1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
|
/* mpfr_pow_ui-- compute the power of a floating-point
by a machine integer
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* sets y to x^n, and return 0 if exact, non-zero otherwise */
int
mpfr_pow_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int n, mp_rnd_t rnd)
{
unsigned long m;
mpfr_t res;
mp_prec_t prec, err;
int inexact;
mp_rnd_t rnd1;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_BLOCK_DECL (flags);
MPFR_LOG_FUNC (("x[%#R]=%R n=%lu rnd=%d", x, x, n, rnd),
("y[%#R]=%R inexact=%d", y, y, inexact));
/* x^0 = 1 for any x, even a NaN */
if (MPFR_UNLIKELY (n == 0))
return mpfr_set_ui (y, 1, rnd);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
/* Inf^n = Inf, (-Inf)^n = Inf for n even, -Inf for n odd */
if (MPFR_IS_NEG (x) && (n & 1) == 1)
MPFR_SET_NEG (y);
else
MPFR_SET_POS (y);
MPFR_SET_INF (y);
MPFR_RET (0);
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
/* 0^n = 0 for any n */
MPFR_SET_ZERO (y);
if (MPFR_IS_POS (x) || (n & 1) == 0)
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET (0);
}
}
else if (MPFR_UNLIKELY (n <= 2))
{
if (n < 2)
/* x^1 = x */
return mpfr_set (y, x, rnd);
else
/* x^2 = sqr(x) */
return mpfr_sqr (y, x, rnd);
}
/* Augment exponent range */
MPFR_SAVE_EXPO_MARK (expo);
/* setup initial precision */
prec = MPFR_PREC (y) + 3 + BITS_PER_MP_LIMB
+ MPFR_INT_CEIL_LOG2 (MPFR_PREC (y));
mpfr_init2 (res, prec);
rnd1 = MPFR_IS_POS (x) ? GMP_RNDU : GMP_RNDD; /* away */
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
int i;
for (m = n, i = 0; m; i++, m >>= 1)
;
/* now 2^(i-1) <= n < 2^i */
MPFR_ASSERTD (prec > (mpfr_prec_t) i);
err = prec - 1 - (mpfr_prec_t) i;
/* First step: compute square from x */
MPFR_BLOCK (flags,
inexact = mpfr_mul (res, x, x, GMP_RNDU);
MPFR_ASSERTD (i >= 2);
if (n & (1UL << (i-2)))
inexact |= mpfr_mul (res, res, x, rnd1);
for (i -= 3; i >= 0 && !MPFR_BLOCK_EXCEP; i--)
{
inexact |= mpfr_mul (res, res, res, GMP_RNDU);
if (n & (1UL << i))
inexact |= mpfr_mul (res, res, x, rnd1);
});
/* let r(n) be the number of roundings: we have r(2)=1, r(3)=2,
and r(2n)=2r(n)+1, r(2n+1)=2r(n)+2, thus r(n)=n-1.
Using Higham's method, to each rounding corresponds a factor
(1-theta) with 0 <= theta <= 2^(1-p), thus at the end the
absolute error is bounded by (n-1)*2^(1-p)*res <= 2*(n-1)*ulp(res)
since 2^(-p)*x <= ulp(x). Since n < 2^i, this gives a maximal
error of 2^(1+i)*ulp(res).
*/
if (MPFR_LIKELY (inexact == 0
|| MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)
|| MPFR_CAN_ROUND (res, err, MPFR_PREC (y), rnd)))
break;
/* Actualisation of the precision */
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (res, prec);
}
MPFR_ZIV_FREE (loop);
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)))
{
mpz_t z;
/* Internal overflow or underflow. However the approximation error has
* not been taken into account. So, let's solve this problem by using
* mpfr_pow_z, which can handle it. This case could be improved in the
* future, without having to use mpfr_pow_z.
*/
MPFR_LOG_MSG (("Internal overflow or underflow,"
" let's use mpfr_pow_z.\n", 0));
mpfr_clear (res);
MPFR_SAVE_EXPO_FREE (expo);
mpz_init (z);
mpz_set_ui (z, n);
inexact = mpfr_pow_z (y, x, z, rnd);
mpz_clear (z);
return inexact;
}
inexact = mpfr_set (y, res, rnd);
mpfr_clear (res);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
|
