * |
| 624 | * Return value: If a valid rational number is found, the function |
| 625 | * returns a pointer to the next char in the string |
| 626 | * following the number. Otherwise it returns NULL. |
| 627 | * |
| 628 | * Side effects: If ator() succeeds, it sets *rp to the result. |
| 629 | * Otherwise, it sets it to 0/1. |
| 630 | */ |
| 631 | |
| 632 | char * |
| 633 | ator(rp, s) |
| 634 | |
| 635 | register RATIONAL *rp; |
| 636 | register char s[]; |
| 637 | |
| 638 | { |
| 639 | register char *p; /* point somewhere in s[] */ |
| 640 | int sign; /* 1 means positive, -1 negative */ |
| 641 | |
| 642 | |
| 643 | /* skip by white space */ |
| 644 | for (p = s; isspace(*p); p++) |
| 645 | ; |
| 646 | |
| 647 | /* init sign to positive; then reverse it if a dash is found */ |
| 648 | sign = 1; |
| 649 | if (p[0] == '-') { |
| 650 | sign = -1; |
| 651 | p++; |
| 652 | } |
| 653 | |
| 654 | /* fail if there are no digits */ |
| 655 | if ( ! isdigit(*p) ) { |
| 656 | *rp = zero; |
| 657 | return(NULL); |
| 658 | } |
| 659 | |
| 660 | /* |
| 661 | * Collect the numerator digits, and defend against overflow. |
| 662 | */ |
| 663 | rp->n = 0; |
| 664 | while ( isdigit(*p) ) { |
| 665 | if (rp->n > MAXLONG / 10) { |
| 666 | *rp = zero; |
| 667 | return(NULL); |
| 668 | } |
| 669 | rp->n *= 10; |
| 670 | if (rp->n > MAXLONG - (*p - '0')) { |
| 671 | *rp = zero; |
| 672 | return(NULL); |
| 673 | } |
| 674 | rp->n += *p++ - '0'; |
| 675 | } |
| 676 | |
| 677 | if (sign < 0) /* make negative if necessary */ |
| 678 | rp->n = -(rp->n); |
| 679 | |
| 680 | |
| 681 | /* |
| 682 | * If there is to be a denominator, collect its digits. Otherwise, |
| 683 | * set it to 1. Defend against overflow. |
| 684 | */ |
| 685 | if (*p == '/') { |
| 686 | p++; |
| 687 | if ( ! isdigit(*p) ) { /* must be digit (no '-' allowed) */ |
| 688 | *rp = zero; |
| 689 | return(NULL); |
| 690 | } |
| 691 | rp->d = 0; |
| 692 | while ( isdigit(*p) ) { |
| 693 | if (rp->d > MAXLONG / 10) { |
| 694 | *rp = zero; |
| 695 | return(NULL); |
| 696 | } |
| 697 | rp->d *= 10; |
| 698 | if (rp->d > MAXLONG - (*p - '0')) { |
| 699 | *rp = zero; |
| 700 | return(NULL); |
| 701 | } |
| 702 | rp->d += *p++ - '0'; |
| 703 | } |
| 704 | if (rp->d == 0) { /* zero denominator is a failure */ |
| 705 | *rp = zero; |
| 706 | return(NULL); |
| 707 | } |
| 708 | } else { |
| 709 | rp->d = 1; /* no denominator; assume 1 */ |
| 710 | } |
| 711 | |
| 712 | rred(rp); /* reduce the fraction */ |
| 713 | |
| 714 | return(p); /* first char after the number */ |
| 715 | } |
| 716 | \f |
| 717 | /* |
| 718 | * rtoa() convert a rational number to an ascii string |
| 719 | * |
| 720 | * This function takes a rational number as input converts it into |
| 721 | * an ascii string. If the denominator is 1, it will not be printed. |
| 722 | * The number must be in standard form. |
| 723 | * |
| 724 | * Parameters: s pointer to where the answer goes |
| 725 | * rp pointer to the rational number |
| 726 | * |
| 727 | * Return value: The function returns a pointer to the next char in |
| 728 | * the string following the number. |
| 729 | * |
| 730 | * Side effects: The function sets s[] to the result. |
| 731 | */ |
| 732 | |
| 733 | char * |
| 734 | rtoa(s, rp) |
| 735 | |
| 736 | register char s[]; |
| 737 | RATIONAL *rp; |
| 738 | |
| 739 | { |
| 740 | register INT32B num, den; /* copy of num and den from *rp */ |
| 741 | register int i; /* index into t[] */ |
| 742 | char t[12]; /* temp answer string */ |
| 743 | |
| 744 | |
| 745 | num = rp->n; /* copy num and den for efficiency */ |
| 746 | den = rp->d; |
| 747 | |
| 748 | if (num < 0) { /* if num is negative */ |
| 749 | *s++ = '-'; /* output minus sign */ |
| 750 | num = -num; /* and make num positive */ |
| 751 | } |
| 752 | |
| 753 | i = 0; |
| 754 | do { /* calc digits in reverse order */ |
| 755 | t[i++] = num % 10 + '0'; |
| 756 | num /= 10; |
| 757 | } while (num > 0); /* always loop at least once so 0="0"*/ |
| 758 | |
| 759 | while (--i >= 0) /* copy digits to answer string */ |
| 760 | *s++ = t[i]; |
| 761 | |
| 762 | if (den != 1) { /* if a denominator is needed */ |
| 763 | *s++ = '/'; /* fraction bar */ |
| 764 | i = 0; |
| 765 | do { /* calc digits in reverse order */ |
| 766 | t[i++] = den % 10 + '0'; |
| 767 | den /= 10; |
| 768 | } while (den > 0); |
| 769 | |
| 770 | while (--i >= 0) /* copy digits to answer string */ |
| 771 | *s++ = t[i]; |
| 772 | } |
| 773 | |
| 774 | return(s); |
| 775 | } |
| 776 | \f |
| 777 | /* |
| 778 | * gtrat() decide whether one rational is greater than another |
| 779 | * |
| 780 | * This function decides whether its first parameter is greater than |
| 781 | * its second. It is used by the macros GT, GE, LT, and LE. |
| 782 | * The numbers must be in standard form. (Actually, all that is |
| 783 | * matters is that the denominators be positive.) |
| 784 | * |
| 785 | * Parameters: x the first rational |
| 786 | * y the second rational |
| 787 | * |
| 788 | * Return value: 1 if x > y, otherwise 0 |
| 789 | * |
| 790 | * Side effects: none |
| 791 | */ |
| 792 | |
| 793 | int |
| 794 | gtrat(x, y) |
| 795 | |
| 796 | RATIONAL x, y; |
| 797 | |
| 798 | { |
| 799 | INT32B a[2]; /* temp holding areas for 64-bit numbers */ |
| 800 | INT32B b[2]; |
| 801 | |
| 802 | |
| 803 | /* if no overflow possible, cross-multiply and return truth value */ |
| 804 | /* note: this depends on positive denominators */ |
| 805 | if (SMALLRAT(x) && SMALLRAT(y)) |
| 806 | return(x.n * y.d > x.d * y.n); |
| 807 | |
| 808 | /* |
| 809 | * The numbers are too big; we have to do it the hard way to avoid |
| 810 | * overflow. Cross-multiply. Note: this depends on positive |
| 811 | * denominators. |
| 812 | */ |
| 813 | mul32_64(a, x.n, y.d); |
| 814 | mul32_64(b, x.d, y.n); |
| 815 | |
| 816 | return(GT64(a, b)); |
| 817 | } |
| 818 | \f |
| 819 | /* |
| 820 | * ratmsg() handle rational error of type "code" |
| 821 | * |
| 822 | * This function sets raterrno. Then calls the user's error handler, |
| 823 | * if there is one, or else prints a message to standard error. |
| 824 | * |
| 825 | * Parameters: code the error code |
| 826 | * |
| 827 | * Return value: None. |
| 828 | * |
| 829 | * Side effects: raterrno is set; then either a message is printed |
| 830 | * to standard error or the user's error handler is |
| 831 | * called. |
| 832 | */ |
| 833 | |
| 834 | static void |
| 835 | ratmsg(code) |
| 836 | |
| 837 | int code; |
| 838 | |
| 839 | { |
| 840 | raterrno = code; /* set global error flag */ |
| 841 | |
| 842 | if (raterrfuncp == 0) { |
| 843 | /* no user trap exists, so print message from here */ |
| 844 | switch (code) { |
| 845 | case RATOVER: |
| 846 | (void)fputs("rational overflow\n", stderr); |
| 847 | break; |
| 848 | |
| 849 | case RATDIV0: |
| 850 | (void)fputs("rational division by zero\n", stderr); |
| 851 | break; |
| 852 | |
| 853 | case RATPARM: |
| 854 | (void)fputs("invalid number passed to rational number routine\n", stderr); |
| 855 | break; |
| 856 | |
| 857 | default: |
| 858 | (void)fputs("error in rational routines\n", stderr); |
| 859 | break; |
| 860 | } |
| 861 | } else { |
| 862 | /* call user trap function to handle the error */ |
| 863 | (*raterrfuncp)(code); |
| 864 | } |
| 865 | } |
| 866 | \f |
| 867 | /* |
| 868 | * add64_64() add 64-bit numbers to get a 64-bit number |
| 869 | * |
| 870 | * This function adds two 64-bit signed numbers to get a 64-bit |
| 871 | * signed number. It is assumed that the result will not overflow. |
| 872 | * Any of the inputs may be the same arrays. |
| 873 | * |
| 874 | * Parameters: a answer goes here |
| 875 | * x the first input |
| 876 | * y the second input |
| 877 | * |
| 878 | * Return value: none |
| 879 | * |
| 880 | * Side effects: a is set to the result. |
| 881 | */ |
| 882 | |
| 883 | static void |
| 884 | add64_64(a, x, y) |
| 885 | |
| 886 | INT32B a[2]; |
| 887 | INT32B x[2]; |
| 888 | INT32B y[2]; |
| 889 | |
| 890 | { |
| 891 | INT32B t[2]; /* temp storage */ |
| 892 | |
| 893 | |
| 894 | /* first add low and high parts separately */ |
| 895 | /* use temp storage in case a[] is the same array as x[] or y[] */ |
| 896 | t[0] = x[0] + y[0]; |
| 897 | t[1] = x[1] + y[1]; |
| 898 | |
| 899 | /* figure out if the low part carries into the high part */ |
| 900 | if (x[0] < 0 && y[0] < 0) { /* both high order bits set */ |
| 901 | t[1]++; /* must be a carry */ |
| 902 | } else if (x[0] < 0 || y[0] < 0) { /* exactly one high bit set */ |
| 903 | if (t[0] >= 0) /* if result high bit clear */ |
| 904 | t[1]++; /* must be a carry */ |
| 905 | } |
| 906 | |
| 907 | a[0] = t[0]; /* copy results */ |
| 908 | a[1] = t[1]; |
| 909 | } |
| 910 | \f |
| 911 | /* |
| 912 | * mul32_64() multiply 32-bit numbers to get a 64-bit number |
| 913 | * |
| 914 | * This function multiplies two 32-bit signed numbers to get a 64-bit |
| 915 | * signed number. The numbers must not equal 0x80000000. Overflow |
| 916 | * cannot occur. |
| 917 | * |
| 918 | * Parameters: a answer goes here |
| 919 | * x the first 32-bit number |
| 920 | * y the second 32-bit number |
| 921 | * |
| 922 | * Return value: none |
| 923 | * |
| 924 | * Side effects: a is set to the result. |
| 925 | */ |
| 926 | |
| 927 | static void |
| 928 | mul32_64(a, x, y) |
| 929 | |
| 930 | INT32B a[2]; |
| 931 | INT32B x; |
| 932 | INT32B y; |
| 933 | |
| 934 | { |
| 935 | INT32B t[2]; /* temp storage for inner terms */ |
| 936 | INT32B xl, xh; /* low and high 16 bits of x */ |
| 937 | INT32B yl, yh; /* low and high 16 bits of y */ |
| 938 | int sign; /* sign of the result */ |
| 939 | |
| 940 | |
| 941 | /* make both numbers positive and determine the sign of the result */ |
| 942 | sign = 1; /* start at positive */ |
| 943 | if (x < 0) { |
| 944 | x = -x; |
| 945 | sign = -sign; |
| 946 | } |
| 947 | if (y < 0) { |
| 948 | y = -y; |
| 949 | sign = -sign; |
| 950 | } |
| 951 | |
| 952 | /* break x and y into high and low pieces */ |
| 953 | xl = x & 0xffff; /* 0 <= xl <= 0xffff */ |
| 954 | xh = x >> 16; /* 0 <= xh <= 0x7fff */ |
| 955 | yl = y & 0xffff; /* 0 <= yl <= 0xffff */ |
| 956 | yh = y >> 16; /* 0 <= yh <= 0x7fff */ |
| 957 | |
| 958 | /* multiply the outer parts */ |
| 959 | a[0] = xl * yl; /* 0 <= a[0] <= 0xfffe0001 */ |
| 960 | a[1] = xh * yh; /* 0 <= a[1] <= 0x3fff0001 */ |
| 961 | |
| 962 | /* multiply the inner parts and break the result in two pieces */ |
| 963 | t[0] = xl * yh + xh * yl; /* 0 <= t[0] <= 0xfffd0002 */ |
| 964 | t[1] = (UINT32B)t[0] >> 16; /* 0 <= t[1] <= 0x0000fffd */ |
| 965 | t[0] <<= 16; /* 0 <= t[0] <= 0xffff0000 */ |
| 966 | |
| 967 | /* add the two partial products */ |
| 968 | add64_64(a, a, t); /* 0 <= a <= 0x3fffffff00000001 */ |
| 969 | |
| 970 | /* if the answer is supposed to be negative, negate it */ |
| 971 | if (sign < 0) |
| 972 | NEG64(a); |
| 973 | } |
| 974 | \f |
| 975 | /* |
| 976 | * divmod64() find quotient and remainder of two 64-bit numbers |
| 977 | * |
| 978 | * This function takes two 64-bit numbers and divides the first by the |
| 979 | * second, to get a quotient and remainder, both 64 bits. It is assumed |
| 980 | * that the first number is nonnegative and the second number is positive. |
| 981 | * q and r must be different arrays. |
| 982 | * |
| 983 | * Parameters: x first number (dividend) |
| 984 | * y second number (divisor) |
| 985 | * q quotient |
| 986 | * r remainder |
| 987 | * |
| 988 | * Return value: none |
| 989 | * |
| 990 | * Side effects: q and r are altered to be the results |
| 991 | * x and y are not altered |
| 992 | */ |
| 993 | |
| 994 | static void |
| 995 | divmod64(x, y, q, r) |
| 996 | |
| 997 | INT32B x[2]; |
| 998 | INT32B y[2]; |
| 999 | INT32B q[2]; |
| 1000 | INT32B r[2]; |
| 1001 | |
| 1002 | { |
| 1003 | INT32B s[2]; /* temp storage for divisor */ |
| 1004 | INT32B t[2]; /* temp storage for scratch */ |
| 1005 | register int shift; /* how far has y been shifted? */ |
| 1006 | |
| 1007 | |
| 1008 | r[0] = x[0]; /* copy dividend to remainder place */ |
| 1009 | r[1] = x[1]; |
| 1010 | s[0] = y[0]; /* copy divisor to temp storage */ |
| 1011 | s[1] = y[1]; |
| 1012 | |
| 1013 | /* shift divisor left until greater than dividend */ |
| 1014 | /* compare as unsigned so no problem if it gets shifted into sign bit */ |
| 1015 | for (shift = 0; LEU64(s, r); shift++) |
| 1016 | SHL1U64(s); |
| 1017 | |
| 1018 | SHR1U64(s); /* shift it back right one, so <= dividend */ |
| 1019 | shift--; |
| 1020 | |
| 1021 | q[0] = 0; /* start quotient at 0 */ |
| 1022 | q[1] = 0; |
| 1023 | |
| 1024 | /* |
| 1025 | * Loop once for each bit shifted. |
| 1026 | */ |
| 1027 | for ( ; shift >= 0; shift--) { |
| 1028 | /* |
| 1029 | * If the current divisor does not exceed what's left of the |
| 1030 | * dividend, subtract it from it, and record that by setting |
| 1031 | * the low order bit of the current quotient. |
| 1032 | */ |
| 1033 | if ( ! GT64(s, r) ) { |
| 1034 | t[0] = s[0]; |
| 1035 | t[1] = s[1]; |
| 1036 | NEG64(t); |
| 1037 | add64_64(r, r, t); |
| 1038 | q[0] |= 1; |
| 1039 | } |
| 1040 | |
| 1041 | /* shift quotient left and divisor right */ |
| 1042 | SHL1U64(q); |
| 1043 | SHR1U64(s); |
| 1044 | } |
| 1045 | |
| 1046 | SHR1U64(q); /* shift quotient right */ |
| 1047 | } |
| 1048 | \f |
| 1049 | /* |
| 1050 | * red64_64() reduce a 64 bit over 64 bit rational to lowest terms |
| 1051 | * |
| 1052 | * This function takes two 64-bit numbers as numerator and denominator |
| 1053 | * of a rational number, and reduces them to lowest terms, with the |
| 1054 | * denominator positive. If the user called this package correctly, the |
| 1055 | * denominator cannot be zero, but to be defensive we check for that, and |
| 1056 | * if it happens, set raterrno to RATPARM and either print a message or |
| 1057 | * call a user-supplied error handler. |
| 1058 | * |
| 1059 | * Parameters: num numerator |
| 1060 | * den denominator |
| 1061 | * |
| 1062 | * Return value: none |
| 1063 | * |
| 1064 | * Side effects: num and den are altered to be the result |
| 1065 | */ |
| 1066 | |
| 1067 | static void |
| 1068 | red64_64(num, den) |
| 1069 | |
| 1070 | INT32B num[2]; |
| 1071 | INT32B den[2]; |
| 1072 | |
| 1073 | { |
| 1074 | INT32B b[2], c[2], r[2]; /* temp variables for Euclidean algorithm */ |
| 1075 | INT32B junk[2]; /* placeholder for calling divmod64 */ |
| 1076 | int sign; /* answer is pos (1) or neg (-1) */ |
| 1077 | |
| 1078 | |
| 1079 | if (den[1] == 0 && den[0] == 0) { /* if den == 0 */ |
| 1080 | /* |
| 1081 | * This is an error. The user must have called a routine with |
| 1082 | * an invalid number for us to get here, in fact a number with |
| 1083 | * a zero denominator, since "den" here always was created as |
| 1084 | * the product of denominators. Report the error and return |
| 1085 | * zero. |
| 1086 | */ |
| 1087 | num[1] = 0; /* set num = 0 */ |
| 1088 | num[0] = 0; |
| 1089 | den[1] = 0; /* set den = 1 */ |
| 1090 | den[0] = 1; |
| 1091 | ratmsg(RATPARM); /* set raterrno, report error */ |
| 1092 | return; |
| 1093 | } |
| 1094 | |
| 1095 | if (num[1] == 0 && num[0] == 0) { /* if num == 0 */ |
| 1096 | den[1] = 0; /* set den = 1; answer is 0/1 */ |
| 1097 | den[0] = 1; |
| 1098 | return; |
| 1099 | } |
| 1100 | |
| 1101 | /* now figure out sign of answer, and make num & den positive */ |
| 1102 | sign = 1; /* init to positive */ |
| 1103 | if (num[1] < 0) { /* if numerator neg */ |
| 1104 | sign = -sign; /* reverse the sign */ |
| 1105 | NEG64(num); |
| 1106 | } |
| 1107 | if (den[1] < 0) { /* if denominator neg */ |
| 1108 | sign = -sign; /* reverse the sign */ |
| 1109 | NEG64(den); |
| 1110 | } |
| 1111 | |
| 1112 | /* now check whether numerator or denominator is larger */ |
| 1113 | if (EQ64(num, den)) { |
| 1114 | num[0] = sign; /* answer is +1 or -1 */ |
| 1115 | |
| 1116 | if (sign < 0) /* set high order word to sign bit */ |
| 1117 | num[1] = -1; |
| 1118 | else |
| 1119 | num[1] = 0; |
| 1120 | |
| 1121 | den[1] = 0; /* set den to 1 */ |
| 1122 | den[0] = 1; |
| 1123 | |
| 1124 | return; |
| 1125 | } |
| 1126 | |
| 1127 | /* set up c and b so that one is num, the other den, and c > b */ |
| 1128 | if (LT64(num, den)) { /* if num < den */ |
| 1129 | c[0] = den[0]; /* c = den */ |
| 1130 | c[1] = den[1]; |
| 1131 | b[0] = num[0]; /* b = num */ |
| 1132 | b[1] = num[1]; |
| 1133 | } else { |
| 1134 | c[0] = num[0]; /* c = num */ |
| 1135 | c[1] = num[1]; |
| 1136 | b[0] = den[0]; /* b = den */ |
| 1137 | b[1] = den[1]; |
| 1138 | } |
| 1139 | |
| 1140 | /* use Euclidean Algorithm to find greatest common divisor of c & b */ |
| 1141 | do { |
| 1142 | divmod64(c, b, junk, r); /* r = c % b */ |
| 1143 | c[0] = b[0]; /* c = b */ |
| 1144 | c[1] = b[1]; |
| 1145 | b[0] = r[0]; /* b = r */ |
| 1146 | b[1] = r[1]; |
| 1147 | } while (r[0] != 0 || r[1] != 0); /* while r != 0 */ |
| 1148 | |
| 1149 | /* now c is the greatest common divisor of num and den */ |
| 1150 | |
| 1151 | /* divide out the greatest common divisor and put the sign in */ |
| 1152 | divmod64(num, c, num, junk); /* num /= c */ |
| 1153 | if (sign < 0) /* if should be negative */ |
| 1154 | NEG64(num); /* negate the numerator */ |
| 1155 | divmod64(den, c, den, junk); /* den /= c */ |
| 1156 | |
| 1157 | return; |
| 1158 | } |