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
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
|
/* powi.c
*
* Real raised to integer power
*
*
*
* SYNOPSIS:
*
* double x, y, powi();
* int n;
*
* y = powi( x, n );
*
*
*
* DESCRIPTION:
*
* Returns argument x raised to the nth power.
* The routine efficiently decomposes n as a sum of powers of
* two. The desired power is a product of two-to-the-kth
* powers of x. Thus to compute the 32767 power of x requires
* 28 multiplications instead of 32767 multiplications.
*
*
*
* ACCURACY:
*
*
* Relative error:
* arithmetic x domain n domain # trials peak rms
* DEC .04,26 -26,26 100000 2.7e-16 4.3e-17
* IEEE .04,26 -26,26 50000 2.0e-15 3.8e-16
* IEEE 1,2 -1022,1023 50000 8.6e-14 1.6e-14
*
* Returns MAXNUM on overflow, zero on underflow.
*
*/
/* powi.c */
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/
#include "mconf.h"
extern double NEGZERO, MAXNUM, MAXLOG, MINLOG, LOGE2;
double powi( x, nn )
double x;
int nn;
{
int n, e, sign, asign, lx;
double w, y, s;
/* See pow.c for these tests. */
if( x == 0.0 )
{
if( nn == 0 )
return( 1.0 );
else if( nn < 0 )
return( NPY_INFINITY );
else
{
if( nn & 1 )
return( x );
else
return( 0.0 );
}
}
if( nn == 0 )
return( 1.0 );
if( nn == -1 )
return( 1.0/x );
if( x < 0.0 )
{
asign = -1;
x = -x;
}
else
asign = 0;
if( nn < 0 )
{
sign = -1;
n = -nn;
}
else
{
sign = 1;
n = nn;
}
/* Even power will be positive. */
if( (n & 1) == 0 )
asign = 0;
/* Overflow detection */
/* Calculate approximate logarithm of answer */
s = frexp( x, &lx );
e = (lx - 1)*n;
if( (e == 0) || (e > 64) || (e < -64) )
{
s = (s - 7.0710678118654752e-1) / (s + 7.0710678118654752e-1);
s = (2.9142135623730950 * s - 0.5 + lx) * nn * LOGE2;
}
else
{
s = LOGE2 * e;
}
if( s > MAXLOG )
{
mtherr( "powi", OVERFLOW );
y = NPY_INFINITY;
goto done;
}
#if DENORMAL
if( s < MINLOG )
{
y = 0.0;
goto done;
}
/* Handle tiny denormal answer, but with less accuracy
* since roundoff error in 1.0/x will be amplified.
* The precise demarcation should be the gradual underflow threshold.
*/
if( (s < (-MAXLOG+2.0)) && (sign < 0) )
{
x = 1.0/x;
sign = -sign;
}
#else
/* do not produce denormal answer */
if( s < -MAXLOG )
return(0.0);
#endif
/* First bit of the power */
if( n & 1 )
y = x;
else
y = 1.0;
w = x;
n >>= 1;
while( n )
{
w = w * w; /* arg to the 2-to-the-kth power */
if( n & 1 ) /* if that bit is set, then include in product */
y *= w;
n >>= 1;
}
if( sign < 0 )
y = 1.0/y;
done:
if( asign )
{
/* odd power of negative number */
if( y == 0.0 )
y = NEGZERO;
else
y = -y;
}
return(y);
}
|
