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
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
|
/* Function powf vectorized with AVX2.
Copyright (C) 2014-2017 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>
#include "svml_s_powf_data.h"
.text
ENTRY(_ZGVdN8vv_powf_avx2)
/*
ALGORITHM DESCRIPTION:
We are using the next identity : pow(x,y) = 2^(y * log2(x)).
1) log2(x) calculation
Here we use the following formula.
Let |x|=2^k1*X1, where k1 is integer, 1<=X1<2.
Let C ~= 1/ln(2),
Rcp1 ~= 1/X1, X2=Rcp1*X1,
Rcp2 ~= 1/X2, X3=Rcp2*X2,
Rcp3 ~= 1/X3, Rcp3C ~= C/X3.
Then
log2|x| = k1 + log2(1/Rcp1) + log2(1/Rcp2) + log2(C/Rcp3C) +
log2(X1*Rcp1*Rcp2*Rcp3C/C),
where X1*Rcp1*Rcp2*Rcp3C = C*(1+q), q is very small.
The values of Rcp1, log2(1/Rcp1), Rcp2, log2(1/Rcp2),
Rcp3C, log2(C/Rcp3C) are taken from tables.
Values of Rcp1, Rcp2, Rcp3C are such that RcpC=Rcp1*Rcp2*Rcp3C
is exactly represented in target precision.
log2(X1*Rcp1*Rcp2*Rcp3C/C) = log2(1+q) = ln(1+q)/ln2 =
= 1/(ln2)*q - 1/(2ln2)*q^2 + 1/(3ln2)*q^3 - ... =
= 1/(C*ln2)*cq - 1/(2*C^2*ln2)*cq^2 + 1/(3*C^3*ln2)*cq^3 - ... =
= (1 + a1)*cq + a2*cq^2 + a3*cq^3 + ...,
where
cq=X1*Rcp1*Rcp2*Rcp3C-C,
a1=1/(C*ln(2))-1 is small,
a2=1/(2*C^2*ln2),
a3=1/(3*C^3*ln2),
...
Log2 result is split by three parts: HH+HL+HLL
2) Calculation of y*log2(x)
Split y into YHi+YLo.
Get high PH and medium PL parts of y*log2|x|.
Get low PLL part of y*log2|x|.
Now we have PH+PL+PLL ~= y*log2|x|.
3) Calculation of 2^(y*log2(x))
Let's represent PH+PL+PLL in the form N + j/2^expK + Z,
where expK=7 in this implementation, N and j are integers,
0<=j<=2^expK-1, |Z|<2^(-expK-1). Hence
2^(PH+PL+PLL) ~= 2^N * 2^(j/2^expK) * 2^Z,
where 2^(j/2^expK) is stored in a table, and
2^Z ~= 1 + B1*Z + B2*Z^2 ... + B5*Z^5.
We compute 2^(PH+PL+PLL) as follows:
Break PH into PHH + PHL, where PHH = N + j/2^expK.
Z = PHL + PL + PLL
Exp2Poly = B1*Z + B2*Z^2 ... + B5*Z^5
Get 2^(j/2^expK) from table in the form THI+TLO.
Now we have 2^(PH+PL+PLL) ~= 2^N * (THI + TLO) * (1 + Exp2Poly).
Get significand of 2^(PH+PL+PLL) in the form ResHi+ResLo:
ResHi := THI
ResLo := THI * Exp2Poly + TLO
Get exponent ERes of the result:
Res := ResHi + ResLo:
Result := ex(Res) + N. */
pushq %rbp
cfi_adjust_cfa_offset (8)
cfi_rel_offset (%rbp, 0)
movq %rsp, %rbp
cfi_def_cfa_register (%rbp)
andq $-64, %rsp
subq $448, %rsp
lea __VPACK_ODD_ind.6357.0.1(%rip), %rcx
vmovups %ymm14, 320(%rsp)
/* hi bits */
lea __VPACK_ODD_ind.6358.0.1(%rip), %rax
vmovups %ymm12, 256(%rsp)
vmovups %ymm9, 96(%rsp)
vmovups %ymm13, 224(%rsp)
vmovups %ymm15, 352(%rsp)
vmovups %ymm11, 384(%rsp)
vmovups %ymm10, 288(%rsp)
vmovups (%rcx), %ymm10
vmovups %ymm8, 160(%rsp)
vmovdqa %ymm1, %ymm9
movq __svml_spow_data@GOTPCREL(%rip), %rdx
vextractf128 $1, %ymm0, %xmm7
vcvtps2pd %xmm0, %ymm14
vcvtps2pd %xmm7, %ymm12
vpsubd _NMINNORM(%rdx), %ymm0, %ymm7
/* preserve mantissa, set input exponent to 2^(-10) */
vandpd _ExpMask(%rdx), %ymm14, %ymm3
vandpd _ExpMask(%rdx), %ymm12, %ymm13
/* exponent bits selection */
vpsrlq $20, %ymm12, %ymm12
vpsrlq $20, %ymm14, %ymm14
vextractf128 $1, %ymm9, %xmm2
vcvtps2pd %xmm9, %ymm1
vpand _ABSMASK(%rdx), %ymm9, %ymm8
vcvtps2pd %xmm2, %ymm6
vorpd _Two10(%rdx), %ymm3, %ymm2
vorpd _Two10(%rdx), %ymm13, %ymm3
/* reciprocal approximation good to at least 11 bits */
vcvtpd2ps %ymm2, %xmm5
vcvtpd2ps %ymm3, %xmm15
vrcpps %xmm5, %xmm4
vrcpps %xmm15, %xmm11
vcvtps2pd %xmm4, %ymm13
vcvtps2pd %xmm11, %ymm4
vpermps %ymm12, %ymm10, %ymm11
/* round reciprocal to nearest integer, will have 1+9 mantissa bits */
vroundpd $0, %ymm13, %ymm12
vpermps %ymm14, %ymm10, %ymm5
vroundpd $0, %ymm4, %ymm14
vmovupd _One(%rdx), %ymm4
/* table lookup */
vpsrlq $40, %ymm12, %ymm10
vfmsub213pd %ymm4, %ymm12, %ymm2
vfmsub213pd %ymm4, %ymm14, %ymm3
vcmpgt_oqpd _Threshold(%rdx), %ymm12, %ymm12
vxorpd %ymm4, %ymm4, %ymm4
vandpd _Bias(%rdx), %ymm12, %ymm12
/* biased exponent in DP format */
vcvtdq2pd %xmm11, %ymm13
vpcmpeqd %ymm11, %ymm11, %ymm11
vgatherqpd %ymm11, _Log2Rcp_lookup(%rdx,%ymm10), %ymm4
vpsrlq $40, %ymm14, %ymm10
vcmpgt_oqpd _Threshold(%rdx), %ymm14, %ymm14
vpcmpeqd %ymm11, %ymm11, %ymm11
vandpd _Bias(%rdx), %ymm14, %ymm14
vcvtdq2pd %xmm5, %ymm15
vxorpd %ymm5, %ymm5, %ymm5
vgatherqpd %ymm11, _Log2Rcp_lookup(%rdx,%ymm10), %ymm5
vorpd _Bias1(%rdx), %ymm12, %ymm11
vorpd _Bias1(%rdx), %ymm14, %ymm10
vsubpd %ymm11, %ymm15, %ymm11
vsubpd %ymm10, %ymm13, %ymm14
vmovupd _poly_coeff_4(%rdx), %ymm15
vmovupd _poly_coeff_3(%rdx), %ymm13
vmulpd %ymm3, %ymm3, %ymm10
vfmadd213pd %ymm15, %ymm3, %ymm13
vmovdqa %ymm15, %ymm12
vfmadd231pd _poly_coeff_3(%rdx), %ymm2, %ymm12
vmulpd %ymm2, %ymm2, %ymm15
/* reconstruction */
vfmadd213pd %ymm3, %ymm10, %ymm13
vfmadd213pd %ymm2, %ymm15, %ymm12
vaddpd %ymm5, %ymm13, %ymm13
vaddpd %ymm4, %ymm12, %ymm2
vfmadd231pd _L2(%rdx), %ymm14, %ymm13
vfmadd132pd _L2(%rdx), %ymm2, %ymm11
vmulpd %ymm6, %ymm13, %ymm2
vmulpd %ymm1, %ymm11, %ymm10
vmulpd __dbInvLn2(%rdx), %ymm2, %ymm6
vmulpd __dbInvLn2(%rdx), %ymm10, %ymm15
/* to round down; if dR is an integer we will get R = 1, which is ok */
vsubpd __dbHALF(%rdx), %ymm6, %ymm3
vsubpd __dbHALF(%rdx), %ymm15, %ymm1
vaddpd __dbShifter(%rdx), %ymm3, %ymm13
vaddpd __dbShifter(%rdx), %ymm1, %ymm14
vsubpd __dbShifter(%rdx), %ymm13, %ymm12
vmovups (%rax), %ymm1
vsubpd __dbShifter(%rdx), %ymm14, %ymm11
/* [0..1) */
vsubpd %ymm12, %ymm6, %ymm6
vpermps %ymm10, %ymm1, %ymm3
vpermps %ymm2, %ymm1, %ymm10
vpcmpgtd _NMAXVAL(%rdx), %ymm7, %ymm4
vpcmpgtd _INF(%rdx), %ymm8, %ymm1
vpcmpeqd _NMAXVAL(%rdx), %ymm7, %ymm7
vpcmpeqd _INF(%rdx), %ymm8, %ymm8
vpor %ymm7, %ymm4, %ymm2
vpor %ymm8, %ymm1, %ymm1
vsubpd %ymm11, %ymm15, %ymm7
vinsertf128 $1, %xmm10, %ymm3, %ymm10
vpor %ymm1, %ymm2, %ymm3
/* iAbsX = iAbsX&iAbsMask */
vandps __iAbsMask(%rdx), %ymm10, %ymm10
/* iRangeMask = (iAbsX>iDomainRange) */
vpcmpgtd __iDomainRange(%rdx), %ymm10, %ymm4
vpor %ymm4, %ymm3, %ymm5
vmulpd __dbC1(%rdx), %ymm7, %ymm4
vmovmskps %ymm5, %ecx
vmulpd __dbC1(%rdx), %ymm6, %ymm5
/* low K bits */
vandps __lbLOWKBITS(%rdx), %ymm14, %ymm6
/* dpP= _dbT+lJ*T_ITEM_GRAN */
vxorpd %ymm7, %ymm7, %ymm7
vpcmpeqd %ymm1, %ymm1, %ymm1
vandps __lbLOWKBITS(%rdx), %ymm13, %ymm2
vxorpd %ymm10, %ymm10, %ymm10
vpcmpeqd %ymm3, %ymm3, %ymm3
vgatherqpd %ymm1, 13952(%rdx,%ymm6,8), %ymm7
vgatherqpd %ymm3, 13952(%rdx,%ymm2,8), %ymm10
vpsrlq $11, %ymm14, %ymm14
vpsrlq $11, %ymm13, %ymm13
vfmadd213pd %ymm7, %ymm4, %ymm7
vfmadd213pd %ymm10, %ymm5, %ymm10
/* NB : including +/- sign for the exponent!! */
vpsllq $52, %ymm14, %ymm8
vpsllq $52, %ymm13, %ymm11
vpaddq %ymm8, %ymm7, %ymm12
vpaddq %ymm11, %ymm10, %ymm1
vcvtpd2ps %ymm12, %xmm15
vcvtpd2ps %ymm1, %xmm2
vinsertf128 $1, %xmm2, %ymm15, %ymm1
testl %ecx, %ecx
jne .LBL_1_3
.LBL_1_2:
cfi_remember_state
vmovups 160(%rsp), %ymm8
vmovups 96(%rsp), %ymm9
vmovups 288(%rsp), %ymm10
vmovups 384(%rsp), %ymm11
vmovups 256(%rsp), %ymm12
vmovups 224(%rsp), %ymm13
vmovups 320(%rsp), %ymm14
vmovups 352(%rsp), %ymm15
vmovdqa %ymm1, %ymm0
movq %rbp, %rsp
cfi_def_cfa_register (%rsp)
popq %rbp
cfi_adjust_cfa_offset (-8)
cfi_restore (%rbp)
ret
.LBL_1_3:
cfi_restore_state
vmovups %ymm0, 64(%rsp)
vmovups %ymm9, 128(%rsp)
vmovups %ymm1, 192(%rsp)
je .LBL_1_2
xorb %dl, %dl
xorl %eax, %eax
movq %rsi, 8(%rsp)
movq %rdi, (%rsp)
movq %r12, 40(%rsp)
cfi_offset_rel_rsp (12, 40)
movb %dl, %r12b
movq %r13, 32(%rsp)
cfi_offset_rel_rsp (13, 32)
movl %ecx, %r13d
movq %r14, 24(%rsp)
cfi_offset_rel_rsp (14, 24)
movl %eax, %r14d
movq %r15, 16(%rsp)
cfi_offset_rel_rsp (15, 16)
cfi_remember_state
.LBL_1_6:
btl %r14d, %r13d
jc .LBL_1_12
.LBL_1_7:
lea 1(%r14), %esi
btl %esi, %r13d
jc .LBL_1_10
.LBL_1_8:
incb %r12b
addl $2, %r14d
cmpb $16, %r12b
jb .LBL_1_6
movq 8(%rsp), %rsi
movq (%rsp), %rdi
movq 40(%rsp), %r12
cfi_restore (%r12)
movq 32(%rsp), %r13
cfi_restore (%r13)
movq 24(%rsp), %r14
cfi_restore (%r14)
movq 16(%rsp), %r15
cfi_restore (%r15)
vmovups 192(%rsp), %ymm1
jmp .LBL_1_2
.LBL_1_10:
cfi_restore_state
movzbl %r12b, %r15d
vmovss 68(%rsp,%r15,8), %xmm0
vmovss 132(%rsp,%r15,8), %xmm1
vzeroupper
call JUMPTARGET(__powf_finite)
vmovss %xmm0, 196(%rsp,%r15,8)
jmp .LBL_1_8
.LBL_1_12:
movzbl %r12b, %r15d
vmovss 64(%rsp,%r15,8), %xmm0
vmovss 128(%rsp,%r15,8), %xmm1
vzeroupper
call JUMPTARGET(__powf_finite)
vmovss %xmm0, 192(%rsp,%r15,8)
jmp .LBL_1_7
END(_ZGVdN8vv_powf_avx2)
.section .rodata, "a"
__VPACK_ODD_ind.6357.0.1:
.long 1
.long 3
.long 5
.long 7
.long 0
.long 0
.long 0
.long 0
.space 32, 0x00
__VPACK_ODD_ind.6358.0.1:
.long 1
.long 3
.long 5
.long 7
.long 0
.long 0
.long 0
.long 0
|