1/* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
2
3Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4Contributed by the Arenaire and Cacao projects, INRIA.
5Contributed by Alain Delplanque and Paul Zimmermann.
6
7This file is part of the GNU MPFR Library.
8
9The GNU MPFR Library is free software; you can redistribute it and/or modify
10it under the terms of the GNU Lesser General Public License as published by
11the Free Software Foundation; either version 3 of the License, or (at your
12option) any later version.
13
14The GNU MPFR Library is distributed in the hope that it will be useful, but
15WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
16or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
17License for more details.
18
19You should have received a copy of the GNU Lesser General Public License
20along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
21http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2251 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
23
24
25#define MPFR_NEED_LONGLONG_H
26#include "mpfr-impl.h"
27
28/* this function computes an approximation of b^e in {a, n}, with exponent
29   stored in exp_r. The computed value is rounded toward zero (truncated).
30   It returns an integer f such that the final error is bounded by 2^f ulps,
31   that is:
32   a*2^exp_r <= b^e <= 2^exp_r (a + 2^f),
33   where a represents {a, n}, i.e. the integer
34   a[0] + a[1]*B + ... + a[n-1]*B^(n-1) where B=2^GMP_NUMB_BITS
35
36   Return -1 is the result is exact.
37   Return -2 if an overflow occurred in the computation of exp_r.
38*/
39
40long
41mpfr_mpn_exp (mp_limb_t *a, mpfr_exp_t *exp_r, int b, mpfr_exp_t e, size_t n)
42{
43  mp_limb_t *c, B;
44  mpfr_exp_t f, h;
45  int i;
46  unsigned long t; /* number of bits in e */
47  unsigned long bits;
48  size_t n1;
49  unsigned int error;           /* (number - 1) of loop a^2b inexact */
50                                 /* error == t means no error */
51  int err_s_a2 = 0;
52  int err_s_ab = 0;              /* number of error when shift A^2, AB */
53  MPFR_TMP_DECL(marker);
54
55  MPFR_ASSERTN(e > 0);
56  MPFR_ASSERTN((2 <= b) && (b <= 62));
57
58  MPFR_TMP_MARK(marker);
59
60  /* initialization of a, b, f, h */
61
62  /* normalize the base */
63  B = (mp_limb_t) b;
64  count_leading_zeros (h, B);
65
66  bits = GMP_NUMB_BITS - h;
67
68  B = B << h;
69  h = - h;
70
71  /* allocate space for A and set it to B */
72  c = (mp_limb_t*) MPFR_TMP_ALLOC(2 * n * BYTES_PER_MP_LIMB);
73  a [n - 1] = B;
74  MPN_ZERO (a, n - 1);
75  /* initial exponent for A: invariant is A = {a, n} * 2^f */
76  f = h - (n - 1) * GMP_NUMB_BITS;
77
78  /* determine number of bits in e */
79  count_leading_zeros (t, (mp_limb_t) e);
80
81  t = GMP_NUMB_BITS - t; /* number of bits of exponent e */
82
83  error = t; /* error <= GMP_NUMB_BITS */
84
85  MPN_ZERO (c, 2 * n);
86
87  for (i = t - 2; i >= 0; i--)
88    {
89
90      /* determine precision needed */
91      bits = n * GMP_NUMB_BITS - mpn_scan1 (a, 0);
92      n1 = (n * GMP_NUMB_BITS - bits) / GMP_NUMB_BITS;
93
94      /* square of A : {c+2n1, 2(n-n1)} = {a+n1, n-n1}^2 */
95      mpn_sqr_n (c + 2 * n1, a + n1, n - n1);
96
97      /* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */
98
99      /* check overflow on f */
100      if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2))
101        {
102        overflow:
103          MPFR_TMP_FREE(marker);
104          return -2;
105        }
106      /* FIXME: Could f = 2*f + n * GMP_NUMB_BITS be used? */
107      f = 2*f;
108      MPFR_SADD_OVERFLOW (f, f, n * GMP_NUMB_BITS,
109                          mpfr_exp_t, mpfr_uexp_t,
110                          MPFR_EXP_MIN, MPFR_EXP_MAX,
111                          goto overflow, goto overflow);
112      if ((c[2*n - 1] & MPFR_LIMB_HIGHBIT) == 0)
113        {
114          /* shift A by one bit to the left */
115          mpn_lshift (a, c + n, n, 1);
116          a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
117          f --;
118          if (error != t)
119            err_s_a2 ++;
120        }
121      else
122        MPN_COPY (a, c + n, n);
123
124      if ((error == t) && (2 * n1 <= n) &&
125          (mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * GMP_NUMB_BITS))
126        error = i;
127
128      if (e & ((mpfr_exp_t) 1 << i))
129        {
130          /* multiply A by B */
131          c[2 * n - 1] = mpn_mul_1 (c + n - 1, a, n, B);
132          f += h + GMP_NUMB_BITS;
133          if ((c[2 * n - 1] & MPFR_LIMB_HIGHBIT) == 0)
134            { /* shift A by one bit to the left */
135              mpn_lshift (a, c + n, n, 1);
136              a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
137              f --;
138            }
139          else
140            {
141              MPN_COPY (a, c + n, n);
142              if (error != t)
143                err_s_ab ++;
144            }
145          if ((error == t) && (c[n - 1] != 0))
146            error = i;
147        }
148    }
149
150  MPFR_TMP_FREE(marker);
151
152  *exp_r = f;
153
154  if (error == t)
155    return -1; /* result is exact */
156  else /* error <= t-2 <= GMP_NUMB_BITS-2
157          err_s_ab, err_s_a2 <= t-1       */
158    {
159      /* if there are p loops after the first inexact result, with
160         j shifts in a^2 and l shifts in a*b, then the final error is
161         at most 2^(p+ceil((j+1)/2)+l+1)*ulp(res).
162         This is bounded by 2^(5/2*t-1/2) where t is the number of bits of e.
163      */
164      error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */
165#if 0
166      if ((error - 1) >= ((n * GMP_NUMB_BITS - 1) / 2))
167        error = n * GMP_NUMB_BITS; /* result is completely wrong:
168                                         this is very unlikely since error is
169                                         at most 5/2*log_2(e), and
170                                         n * GMP_NUMB_BITS is at least
171                                         3*log_2(e) */
172#endif
173      return error;
174    }
175}
176