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
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
|
/* Single-precision pow function.
Copyright (c) 2017 ARM Ltd. All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. The name of the company may not be used to endorse or promote
products derived from this software without specific prior written
permission.
THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS AND ANY EXPRESS OR IMPLIED
WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
#include "fdlibm.h"
#if !__OBSOLETE_MATH
#include <math.h>
#include <stdint.h>
#include "math_config.h"
/*
POWF_LOG2_POLY_ORDER = 5
EXP2F_TABLE_BITS = 5
ULP error: 0.82 (~ 0.5 + relerr*2^24)
relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
*/
#define N (1 << POWF_LOG2_TABLE_BITS)
#define T __powf_log2_data.tab
#define A __powf_log2_data.poly
#define OFF 0x3f330000
/* Subnormal input is normalized so ix has negative biased exponent.
Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
static inline double_t
log2_inline (uint32_t ix)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t z, r, r2, r4, p, q, y, y0, invc, logc;
uint32_t iz, top, tmp;
int k, i;
/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
top = tmp & 0xff800000;
iz = ix - top;
k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
invc = T[i].invc;
logc = T[i].logc;
z = (double_t) asfloat (iz);
/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
r = z * invc - 1;
y0 = logc + (double_t) k;
/* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
r2 = r * r;
y = A[0] * r + A[1];
p = A[2] * r + A[3];
r4 = r2 * r2;
q = A[4] * r + y0;
q = p * r2 + q;
y = y * r4 + q;
return y;
}
#undef N
#undef T
#define N (1 << EXP2F_TABLE_BITS)
#define T __exp2f_data.tab
#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
/* The output of log2 and thus the input of exp2 is either scaled by N
(in case of fast toint intrinsics) or not. The unscaled xd must be
in [-1021,1023], sign_bias sets the sign of the result. */
static inline double_t
exp2_inline (double_t xd, unsigned long sign_bias)
{
uint64_t ki, ski, t;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, y, s;
#if TOINT_INTRINSICS
# define C __exp2f_data.poly_scaled
/* N*x = k + r with r in [-1/2, 1/2] */
kd = roundtoint (xd); /* k */
ki = converttoint (xd);
#else
# define C __exp2f_data.poly
# define SHIFT __exp2f_data.shift_scaled
/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */
ki = asuint64 (kd);
kd -= SHIFT; /* k/N */
#endif
r = xd - kd;
/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
t = T[ki % N];
ski = ki + sign_bias;
t += ski << (52 - EXP2F_TABLE_BITS);
s = asdouble (t);
z = C[0] * r + C[1];
r2 = r * r;
y = C[2] * r + 1;
y = z * r2 + y;
y = y * s;
return y;
}
/* Returns 0 if not int, 1 if odd int, 2 if even int. */
static inline int
checkint (uint32_t iy)
{
int e = iy >> 23 & 0xff;
if (e < 0x7f)
return 0;
if (e > 0x7f + 23)
return 2;
if (iy & ((1 << (0x7f + 23 - e)) - 1))
return 0;
if (iy & (1 << (0x7f + 23 - e)))
return 1;
return 2;
}
static inline int
zeroinfnan (uint32_t ix)
{
return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
}
float
powf (float x, float y)
{
unsigned long sign_bias = 0;
uint32_t ix, iy;
ix = asuint (x);
iy = asuint (y);
if (__builtin_expect (ix - 0x00800000 >= 0x7f800000 - 0x00800000
|| zeroinfnan (iy),
0))
{
/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
if (__builtin_expect (zeroinfnan (iy), 0))
{
if (2 * iy == 0)
return issignalingf_inline (x) ? x + y : 1.0f;
if (ix == 0x3f800000)
return issignalingf_inline (y) ? x + y : 1.0f;
if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
return x + y;
if (2 * ix == 2 * 0x3f800000)
return 1.0f;
if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
return y * y;
}
if (__builtin_expect (zeroinfnan (ix), 0))
{
float_t x2 = x * x;
if (ix & 0x80000000 && checkint (iy) == 1)
{
x2 = -x2;
sign_bias = 1;
}
#if WANT_ERRNO
if (2 * ix == 0 && iy & 0x80000000)
return __math_divzerof (sign_bias);
#endif
return iy & 0x80000000 ? 1 / x2 : x2;
}
/* x and y are non-zero finite. */
if (ix & 0x80000000)
{
/* Finite x < 0. */
int yint = checkint (iy);
if (yint == 0)
return __math_invalidf (x);
if (yint == 1)
sign_bias = SIGN_BIAS;
ix &= 0x7fffffff;
}
if (ix < 0x00800000)
{
/* Normalize subnormal x so exponent becomes negative. */
ix = asuint (x * 0x1p23f);
ix &= 0x7fffffff;
ix -= 23 << 23;
}
}
double_t logx = log2_inline (ix);
double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
if (__builtin_expect ((asuint64 (ylogx) >> 47 & 0xffff)
>= asuint64 (126.0 * POWF_SCALE) >> 47,
0))
{
/* |y*log(x)| >= 126. */
if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
return __math_oflowf (sign_bias);
if (ylogx <= -150.0 * POWF_SCALE)
return __math_uflowf (sign_bias);
#if WANT_ERRNO_UFLOW
if (ylogx < -149.0 * POWF_SCALE)
return __math_may_uflowf (sign_bias);
#endif
}
return (float) exp2_inline (ylogx, sign_bias);
}
#endif /* !__OBSOLETE_MATH */
|