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 MUTABLE 6 INTERFACES. IT IS ONLY SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES. 7 IN FACT, IT IS ALMOST GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A 8 FUTURE GNU MP RELEASE. 9 10 Copyright 1991-2017 Free Software Foundation, Inc. 11 12 This file is part of the GNU MP Library. 13 14 The GNU MP Library is free software; you can redistribute it and/or modify 15 it under the terms of either: 16 17 * the GNU Lesser General Public License as published by the Free 18 Software Foundation; either version 3 of the License, or (at your 19 option) any later version. 20 21 or 22 23 * the GNU General Public License as published by the Free Software 24 Foundation; either version 2 of the License, or (at your option) any 25 later version. 26 27 or both in parallel, as here. 28 29 The GNU MP Library is distributed in the hope that it will be useful, but 30 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 31 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 32 for more details. 33 34 You should have received copies of the GNU General Public License and the 35 GNU Lesser General Public License along with the GNU MP Library. If not, 36 see https://www.gnu.org/licenses/. */ 37 38 #include "gmp-impl.h" 39 #include "longlong.h" 40 41 /* Conversion of U {up,un} to a string in base b. Internally, we convert to 42 base B = b^m, the largest power of b that fits a limb. Basic algorithms: 43 44 A) Divide U repeatedly by B, generating a quotient and remainder, until the 45 quotient becomes zero. The remainders hold the converted digits. Digits 46 come out from right to left. (Used in mpn_bc_get_str.) 47 48 B) Divide U by b^g, for g such that 1/b <= U/b^g < 1, generating a fraction. 49 Then develop digits by multiplying the fraction repeatedly by b. Digits 50 come out from left to right. (Currently not used herein, except for in 51 code for converting single limbs to individual digits.) 52 53 C) Compute B^1, B^2, B^4, ..., B^s, for s such that B^s is just above 54 sqrt(U). Then divide U by B^s, generating quotient and remainder. 55 Recursively convert the quotient, then the remainder, using the 56 precomputed powers. Digits come out from left to right. (Used in 57 mpn_dc_get_str.) 58 59 When using algorithm C, algorithm B might be suitable for basecase code, 60 since the required b^g power will be readily accessible. 61 62 Optimization ideas: 63 1. The recursive function of (C) could use less temporary memory. The powtab 64 allocation could be trimmed with some computation, and the tmp area could 65 be reduced, or perhaps eliminated if up is reused for both quotient and 66 remainder (it is currently used just for remainder). 67 2. Store the powers of (C) in normalized form, with the normalization count. 68 Quotients will usually need to be left-shifted before each divide, and 69 remainders will either need to be left-shifted of right-shifted. 70 3. In the code for developing digits from a single limb, we could avoid using 71 a full umul_ppmm except for the first (or first few) digits, provided base 72 is even. Subsequent digits can be developed using plain multiplication. 73 (This saves on register-starved machines (read x86) and on all machines 74 that generate the upper product half using a separate instruction (alpha, 75 powerpc, IA-64) or lacks such support altogether (sparc64, hppa64). 76 4. Separate mpn_dc_get_str basecase code from code for small conversions. The 77 former code will have the exact right power readily available in the 78 powtab parameter for dividing the current number into a fraction. Convert 79 that using algorithm B. 80 5. Completely avoid division. Compute the inverses of the powers now in 81 powtab instead of the actual powers. 82 6. Decrease powtab allocation for even bases. E.g. for base 10 we could save 83 about 30% (1-log(5)/log(10)). 84 85 Basic structure of (C): 86 mpn_get_str: 87 if POW2_P (n) 88 ... 89 else 90 if (un < GET_STR_PRECOMPUTE_THRESHOLD) 91 mpn_bx_get_str (str, base, up, un); 92 else 93 precompute_power_tables 94 mpn_dc_get_str 95 96 mpn_dc_get_str: 97 mpn_tdiv_qr 98 if (qn < GET_STR_DC_THRESHOLD) 99 mpn_bc_get_str 100 else 101 mpn_dc_get_str 102 if (rn < GET_STR_DC_THRESHOLD) 103 mpn_bc_get_str 104 else 105 mpn_dc_get_str 106 107 108 The reason for the two threshold values is the cost of 109 precompute_power_tables. GET_STR_PRECOMPUTE_THRESHOLD will be 110 considerably larger than GET_STR_DC_THRESHOLD. */ 111 112 113 /* The x86s and m68020 have a quotient and remainder "div" instruction and 114 gcc recognises an adjacent "/" and "%" can be combined using that. 115 Elsewhere "/" and "%" are either separate instructions, or separate 116 libgcc calls (which unfortunately gcc as of version 3.0 doesn't combine). 117 A multiply and subtract should be faster than a "%" in those cases. */ 118 #if HAVE_HOST_CPU_FAMILY_x86 \ 119 || HAVE_HOST_CPU_m68020 \ 120 || HAVE_HOST_CPU_m68030 \ 121 || HAVE_HOST_CPU_m68040 \ 122 || HAVE_HOST_CPU_m68060 \ 123 || HAVE_HOST_CPU_m68360 /* CPU32 */ 124 #define udiv_qrnd_unnorm(q,r,n,d) \ 125 do { \ 126 mp_limb_t __q = (n) / (d); \ 127 mp_limb_t __r = (n) % (d); \ 128 (q) = __q; \ 129 (r) = __r; \ 130 } while (0) 131 #else 132 #define udiv_qrnd_unnorm(q,r,n,d) \ 133 do { \ 134 mp_limb_t __q = (n) / (d); \ 135 mp_limb_t __r = (n) - __q*(d); \ 136 (q) = __q; \ 137 (r) = __r; \ 138 } while (0) 139 #endif 140 141 142 /* Convert {up,un} to a string in base base, and put the result in str. 143 Generate len characters, possibly padding with zeros to the left. If len is 144 zero, generate as many characters as required. Return a pointer immediately 145 after the last digit of the result string. Complexity is O(un^2); intended 146 for small conversions. */ 147 static unsigned char * 148 mpn_bc_get_str (unsigned char *str, size_t len, 149 mp_ptr up, mp_size_t un, int base) 150 { 151 mp_limb_t rl, ul; 152 unsigned char *s; 153 size_t l; 154 /* Allocate memory for largest possible string, given that we only get here 155 for operands with un < GET_STR_PRECOMPUTE_THRESHOLD and that the smallest 156 base is 3. 7/11 is an approximation to 1/log2(3). */ 157 #if TUNE_PROGRAM_BUILD 158 #define BUF_ALLOC (GET_STR_THRESHOLD_LIMIT * GMP_LIMB_BITS * 7 / 11) 159 #else 160 #define BUF_ALLOC (GET_STR_PRECOMPUTE_THRESHOLD * GMP_LIMB_BITS * 7 / 11) 161 #endif 162 unsigned char buf[BUF_ALLOC]; 163 #if TUNE_PROGRAM_BUILD 164 mp_limb_t rp[GET_STR_THRESHOLD_LIMIT]; 165 #else 166 mp_limb_t rp[GET_STR_PRECOMPUTE_THRESHOLD]; 167 #endif 168 169 if (base == 10) 170 { 171 /* Special case code for base==10 so that the compiler has a chance to 172 optimize things. */ 173 174 MPN_COPY (rp + 1, up, un); 175 176 s = buf + BUF_ALLOC; 177 while (un > 1) 178 { 179 int i; 180 mp_limb_t frac, digit; 181 MPN_DIVREM_OR_PREINV_DIVREM_1 (rp, (mp_size_t) 1, rp + 1, un, 182 MP_BASES_BIG_BASE_10, 183 MP_BASES_BIG_BASE_INVERTED_10, 184 MP_BASES_NORMALIZATION_STEPS_10); 185 un -= rp[un] == 0; 186 frac = (rp[0] + 1) << GMP_NAIL_BITS; 187 s -= MP_BASES_CHARS_PER_LIMB_10; 188 #if HAVE_HOST_CPU_FAMILY_x86 189 /* The code below turns out to be a bit slower for x86 using gcc. 190 Use plain code. */ 191 i = MP_BASES_CHARS_PER_LIMB_10; 192 do 193 { 194 umul_ppmm (digit, frac, frac, 10); 195 *s++ = digit; 196 } 197 while (--i); 198 #else 199 /* Use the fact that 10 in binary is 1010, with the lowest bit 0. 200 After a few umul_ppmm, we will have accumulated enough low zeros 201 to use a plain multiply. */ 202 if (MP_BASES_NORMALIZATION_STEPS_10 == 0) 203 { 204 umul_ppmm (digit, frac, frac, 10); 205 *s++ = digit; 206 } 207 if (MP_BASES_NORMALIZATION_STEPS_10 <= 1) 208 { 209 umul_ppmm (digit, frac, frac, 10); 210 *s++ = digit; 211 } 212 if (MP_BASES_NORMALIZATION_STEPS_10 <= 2) 213 { 214 umul_ppmm (digit, frac, frac, 10); 215 *s++ = digit; 216 } 217 if (MP_BASES_NORMALIZATION_STEPS_10 <= 3) 218 { 219 umul_ppmm (digit, frac, frac, 10); 220 *s++ = digit; 221 } 222 i = (MP_BASES_CHARS_PER_LIMB_10 - ((MP_BASES_NORMALIZATION_STEPS_10 < 4) 223 ? (4-MP_BASES_NORMALIZATION_STEPS_10) 224 : 0)); 225 frac = (frac + 0xf) >> 4; 226 do 227 { 228 frac *= 10; 229 digit = frac >> (GMP_LIMB_BITS - 4); 230 *s++ = digit; 231 frac &= (~(mp_limb_t) 0) >> 4; 232 } 233 while (--i); 234 #endif 235 s -= MP_BASES_CHARS_PER_LIMB_10; 236 } 237 238 ul = rp[1]; 239 while (ul != 0) 240 { 241 udiv_qrnd_unnorm (ul, rl, ul, 10); 242 *--s = rl; 243 } 244 } 245 else /* not base 10 */ 246 { 247 unsigned chars_per_limb; 248 mp_limb_t big_base, big_base_inverted; 249 unsigned normalization_steps; 250 251 chars_per_limb = mp_bases[base].chars_per_limb; 252 big_base = mp_bases[base].big_base; 253 big_base_inverted = mp_bases[base].big_base_inverted; 254 count_leading_zeros (normalization_steps, big_base); 255 256 MPN_COPY (rp + 1, up, un); 257 258 s = buf + BUF_ALLOC; 259 while (un > 1) 260 { 261 int i; 262 mp_limb_t frac; 263 MPN_DIVREM_OR_PREINV_DIVREM_1 (rp, (mp_size_t) 1, rp + 1, un, 264 big_base, big_base_inverted, 265 normalization_steps); 266 un -= rp[un] == 0; 267 frac = (rp[0] + 1) << GMP_NAIL_BITS; 268 s -= chars_per_limb; 269 i = chars_per_limb; 270 do 271 { 272 mp_limb_t digit; 273 umul_ppmm (digit, frac, frac, base); 274 *s++ = digit; 275 } 276 while (--i); 277 s -= chars_per_limb; 278 } 279 280 ul = rp[1]; 281 while (ul != 0) 282 { 283 udiv_qrnd_unnorm (ul, rl, ul, base); 284 *--s = rl; 285 } 286 } 287 288 l = buf + BUF_ALLOC - s; 289 while (l < len) 290 { 291 *str++ = 0; 292 len--; 293 } 294 while (l != 0) 295 { 296 *str++ = *s++; 297 l--; 298 } 299 return str; 300 } 301 302 303 /* Convert {UP,UN} to a string with a base as represented in POWTAB, and put 304 the string in STR. Generate LEN characters, possibly padding with zeros to 305 the left. If LEN is zero, generate as many characters as required. 306 Return a pointer immediately after the last digit of the result string. 307 This uses divide-and-conquer and is intended for large conversions. */ 308 static unsigned char * 309 mpn_dc_get_str (unsigned char *str, size_t len, 310 mp_ptr up, mp_size_t un, 311 const powers_t *powtab, mp_ptr tmp) 312 { 313 if (BELOW_THRESHOLD (un, GET_STR_DC_THRESHOLD)) 314 { 315 if (un != 0) 316 str = mpn_bc_get_str (str, len, up, un, powtab->base); 317 else 318 { 319 while (len != 0) 320 { 321 *str++ = 0; 322 len--; 323 } 324 } 325 } 326 else 327 { 328 mp_ptr pwp, qp, rp; 329 mp_size_t pwn, qn; 330 mp_size_t sn; 331 332 pwp = powtab->p; 333 pwn = powtab->n; 334 sn = powtab->shift; 335 336 if (un < pwn + sn || (un == pwn + sn && mpn_cmp (up + sn, pwp, un - sn) < 0)) 337 { 338 str = mpn_dc_get_str (str, len, up, un, powtab - 1, tmp); 339 } 340 else 341 { 342 qp = tmp; /* (un - pwn + 1) limbs for qp */ 343 rp = up; /* pwn limbs for rp; overwrite up area */ 344 345 mpn_tdiv_qr (qp, rp + sn, 0L, up + sn, un - sn, pwp, pwn); 346 qn = un - sn - pwn; qn += qp[qn] != 0; /* quotient size */ 347 348 ASSERT (qn < pwn + sn || (qn == pwn + sn && mpn_cmp (qp + sn, pwp, pwn) < 0)); 349 350 if (len != 0) 351 len = len - powtab->digits_in_base; 352 353 str = mpn_dc_get_str (str, len, qp, qn, powtab - 1, tmp + qn); 354 str = mpn_dc_get_str (str, powtab->digits_in_base, rp, pwn + sn, powtab - 1, tmp); 355 } 356 } 357 return str; 358 } 359 360 /* There are no leading zeros on the digits generated at str, but that's not 361 currently a documented feature. The current mpz_out_str and mpz_get_str 362 rely on it. */ 363 364 size_t 365 mpn_get_str (unsigned char *str, int base, mp_ptr up, mp_size_t un) 366 { 367 mp_ptr powtab_mem; 368 powers_t powtab[GMP_LIMB_BITS]; 369 int pi; 370 size_t out_len; 371 mp_ptr tmp; 372 size_t ndig; 373 mp_size_t xn; 374 TMP_DECL; 375 376 /* Special case zero, as the code below doesn't handle it. */ 377 if (un == 0) 378 { 379 str[0] = 0; 380 return 1; 381 } 382 383 if (POW2_P (base)) 384 { 385 /* The base is a power of 2. Convert from most significant end. */ 386 mp_limb_t n1, n0; 387 int bits_per_digit = mp_bases[base].big_base; 388 int cnt; 389 int bit_pos; 390 mp_size_t i; 391 unsigned char *s = str; 392 mp_bitcnt_t bits; 393 394 n1 = up[un - 1]; 395 count_leading_zeros (cnt, n1); 396 397 /* BIT_POS should be R when input ends in least significant nibble, 398 R + bits_per_digit * n when input ends in nth least significant 399 nibble. */ 400 401 bits = (mp_bitcnt_t) GMP_NUMB_BITS * un - cnt + GMP_NAIL_BITS; 402 cnt = bits % bits_per_digit; 403 if (cnt != 0) 404 bits += bits_per_digit - cnt; 405 bit_pos = bits - (mp_bitcnt_t) (un - 1) * GMP_NUMB_BITS; 406 407 /* Fast loop for bit output. */ 408 i = un - 1; 409 for (;;) 410 { 411 bit_pos -= bits_per_digit; 412 while (bit_pos >= 0) 413 { 414 *s++ = (n1 >> bit_pos) & ((1 << bits_per_digit) - 1); 415 bit_pos -= bits_per_digit; 416 } 417 i--; 418 if (i < 0) 419 break; 420 n0 = (n1 << -bit_pos) & ((1 << bits_per_digit) - 1); 421 n1 = up[i]; 422 bit_pos += GMP_NUMB_BITS; 423 *s++ = n0 | (n1 >> bit_pos); 424 } 425 426 return s - str; 427 } 428 429 /* General case. The base is not a power of 2. */ 430 431 if (BELOW_THRESHOLD (un, GET_STR_PRECOMPUTE_THRESHOLD)) 432 return mpn_bc_get_str (str, (size_t) 0, up, un, base) - str; 433 434 TMP_MARK; 435 436 /* Allocate one large block for the powers of big_base. */ 437 powtab_mem = TMP_BALLOC_LIMBS (mpn_str_powtab_alloc (un)); 438 439 /* Compute a table of powers, were the largest power is >= sqrt(U). */ 440 DIGITS_IN_BASE_PER_LIMB (ndig, un, base); 441 xn = 1 + ndig / mp_bases[base].chars_per_limb; /* FIXME: scalar integer division */ 442 443 pi = 1 + mpn_compute_powtab (powtab, powtab_mem, xn, base); 444 445 /* Using our precomputed powers, now in powtab[], convert our number. */ 446 tmp = TMP_BALLOC_LIMBS (mpn_dc_get_str_itch (un)); 447 out_len = mpn_dc_get_str (str, 0, up, un, powtab + (pi - 1), tmp) - str; 448 TMP_FREE; 449 450 return out_len; 451 } 452