/* Function pow vectorized with AVX2.
Copyright (C) 2014-2018 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 JUMPTARGET(__pow_finite)
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 JUMPTARGET(__pow_finite)
vmovsd %xmm0, 320(%rsp,%r15)
jmp .LBL_1_7
END (_ZGVdN4vv_pow_avx2)