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 TMP_DECL;
373
374 /* Special case zero, as the code below doesn't handle it. */
375 if (un == 0)
376 {
377 str[0] = 0;
378 return 1;
379 }
380
381 if (POW2_P (base))
382 {
383 /* The base is a power of 2. Convert from most significant end. */
384 mp_limb_t n1, n0;
385 int bits_per_digit = mp_bases[base].big_base;
386 int cnt;
387 int bit_pos;
388 mp_size_t i;
389 unsigned char *s = str;
390 mp_bitcnt_t bits;
391
392 n1 = up[un - 1];
393 count_leading_zeros (cnt, n1);
394
395 /* BIT_POS should be R when input ends in least significant nibble,
396 R + bits_per_digit * n when input ends in nth least significant
397 nibble. */
398
399 bits = (mp_bitcnt_t) GMP_NUMB_BITS * un - cnt + GMP_NAIL_BITS;
400 cnt = bits % bits_per_digit;
401 if (cnt != 0)
402 bits += bits_per_digit - cnt;
403 bit_pos = bits - (mp_bitcnt_t) (un - 1) * GMP_NUMB_BITS;
404
405 /* Fast loop for bit output. */
406 i = un - 1;
407 for (;;)
408 {
409 bit_pos -= bits_per_digit;
410 while (bit_pos >= 0)
411 {
412 *s++ = (n1 >> bit_pos) & ((1 << bits_per_digit) - 1);
413 bit_pos -= bits_per_digit;
414 }
415 i--;
416 if (i < 0)
417 break;
418 n0 = (n1 << -bit_pos) & ((1 << bits_per_digit) - 1);
419 n1 = up[i];
420 bit_pos += GMP_NUMB_BITS;
421 *s++ = n0 | (n1 >> bit_pos);
422 }
423
424 return s - str;
425 }
426
427 /* General case. The base is not a power of 2. */
428
429 if (BELOW_THRESHOLD (un, GET_STR_PRECOMPUTE_THRESHOLD))
430 return mpn_bc_get_str (str, (size_t) 0, up, un, base) - str;
431
432 TMP_MARK;
433
434 /* Allocate one large block for the powers of big_base. */
435 powtab_mem = TMP_BALLOC_LIMBS (mpn_str_powtab_alloc (un));
436
437 /* Compute a table of powers, were the largest power is >= sqrt(U). */
438 size_t ndig;
439 mp_size_t xn;
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 }