1 /* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
3 Copyright 1999-2015 Free Software Foundation, Inc.
4 Contributed by the AriC 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