contrib/mpfr/src/mpn_exp.c

   1 /* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
   2
   3 Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
   4 Contributed by the AriC and Caramel projects, INRIA.
   5
   6 This file is part of the GNU MPFR Library.
   7
   8 The GNU MPFR Library is free software; you can redistribute it and/or modify
   9 it under the terms of the GNU Lesser General Public License as published by
  10 the Free Software Foundation; either version 3 of the License, or (at your
  11 option) any later version.
  12
  13 The GNU MPFR Library is distributed in the hope that it will be useful, but
  14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  15 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
  16 License for more details.
  17
  18 You should have received a copy of the GNU Lesser General Public License
  19 along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
  20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
  21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
  22
  23
  24 #define MPFR_NEED_LONGLONG_H
  25 #include "mpfr-impl.h"
  26
  27 /* this function computes an approximation of b^e in {a, n}, with exponent
  28    stored in exp_r. The computed value is rounded toward zero (truncated).
  29    It returns an integer f such that the final error is bounded by 2^f ulps,
  30    that is:
  31    a*2^exp_r <= b^e <= 2^exp_r (a + 2^f),
  32    where a represents {a, n}, i.e. the integer
  33    a[0] + a[1]*B + ... + a[n-1]*B^(n-1) where B=2^GMP_NUMB_BITS
  34
  35    Return -1 is the result is exact.
  36    Return -2 if an overflow occurred in the computation of exp_r.
  37 */
  38
  39 long
  40 mpfr_mpn_exp (mp_limb_t *a, mpfr_exp_t *exp_r, int b, mpfr_exp_t e, size_t n)
  41 {
  42   mp_limb_t *c, B;
  43   mpfr_exp_t f, h;
  44   int i;
  45   unsigned long t; /* number of bits in e */
  46   unsigned long bits;
  47   size_t n1;
  48   unsigned int error;           /* (number - 1) of loop a^2b inexact */
  49                                  /* error == t means no error */
  50   int err_s_a2 = 0;
  51   int err_s_ab = 0;              /* number of error when shift A^2, AB */
  52   MPFR_TMP_DECL(marker);
  53
  54   MPFR_ASSERTN(e > 0);
  55   MPFR_ASSERTN((2 <= b) && (b <= 62));
  56
  57   MPFR_TMP_MARK(marker);
  58
  59   /* initialization of a, b, f, h */
  60
  61   /* normalize the base */
  62   B = (mp_limb_t) b;
  63   count_leading_zeros (h, B);
  64
  65   bits = GMP_NUMB_BITS - h;
  66
  67   B = B << h;
  68   h = - h;
  69
  70   /* allocate space for A and set it to B */
  71   c = MPFR_TMP_LIMBS_ALLOC (2 * n);
  72   a [n - 1] = B;
  73   MPN_ZERO (a, n - 1);
  74   /* initial exponent for A: invariant is A = {a, n} * 2^f */
  75   f = h - (n - 1) * GMP_NUMB_BITS;
  76
  77   /* determine number of bits in e */
  78   count_leading_zeros (t, (mp_limb_t) e);
  79
  80   t = GMP_NUMB_BITS - t; /* number of bits of exponent e */
  81
  82   error = t; /* error <= GMP_NUMB_BITS */
  83
  84   MPN_ZERO (c, 2 * n);
  85
  86   for (i = t - 2; i >= 0; i--)
  87     {
  88
  89       /* determine precision needed */
  90       bits = n * GMP_NUMB_BITS - mpn_scan1 (a, 0);
  91       n1 = (n * GMP_NUMB_BITS - bits) / GMP_NUMB_BITS;
  92
  93       /* square of A : {c+2n1, 2(n-n1)} = {a+n1, n-n1}^2 */
  94       mpn_sqr_n (c + 2 * n1, a + n1, n - n1);
  95
  96       /* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */
  97
  98       /* check overflow on f */
  99       if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2))
 100         {
 101         overflow:
 102           MPFR_TMP_FREE(marker);
 103           return -2;
 104         }
 105       /* FIXME: Could f = 2*f + n * GMP_NUMB_BITS be used? */
 106       f = 2*f;
 107       MPFR_SADD_OVERFLOW (f, f, n * GMP_NUMB_BITS,
 108                           mpfr_exp_t, mpfr_uexp_t,
 109                           MPFR_EXP_MIN, MPFR_EXP_MAX,
 110                           goto overflow, goto overflow);
 111       if ((c[2*n - 1] & MPFR_LIMB_HIGHBIT) == 0)
 112         {
 113           /* shift A by one bit to the left */
 114           mpn_lshift (a, c + n, n, 1);
 115           a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
 116           f --;
 117           if (error != t)
 118             err_s_a2 ++;
 119         }
 120       else
 121         MPN_COPY (a, c + n, n);
 122
 123       if ((error == t) && (2 * n1 <= n) &&
 124           (mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * GMP_NUMB_BITS))
 125         error = i;
 126
 127       if (e & ((mpfr_exp_t) 1 << i))
 128         {
 129           /* multiply A by B */
 130           c[2 * n - 1] = mpn_mul_1 (c + n - 1, a, n, B);
 131           f += h + GMP_NUMB_BITS;
 132           if ((c[2 * n - 1] & MPFR_LIMB_HIGHBIT) == 0)
 133             { /* shift A by one bit to the left */
 134               mpn_lshift (a, c + n, n, 1);
 135               a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
 136               f --;
 137             }
 138           else
 139             {
 140               MPN_COPY (a, c + n, n);
 141               if (error != t)
 142                 err_s_ab ++;
 143             }
 144           if ((error == t) && (c[n - 1] != 0))
 145             error = i;
 146         }
 147     }
 148
 149   MPFR_TMP_FREE(marker);
 150
 151   *exp_r = f;
 152
 153   if (error == t)
 154     return -1; /* result is exact */
 155   else /* error <= t-2 <= GMP_NUMB_BITS-2
 156           err_s_ab, err_s_a2 <= t-1       */
 157     {
 158       /* if there are p loops after the first inexact result, with
 159          j shifts in a^2 and l shifts in a*b, then the final error is
 160          at most 2^(p+ceil((j+1)/2)+l+1)*ulp(res).
 161          This is bounded by 2^(5/2*t-1/2) where t is the number of bits of e.
 162       */
 163       error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */
 164 #if 0
 165       if ((error - 1) >= ((n * GMP_NUMB_BITS - 1) / 2))
 166         error = n * GMP_NUMB_BITS; /* result is completely wrong:
 167                                          this is very unlikely since error is
 168                                          at most 5/2*log_2(e), and
 169                                          n * GMP_NUMB_BITS is at least
 170                                          3*log_2(e) */
 171 #endif
 172       return error;
 173     }
 174 }