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
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
|
/* Optimized expf(). PowerPC64/POWER8 version.
Copyright (C) 2016 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C 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 C 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 C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <sysdep.h>
/* Short algorithm description:
*
* Let K = 64 (table size).
* e^x = 2^(x/log(2)) = 2^n * T[j] * (1 + P(y))
* where:
* x = m*log(2)/K + y, y in [0.0..log(2)/K]
* m = n*K + j, m,n,j - signed integer, j in [0..K-1]
* values of 2^(j/K) are tabulated as T[j].
*
* P(y) is a minimax polynomial approximation of expf(y)-1
* on small interval [0.0..log(2)/K].
*
* P(y) = P3*y*y*y*y + P2*y*y*y + P1*y*y + P0*y, calculated as
* z = y*y; P(y) = (P3*z + P1)*z + (P2*z + P0)*y
*
* Special cases:
* expf(NaN) = NaN
* expf(+INF) = +INF
* expf(-INF) = 0
* expf(x) = 1 for subnormals
* for finite argument, only expf(0)=1 is exact
* expf(x) overflows if x>88.7228317260742190
* expf(x) underflows if x<-103.972076416015620
*/
#define C1 0x42ad496b /* Single precision 125*log(2). */
#define C2 0x31800000 /* Single precision 2^(-28). */
#define SP_INF 0x7f800000 /* Single precision Inf. */
#define SP_EXP_BIAS 0x1fc0 /* Single precision exponent bias. */
#define DATA_OFFSET r9
/* Implements the function
float [fp1] expf (float [fp1] x) */
.machine power8
EALIGN(__ieee754_expf, 4, 0)
addis DATA_OFFSET,r2,.Lanchor@toc@ha
addi DATA_OFFSET,DATA_OFFSET,.Lanchor@toc@l
xscvdpspn v0,v1
mfvsrd r8,v0 /* r8 = x */
lfd fp2,(.KLN2-.Lanchor)(DATA_OFFSET)
lfd fp3,(.P2-.Lanchor)(DATA_OFFSET)
rldicl r3,r8,32,33 /* r3 = |x| */
lis r4,C1@ha /* r4 = 125*log(2) */
ori r4,r4,C1@l
cmpw r3,r4
lfd fp5,(.P3-.Lanchor)(DATA_OFFSET)
lfd fp4,(.RS-.Lanchor)(DATA_OFFSET)
fmadd fp2,fp1,fp2,fp4 /* fp2 = x * K/log(2) + (2^23 + 2^22) */
bge L(special_paths) /* |x| >= 125*log(2) ? */
lis r4,C2@ha
ori r4,r4,C2@l
cmpw r3,r4
blt L(small_args) /* |x| < 2^(-28) ? */
/* Main path: here if 2^(-28) <= |x| < 125*log(2) */
frsp fp6,fp2
xscvdpsp v2,v2
mfvsrd r8,v2
mr r3,r8 /* r3 = m */
rldicl r8,r8,32,58 /* r8 = j */
lfs fp4,(.SP_RS-.Lanchor)(DATA_OFFSET)
fsubs fp2,fp6,fp4 /* fp2 = m = x * K/log(2) */
srdi r3,r3,32
clrrwi r3,r3,6 /* r3 = n */
lfd fp6,(.NLN2K-.Lanchor)(DATA_OFFSET)
fmadd fp0,fp2,fp6,fp1 /* fp0 = y = x - m*log(2)/K */
fmul fp2,fp0,fp0 /* fp2 = z = y^2 */
lfd fp4,(.P1-.Lanchor)(DATA_OFFSET)
lfd fp6,(.P0-.Lanchor)(DATA_OFFSET)
lis r4,SP_EXP_BIAS@ha
ori r4,r4,SP_EXP_BIAS@l
add r3,r3,r4
rldic r3,r3,49,1 /* r3 = 2^n */
fmadd fp4,fp5,fp2,fp4 /* fp4 = P3 * z + P1 */
fmadd fp6,fp3,fp2,fp6 /* fp6 = P2 * z + P0 */
mtvsrd v1,r3
xscvspdp v1,v1
fmul fp4,fp4,fp2 /* fp4 = (P3 * z + P1)*z */
fmadd fp0,fp0,fp6,fp4 /* fp0 = P(y) */
sldi r8,r8,3 /* Access doublewords from T[j]. */
addi r6,DATA_OFFSET,(.Ttable-.Lanchor)
lfdx fp3,r6,r8
fmadd fp0,fp0,fp3,fp3 /* fp0 = T[j] * (1 + P(y)) */
fmul fp1,fp1,fp0 /* fp1 = 2^n * T[j] * (1 + P(y)) */
frsp fp1,fp1
blr
.align 4
/* x is either underflow, overflow, infinite or NaN. */
L(special_paths):
srdi r8,r8,32
rlwinm r8,r8,3,29,29 /* r8 = 0, if x positive.
r8 = 4, otherwise. */
addi r6,DATA_OFFSET,(.SPRANGE-.Lanchor)
lwzx r4,r6,r8 /* r4 = .SPRANGE[signbit(x)] */
cmpw r3,r4
/* |x| <= .SPRANGE[signbit(x)] */
ble L(near_under_or_overflow)
lis r4,SP_INF@ha
ori r4,r4,SP_INF@l
cmpw r3,r4
bge L(arg_inf_or_nan) /* |x| > Infinite ? */
addi r6,DATA_OFFSET,(.SPLARGE_SMALL-.Lanchor)
lfsx fp1,r6,r8
fmuls fp1,fp1,fp1
blr
.align 4
L(small_args):
/* expf(x) = 1.0, where |x| < |2^(-28)| */
lfs fp2,(.SPone-.Lanchor)(DATA_OFFSET)
fadds fp1,fp1,fp2
blr
.align 4
L(arg_inf_or_nan:)
bne L(arg_nan)
/* expf(+INF) = +INF
expf(-INF) = 0 */
addi r6,DATA_OFFSET,(.INF_ZERO-.Lanchor)
lfsx fp1,r6,r8
blr
.align 4
L(arg_nan):
/* expf(NaN) = NaN */
fadd fp1,fp1,fp1
frsp fp1,fp1
blr
.align 4
L(near_under_or_overflow):
frsp fp6,fp2
xscvdpsp v2,v2
mfvsrd r8,v2
mr r3,r8 /* r3 = m */
rldicl r8,r8,32,58 /* r8 = j */
lfs fp4,(.SP_RS-.Lanchor)(DATA_OFFSET)
fsubs fp2,fp6,fp4 /* fp2 = m = x * K/log(2) */
srdi r3,r3,32
clrrwi r3,r3,6 /* r3 = n */
lfd fp6,(.NLN2K-.Lanchor)(DATA_OFFSET)
fmadd fp0,fp2,fp6,fp1 /* fp0 = y = x - m*log(2)/K */
fmul fp2,fp0,fp0 /* fp2 = z = y^2 */
lfd fp4,(.P1-.Lanchor)(DATA_OFFSET)
lfd fp6,(.P0-.Lanchor)(DATA_OFFSET)
ld r4,(.DP_EXP_BIAS-.Lanchor)(DATA_OFFSET)
add r3,r3,r4
rldic r3,r3,46,1 /* r3 = 2 */
fmadd fp4,fp5,fp2,fp4 /* fp4 = P3 * z + P1 */
fmadd fp6,fp3,fp2,fp6 /* fp6 = P2 * z + P0 */
mtvsrd v1,r3
fmul fp4,fp4,fp2 /* fp4 = (P3*z + P1)*z */
fmadd fp0,fp0,fp6,fp4 /* fp0 = P(y) */
sldi r8,r8,3 /* Access doublewords from T[j]. */
addi r6,DATA_OFFSET,(.Ttable-.Lanchor)
lfdx fp3,r6,r8
fmadd fp0,fp0,fp3,fp3 /* fp0 = T[j] * (1 + T[j]) */
fmul fp1,fp1,fp0 /* fp1 = 2^n * T[j] * (1 + T[j]) */
frsp fp1,fp1
blr
END(__ieee754_expf)
.section .rodata, "a",@progbits
.Lanchor:
.balign 8
/* Table T[j] = 2^(j/K). Double precision. */
.Ttable:
.8byte 0x3ff0000000000000
.8byte 0x3ff02c9a3e778061
.8byte 0x3ff059b0d3158574
.8byte 0x3ff0874518759bc8
.8byte 0x3ff0b5586cf9890f
.8byte 0x3ff0e3ec32d3d1a2
.8byte 0x3ff11301d0125b51
.8byte 0x3ff1429aaea92de0
.8byte 0x3ff172b83c7d517b
.8byte 0x3ff1a35beb6fcb75
.8byte 0x3ff1d4873168b9aa
.8byte 0x3ff2063b88628cd6
.8byte 0x3ff2387a6e756238
.8byte 0x3ff26b4565e27cdd
.8byte 0x3ff29e9df51fdee1
.8byte 0x3ff2d285a6e4030b
.8byte 0x3ff306fe0a31b715
.8byte 0x3ff33c08b26416ff
.8byte 0x3ff371a7373aa9cb
.8byte 0x3ff3a7db34e59ff7
.8byte 0x3ff3dea64c123422
.8byte 0x3ff4160a21f72e2a
.8byte 0x3ff44e086061892d
.8byte 0x3ff486a2b5c13cd0
.8byte 0x3ff4bfdad5362a27
.8byte 0x3ff4f9b2769d2ca7
.8byte 0x3ff5342b569d4f82
.8byte 0x3ff56f4736b527da
.8byte 0x3ff5ab07dd485429
.8byte 0x3ff5e76f15ad2148
.8byte 0x3ff6247eb03a5585
.8byte 0x3ff6623882552225
.8byte 0x3ff6a09e667f3bcd
.8byte 0x3ff6dfb23c651a2f
.8byte 0x3ff71f75e8ec5f74
.8byte 0x3ff75feb564267c9
.8byte 0x3ff7a11473eb0187
.8byte 0x3ff7e2f336cf4e62
.8byte 0x3ff82589994cce13
.8byte 0x3ff868d99b4492ed
.8byte 0x3ff8ace5422aa0db
.8byte 0x3ff8f1ae99157736
.8byte 0x3ff93737b0cdc5e5
.8byte 0x3ff97d829fde4e50
.8byte 0x3ff9c49182a3f090
.8byte 0x3ffa0c667b5de565
.8byte 0x3ffa5503b23e255d
.8byte 0x3ffa9e6b5579fdbf
.8byte 0x3ffae89f995ad3ad
.8byte 0x3ffb33a2b84f15fb
.8byte 0x3ffb7f76f2fb5e47
.8byte 0x3ffbcc1e904bc1d2
.8byte 0x3ffc199bdd85529c
.8byte 0x3ffc67f12e57d14b
.8byte 0x3ffcb720dcef9069
.8byte 0x3ffd072d4a07897c
.8byte 0x3ffd5818dcfba487
.8byte 0x3ffda9e603db3285
.8byte 0x3ffdfc97337b9b5f
.8byte 0x3ffe502ee78b3ff6
.8byte 0x3ffea4afa2a490da
.8byte 0x3ffefa1bee615a27
.8byte 0x3fff50765b6e4540
.8byte 0x3fffa7c1819e90d8
.KLN2:
.8byte 0x40571547652b82fe /* Double precision K/log(2). */
/* Double precision polynomial coefficients. */
.P0:
.8byte 0x3fefffffffffe7c6
.P1:
.8byte 0x3fe00000008d6118
.P2:
.8byte 0x3fc55550da752d4f
.P3:
.8byte 0x3fa56420eb78fa85
.RS:
.8byte 0x4168000000000000 /* Double precision 2^23 + 2^22. */
.NLN2K:
.8byte 0xbf862e42fefa39ef /* Double precision -log(2)/K. */
.DP_EXP_BIAS:
.8byte 0x000000000000ffc0 /* Double precision exponent bias. */
.balign 4
.SPone:
.4byte 0x3f800000 /* Single precision 1.0. */
.SP_RS:
.4byte 0x4b400000 /* Single precision 2^23 + 2^22. */
.SPRANGE: /* Single precision overflow/underflow bounds. */
.4byte 0x42b17217 /* if x>this bound, then result overflows. */
.4byte 0x42cff1b4 /* if x<this bound, then result underflows. */
.SPLARGE_SMALL:
.4byte 0x71800000 /* 2^100. */
.4byte 0x0d800000 /* 2^-100. */
.INF_ZERO:
.4byte 0x7f800000 /* Single precision Inf. */
.4byte 0 /* Single precision zero. */
strong_alias (__ieee754_expf, __expf_finite)
|
