1/* Function pow vectorized with AVX2.
2 Copyright (C) 2014-2024 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19#include <sysdep.h>
20#include "svml_d_pow_data.h"
21
22 .section .text.avx2, "ax", @progbits
23ENTRY (_ZGVdN4vv_pow_avx2)
24/*
25 ALGORITHM DESCRIPTION:
26
27 1) Calculating log2|x|
28 Here we use the following formula.
29 Let |x|=2^k1*X1, where k1 is integer, 1<=X1<2.
30 Let C ~= 1/ln(2),
31 Rcp1 ~= 1/X1, X2=Rcp1*X1,
32 Rcp2 ~= 1/X2, X3=Rcp2*X2,
33 Rcp3 ~= 1/X3, Rcp3C ~= C/X3.
34 Then
35 log2|x| = k1 + log2(1/Rcp1) + log2(1/Rcp2) + log2(C/Rcp3C) +
36 log2(X1*Rcp1*Rcp2*Rcp3C/C),
37 where X1*Rcp1*Rcp2*Rcp3C = C*(1+q), q is very small.
38
39 The values of Rcp1, log2(1/Rcp1), Rcp2, log2(1/Rcp2),
40 Rcp3C, log2(C/Rcp3C) are taken from tables.
41 Values of Rcp1, Rcp2, Rcp3C are such that RcpC=Rcp1*Rcp2*Rcp3C
42 is exactly represented in target precision.
43
44 log2(X1*Rcp1*Rcp2*Rcp3C/C) = log2(1+q) = ln(1+q)/ln2 =
45 = 1/(ln2)*q - 1/(2ln2)*q^2 + 1/(3ln2)*q^3 - ... =
46 = 1/(C*ln2)*cq - 1/(2*C^2*ln2)*cq^2 + 1/(3*C^3*ln2)*cq^3 - ... =
47 = (1 + a1)*cq + a2*cq^2 + a3*cq^3 + ...,
48 where cq = X1*Rcp1*Rcp2*Rcp3C-C,
49 a1=1/(C*ln(2))-1 is small,
50 a2=1/(2*C^2*ln2),
51 a3=1/(3*C^3*ln2),
52 ...
53 We get 3 parts of log2 result: HH+HL+HLL ~= log2|x|.
54
55 2) Calculation of y*(HH+HL+HLL).
56 Split y into YHi+YLo.
57 Get high PH and medium PL parts of y*log2|x|.
58 Get low PLL part of y*log2|x|.
59 Now we have PH+PL+PLL ~= y*log2|x|.
60
61 3) Calculation of 2^(PH+PL+PLL).
62 Mathematical idea of computing 2^(PH+PL+PLL) is the following.
63 Let's represent PH+PL+PLL in the form N + j/2^expK + Z,
64 where expK=7 in this implementation, N and j are integers,
65 0<=j<=2^expK-1, |Z|<2^(-expK-1).
66 Hence 2^(PH+PL+PLL) ~= 2^N * 2^(j/2^expK) * 2^Z,
67 where 2^(j/2^expK) is stored in a table, and
68 2^Z ~= 1 + B1*Z + B2*Z^2 ... + B5*Z^5.
69
70 We compute 2^(PH+PL+PLL) as follows.
71 Break PH into PHH + PHL, where PHH = N + j/2^expK.
72 Z = PHL + PL + PLL
73 Exp2Poly = B1*Z + B2*Z^2 ... + B5*Z^5
74 Get 2^(j/2^expK) from table in the form THI+TLO.
75 Now we have 2^(PH+PL+PLL) ~= 2^N * (THI + TLO) * (1 + Exp2Poly).
76
77 Get significand of 2^(PH+PL+PLL) in the form ResHi+ResLo:
78 ResHi := THI
79 ResLo := THI * Exp2Poly + TLO
80
81 Get exponent ERes of the result:
82 Res := ResHi + ResLo:
83 Result := ex(Res) + N. */
84
85 pushq %rbp
86 cfi_adjust_cfa_offset (8)
87 cfi_rel_offset (%rbp, 0)
88 movq %rsp, %rbp
89 cfi_def_cfa_register (%rbp)
90 andq $-64, %rsp
91 subq $448, %rsp
92 movq __svml_dpow_data@GOTPCREL(%rip), %rax
93 vmovups %ymm11, 160(%rsp)
94 vmovups %ymm8, 224(%rsp)
95 vmovups %ymm10, 352(%rsp)
96 vmovups %ymm9, 384(%rsp)
97 vmovups %ymm13, 288(%rsp)
98 vmovapd %ymm1, %ymm11
99 vxorpd %ymm1, %ymm1, %ymm1
100 vextracti128 $1, %ymm0, %xmm5
101 vshufps $221, %xmm5, %xmm0, %xmm5
102
103/* i = (((Hi(x) & 0x000ffe00) + 0x00000200) >> 10); -> i = (b1..b11 + 1) / 2 */
104 vandps _iIndexMask(%rax), %xmm5, %xmm3
105 vpaddd _iIndexAdd(%rax), %xmm3, %xmm6
106 vpsrld $10, %xmm6, %xmm8
107
108/* Index for reciprocal table */
109 vpslld $3, %xmm8, %xmm9
110
111/* Index for log2 table */
112 vpslld $4, %xmm8, %xmm6
113
114/* x1 = x; Hi(x1) = (Hi(x1)&0x000fffff)|0x3ff00000 */
115 vandpd _iMantissaMask(%rax), %ymm0, %ymm4
116 vorpd _dbOne(%rax), %ymm4, %ymm13
117 vpcmpeqd %ymm4, %ymm4, %ymm4
118 vpcmpeqd %ymm8, %ymm8, %ymm8
119
120/* k = Hi(x); k = k - 0x3fe7fe00; k = k >> 20 */
121 vpsubd _i3fe7fe00(%rax), %xmm5, %xmm3
122 vpaddd _HIDELTA(%rax), %xmm5, %xmm5
123 vextracti128 $1, %ymm11, %xmm7
124 vshufps $221, %xmm7, %xmm11, %xmm2
125 vpand _ABSMASK(%rax), %xmm2, %xmm10
126 vpcmpeqd %ymm2, %ymm2, %ymm2
127 vgatherdpd %ymm2, 11712(%rax,%xmm9), %ymm1
128 vmovups _LORANGE(%rax), %xmm7
129 vxorpd %ymm2, %ymm2, %ymm2
130 vgatherdpd %ymm4, 19968(%rax,%xmm6), %ymm2
131 vxorpd %ymm4, %ymm4, %ymm4
132 vgatherdpd %ymm8, 19976(%rax,%xmm6), %ymm4
133 vpsrad $20, %xmm3, %xmm6
134 vpaddd _i2p20_2p19(%rax), %xmm6, %xmm9
135 vpshufd $80, %xmm9, %xmm8
136 vpshufd $250, %xmm9, %xmm3
137
138/* x1Hi=x1; Lo(x1Hi)&=0xf8000000; x1Lo = x1-x1Hi */
139 vandpd _iHighMask(%rax), %ymm13, %ymm9
140 vinserti128 $1, %xmm3, %ymm8, %ymm6
141 vandpd _iffffffff00000000(%rax), %ymm6, %ymm8
142
143/* r1 = x1*rcp1 */
144 vmulpd %ymm1, %ymm13, %ymm6
145 vsubpd %ymm9, %ymm13, %ymm3
146 vsubpd _db2p20_2p19(%rax), %ymm8, %ymm8
147
148/* cq = c+r1 */
149 vaddpd _LHN(%rax), %ymm6, %ymm13
150
151/* E = -r1+__fence(x1Hi*rcp1) */
152 vfmsub213pd %ymm6, %ymm1, %ymm9
153
154/* E=E+x1Lo*rcp1 */
155 vfmadd213pd %ymm9, %ymm1, %ymm3
156
157/* T = k + L1hi */
158 vaddpd %ymm2, %ymm8, %ymm1
159
160/* T_Rh = T + cq */
161 vaddpd %ymm13, %ymm1, %ymm8
162
163/* Rl = T-T_Rh; -> -Rh */
164 vsubpd %ymm8, %ymm1, %ymm6
165
166/* Rl=Rl+cq */
167 vaddpd %ymm6, %ymm13, %ymm1
168
169/* T_Rh_Eh = T_Rh + E */
170 vaddpd %ymm3, %ymm8, %ymm6
171
172/* cq = cq + E */
173 vaddpd %ymm3, %ymm13, %ymm13
174
175/* HLL = T_Rh - T_Rh_Eh; -> -Eh */
176 vsubpd %ymm6, %ymm8, %ymm9
177
178/* HLL+=E; -> El */
179 vaddpd %ymm9, %ymm3, %ymm2
180
181/* HLL+=Rl */
182 vaddpd %ymm1, %ymm2, %ymm8
183
184/* HLL+=L1lo */
185 vaddpd %ymm4, %ymm8, %ymm4
186 vmovupd _clv_2(%rax), %ymm8
187
188/* HLL = HLL + (((((((a7)*cq+a6)*cq+a5)*cq+a4)*cq+a3)*cq+a2)*cq+a1)*cq */
189 vfmadd213pd _clv_3(%rax), %ymm13, %ymm8
190 vfmadd213pd _clv_4(%rax), %ymm13, %ymm8
191 vfmadd213pd _clv_5(%rax), %ymm13, %ymm8
192 vfmadd213pd _clv_6(%rax), %ymm13, %ymm8
193 vfmadd213pd _clv_7(%rax), %ymm13, %ymm8
194 vfmadd213pd %ymm4, %ymm13, %ymm8
195
196/* T_Rh_Eh_HLLhi = T_Rh_Eh + HLL */
197 vaddpd %ymm8, %ymm6, %ymm9
198
199/* HH = T_Rh_Eh_HLLhi; Lo(HH)&=0xf8000000 */
200 vandpd _iHighMask(%rax), %ymm9, %ymm2
201
202/*
203 2^(y*(HH+HL+HLL)) starts here:
204 yH = y; Lo(yH)&=0xf8000000;
205 */
206 vandpd _iHighMask(%rax), %ymm11, %ymm1
207
208/* HLLhi = T_Rh_Eh_HLLhi - T_Rh_Eh */
209 vsubpd %ymm6, %ymm9, %ymm13
210
211/* HL = T_Rh_Eh_HLLhi-HH */
212 vsubpd %ymm2, %ymm9, %ymm4
213
214/* pH = yH*HH */
215 vmulpd %ymm2, %ymm1, %ymm9
216
217/* HLL = HLL - HLLhi */
218 vsubpd %ymm13, %ymm8, %ymm6
219
220/* yL = y-yH */
221 vsubpd %ymm1, %ymm11, %ymm8
222 vextracti128 $1, %ymm9, %xmm3
223 vshufps $221, %xmm3, %xmm9, %xmm13
224 vpand _ABSMASK(%rax), %xmm13, %xmm3
225 vpcmpgtd %xmm5, %xmm7, %xmm13
226 vpcmpgtd _INF(%rax), %xmm10, %xmm7
227 vpcmpeqd _INF(%rax), %xmm10, %xmm10
228 vpor %xmm10, %xmm7, %xmm7
229 vpor %xmm7, %xmm13, %xmm5
230
231/* pL=yL*HL+yH*HL; pL+=yL*HH */
232 vmulpd %ymm4, %ymm8, %ymm7
233 vpcmpgtd _DOMAINRANGE(%rax), %xmm3, %xmm13
234 vpcmpeqd _DOMAINRANGE(%rax), %xmm3, %xmm10
235 vpor %xmm10, %xmm13, %xmm3
236 vpor %xmm3, %xmm5, %xmm13
237 vfmadd213pd %ymm7, %ymm4, %ymm1
238
239/* pLL = y*HLL;
240 pHH = pH + *(double*)&db2p45_2p44
241 */
242 vaddpd _db2p45_2p44(%rax), %ymm9, %ymm7
243 vmovmskps %xmm13, %ecx
244 vfmadd213pd %ymm1, %ymm2, %ymm8
245
246/* t=pL+pLL; t+=pHL */
247 vfmadd231pd %ymm11, %ymm6, %ymm8
248 vextracti128 $1, %ymm7, %xmm1
249 vshufps $136, %xmm1, %xmm7, %xmm10
250
251/* _n = Lo(pHH);
252 _n = _n & 0xffffff80;
253 _n = _n >> 7;
254 Hi(_2n) = (0x3ff+_n)<<20; Lo(_2n) = 0; -> 2^n
255 */
256 vpslld $13, %xmm10, %xmm2
257 vpaddd _iOne(%rax), %xmm2, %xmm13
258 vpshufd $80, %xmm13, %xmm4
259 vpshufd $250, %xmm13, %xmm1
260
261/* j = Lo(pHH)&0x0000007f */
262 vandps _jIndexMask(%rax), %xmm10, %xmm3
263
264/* T1 = ((double*)exp2_tbl)[ 2*j ] */
265 vpcmpeqd %ymm10, %ymm10, %ymm10
266 vpslld $4, %xmm3, %xmm5
267
268/* pHH = pHH - *(double*)&db2p45_2p44 */
269 vsubpd _db2p45_2p44(%rax), %ymm7, %ymm7
270
271/* pHL = pH - pHH */
272 vsubpd %ymm7, %ymm9, %ymm9
273 vaddpd %ymm9, %ymm8, %ymm6
274 vinserti128 $1, %xmm1, %ymm4, %ymm2
275 vxorpd %ymm1, %ymm1, %ymm1
276 vgatherdpd %ymm10, 36416(%rax,%xmm5), %ymm1
277 vandpd _ifff0000000000000(%rax), %ymm2, %ymm13
278 vmovupd _cev_1(%rax), %ymm2
279 vmulpd %ymm1, %ymm13, %ymm1
280 vfmadd213pd _cev_2(%rax), %ymm6, %ymm2
281 vmulpd %ymm6, %ymm1, %ymm8
282 vfmadd213pd _cev_3(%rax), %ymm6, %ymm2
283 vfmadd213pd _cev_4(%rax), %ymm6, %ymm2
284 vfmadd213pd _cev_5(%rax), %ymm6, %ymm2
285 vfmadd213pd %ymm1, %ymm8, %ymm2
286 testl %ecx, %ecx
287 jne .LBL_1_3
288
289.LBL_1_2:
290 cfi_remember_state
291 vmovups 224(%rsp), %ymm8
292 vmovups 384(%rsp), %ymm9
293 vmovups 352(%rsp), %ymm10
294 vmovups 160(%rsp), %ymm11
295 vmovups 288(%rsp), %ymm13
296 vmovdqa %ymm2, %ymm0
297 movq %rbp, %rsp
298 cfi_def_cfa_register (%rsp)
299 popq %rbp
300 cfi_adjust_cfa_offset (-8)
301 cfi_restore (%rbp)
302 ret
303
304.LBL_1_3:
305 cfi_restore_state
306 vmovupd %ymm0, 192(%rsp)
307 vmovupd %ymm11, 256(%rsp)
308 vmovupd %ymm2, 320(%rsp)
309 je .LBL_1_2
310
311 xorb %dl, %dl
312 xorl %eax, %eax
313 vmovups %ymm12, 64(%rsp)
314 vmovups %ymm14, 32(%rsp)
315 vmovups %ymm15, (%rsp)
316 movq %rsi, 104(%rsp)
317 movq %rdi, 96(%rsp)
318 movq %r12, 136(%rsp)
319 cfi_offset_rel_rsp (12, 136)
320 movb %dl, %r12b
321 movq %r13, 128(%rsp)
322 cfi_offset_rel_rsp (13, 128)
323 movl %ecx, %r13d
324 movq %r14, 120(%rsp)
325 cfi_offset_rel_rsp (14, 120)
326 movl %eax, %r14d
327 movq %r15, 112(%rsp)
328 cfi_offset_rel_rsp (15, 112)
329 cfi_remember_state
330
331.LBL_1_6:
332 btl %r14d, %r13d
333 jc .LBL_1_12
334
335.LBL_1_7:
336 lea 1(%r14), %esi
337 btl %esi, %r13d
338 jc .LBL_1_10
339
340.LBL_1_8:
341 incb %r12b
342 addl $2, %r14d
343 cmpb $16, %r12b
344 jb .LBL_1_6
345
346 vmovups 64(%rsp), %ymm12
347 vmovups 32(%rsp), %ymm14
348 vmovups (%rsp), %ymm15
349 vmovupd 320(%rsp), %ymm2
350 movq 104(%rsp), %rsi
351 movq 96(%rsp), %rdi
352 movq 136(%rsp), %r12
353 cfi_restore (%r12)
354 movq 128(%rsp), %r13
355 cfi_restore (%r13)
356 movq 120(%rsp), %r14
357 cfi_restore (%r14)
358 movq 112(%rsp), %r15
359 cfi_restore (%r15)
360 jmp .LBL_1_2
361
362.LBL_1_10:
363 cfi_restore_state
364 movzbl %r12b, %r15d
365 shlq $4, %r15
366 vmovsd 200(%rsp,%r15), %xmm0
367 vmovsd 264(%rsp,%r15), %xmm1
368 vzeroupper
369
370 call JUMPTARGET(pow)
371
372 vmovsd %xmm0, 328(%rsp,%r15)
373 jmp .LBL_1_8
374
375.LBL_1_12:
376 movzbl %r12b, %r15d
377 shlq $4, %r15
378 vmovsd 192(%rsp,%r15), %xmm0
379 vmovsd 256(%rsp,%r15), %xmm1
380 vzeroupper
381
382 call JUMPTARGET(pow)
383
384 vmovsd %xmm0, 320(%rsp,%r15)
385 jmp .LBL_1_7
386
387END (_ZGVdN4vv_pow_avx2)
388

source code of glibc/sysdeps/x86_64/fpu/multiarch/svml_d_pow4_core_avx2.S