contrib/mpfr/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 Free Software Foundation, Inc.
   4 Contributed by the Arenaire and Cacao projects, INRIA.
   5 Contributed by Alain Delplanque and Paul Zimmermann.
   6
   7 This file is part of the GNU MPFR Library.
   8
   9 The GNU MPFR Library is free software; you can redistribute it and/or modify
  10 it under the terms of the GNU Lesser General Public License as published by
  11 the Free Software Foundation; either version 2.1 of the License, or (at your
  12 option) any later version.
  13
  14 The GNU MPFR Library is distributed in the hope that it will be useful, but
  15 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
  16 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
  17 License for more details.
  18
  19 You should have received a copy of the GNU Lesser General Public License
  20 along with the GNU MPFR Library; see the file COPYING.LIB.  If not, write to
  21 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
  22 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 towards 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^BITS_PER_MP_LIMB
  35
  36    Return -1 is the result is exact.
  37    Return -2 if an overflow occurred in the computation of exp_r.
  38 */
  39
  40 long
  41 mpfr_mpn_exp (mp_limb_t *a, mp_exp_t *exp_r, int b, mp_exp_t e, size_t n)
  42 {
  43   mp_limb_t *c, B;
  44   mp_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 <= 36));
  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 = BITS_PER_MP_LIMB - 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) * BITS_PER_MP_LIMB;
  77
  78   /* determine number of bits in e */
  79   count_leading_zeros (t, (mp_limb_t) e);
  80
  81   t = BITS_PER_MP_LIMB - t; /* number of bits of exponent e */
  82
  83   error = t; /* error <= BITS_PER_MP_LIMB */
  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 * BITS_PER_MP_LIMB - mpn_scan1 (a, 0);
  92       n1 = (n * BITS_PER_MP_LIMB - bits) / BITS_PER_MP_LIMB;
  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 * BITS_PER_MP_LIMB be used? */
 107       f = 2*f;
 108       MPFR_SADD_OVERFLOW (f, f, n * BITS_PER_MP_LIMB,
 109                           mp_exp_t, mp_exp_unsigned_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) * BITS_PER_MP_LIMB))
 126         error = i;
 127
 128       if (e & ((mp_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 + BITS_PER_MP_LIMB;
 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 <= BITS_PER_MP_LIMB-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 * BITS_PER_MP_LIMB - 1) / 2))
 167         error = n * BITS_PER_MP_LIMB; /* result is completely wrong:
 168                                          this is very unlikely since error is
 169                                          at most 5/2*log_2(e), and
 170                                          n * BITS_PER_MP_LIMB is at least
 171                                          3*log_2(e) */
 172 #endif
 173       return error;
 174     }
 175 }