summaryrefslogtreecommitdiff
path: root/Build/source/libs/mpfr/mpfr-3.1.2/src/pow_z.c
blob: df356ee0aa486c7de257ef928aff730be6322089 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
/* mpfr_pow_z -- power function x^z with z a MPZ

Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.

This file is part of the GNU MPFR Library.

The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"

/* y <- x^|z| with z != 0
   if cr=1: ensures correct rounding of y
   if cr=0: does not ensure correct rounding, but avoid spurious overflow
   or underflow, and uses the precision of y as working precision (warning,
   y and x might be the same variable). */
static int
mpfr_pow_pos_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mpfr_rnd_t rnd, int cr)
{
  mpfr_t res;
  mpfr_prec_t prec, err;
  int inexact;
  mpfr_rnd_t rnd1, rnd2;
  mpz_t absz;
  mp_size_t size_z;
  MPFR_ZIV_DECL (loop);
  MPFR_BLOCK_DECL (flags);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg z=%Zd rnd=%d cr=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, z, rnd, cr),
     ("y[%Pu]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  MPFR_ASSERTD (mpz_sgn (z) != 0);

  if (MPFR_UNLIKELY (mpz_cmpabs_ui (z, 1) == 0))
    return mpfr_set (y, x, rnd);

  absz[0] = z[0];
  SIZ (absz) = ABS(SIZ(absz)); /* Hack to get abs(z) */
  MPFR_MPZ_SIZEINBASE2 (size_z, z);

  /* round toward 1 (or -1) to avoid spurious overflow or underflow,
     i.e. if an overflow or underflow occurs, it is a real exception
     and is not just due to the rounding error. */
  rnd1 = (MPFR_EXP(x) >= 1) ? MPFR_RNDZ
    : (MPFR_IS_POS(x) ? MPFR_RNDU : MPFR_RNDD);
  rnd2 = (MPFR_EXP(x) >= 1) ? MPFR_RNDD : MPFR_RNDU;

  if (cr != 0)
    prec = MPFR_PREC (y) + 3 + size_z + MPFR_INT_CEIL_LOG2 (MPFR_PREC (y));
  else
    prec = MPFR_PREC (y);
  mpfr_init2 (res, prec);

  MPFR_ZIV_INIT (loop, prec);
  for (;;)
    {
      unsigned int inexmul;  /* will be non-zero if res may be inexact */
      mp_size_t i = size_z;

      /* now 2^(i-1) <= z < 2^i */
      /* see below (case z < 0) for the error analysis, which is identical,
         except if z=n, the maximal relative error is here 2(n-1)2^(-prec)
         instead of 2(2n-1)2^(-prec) for z<0. */
      MPFR_ASSERTD (prec > (mpfr_prec_t) i);
      err = prec - 1 - (mpfr_prec_t) i;

      MPFR_BLOCK (flags,
                  inexmul = mpfr_mul (res, x, x, rnd2);
                  MPFR_ASSERTD (i >= 2);
                  if (mpz_tstbit (absz, i - 2))
                    inexmul |= mpfr_mul (res, res, x, rnd1);
                  for (i -= 3; i >= 0 && !MPFR_BLOCK_EXCEP; i--)
                    {
                      inexmul |= mpfr_mul (res, res, res, rnd2);
                      if (mpz_tstbit (absz, i))
                        inexmul |= mpfr_mul (res, res, x, rnd1);
                    });
      if (MPFR_LIKELY (inexmul == 0 || cr == 0
                       || MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)
                       || MPFR_CAN_ROUND (res, err, MPFR_PREC (y), rnd)))
        break;
      /* Can't decide correct rounding, increase the precision */
      MPFR_ZIV_NEXT (loop, prec);
      mpfr_set_prec (res, prec);
    }
  MPFR_ZIV_FREE (loop);

  /* Check Overflow */
  if (MPFR_OVERFLOW (flags))
    {
      MPFR_LOG_MSG (("overflow\n", 0));
      inexact = mpfr_overflow (y, rnd, mpz_odd_p (absz) ?
                               MPFR_SIGN (x) : MPFR_SIGN_POS);
    }
  /* Check Underflow */
  else if (MPFR_UNDERFLOW (flags))
    {
      MPFR_LOG_MSG (("underflow\n", 0));
      if (rnd == MPFR_RNDN)
        {
          mpfr_t y2, zz;

          /* We cannot decide now whether the result should be rounded
             toward zero or +Inf. So, let's use the general case of
             mpfr_pow, which can do that. But the problem is that the
             result can be exact! However, it is sufficient to try to
             round on 2 bits (the precision does not matter in case of
             underflow, since MPFR does not have subnormals), in which
             case, the result cannot be exact due to previous filtering
             of trivial cases. */
          MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
                                          MPFR_EXP (x) - 1) != 0);
          mpfr_init2 (y2, 2);
          mpfr_init2 (zz, ABS (SIZ (z)) * GMP_NUMB_BITS);
          inexact = mpfr_set_z (zz, z, MPFR_RNDN);
          MPFR_ASSERTN (inexact == 0);
          inexact = mpfr_pow_general (y2, x, zz, rnd, 1,
                                      (mpfr_save_expo_t *) NULL);
          mpfr_clear (zz);
          mpfr_set (y, y2, MPFR_RNDN);
          mpfr_clear (y2);
          __gmpfr_flags = MPFR_FLAGS_INEXACT | MPFR_FLAGS_UNDERFLOW;
        }
      else
        {
          inexact = mpfr_underflow (y, rnd, mpz_odd_p (absz) ?
                                    MPFR_SIGN (x) : MPFR_SIGN_POS);
        }
    }
  else
    inexact = mpfr_set (y, res, rnd);

  mpfr_clear (res);
  return inexact;
}

/* The computation of y = pow(x,z) is done by
 *    y = set_ui(1)      if z = 0
 *    y = pow_ui(x,z)    if z > 0
 *    y = pow_ui(1/x,-z) if z < 0
 *
 * Note: in case z < 0, we could also compute 1/pow_ui(x,-z). However, in
 * case MAX < 1/MIN, where MAX is the largest positive value, i.e.,
 * MAX = nextbelow(+Inf), and MIN is the smallest positive value, i.e.,
 * MIN = nextabove(+0), then x^(-z) might produce an overflow, whereas
 * x^z is representable.
 */

int
mpfr_pow_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mpfr_rnd_t rnd)
{
  int   inexact;
  mpz_t tmp;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg z=%Zd rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, z, rnd),
     ("y[%Pu]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  /* x^0 = 1 for any x, even a NaN */
  if (MPFR_UNLIKELY (mpz_sgn (z) == 0))
    return mpfr_set_ui (y, 1, rnd);

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x))
        {
          /* Inf^n = Inf, (-Inf)^n = Inf for n even, -Inf for n odd */
          /* Inf ^(-n) = 0, sign = + if x>0 or z even */
          if (mpz_sgn (z) > 0)
            MPFR_SET_INF (y);
          else
            MPFR_SET_ZERO (y);
          if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && mpz_odd_p (z)))
            MPFR_SET_NEG (y);
          else
            MPFR_SET_POS (y);
          MPFR_RET (0);
        }
      else /* x is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO(x));
          if (mpz_sgn (z) > 0)
            /* 0^n = +/-0 for any n */
            MPFR_SET_ZERO (y);
          else
            {
              /* 0^(-n) if +/- INF */
              MPFR_SET_INF (y);
              mpfr_set_divby0 ();
            }
          if (MPFR_LIKELY (MPFR_IS_POS (x) || mpz_even_p (z)))
            MPFR_SET_POS (y);
          else
            MPFR_SET_NEG (y);
          MPFR_RET(0);
        }
    }

  MPFR_SAVE_EXPO_MARK (expo);

  /* detect exact powers: x^-n is exact iff x is a power of 2
     Do it if n > 0 too as this is faster and this filtering is
     needed in case of underflow. */
  if (MPFR_UNLIKELY (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
                                       MPFR_EXP (x) - 1) == 0))
    {
      mpfr_exp_t expx = MPFR_EXP (x); /* warning: x and y may be the same
                                         variable */

      MPFR_LOG_MSG (("x^n with x power of two\n", 0));
      mpfr_set_si (y, mpz_odd_p (z) ? MPFR_INT_SIGN(x) : 1, rnd);
      MPFR_ASSERTD (MPFR_IS_FP (y));
      mpz_init (tmp);
      mpz_mul_si (tmp, z, expx - 1);
      MPFR_ASSERTD (MPFR_GET_EXP (y) == 1);
      mpz_add_ui (tmp, tmp, 1);
      inexact = 0;
      if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emin) < 0))
        {
          MPFR_LOG_MSG (("underflow\n", 0));
          /* |y| is a power of two, thus |y| <= 2^(emin-2), and in
             rounding to nearest, the value must be rounded to 0. */
          if (rnd == MPFR_RNDN)
            rnd = MPFR_RNDZ;
          inexact = mpfr_underflow (y, rnd, MPFR_SIGN (y));
        }
      else if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emax) > 0))
        {
          MPFR_LOG_MSG (("overflow\n", 0));
          inexact = mpfr_overflow (y, rnd, MPFR_SIGN (y));
        }
      else
        MPFR_SET_EXP (y, mpz_get_si (tmp));
      mpz_clear (tmp);
      MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
    }
  else if (mpz_sgn (z) > 0)
    {
      inexact = mpfr_pow_pos_z (y, x, z, rnd, 1);
      MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
    }
  else
    {
      /* Declaration of the intermediary variable */
      mpfr_t t;
      mpfr_prec_t Nt;   /* Precision of the intermediary variable */
      mpfr_rnd_t rnd1;
      mp_size_t size_z;
      MPFR_ZIV_DECL (loop);

      MPFR_MPZ_SIZEINBASE2 (size_z, z);

      /* initial working precision */
      Nt = MPFR_PREC (y);
      Nt = Nt + size_z + 3 + MPFR_INT_CEIL_LOG2 (Nt);
      /* ensures Nt >= bits(z)+2 */

      /* initialise of intermediary variable */
      mpfr_init2 (t, Nt);

      /* We will compute rnd(rnd1(1/x) ^ (-z)), where rnd1 is the rounding
         toward sign(x), to avoid spurious overflow or underflow. */
      rnd1 = MPFR_EXP (x) < 1 ? MPFR_RNDZ :
        (MPFR_SIGN (x) > 0 ? MPFR_RNDU : MPFR_RNDD);

      MPFR_ZIV_INIT (loop, Nt);
      for (;;)
        {
          MPFR_BLOCK_DECL (flags);

          /* compute (1/x)^(-z), -z>0 */
          /* As emin = -emax, an underflow cannot occur in the division.
             And if an overflow occurs, then this means that x^z overflows
             too (since we have rounded toward 1 or -1). */
          MPFR_BLOCK (flags, mpfr_ui_div (t, 1, x, rnd1));
          MPFR_ASSERTD (! MPFR_UNDERFLOW (flags));
          /* t = (1/x)*(1+theta) where |theta| <= 2^(-Nt) */
          if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
            goto overflow;
          MPFR_BLOCK (flags, mpfr_pow_pos_z (t, t, z, rnd, 0));
          /* Now if z=-n, t = x^z*(1+theta)^(2n-1) where |theta| <= 2^(-Nt),
             with theta maybe different from above. If (2n-1)*2^(-Nt) <= 1/2,
             which is satisfied as soon as Nt >= bits(z)+2, then we can use
             Lemma \ref{lemma_graillat} from algorithms.tex, which yields
             t = x^z*(1+theta) with |theta| <= 2(2n-1)*2^(-Nt), thus the
             error is bounded by 2(2n-1) ulps <= 2^(bits(z)+2) ulps. */
          if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
            {
            overflow:
              MPFR_ZIV_FREE (loop);
              mpfr_clear (t);
              MPFR_SAVE_EXPO_FREE (expo);
              MPFR_LOG_MSG (("overflow\n", 0));
              return mpfr_overflow (y, rnd,
                                    mpz_odd_p (z) ? MPFR_SIGN (x) :
                                    MPFR_SIGN_POS);
            }
          if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
            {
              MPFR_ZIV_FREE (loop);
              mpfr_clear (t);
              MPFR_LOG_MSG (("underflow\n", 0));
              if (rnd == MPFR_RNDN)
                {
                  mpfr_t y2, zz;

                  /* We cannot decide now whether the result should be
                     rounded toward zero or away from zero. So, like
                     in mpfr_pow_pos_z, let's use the general case of
                     mpfr_pow in precision 2. */
                  MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
                                                  MPFR_EXP (x) - 1) != 0);
                  mpfr_init2 (y2, 2);
                  mpfr_init2 (zz, ABS (SIZ (z)) * GMP_NUMB_BITS);
                  inexact = mpfr_set_z (zz, z, MPFR_RNDN);
                  MPFR_ASSERTN (inexact == 0);
                  inexact = mpfr_pow_general (y2, x, zz, rnd, 1,
                                              (mpfr_save_expo_t *) NULL);
                  mpfr_clear (zz);
                  mpfr_set (y, y2, MPFR_RNDN);
                  mpfr_clear (y2);
                  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
                  goto end;
                }
              else
                {
                  MPFR_SAVE_EXPO_FREE (expo);
                  return mpfr_underflow (y, rnd, mpz_odd_p (z) ?
                                         MPFR_SIGN (x) : MPFR_SIGN_POS);
                }
            }
          if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_z - 2, MPFR_PREC (y),
                                           rnd)))
            break;
          /* actualisation of the precision */
          MPFR_ZIV_NEXT (loop, Nt);
          mpfr_set_prec (t, Nt);
        }
      MPFR_ZIV_FREE (loop);

      inexact = mpfr_set (y, t, rnd);
      mpfr_clear (t);
    }

 end:
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd);
}