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
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
|
/* Function pow vectorized with AVX2.
Copyright (C) 2014-2016 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>
#include "svml_d_pow_data.h"
.text
ENTRY (_ZGVdN4vv_pow_avx2)
/*
ALGORITHM DESCRIPTION:
1) Calculating log2|x|
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),
...
We get 3 parts of log2 result: HH+HL+HLL ~= log2|x|.
2) Calculation of y*(HH+HL+HLL).
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^(PH+PL+PLL).
Mathematical idea of computing 2^(PH+PL+PLL) is the following.
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
movq __svml_dpow_data@GOTPCREL(%rip), %rax
vmovups %ymm11, 160(%rsp)
vmovups %ymm8, 224(%rsp)
vmovups %ymm10, 352(%rsp)
vmovups %ymm9, 384(%rsp)
vmovups %ymm13, 288(%rsp)
vmovapd %ymm1, %ymm11
vxorpd %ymm1, %ymm1, %ymm1
vextracti128 $1, %ymm0, %xmm5
vshufps $221, %xmm5, %xmm0, %xmm5
/* i = (((Hi(x) & 0x000ffe00) + 0x00000200) >> 10); -> i = (b1..b11 + 1) / 2 */
vandps _iIndexMask(%rax), %xmm5, %xmm3
vpaddd _iIndexAdd(%rax), %xmm3, %xmm6
vpsrld $10, %xmm6, %xmm8
/* Index for reciprocal table */
vpslld $3, %xmm8, %xmm9
/* Index for log2 table */
vpslld $4, %xmm8, %xmm6
/* x1 = x; Hi(x1) = (Hi(x1)&0x000fffff)|0x3ff00000 */
vandpd _iMantissaMask(%rax), %ymm0, %ymm4
vorpd _dbOne(%rax), %ymm4, %ymm13
vpcmpeqd %ymm4, %ymm4, %ymm4
vpcmpeqd %ymm8, %ymm8, %ymm8
/* k = Hi(x); k = k - 0x3fe7fe00; k = k >> 20 */
vpsubd _i3fe7fe00(%rax), %xmm5, %xmm3
vpaddd _HIDELTA(%rax), %xmm5, %xmm5
vextracti128 $1, %ymm11, %xmm7
vshufps $221, %xmm7, %xmm11, %xmm2
vpand _ABSMASK(%rax), %xmm2, %xmm10
vpcmpeqd %ymm2, %ymm2, %ymm2
vgatherdpd %ymm2, 11712(%rax,%xmm9), %ymm1
vmovups _LORANGE(%rax), %xmm7
vxorpd %ymm2, %ymm2, %ymm2
vgatherdpd %ymm4, 19968(%rax,%xmm6), %ymm2
vxorpd %ymm4, %ymm4, %ymm4
vgatherdpd %ymm8, 19976(%rax,%xmm6), %ymm4
vpsrad $20, %xmm3, %xmm6
vpaddd _i2p20_2p19(%rax), %xmm6, %xmm9
vpshufd $80, %xmm9, %xmm8
vpshufd $250, %xmm9, %xmm3
/* x1Hi=x1; Lo(x1Hi)&=0xf8000000; x1Lo = x1-x1Hi */
vandpd _iHighMask(%rax), %ymm13, %ymm9
vinserti128 $1, %xmm3, %ymm8, %ymm6
vandpd _iffffffff00000000(%rax), %ymm6, %ymm8
/* r1 = x1*rcp1 */
vmulpd %ymm1, %ymm13, %ymm6
vsubpd %ymm9, %ymm13, %ymm3
vsubpd _db2p20_2p19(%rax), %ymm8, %ymm8
/* cq = c+r1 */
vaddpd _LHN(%rax), %ymm6, %ymm13
/* E = -r1+__fence(x1Hi*rcp1) */
vfmsub213pd %ymm6, %ymm1, %ymm9
/* E=E+x1Lo*rcp1 */
vfmadd213pd %ymm9, %ymm1, %ymm3
/* T = k + L1hi */
vaddpd %ymm2, %ymm8, %ymm1
/* T_Rh = T + cq */
vaddpd %ymm13, %ymm1, %ymm8
/* Rl = T-T_Rh; -> -Rh */
vsubpd %ymm8, %ymm1, %ymm6
/* Rl=Rl+cq */
vaddpd %ymm6, %ymm13, %ymm1
/* T_Rh_Eh = T_Rh + E */
vaddpd %ymm3, %ymm8, %ymm6
/* cq = cq + E */
vaddpd %ymm3, %ymm13, %ymm13
/* HLL = T_Rh - T_Rh_Eh; -> -Eh */
vsubpd %ymm6, %ymm8, %ymm9
/* HLL+=E; -> El */
vaddpd %ymm9, %ymm3, %ymm2
/* HLL+=Rl */
vaddpd %ymm1, %ymm2, %ymm8
/* HLL+=L1lo */
vaddpd %ymm4, %ymm8, %ymm4
vmovupd _clv_2(%rax), %ymm8
/* HLL = HLL + (((((((a7)*cq+a6)*cq+a5)*cq+a4)*cq+a3)*cq+a2)*cq+a1)*cq */
vfmadd213pd _clv_3(%rax), %ymm13, %ymm8
vfmadd213pd _clv_4(%rax), %ymm13, %ymm8
vfmadd213pd _clv_5(%rax), %ymm13, %ymm8
vfmadd213pd _clv_6(%rax), %ymm13, %ymm8
vfmadd213pd _clv_7(%rax), %ymm13, %ymm8
vfmadd213pd %ymm4, %ymm13, %ymm8
/* T_Rh_Eh_HLLhi = T_Rh_Eh + HLL */
vaddpd %ymm8, %ymm6, %ymm9
/* HH = T_Rh_Eh_HLLhi; Lo(HH)&=0xf8000000 */
vandpd _iHighMask(%rax), %ymm9, %ymm2
/*
2^(y*(HH+HL+HLL)) starts here:
yH = y; Lo(yH)&=0xf8000000;
*/
vandpd _iHighMask(%rax), %ymm11, %ymm1
/* HLLhi = T_Rh_Eh_HLLhi - T_Rh_Eh */
vsubpd %ymm6, %ymm9, %ymm13
/* HL = T_Rh_Eh_HLLhi-HH */
vsubpd %ymm2, %ymm9, %ymm4
/* pH = yH*HH */
vmulpd %ymm2, %ymm1, %ymm9
/* HLL = HLL - HLLhi */
vsubpd %ymm13, %ymm8, %ymm6
/* yL = y-yH */
vsubpd %ymm1, %ymm11, %ymm8
vextracti128 $1, %ymm9, %xmm3
vshufps $221, %xmm3, %xmm9, %xmm13
vpand _ABSMASK(%rax), %xmm13, %xmm3
vpcmpgtd %xmm5, %xmm7, %xmm13
vpcmpgtd _INF(%rax), %xmm10, %xmm7
vpcmpeqd _INF(%rax), %xmm10, %xmm10
vpor %xmm10, %xmm7, %xmm7
vpor %xmm7, %xmm13, %xmm5
/* pL=yL*HL+yH*HL; pL+=yL*HH */
vmulpd %ymm4, %ymm8, %ymm7
vpcmpgtd _DOMAINRANGE(%rax), %xmm3, %xmm13
vpcmpeqd _DOMAINRANGE(%rax), %xmm3, %xmm10
vpor %xmm10, %xmm13, %xmm3
vpor %xmm3, %xmm5, %xmm13
vfmadd213pd %ymm7, %ymm4, %ymm1
/* pLL = y*HLL;
pHH = pH + *(double*)&db2p45_2p44
*/
vaddpd _db2p45_2p44(%rax), %ymm9, %ymm7
vmovmskps %xmm13, %ecx
vfmadd213pd %ymm1, %ymm2, %ymm8
/* t=pL+pLL; t+=pHL */
vfmadd231pd %ymm11, %ymm6, %ymm8
vextracti128 $1, %ymm7, %xmm1
vshufps $136, %xmm1, %xmm7, %xmm10
/* _n = Lo(pHH);
_n = _n & 0xffffff80;
_n = _n >> 7;
Hi(_2n) = (0x3ff+_n)<<20; Lo(_2n) = 0; -> 2^n
*/
vpslld $13, %xmm10, %xmm2
vpaddd _iOne(%rax), %xmm2, %xmm13
vpshufd $80, %xmm13, %xmm4
vpshufd $250, %xmm13, %xmm1
/* j = Lo(pHH)&0x0000007f */
vandps _jIndexMask(%rax), %xmm10, %xmm3
/* T1 = ((double*)exp2_tbl)[ 2*j ] */
vpcmpeqd %ymm10, %ymm10, %ymm10
vpslld $4, %xmm3, %xmm5
/* pHH = pHH - *(double*)&db2p45_2p44 */
vsubpd _db2p45_2p44(%rax), %ymm7, %ymm7
/* pHL = pH - pHH */
vsubpd %ymm7, %ymm9, %ymm9
vaddpd %ymm9, %ymm8, %ymm6
vinserti128 $1, %xmm1, %ymm4, %ymm2
vxorpd %ymm1, %ymm1, %ymm1
vgatherdpd %ymm10, 36416(%rax,%xmm5), %ymm1
vandpd _ifff0000000000000(%rax), %ymm2, %ymm13
vmovupd _cev_1(%rax), %ymm2
vmulpd %ymm1, %ymm13, %ymm1
vfmadd213pd _cev_2(%rax), %ymm6, %ymm2
vmulpd %ymm6, %ymm1, %ymm8
vfmadd213pd _cev_3(%rax), %ymm6, %ymm2
vfmadd213pd _cev_4(%rax), %ymm6, %ymm2
vfmadd213pd _cev_5(%rax), %ymm6, %ymm2
vfmadd213pd %ymm1, %ymm8, %ymm2
testl %ecx, %ecx
jne .LBL_1_3
.LBL_1_2:
cfi_remember_state
vmovups 224(%rsp), %ymm8
vmovups 384(%rsp), %ymm9
vmovups 352(%rsp), %ymm10
vmovups 160(%rsp), %ymm11
vmovups 288(%rsp), %ymm13
vmovdqa %ymm2, %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
vmovupd %ymm0, 192(%rsp)
vmovupd %ymm11, 256(%rsp)
vmovupd %ymm2, 320(%rsp)
je .LBL_1_2
xorb %dl, %dl
xorl %eax, %eax
vmovups %ymm12, 64(%rsp)
vmovups %ymm14, 32(%rsp)
vmovups %ymm15, (%rsp)
movq %rsi, 104(%rsp)
movq %rdi, 96(%rsp)
movq %r12, 136(%rsp)
cfi_offset_rel_rsp (12, 136)
movb %dl, %r12b
movq %r13, 128(%rsp)
cfi_offset_rel_rsp (13, 128)
movl %ecx, %r13d
movq %r14, 120(%rsp)
cfi_offset_rel_rsp (14, 120)
movl %eax, %r14d
movq %r15, 112(%rsp)
cfi_offset_rel_rsp (15, 112)
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
vmovups 64(%rsp), %ymm12
vmovups 32(%rsp), %ymm14
vmovups (%rsp), %ymm15
vmovupd 320(%rsp), %ymm2
movq 104(%rsp), %rsi
movq 96(%rsp), %rdi
movq 136(%rsp), %r12
cfi_restore (%r12)
movq 128(%rsp), %r13
cfi_restore (%r13)
movq 120(%rsp), %r14
cfi_restore (%r14)
movq 112(%rsp), %r15
cfi_restore (%r15)
jmp .LBL_1_2
.LBL_1_10:
cfi_restore_state
movzbl %r12b, %r15d
shlq $4, %r15
vmovsd 200(%rsp,%r15), %xmm0
vmovsd 264(%rsp,%r15), %xmm1
vzeroupper
call pow@PLT
vmovsd %xmm0, 328(%rsp,%r15)
jmp .LBL_1_8
.LBL_1_12:
movzbl %r12b, %r15d
shlq $4, %r15
vmovsd 192(%rsp,%r15), %xmm0
vmovsd 256(%rsp,%r15), %xmm1
vzeroupper
call pow@PLT
vmovsd %xmm0, 320(%rsp,%r15)
jmp .LBL_1_7
END (_ZGVdN4vv_pow_avx2)
|
