1/* Function pow vectorized with SSE4.
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.sse4, "ax", @progbits
23ENTRY (_ZGVbN2vv_pow_sse4)
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 $256, %rsp
92 movq __svml_dpow_data@GOTPCREL(%rip), %rdx
93 movups %xmm14, 80(%rsp)
94 movups %xmm9, 176(%rsp)
95 movaps %xmm1, %xmm9
96 pshufd $221, %xmm0, %xmm1
97 movq _iIndexMask(%rdx), %xmm14
98 movq _iIndexAdd(%rdx), %xmm6
99
100/* i = (((Hi(x) & 0x000ffe00) + 0x00000200) >> 10); -> i = (b1..b11 + 1) / 2 */
101 pand %xmm1, %xmm14
102 paddd %xmm6, %xmm14
103 psrld $10, %xmm14
104 movups %xmm13, 96(%rsp)
105
106/* Index for reciprocal table */
107 movdqa %xmm14, %xmm13
108 pslld $3, %xmm13
109
110/* Index for log2 table */
111 pslld $4, %xmm14
112 movd %xmm13, %eax
113 movups %xmm10, 160(%rsp)
114 movups _iMantissaMask(%rdx), %xmm10
115 movslq %eax, %rax
116
117/* x1 = x; Hi(x1) = (Hi(x1)&0x000fffff)|0x3ff00000 */
118 andps %xmm0, %xmm10
119 pextrd $1, %xmm13, %ecx
120 movslq %ecx, %rcx
121 movups %xmm0, (%rsp)
122 movdqa %xmm1, %xmm0
123
124/* k = Hi(x); k = k - 0x3fe7fe00; k = k >> 20 */
125 movq _i3fe7fe00(%rdx), %xmm6
126 psubd %xmm6, %xmm0
127 movups _iHighMask(%rdx), %xmm6
128 psrad $20, %xmm0
129 movups %xmm15, 48(%rsp)
130 movups %xmm12, 112(%rsp)
131 orps _dbOne(%rdx), %xmm10
132 movsd 11712(%rdx,%rax), %xmm12
133 movd %xmm14, %r8d
134 movq _i2p20_2p19(%rdx), %xmm15
135 movhpd 11712(%rdx,%rcx), %xmm12
136 paddd %xmm15, %xmm0
137 pextrd $1, %xmm14, %r9d
138
139/* x1Hi=x1; Lo(x1Hi)&=0xf8000000; x1Lo = x1-x1Hi */
140 movaps %xmm6, %xmm14
141 andps %xmm10, %xmm14
142 movaps %xmm10, %xmm15
143 subpd %xmm14, %xmm15
144
145/* r1 = x1*rcp1 */
146 mulpd %xmm12, %xmm10
147
148/* E = -r1+__fence(x1Hi*rcp1) */
149 mulpd %xmm12, %xmm14
150
151/* E=E+x1Lo*rcp1 */
152 mulpd %xmm15, %xmm12
153 subpd %xmm10, %xmm14
154 pshufd $80, %xmm0, %xmm0
155 movslq %r8d, %r8
156 andps _iffffffff00000000(%rdx), %xmm0
157 subpd _db2p20_2p19(%rdx), %xmm0
158 addpd %xmm12, %xmm14
159 movslq %r9d, %r9
160
161/* T_Rh_Eh = T_Rh + E */
162 movaps %xmm14, %xmm15
163 movups %xmm8, 208(%rsp)
164 movups 19968(%rdx,%r8), %xmm8
165 movups %xmm11, 144(%rsp)
166 movaps %xmm8, %xmm11
167
168/* cq = c+r1 */
169 movups _LHN(%rdx), %xmm13
170 movhpd 19968(%rdx,%r9), %xmm11
171 addpd %xmm10, %xmm13
172
173/* T = k + L1hi */
174 addpd %xmm0, %xmm11
175
176/* T_Rh = T + cq */
177 movaps %xmm13, %xmm12
178 addpd %xmm11, %xmm12
179 addpd %xmm12, %xmm15
180
181/* Rl = T-T_Rh; -> -Rh */
182 subpd %xmm12, %xmm11
183
184/* HLL = T_Rh - T_Rh_Eh; -> -Eh */
185 subpd %xmm15, %xmm12
186
187/* Rl=Rl+cq; */
188 addpd %xmm13, %xmm11
189
190/* cq = cq + E */
191 addpd %xmm14, %xmm13
192
193/* HLL+=E; -> El */
194 addpd %xmm14, %xmm12
195
196/* HLL+=Rl */
197 addpd %xmm12, %xmm11
198 unpckhpd 19968(%rdx,%r9), %xmm8
199
200/* T_Rh_Eh_HLLhi = T_Rh_Eh + HLL */
201 movaps %xmm15, %xmm14
202
203/* HLL+=L1lo; */
204 addpd %xmm11, %xmm8
205 movups _clv_2(%rdx), %xmm11
206
207/* HH = T_Rh_Eh_HLLhi; Lo(HH)&=0xf8000000 */
208 movaps %xmm6, %xmm12
209
210/* HLL = HLL + (((((((a7)*cq+a6)*cq+a5)*cq+a4)*cq+a3)*cq+a2)*cq+a1)*cq */
211 mulpd %xmm13, %xmm11
212 addpd _clv_3(%rdx), %xmm11
213 mulpd %xmm13, %xmm11
214 addpd _clv_4(%rdx), %xmm11
215 mulpd %xmm13, %xmm11
216 addpd _clv_5(%rdx), %xmm11
217 mulpd %xmm13, %xmm11
218 addpd _clv_6(%rdx), %xmm11
219 mulpd %xmm13, %xmm11
220 addpd _clv_7(%rdx), %xmm11
221 mulpd %xmm11, %xmm13
222 addpd %xmm13, %xmm8
223 addpd %xmm8, %xmm14
224
225/*
226 2^(y*(HH+HL+HLL)) starts here:
227 yH = y; Lo(yH)&=0xf8000000
228 */
229 andps %xmm9, %xmm6
230
231/* yL = y-yH; */
232 movaps %xmm9, %xmm11
233 subpd %xmm6, %xmm11
234 andps %xmm14, %xmm12
235
236/* HLLhi = T_Rh_Eh_HLLhi - T_Rh_Eh */
237 movaps %xmm14, %xmm10
238
239/* HL = T_Rh_Eh_HLLhi-HH; */
240 subpd %xmm12, %xmm14
241 subpd %xmm15, %xmm10
242 movq _HIDELTA(%rdx), %xmm2
243
244/* pH = yH*HH; */
245 movaps %xmm6, %xmm13
246 movq _LORANGE(%rdx), %xmm3
247 paddd %xmm2, %xmm1
248 pcmpgtd %xmm1, %xmm3
249
250/* pL=yL*HL+yH*HL; pL+=yL*HH; */
251 movaps %xmm11, %xmm1
252 mulpd %xmm14, %xmm1
253 mulpd %xmm14, %xmm6
254 mulpd %xmm12, %xmm13
255 mulpd %xmm11, %xmm12
256 addpd %xmm6, %xmm1
257
258/* HLL = HLL - HLLhi */
259 subpd %xmm10, %xmm8
260 addpd %xmm12, %xmm1
261
262/* pLL = y*HLL */
263 mulpd %xmm9, %xmm8
264 movups _db2p45_2p44(%rdx), %xmm11
265
266/* pHH = pH + *(double*)&db2p45_2p44 */
267 movaps %xmm11, %xmm12
268 addpd %xmm13, %xmm12
269
270/* t=pL+pLL; t+=pHL */
271 addpd %xmm8, %xmm1
272 movq _ABSMASK(%rdx), %xmm5
273 pshufd $221, %xmm9, %xmm4
274 pand %xmm5, %xmm4
275 movq _INF(%rdx), %xmm7
276 movdqa %xmm4, %xmm2
277 pcmpgtd %xmm7, %xmm2
278 pcmpeqd %xmm7, %xmm4
279 pshufd $136, %xmm12, %xmm7
280 por %xmm4, %xmm2
281
282/* pHH = pHH - *(double*)&db2p45_2p44 */
283 subpd %xmm11, %xmm12
284 pshufd $221, %xmm13, %xmm10
285 por %xmm2, %xmm3
286
287/* pHL = pH - pHH; */
288 subpd %xmm12, %xmm13
289 pand %xmm5, %xmm10
290 movq _DOMAINRANGE(%rdx), %xmm5
291 movdqa %xmm10, %xmm4
292 addpd %xmm1, %xmm13
293 pcmpgtd %xmm5, %xmm4
294 pcmpeqd %xmm5, %xmm10
295 por %xmm10, %xmm4
296 movq _jIndexMask(%rdx), %xmm6
297 por %xmm4, %xmm3
298 movmskps %xmm3, %eax
299
300/* j = Lo(pHH)&0x0000007f */
301 pand %xmm7, %xmm6
302 movq _iOne(%rdx), %xmm3
303
304/* _n = Lo(pHH);
305 _n = _n & 0xffffff80;
306 _n = _n >> 7;
307 Hi(_2n) = (0x3ff+_n)<<20; Lo(_2n) = 0; -> 2^n
308 */
309 pslld $13, %xmm7
310 paddd %xmm3, %xmm7
311 pslld $4, %xmm6
312 movups _cev_1(%rdx), %xmm3
313 movaps %xmm13, %xmm4
314 mulpd %xmm13, %xmm3
315
316/* T1 = ((double*)exp2_tbl)[ 2*j ] */
317 movd %xmm6, %r10d
318 pshufd $80, %xmm7, %xmm0
319 andps _ifff0000000000000(%rdx), %xmm0
320 addpd _cev_2(%rdx), %xmm3
321 mulpd %xmm13, %xmm3
322 addpd _cev_3(%rdx), %xmm3
323 mulpd %xmm13, %xmm3
324 movslq %r10d, %r10
325 andl $3, %eax
326 pextrd $1, %xmm6, %r11d
327 movslq %r11d, %r11
328 addpd _cev_4(%rdx), %xmm3
329 movsd 36416(%rdx,%r10), %xmm2
330 movhpd 36416(%rdx,%r11), %xmm2
331 mulpd %xmm2, %xmm0
332 mulpd %xmm3, %xmm13
333 mulpd %xmm0, %xmm4
334 addpd _cev_5(%rdx), %xmm13
335 mulpd %xmm4, %xmm13
336 addpd %xmm13, %xmm0
337 jne .LBL_1_3
338
339.LBL_1_2:
340 cfi_remember_state
341 movups 208(%rsp), %xmm8
342 movups 176(%rsp), %xmm9
343 movups 160(%rsp), %xmm10
344 movups 144(%rsp), %xmm11
345 movups 112(%rsp), %xmm12
346 movups 96(%rsp), %xmm13
347 movups 80(%rsp), %xmm14
348 movups 48(%rsp), %xmm15
349 movq %rbp, %rsp
350 cfi_def_cfa_register (%rsp)
351 popq %rbp
352 cfi_adjust_cfa_offset (-8)
353 cfi_restore (%rbp)
354 ret
355
356.LBL_1_3:
357 cfi_restore_state
358 movups (%rsp), %xmm1
359 movups %xmm1, 64(%rsp)
360 movups %xmm9, 128(%rsp)
361 movups %xmm0, 192(%rsp)
362 je .LBL_1_2
363
364 xorb %cl, %cl
365 xorl %edx, %edx
366 movq %rsi, 8(%rsp)
367 movq %rdi, (%rsp)
368 movq %r12, 40(%rsp)
369 cfi_offset_rel_rsp (12, 40)
370 movb %cl, %r12b
371 movq %r13, 32(%rsp)
372 cfi_offset_rel_rsp (13, 32)
373 movl %eax, %r13d
374 movq %r14, 24(%rsp)
375 cfi_offset_rel_rsp (14, 24)
376 movl %edx, %r14d
377 movq %r15, 16(%rsp)
378 cfi_offset_rel_rsp (15, 16)
379 cfi_remember_state
380
381.LBL_1_6:
382 btl %r14d, %r13d
383 jc .LBL_1_12
384
385.LBL_1_7:
386 lea 1(%r14), %esi
387 btl %esi, %r13d
388 jc .LBL_1_10
389
390.LBL_1_8:
391 incb %r12b
392 addl $2, %r14d
393 cmpb $16, %r12b
394 jb .LBL_1_6
395
396 movq 8(%rsp), %rsi
397 movq (%rsp), %rdi
398 movq 40(%rsp), %r12
399 cfi_restore (%r12)
400 movq 32(%rsp), %r13
401 cfi_restore (%r13)
402 movq 24(%rsp), %r14
403 cfi_restore (%r14)
404 movq 16(%rsp), %r15
405 cfi_restore (%r15)
406 movups 192(%rsp), %xmm0
407 jmp .LBL_1_2
408
409.LBL_1_10:
410 cfi_restore_state
411 movzbl %r12b, %r15d
412 shlq $4, %r15
413 movsd 72(%rsp,%r15), %xmm0
414 movsd 136(%rsp,%r15), %xmm1
415
416 call JUMPTARGET(pow)
417
418 movsd %xmm0, 200(%rsp,%r15)
419 jmp .LBL_1_8
420
421.LBL_1_12:
422 movzbl %r12b, %r15d
423 shlq $4, %r15
424 movsd 64(%rsp,%r15), %xmm0
425 movsd 128(%rsp,%r15), %xmm1
426
427 call JUMPTARGET(pow)
428
429 movsd %xmm0, 192(%rsp,%r15)
430 jmp .LBL_1_7
431
432END (_ZGVbN2vv_pow_sse4)
433

source code of glibc/sysdeps/x86_64/fpu/multiarch/svml_d_pow2_core_sse4.S