1/* mpfr_exp2 -- power of 2 function 2^y
2
3Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4Contributed by the AriC and Caramel projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23#define MPFR_NEED_LONGLONG_H
24#include "mpfr-impl.h"
25
26/* The computation of y = 2^z is done by                           *
27 *     y = exp(z*log(2)). The result is exact iff z is an integer. */
28
29int
30mpfr_exp2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
31{
32  int inexact;
33  long xint;
34  mpfr_t xfrac;
35  MPFR_SAVE_EXPO_DECL (expo);
36
37  MPFR_LOG_FUNC
38    (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
39     ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
40      inexact));
41
42  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
43    {
44      if (MPFR_IS_NAN (x))
45        {
46          MPFR_SET_NAN (y);
47          MPFR_RET_NAN;
48        }
49      else if (MPFR_IS_INF (x))
50        {
51          if (MPFR_IS_POS (x))
52            MPFR_SET_INF (y);
53          else
54            MPFR_SET_ZERO (y);
55          MPFR_SET_POS (y);
56          MPFR_RET (0);
57        }
58      else /* 2^0 = 1 */
59        {
60          MPFR_ASSERTD (MPFR_IS_ZERO(x));
61          return mpfr_set_ui (y, 1, rnd_mode);
62        }
63    }
64
65  /* since the smallest representable non-zero float is 1/2*2^__gmpfr_emin,
66     if x < __gmpfr_emin - 1, the result is either 1/2*2^__gmpfr_emin or 0 */
67  MPFR_ASSERTN (MPFR_EMIN_MIN >= LONG_MIN + 2);
68  if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emin - 1) < 0))
69    {
70      mpfr_rnd_t rnd2 = rnd_mode;
71      /* in round to nearest mode, round to zero when x <= __gmpfr_emin-2 */
72      if (rnd_mode == MPFR_RNDN &&
73          mpfr_cmp_si_2exp (x, __gmpfr_emin - 2, 0) <= 0)
74        rnd2 = MPFR_RNDZ;
75      return mpfr_underflow (y, rnd2, 1);
76    }
77
78  MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
79  if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emax) >= 0))
80    return mpfr_overflow (y, rnd_mode, 1);
81
82  /* We now know that emin - 1 <= x < emax. */
83
84  MPFR_SAVE_EXPO_MARK (expo);
85
86  /* 2^x = 1 + x*log(2) + O(x^2) for x near zero, and for |x| <= 1 we have
87     |2^x - 1| <= x < 2^EXP(x). If x > 0 we must round away from 0 (dir=1);
88     if x < 0 we must round toward 0 (dir=0). */
89  MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, - MPFR_GET_EXP (x), 0,
90                                    MPFR_SIGN(x) > 0, rnd_mode, expo, {});
91
92  xint = mpfr_get_si (x, MPFR_RNDZ);
93  mpfr_init2 (xfrac, MPFR_PREC (x));
94  mpfr_sub_si (xfrac, x, xint, MPFR_RNDN); /* exact */
95
96  if (MPFR_IS_ZERO (xfrac))
97    {
98      mpfr_set_ui (y, 1, MPFR_RNDN);
99      inexact = 0;
100    }
101  else
102    {
103      /* Declaration of the intermediary variable */
104      mpfr_t t;
105
106      /* Declaration of the size variable */
107      mpfr_prec_t Ny = MPFR_PREC(y);              /* target precision */
108      mpfr_prec_t Nt;                             /* working precision */
109      mpfr_exp_t err;                             /* error */
110      MPFR_ZIV_DECL (loop);
111
112      /* compute the precision of intermediary variable */
113      /* the optimal number of bits : see algorithms.tex */
114      Nt = Ny + 5 + MPFR_INT_CEIL_LOG2 (Ny);
115
116      /* initialise of intermediary variable */
117      mpfr_init2 (t, Nt);
118
119      /* First computation */
120      MPFR_ZIV_INIT (loop, Nt);
121      for (;;)
122        {
123          /* compute exp(x*ln(2))*/
124          mpfr_const_log2 (t, MPFR_RNDU);       /* ln(2) */
125          mpfr_mul (t, xfrac, t, MPFR_RNDU);    /* xfrac * ln(2) */
126          err = Nt - (MPFR_GET_EXP (t) + 2);   /* Estimate of the error */
127          mpfr_exp (t, t, MPFR_RNDN);           /* exp(xfrac * ln(2)) */
128
129          if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
130            break;
131
132          /* Actualisation of the precision */
133          MPFR_ZIV_NEXT (loop, Nt);
134          mpfr_set_prec (t, Nt);
135        }
136      MPFR_ZIV_FREE (loop);
137
138      inexact = mpfr_set (y, t, rnd_mode);
139
140      mpfr_clear (t);
141    }
142
143  mpfr_clear (xfrac);
144  mpfr_clear_flags ();
145  mpfr_mul_2si (y, y, xint, MPFR_RNDN); /* exact or overflow */
146  /* Note: We can have an overflow only when t was rounded up to 2. */
147  MPFR_ASSERTD (MPFR_IS_PURE_FP (y) || inexact > 0);
148  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
149  MPFR_SAVE_EXPO_FREE (expo);
150  return mpfr_check_range (y, inexact, rnd_mode);
151}
152