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
|
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2013 Free Software Foundation, Inc.
*
* This program 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.
*
* This program 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 this program; if not, see <http://www.gnu.org/licenses/>.
*/
/*************************************************************************/
/* MODULE_NAME:slowpow.c */
/* */
/* FUNCTION:slowpow */
/* */
/*FILES NEEDED:mpa.h */
/* mpa.c mpexp.c mplog.c halfulp.c */
/* */
/* Given two IEEE double machine numbers y,x , routine computes the */
/* correctly rounded (to nearest) value of x^y. Result calculated by */
/* multiplication (in halfulp.c) or if result isn't accurate enough */
/* then routine converts x and y into multi-precision doubles and */
/* calls to mpexp routine */
/*************************************************************************/
#include "mpa.h"
#include <math_private.h>
#ifndef SECTION
# define SECTION
#endif
void __mpexp (mp_no *x, mp_no *y, int p);
void __mplog (mp_no *x, mp_no *y, int p);
double ulog (double);
double __halfulp (double x, double y);
double
SECTION
__slowpow (double x, double y, double z)
{
double res, res1;
mp_no mpx, mpy, mpz, mpw, mpp, mpr, mpr1;
static const mp_no eps = {-3, {1.0, 4.0}};
int p;
/* __HALFULP returns -10 or X^Y. */
res = __halfulp (x, y);
/* Return if the result was computed by __HALFULP. */
if (res >= 0)
return res;
/* Compute pow as long double. This is currently only used by powerpc, where
one may get 106 bits of accuracy. */
#ifdef USE_LONG_DOUBLE_FOR_MP
long double ldw, ldz, ldpp;
static const long double ldeps = 0x4.0p-96;
ldz = __ieee754_logl ((long double) x);
ldw = (long double) y *ldz;
ldpp = __ieee754_expl (ldw);
res = (double) (ldpp + ldeps);
res1 = (double) (ldpp - ldeps);
/* Return the result if it is accurate enough. */
if (res == res1)
return res;
#endif
/* Or else, calculate using multiple precision. P = 10 implies accuracy of
240 bits accuracy, since MP_NO has a radix of 2^24. */
p = 10;
__dbl_mp (x, &mpx, p);
__dbl_mp (y, &mpy, p);
__dbl_mp (z, &mpz, p);
/* z = x ^ y
log (z) = y * log (x)
z = exp (y * log (x)) */
__mplog (&mpx, &mpz, p);
__mul (&mpy, &mpz, &mpw, p);
__mpexp (&mpw, &mpp, p);
/* Add and subtract EPS to ensure that the result remains unchanged, i.e. we
have last bit accuracy. */
__add (&mpp, &eps, &mpr, p);
__mp_dbl (&mpr, &res, p);
__sub (&mpp, &eps, &mpr1, p);
__mp_dbl (&mpr1, &res1, p);
if (res == res1)
return res;
/* If we don't, then we repeat using a higher precision. 768 bits of
precision ought to be enough for anybody. */
p = 32;
__dbl_mp (x, &mpx, p);
__dbl_mp (y, &mpy, p);
__dbl_mp (z, &mpz, p);
__mplog (&mpx, &mpz, p);
__mul (&mpy, &mpz, &mpw, p);
__mpexp (&mpw, &mpp, p);
__mp_dbl (&mpp, &res, p);
return res;
}
|
