1 /* mpn_get_str -- Convert {UP,USIZE} to a base BASE string in STR. 2 3 Contributed to the GNU project by Torbjorn Granlund. 4 5 THE FUNCTIONS IN THIS FILE, EXCEPT mpn_get_str, ARE INTERNAL WITH A MUTABLE 6 INTERFACE. IT IS ONLY SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES. IN 7 FACT, IT IS ALMOST GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A FUTURE 8 GNU MP RELEASE. 9 10 Copyright 1991, 1992, 1993, 1994, 1996, 2000, 2001, 2002, 2004, 2006, 2007, 11 2008 Free Software Foundation, Inc. 12 13 This file is part of the GNU MP Library. 14 15 The GNU MP Library is free software; you can redistribute it and/or modify 16 it under the terms of the GNU Lesser General Public License as published by 17 the Free Software Foundation; either version 3 of the License, or (at your 18 option) any later version. 19 20 The GNU MP Library is distributed in the hope that it will be useful, but 21 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 22 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 23 License for more details. 24 25 You should have received a copy of the GNU Lesser General Public License 26 along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */ 27 28 #include "gmp.h" 29 #include "gmp-impl.h" 30 #include "longlong.h" 31 32 /* Conversion of U {up,un} to a string in base b. Internally, we convert to 33 base B = b^m, the largest power of b that fits a limb. Basic algorithms: 34 35 A) Divide U repeatedly by B, generating a quotient and remainder, until the 36 quotient becomes zero. The remainders hold the converted digits. Digits 37 come out from right to left. (Used in mpn_sb_get_str.) 38 39 B) Divide U by b^g, for g such that 1/b <= U/b^g < 1, generating a fraction. 40 Then develop digits by multiplying the fraction repeatedly by b. Digits 41 come out from left to right. (Currently not used herein, except for in 42 code for converting single limbs to individual digits.) 43 44 C) Compute B^1, B^2, B^4, ..., B^s, for s such that B^s is just above 45 sqrt(U). Then divide U by B^s, generating quotient and remainder. 46 Recursively convert the quotient, then the remainder, using the 47 precomputed powers. Digits come out from left to right. (Used in 48 mpn_dc_get_str.) 49 50 When using algorithm C, algorithm B might be suitable for basecase code, 51 since the required b^g power will be readily accessible. 52 53 Optimization ideas: 54 1. The recursive function of (C) could use less temporary memory. The powtab 55 allocation could be trimmed with some computation, and the tmp area could 56 be reduced, or perhaps eliminated if up is reused for both quotient and 57 remainder (it is currently used just for remainder). 58 2. Store the powers of (C) in normalized form, with the normalization count. 59 Quotients will usually need to be left-shifted before each divide, and 60 remainders will either need to be left-shifted of right-shifted. 61 3. In the code for developing digits from a single limb, we could avoid using 62 a full umul_ppmm except for the first (or first few) digits, provided base 63 is even. Subsequent digits can be developed using plain multiplication. 64 (This saves on register-starved machines (read x86) and on all machines 65 that generate the upper product half using a separate instruction (alpha, 66 powerpc, IA-64) or lacks such support altogether (sparc64, hppa64). 67 4. Separate mpn_dc_get_str basecase code from code for small conversions. The 68 former code will have the exact right power readily available in the 69 powtab parameter for dividing the current number into a fraction. Convert 70 that using algorithm B. 71 5. Completely avoid division. Compute the inverses of the powers now in 72 powtab instead of the actual powers. 73 6. Decrease powtab allocation for even bases. E.g. for base 10 we could save 74 about 30% (1-log(5)/log(10)). 75 76 Basic structure of (C): 77 mpn_get_str: 78 if POW2_P (n) 79 ... 80 else 81 if (un < GET_STR_PRECOMPUTE_THRESHOLD) 82 mpn_sb_get_str (str, base, up, un); 83 else 84 precompute_power_tables 85 mpn_dc_get_str 86 87 mpn_dc_get_str: 88 mpn_tdiv_qr 89 if (qn < GET_STR_DC_THRESHOLD) 90 mpn_sb_get_str 91 else 92 mpn_dc_get_str 93 if (rn < GET_STR_DC_THRESHOLD) 94 mpn_sb_get_str 95 else 96 mpn_dc_get_str 97 98 99 The reason for the two threshold values is the cost of 100 precompute_power_tables. GET_STR_PRECOMPUTE_THRESHOLD will be considerably 101 larger than GET_STR_PRECOMPUTE_THRESHOLD. */ 102 103 104 /* The x86s and m68020 have a quotient and remainder "div" instruction and 105 gcc recognises an adjacent "/" and "%" can be combined using that. 106 Elsewhere "/" and "%" are either separate instructions, or separate 107 libgcc calls (which unfortunately gcc as of version 3.0 doesn't combine). 108 A multiply and subtract should be faster than a "%" in those cases. */ 109 #if HAVE_HOST_CPU_FAMILY_x86 \ 110 || HAVE_HOST_CPU_m68020 \ 111 || HAVE_HOST_CPU_m68030 \ 112 || HAVE_HOST_CPU_m68040 \ 113 || HAVE_HOST_CPU_m68060 \ 114 || HAVE_HOST_CPU_m68360 /* CPU32 */ 115 #define udiv_qrnd_unnorm(q,r,n,d) \ 116 do { \ 117 mp_limb_t __q = (n) / (d); \ 118 mp_limb_t __r = (n) % (d); \ 119 (q) = __q; \ 120 (r) = __r; \ 121 } while (0) 122 #else 123 #define udiv_qrnd_unnorm(q,r,n,d) \ 124 do { \ 125 mp_limb_t __q = (n) / (d); \ 126 mp_limb_t __r = (n) - __q*(d); \ 127 (q) = __q; \ 128 (r) = __r; \ 129 } while (0) 130 #endif 131 132 133 /* Convert {up,un} to a string in base base, and put the result in str. 134 Generate len characters, possibly padding with zeros to the left. If len is 135 zero, generate as many characters as required. Return a pointer immediately 136 after the last digit of the result string. Complexity is O(un^2); intended 137 for small conversions. */ 138 static unsigned char * 139 mpn_sb_get_str (unsigned char *str, size_t len, 140 mp_ptr up, mp_size_t un, int base) 141 { 142 mp_limb_t rl, ul; 143 unsigned char *s; 144 size_t l; 145 /* Allocate memory for largest possible string, given that we only get here 146 for operands with un < GET_STR_PRECOMPUTE_THRESHOLD and that the smallest 147 base is 3. 7/11 is an approximation to 1/log2(3). */ 148 #if TUNE_PROGRAM_BUILD 149 #define BUF_ALLOC (GET_STR_THRESHOLD_LIMIT * GMP_LIMB_BITS * 7 / 11) 150 #else 151 #define BUF_ALLOC (GET_STR_PRECOMPUTE_THRESHOLD * GMP_LIMB_BITS * 7 / 11) 152 #endif 153 unsigned char buf[BUF_ALLOC]; 154 #if TUNE_PROGRAM_BUILD 155 mp_limb_t rp[GET_STR_THRESHOLD_LIMIT]; 156 #else 157 mp_limb_t rp[GET_STR_PRECOMPUTE_THRESHOLD]; 158 #endif 159 160 if (base == 10) 161 { 162 /* Special case code for base==10 so that the compiler has a chance to 163 optimize things. */ 164 165 MPN_COPY (rp + 1, up, un); 166 167 s = buf + BUF_ALLOC; 168 while (un > 1) 169 { 170 int i; 171 mp_limb_t frac, digit; 172 MPN_DIVREM_OR_PREINV_DIVREM_1 (rp, (mp_size_t) 1, rp + 1, un, 173 MP_BASES_BIG_BASE_10, 174 MP_BASES_BIG_BASE_INVERTED_10, 175 MP_BASES_NORMALIZATION_STEPS_10); 176 un -= rp[un] == 0; 177 frac = (rp[0] + 1) << GMP_NAIL_BITS; 178 s -= MP_BASES_CHARS_PER_LIMB_10; 179 #if HAVE_HOST_CPU_FAMILY_x86 180 /* The code below turns out to be a bit slower for x86 using gcc. 181 Use plain code. */ 182 i = MP_BASES_CHARS_PER_LIMB_10; 183 do 184 { 185 umul_ppmm (digit, frac, frac, 10); 186 *s++ = digit; 187 } 188 while (--i); 189 #else 190 /* Use the fact that 10 in binary is 1010, with the lowest bit 0. 191 After a few umul_ppmm, we will have accumulated enough low zeros 192 to use a plain multiply. */ 193 if (MP_BASES_NORMALIZATION_STEPS_10 == 0) 194 { 195 umul_ppmm (digit, frac, frac, 10); 196 *s++ = digit; 197 } 198 if (MP_BASES_NORMALIZATION_STEPS_10 <= 1) 199 { 200 umul_ppmm (digit, frac, frac, 10); 201 *s++ = digit; 202 } 203 if (MP_BASES_NORMALIZATION_STEPS_10 <= 2) 204 { 205 umul_ppmm (digit, frac, frac, 10); 206 *s++ = digit; 207 } 208 if (MP_BASES_NORMALIZATION_STEPS_10 <= 3) 209 { 210 umul_ppmm (digit, frac, frac, 10); 211 *s++ = digit; 212 } 213 i = (MP_BASES_CHARS_PER_LIMB_10 - ((MP_BASES_NORMALIZATION_STEPS_10 < 4) 214 ? (4-MP_BASES_NORMALIZATION_STEPS_10) 215 : 0)); 216 frac = (frac + 0xf) >> 4; 217 do 218 { 219 frac *= 10; 220 digit = frac >> (GMP_LIMB_BITS - 4); 221 *s++ = digit; 222 frac &= (~(mp_limb_t) 0) >> 4; 223 } 224 while (--i); 225 #endif 226 s -= MP_BASES_CHARS_PER_LIMB_10; 227 } 228 229 ul = rp[1]; 230 while (ul != 0) 231 { 232 udiv_qrnd_unnorm (ul, rl, ul, 10); 233 *--s = rl; 234 } 235 } 236 else /* not base 10 */ 237 { 238 unsigned chars_per_limb; 239 mp_limb_t big_base, big_base_inverted; 240 unsigned normalization_steps; 241 242 chars_per_limb = mp_bases[base].chars_per_limb; 243 big_base = mp_bases[base].big_base; 244 big_base_inverted = mp_bases[base].big_base_inverted; 245 count_leading_zeros (normalization_steps, big_base); 246 247 MPN_COPY (rp + 1, up, un); 248 249 s = buf + BUF_ALLOC; 250 while (un > 1) 251 { 252 int i; 253 mp_limb_t frac; 254 MPN_DIVREM_OR_PREINV_DIVREM_1 (rp, (mp_size_t) 1, rp + 1, un, 255 big_base, big_base_inverted, 256 normalization_steps); 257 un -= rp[un] == 0; 258 frac = (rp[0] + 1) << GMP_NAIL_BITS; 259 s -= chars_per_limb; 260 i = chars_per_limb; 261 do 262 { 263 mp_limb_t digit; 264 umul_ppmm (digit, frac, frac, base); 265 *s++ = digit; 266 } 267 while (--i); 268 s -= chars_per_limb; 269 } 270 271 ul = rp[1]; 272 while (ul != 0) 273 { 274 udiv_qrnd_unnorm (ul, rl, ul, base); 275 *--s = rl; 276 } 277 } 278 279 l = buf + BUF_ALLOC - s; 280 while (l < len) 281 { 282 *str++ = 0; 283 len--; 284 } 285 while (l != 0) 286 { 287 *str++ = *s++; 288 l--; 289 } 290 return str; 291 } 292 293 294 /* Convert {UP,UN} to a string with a base as represented in POWTAB, and put 295 the string in STR. Generate LEN characters, possibly padding with zeros to 296 the left. If LEN is zero, generate as many characters as required. 297 Return a pointer immediately after the last digit of the result string. 298 This uses divide-and-conquer and is intended for large conversions. */ 299 static unsigned char * 300 mpn_dc_get_str (unsigned char *str, size_t len, 301 mp_ptr up, mp_size_t un, 302 const powers_t *powtab, mp_ptr tmp) 303 { 304 if (BELOW_THRESHOLD (un, GET_STR_DC_THRESHOLD)) 305 { 306 if (un != 0) 307 str = mpn_sb_get_str (str, len, up, un, powtab->base); 308 else 309 { 310 while (len != 0) 311 { 312 *str++ = 0; 313 len--; 314 } 315 } 316 } 317 else 318 { 319 mp_ptr pwp, qp, rp; 320 mp_size_t pwn, qn; 321 mp_size_t sn; 322 323 pwp = powtab->p; 324 pwn = powtab->n; 325 sn = powtab->shift; 326 327 if (un < pwn + sn || (un == pwn + sn && mpn_cmp (up + sn, pwp, un - sn) < 0)) 328 { 329 str = mpn_dc_get_str (str, len, up, un, powtab - 1, tmp); 330 } 331 else 332 { 333 qp = tmp; /* (un - pwn + 1) limbs for qp */ 334 rp = up; /* pwn limbs for rp; overwrite up area */ 335 336 mpn_tdiv_qr (qp, rp + sn, 0L, up + sn, un - sn, pwp, pwn); 337 qn = un - sn - pwn; qn += qp[qn] != 0; /* quotient size */ 338 339 ASSERT (qn < pwn + sn || (qn == pwn + sn && mpn_cmp (qp + sn, pwp, pwn) < 0)); 340 341 if (len != 0) 342 len = len - powtab->digits_in_base; 343 344 str = mpn_dc_get_str (str, len, qp, qn, powtab - 1, tmp + qn); 345 str = mpn_dc_get_str (str, powtab->digits_in_base, rp, pwn + sn, powtab - 1, tmp); 346 } 347 } 348 return str; 349 } 350 351 352 /* There are no leading zeros on the digits generated at str, but that's not 353 currently a documented feature. */ 354 355 size_t 356 mpn_get_str (unsigned char *str, int base, mp_ptr up, mp_size_t un) 357 { 358 mp_ptr powtab_mem, powtab_mem_ptr; 359 mp_limb_t big_base; 360 size_t digits_in_base; 361 powers_t powtab[GMP_LIMB_BITS]; 362 int pi; 363 mp_size_t n; 364 mp_ptr p, t; 365 size_t out_len; 366 mp_ptr tmp; 367 TMP_DECL; 368 369 /* Special case zero, as the code below doesn't handle it. */ 370 if (un == 0) 371 { 372 str[0] = 0; 373 return 1; 374 } 375 376 if (POW2_P (base)) 377 { 378 /* The base is a power of 2. Convert from most significant end. */ 379 mp_limb_t n1, n0; 380 int bits_per_digit = mp_bases[base].big_base; 381 int cnt; 382 int bit_pos; 383 mp_size_t i; 384 unsigned char *s = str; 385 mp_bitcnt_t bits; 386 387 n1 = up[un - 1]; 388 count_leading_zeros (cnt, n1); 389 390 /* BIT_POS should be R when input ends in least significant nibble, 391 R + bits_per_digit * n when input ends in nth least significant 392 nibble. */ 393 394 bits = (mp_bitcnt_t) GMP_NUMB_BITS * un - cnt + GMP_NAIL_BITS; 395 cnt = bits % bits_per_digit; 396 if (cnt != 0) 397 bits += bits_per_digit - cnt; 398 bit_pos = bits - (mp_bitcnt_t) (un - 1) * GMP_NUMB_BITS; 399 400 /* Fast loop for bit output. */ 401 i = un - 1; 402 for (;;) 403 { 404 bit_pos -= bits_per_digit; 405 while (bit_pos >= 0) 406 { 407 *s++ = (n1 >> bit_pos) & ((1 << bits_per_digit) - 1); 408 bit_pos -= bits_per_digit; 409 } 410 i--; 411 if (i < 0) 412 break; 413 n0 = (n1 << -bit_pos) & ((1 << bits_per_digit) - 1); 414 n1 = up[i]; 415 bit_pos += GMP_NUMB_BITS; 416 *s++ = n0 | (n1 >> bit_pos); 417 } 418 419 return s - str; 420 } 421 422 /* General case. The base is not a power of 2. */ 423 424 if (BELOW_THRESHOLD (un, GET_STR_PRECOMPUTE_THRESHOLD)) 425 return mpn_sb_get_str (str, (size_t) 0, up, un, base) - str; 426 427 TMP_MARK; 428 429 /* Allocate one large block for the powers of big_base. */ 430 powtab_mem = TMP_BALLOC_LIMBS (mpn_dc_get_str_powtab_alloc (un)); 431 powtab_mem_ptr = powtab_mem; 432 433 /* Compute a table of powers, were the largest power is >= sqrt(U). */ 434 435 big_base = mp_bases[base].big_base; 436 digits_in_base = mp_bases[base].chars_per_limb; 437 438 { 439 mp_size_t n_pows, xn, pn, exptab[GMP_LIMB_BITS], bexp; 440 mp_limb_t cy; 441 mp_size_t shift; 442 443 n_pows = 0; 444 xn = 1 + un*(mp_bases[base].chars_per_bit_exactly*GMP_NUMB_BITS)/mp_bases[base].chars_per_limb; 445 for (pn = xn; pn != 1; pn = (pn + 1) >> 1) 446 { 447 exptab[n_pows] = pn; 448 n_pows++; 449 } 450 exptab[n_pows] = 1; 451 452 powtab[0].p = &big_base; 453 powtab[0].n = 1; 454 powtab[0].digits_in_base = digits_in_base; 455 powtab[0].base = base; 456 powtab[0].shift = 0; 457 458 powtab[1].p = powtab_mem_ptr; powtab_mem_ptr += 2; 459 powtab[1].p[0] = big_base; 460 powtab[1].n = 1; 461 powtab[1].digits_in_base = digits_in_base; 462 powtab[1].base = base; 463 powtab[1].shift = 0; 464 465 n = 1; 466 p = &big_base; 467 bexp = 1; 468 shift = 0; 469 for (pi = 2; pi < n_pows; pi++) 470 { 471 t = powtab_mem_ptr; 472 powtab_mem_ptr += 2 * n + 2; 473 474 ASSERT_ALWAYS (powtab_mem_ptr < powtab_mem + mpn_dc_get_str_powtab_alloc (un)); 475 476 mpn_sqr (t, p, n); 477 478 digits_in_base *= 2; 479 n *= 2; n -= t[n - 1] == 0; 480 bexp *= 2; 481 482 if (bexp + 1 < exptab[n_pows - pi]) 483 { 484 digits_in_base += mp_bases[base].chars_per_limb; 485 cy = mpn_mul_1 (t, t, n, big_base); 486 t[n] = cy; 487 n += cy != 0; 488 bexp += 1; 489 } 490 shift *= 2; 491 /* Strip low zero limbs. */ 492 while (t[0] == 0) 493 { 494 t++; 495 n--; 496 shift++; 497 } 498 p = t; 499 powtab[pi].p = p; 500 powtab[pi].n = n; 501 powtab[pi].digits_in_base = digits_in_base; 502 powtab[pi].base = base; 503 powtab[pi].shift = shift; 504 } 505 506 for (pi = 1; pi < n_pows; pi++) 507 { 508 t = powtab[pi].p; 509 n = powtab[pi].n; 510 cy = mpn_mul_1 (t, t, n, big_base); 511 t[n] = cy; 512 n += cy != 0; 513 if (t[0] == 0) 514 { 515 powtab[pi].p = t + 1; 516 n--; 517 powtab[pi].shift++; 518 } 519 powtab[pi].n = n; 520 powtab[pi].digits_in_base += mp_bases[base].chars_per_limb; 521 } 522 523 #if 0 524 { int i; 525 printf ("Computed table values for base=%d, un=%d, xn=%d:\n", base, un, xn); 526 for (i = 0; i < n_pows; i++) 527 printf ("%2d: %10ld %10ld %11ld %ld\n", i, exptab[n_pows-i], powtab[i].n, powtab[i].digits_in_base, powtab[i].shift); 528 } 529 #endif 530 } 531 532 /* Using our precomputed powers, now in powtab[], convert our number. */ 533 tmp = TMP_BALLOC_LIMBS (mpn_dc_get_str_itch (un)); 534 out_len = mpn_dc_get_str (str, 0, up, un, powtab - 1 + pi, tmp) - str; 535 TMP_FREE; 536 537 return out_len; 538 } 539