1 /* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
3 Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Caramel projects, INRIA.
6 This file is part of the GNU MPFR Library.
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.
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.
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. */
24 #define MPFR_NEED_LONGLONG_H
25 #include "mpfr-impl.h"
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,
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
35 Return -1 is the result is exact.
36 Return -2 if an overflow occurred in the computation of exp_r.
40 mpfr_mpn_exp (mp_limb_t *a, mpfr_exp_t *exp_r, int b, mpfr_exp_t e, size_t n)
45 unsigned long t; /* number of bits in e */
48 unsigned int error; /* (number - 1) of loop a^2b inexact */
49 /* error == t means no error */
51 int err_s_ab = 0; /* number of error when shift A^2, AB */
52 MPFR_TMP_DECL(marker);
55 MPFR_ASSERTN((2 <= b) && (b <= 62));
57 MPFR_TMP_MARK(marker);
59 /* initialization of a, b, f, h */
61 /* normalize the base */
63 count_leading_zeros (h, B);
65 bits = GMP_NUMB_BITS - h;
70 /* allocate space for A and set it to B */
71 c = MPFR_TMP_LIMBS_ALLOC (2 * n);
74 /* initial exponent for A: invariant is A = {a, n} * 2^f */
75 f = h - (n - 1) * GMP_NUMB_BITS;
77 /* determine number of bits in e */
78 count_leading_zeros (t, (mp_limb_t) e);
80 t = GMP_NUMB_BITS - t; /* number of bits of exponent e */
82 error = t; /* error <= GMP_NUMB_BITS */
86 for (i = t - 2; i >= 0; i--)
89 /* determine precision needed */
90 bits = n * GMP_NUMB_BITS - mpn_scan1 (a, 0);
91 n1 = (n * GMP_NUMB_BITS - bits) / GMP_NUMB_BITS;
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);
96 /* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */
98 /* check overflow on f */
99 if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2))
102 MPFR_TMP_FREE(marker);
105 /* FIXME: Could f = 2*f + n * GMP_NUMB_BITS be used? */
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)
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);
121 MPN_COPY (a, c + n, n);
123 if ((error == t) && (2 * n1 <= n) &&
124 (mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * GMP_NUMB_BITS))
127 if (e & ((mpfr_exp_t) 1 << i))
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);
140 MPN_COPY (a, c + n, n);
144 if ((error == t) && (c[n - 1] != 0))
149 MPFR_TMP_FREE(marker);
154 return -1; /* result is exact */
155 else /* error <= t-2 <= GMP_NUMB_BITS-2
156 err_s_ab, err_s_a2 <= t-1 */
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.
163 error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */
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