avx_mathfun.h

//  Copyright (c) 2018 PaddlePaddle Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//    http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
/*
   AVX implementation of sin, cos, sincos, exp and log

   Based on "sse_mathfun.h", by Julien Pommier
   http://gruntthepeon.free.fr/ssemath/

   Copyright (C) 2012 Giovanni Garberoglio
   Interdisciplinary Laboratory for Computational Science (LISC)
   Fondazione Bruno Kessler and University of Trento
   via Sommarive, 18
   I-38123 Trento (Italy)

  This software is provided 'as-is', without any express or implied
  warranty.  In no event will the authors be held liable for any damages
  arising from the use of this software.

  Permission is granted to anyone to use this software for any purpose,
  including commercial applications, and to alter it and redistribute it
  freely, subject to the following restrictions:

  1. The origin of this software must not be misrepresented; you must not
     claim that you wrote the original software. If you use this software
     in a product, an acknowledgment in the product documentation would be
     appreciated but is not required.
  2. Altered source versions must be plainly marked as such, and must not be
     misrepresented as being the original software.
  3. This notice may not be removed or altered from any source distribution.

  (this is the zlib license)
*/
#pragma once
#include "paddle/phi/backends/cpu/cpu_info.h"

/* __m128 is ugly to write */
typedef __m256 v8sf;   // vector of 8 float (avx)
typedef __m256i v8si;  // vector of 8 int   (avx)
typedef __m128i v4si;  // vector of 8 int   (avx)

#define _PI32AVX_CONST(Name, Val)                                 \
  static const ALIGN32_BEG int _pi32avx_##Name[4] ALIGN32_END = { \
      Val, Val, Val, Val}

_PI32AVX_CONST(1, 1);
_PI32AVX_CONST(inv1, ~1);
_PI32AVX_CONST(2, 2);
_PI32AVX_CONST(4, 4);

/* declare some AVX constants -- why can't I figure a better way to do that? */
#define _PS256_CONST(Name, Val)                                   \
  static const ALIGN32_BEG float _ps256_##Name[8] ALIGN32_END = { \
      Val, Val, Val, Val, Val, Val, Val, Val}
#define _PI32_CONST256(Name, Val)                                  \
  static const ALIGN32_BEG int _pi32_256_##Name[8] ALIGN32_END = { \
      Val, Val, Val, Val, Val, Val, Val, Val}
#define _PS256_CONST_TYPE(Name, Type, Val)                       \
  static const ALIGN32_BEG Type _ps256_##Name[8] ALIGN32_END = { \
      Val, Val, Val, Val, Val, Val, Val, Val}

_PS256_CONST(1, 1.0f);
_PS256_CONST(0p5, 0.5f);
/* the smallest non denormalized float number */
_PS256_CONST_TYPE(min_norm_pos, int, 0x00800000);
_PS256_CONST_TYPE(mant_mask, int, 0x7f800000);
_PS256_CONST_TYPE(inv_mant_mask, int, ~0x7f800000);

_PS256_CONST_TYPE(sign_mask, int, (int)0x80000000);
_PS256_CONST_TYPE(inv_sign_mask, int, ~0x80000000);

_PI32_CONST256(0, 0);
_PI32_CONST256(1, 1);
_PI32_CONST256(inv1, ~1);
_PI32_CONST256(2, 2);
_PI32_CONST256(4, 4);
_PI32_CONST256(0x7f, 0x7f);

_PS256_CONST(cephes_SQRTHF, 0.707106781186547524);
_PS256_CONST(cephes_log_p0, 7.0376836292E-2);
_PS256_CONST(cephes_log_p1, -1.1514610310E-1);
_PS256_CONST(cephes_log_p2, 1.1676998740E-1);
_PS256_CONST(cephes_log_p3, -1.2420140846E-1);
_PS256_CONST(cephes_log_p4, +1.4249322787E-1);
_PS256_CONST(cephes_log_p5, -1.6668057665E-1);
_PS256_CONST(cephes_log_p6, +2.0000714765E-1);
_PS256_CONST(cephes_log_p7, -2.4999993993E-1);
_PS256_CONST(cephes_log_p8, +3.3333331174E-1);
_PS256_CONST(cephes_log_q1, -2.12194440e-4);
_PS256_CONST(cephes_log_q2, 0.693359375);

#ifndef __AVX2__

typedef union imm_xmm_union {
  v8si imm;
  v4si xmm[2];
} imm_xmm_union;

#define COPY_IMM_TO_XMM(imm_, xmm0_, xmm1_)  \
  {                                          \
    imm_xmm_union ALIGN32_BEG u ALIGN32_END; \
    u.imm = imm_;                            \
    xmm0_ = u.xmm[0];                        \
    xmm1_ = u.xmm[1];                        \
  }

#define COPY_XMM_TO_IMM(xmm0_, xmm1_, imm_)  \
  {                                          \
    imm_xmm_union ALIGN32_BEG u ALIGN32_END; \
    u.xmm[0] = xmm0_;                        \
    u.xmm[1] = xmm1_;                        \
    imm_ = u.imm;                            \
  }

#define AVX2_BITOP_USING_SSE2(fn)                        \
  static inline v8si avx2_mm256_##fn(v8si x, int a) {    \
    /* use SSE2 instruction to perform the bitop AVX2 */ \
    v4si x1, x2;                                         \
    v8si ret;                                            \
    COPY_IMM_TO_XMM(x, x1, x2);                          \
    x1 = _mm_##fn(x1, a);                                \
    x2 = _mm_##fn(x2, a);                                \
    COPY_XMM_TO_IMM(x1, x2, ret);                        \
    return (ret);                                        \
  }

// #warning "Using SSE2 to perform AVX2 bitshift ops"
AVX2_BITOP_USING_SSE2(slli_epi32)
AVX2_BITOP_USING_SSE2(srli_epi32)

#define AVX2_INTOP_USING_SSE2(fn)                                     \
  static inline v8si avx2_mm256_##fn(v8si x, v8si y) {                \
    /* use SSE2 instructions to perform the AVX2 integer operation */ \
    v4si x1, x2;                                                      \
    v4si y1, y2;                                                      \
    v8si ret;                                                         \
    COPY_IMM_TO_XMM(x, x1, x2);                                       \
    COPY_IMM_TO_XMM(y, y1, y2);                                       \
    x1 = _mm_##fn(x1, y1);                                            \
    x2 = _mm_##fn(x2, y2);                                            \
    COPY_XMM_TO_IMM(x1, x2, ret);                                     \
    return (ret);                                                     \
  }

// #warning "Using SSE2 to perform AVX2 integer ops"
AVX2_INTOP_USING_SSE2(and_si128)
AVX2_INTOP_USING_SSE2(andnot_si128)
AVX2_INTOP_USING_SSE2(cmpeq_epi32)
AVX2_INTOP_USING_SSE2(sub_epi32)
AVX2_INTOP_USING_SSE2(add_epi32)
#define avx2_mm256_and_si256 avx2_mm256_and_si128
#define avx2_mm256_andnot_si256 avx2_mm256_andnot_si128
#else
#define avx2_mm256_slli_epi32 _mm256_slli_epi32
#define avx2_mm256_srli_epi32 _mm256_srli_epi32
#define avx2_mm256_and_si256 _mm256_and_si256
#define avx2_mm256_andnot_si256 _mm256_andnot_si256
#define avx2_mm256_cmpeq_epi32 _mm256_cmpeq_epi32
#define avx2_mm256_sub_epi32 _mm256_sub_epi32
#define avx2_mm256_add_epi32 _mm256_add_epi32
#endif /* __AVX2__ */

/* natural logarithm computed for 8 simultaneous float
   return NaN for x <= 0
*/
v8sf log256_ps(v8sf x) {
  v8si imm0;
  v8sf one = *reinterpret_cast<const v8sf *>(_ps256_1);

  // v8sf invalid_mask = _mm256_cmple_ps(x, _mm256_setzero_ps());
  v8sf invalid_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_LE_OS);

  /* cut off denormalized stuff */
  x = _mm256_max_ps(x, *reinterpret_cast<const v8sf *>(_ps256_min_norm_pos));

  // can be done with AVX2
  imm0 = avx2_mm256_srli_epi32(_mm256_castps_si256(x), 23);

  /* keep only the fractional part */
  x = _mm256_and_ps(x, *reinterpret_cast<const v8sf *>(_ps256_inv_mant_mask));
  x = _mm256_or_ps(x, *reinterpret_cast<const v8sf *>(_ps256_0p5));

  // this is again another AVX2 instruction
  imm0 = avx2_mm256_sub_epi32(imm0,
                              *reinterpret_cast<const v8si *>(_pi32_256_0x7f));
  v8sf e = _mm256_cvtepi32_ps(imm0);

  e = _mm256_add_ps(e, one);

  /* part2:
     if( x < SQRTHF ) {
       e -= 1;
       x = x + x - 1.0;
     } else { x = x - 1.0; }
  */
  // v8sf mask = _mm256_cmplt_ps(x, *(v8sf*)_ps256_cephes_SQRTHF);
  v8sf mask = _mm256_cmp_ps(
      x, *reinterpret_cast<const v8sf *>(_ps256_cephes_SQRTHF), _CMP_LT_OS);
  v8sf tmp = _mm256_and_ps(x, mask);
  x = _mm256_sub_ps(x, one);
  e = _mm256_sub_ps(e, _mm256_and_ps(one, mask));
  x = _mm256_add_ps(x, tmp);

  v8sf z = _mm256_mul_ps(x, x);

  v8sf y = *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p0);
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p1));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p2));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p3));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p4));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p5));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p6));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p7));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_p8));
  y = _mm256_mul_ps(y, x);

  y = _mm256_mul_ps(y, z);

  tmp = _mm256_mul_ps(e, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_q1));
  y = _mm256_add_ps(y, tmp);

  tmp = _mm256_mul_ps(z, *reinterpret_cast<const v8sf *>(_ps256_0p5));
  y = _mm256_sub_ps(y, tmp);

  tmp = _mm256_mul_ps(e, *reinterpret_cast<const v8sf *>(_ps256_cephes_log_q2));
  x = _mm256_add_ps(x, y);
  x = _mm256_add_ps(x, tmp);
  x = _mm256_or_ps(x, invalid_mask);  // negative arg will be NAN
  return x;
}

_PS256_CONST(exp_hi, 88.3762626647949f);
_PS256_CONST(exp_lo, -88.3762626647949f);

_PS256_CONST(cephes_LOG2EF, 1.44269504088896341);
_PS256_CONST(cephes_exp_C1, 0.693359375);
_PS256_CONST(cephes_exp_C2, -2.12194440e-4);

_PS256_CONST(cephes_exp_p0, 1.9875691500E-4);
_PS256_CONST(cephes_exp_p1, 1.3981999507E-3);
_PS256_CONST(cephes_exp_p2, 8.3334519073E-3);
_PS256_CONST(cephes_exp_p3, 4.1665795894E-2);
_PS256_CONST(cephes_exp_p4, 1.6666665459E-1);
_PS256_CONST(cephes_exp_p5, 5.0000001201E-1);

v8sf exp256_ps(v8sf x) {
  v8sf tmp = _mm256_setzero_ps(), fx;
  v8si imm0;
  v8sf one = *reinterpret_cast<const v8sf *>(_ps256_1);

  x = _mm256_min_ps(x, *reinterpret_cast<const v8sf *>(_ps256_exp_hi));
  x = _mm256_max_ps(x, *reinterpret_cast<const v8sf *>(_ps256_exp_lo));

  /* express exp(x) as exp(g + n*log(2)) */
  fx = _mm256_mul_ps(x, *reinterpret_cast<const v8sf *>(_ps256_cephes_LOG2EF));
  fx = _mm256_add_ps(fx, *reinterpret_cast<const v8sf *>(_ps256_0p5));

  /* how to perform a floorf with SSE: just below */
  // imm0 = _mm256_cvttps_epi32(fx);
  // tmp  = _mm256_cvtepi32_ps(imm0);

  tmp = _mm256_floor_ps(fx);

  /* if greater, substract 1 */
  // v8sf mask = _mm256_cmpgt_ps(tmp, fx);
  v8sf mask = _mm256_cmp_ps(tmp, fx, _CMP_GT_OS);
  mask = _mm256_and_ps(mask, one);
  fx = _mm256_sub_ps(tmp, mask);

  tmp =
      _mm256_mul_ps(fx, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_C1));
  v8sf z =
      _mm256_mul_ps(fx, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_C2));
  x = _mm256_sub_ps(x, tmp);
  x = _mm256_sub_ps(x, z);

  z = _mm256_mul_ps(x, x);

  v8sf y = *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p0);
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p1));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p2));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p3));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p4));
  y = _mm256_mul_ps(y, x);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_cephes_exp_p5));
  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, x);
  y = _mm256_add_ps(y, one);

  /* build 2^n */
  imm0 = _mm256_cvttps_epi32(fx);
  // another two AVX2 instructions
  imm0 = avx2_mm256_add_epi32(imm0,
                              *reinterpret_cast<const v8si *>(_pi32_256_0x7f));
  imm0 = avx2_mm256_slli_epi32(imm0, 23);
  v8sf pow2n = _mm256_castsi256_ps(imm0);
  y = _mm256_mul_ps(y, pow2n);
  return y;
}

_PS256_CONST(minus_cephes_DP1, -0.78515625);
_PS256_CONST(minus_cephes_DP2, -2.4187564849853515625e-4);
_PS256_CONST(minus_cephes_DP3, -3.77489497744594108e-8);
_PS256_CONST(sincof_p0, -1.9515295891E-4);
_PS256_CONST(sincof_p1, 8.3321608736E-3);
_PS256_CONST(sincof_p2, -1.6666654611E-1);
_PS256_CONST(coscof_p0, 2.443315711809948E-005);
_PS256_CONST(coscof_p1, -1.388731625493765E-003);
_PS256_CONST(coscof_p2, 4.166664568298827E-002);
_PS256_CONST(cephes_FOPI, 1.27323954473516);  // 4 / M_PI

/* evaluation of 8 sines at onces using AVX intrisics

   The code is the exact rewriting of the cephes sinf function.
   Precision is excellent as long as x < 8192 (I did not bother to
   take into account the special handling they have for greater values
   -- it does not return garbage for arguments over 8192, though, but
   the extra precision is missing).

   Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the
   surprising but correct result.

*/
v8sf sin256_ps(v8sf x) {  // any x
  v8sf xmm1, xmm2 = _mm256_setzero_ps(), xmm3, sign_bit, y;
  v8si imm0, imm2;

#ifndef __AVX2__
  v4si imm0_1, imm0_2;
  v4si imm2_1, imm2_2;
#endif

  sign_bit = x;
  /* take the absolute value */
  x = _mm256_and_ps(x, *reinterpret_cast<const v8sf *>(_ps256_inv_sign_mask));
  /* extract the sign bit (upper one) */
  sign_bit = _mm256_and_ps(sign_bit,
                           *reinterpret_cast<const v8sf *>(_ps256_sign_mask));

  /* scale by 4/Pi */
  y = _mm256_mul_ps(x, *reinterpret_cast<const v8sf *>(_ps256_cephes_FOPI));

  /*
    Here we start a series of integer operations, which are in the
    realm of AVX2.
    If we don't have AVX, let's perform them using SSE2 directives
  */

#ifdef __AVX2__
  /* store the integer part of y in mm0 */
  imm2 = _mm256_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  // another two AVX2 instruction
  imm2 =
      avx2_mm256_add_epi32(imm2, *reinterpret_cast<const v8si *>(_pi32_256_1));
  imm2 = avx2_mm256_and_si256(imm2,
                              *reinterpret_cast<const v8si *>(_pi32_256_inv1));
  y = _mm256_cvtepi32_ps(imm2);

  /* get the swap sign flag */
  imm0 =
      avx2_mm256_and_si256(imm2, *reinterpret_cast<const v8si *>(_pi32_256_4));
  imm0 = avx2_mm256_slli_epi32(imm0, 29);
  /* get the polynom selection mask
     there is one polynom for 0 <= x <= Pi/4
     and another one for Pi/4<x<=Pi/2

     Both branches will be computed.
  */
  imm2 =
      avx2_mm256_and_si256(imm2, *reinterpret_cast<const v8si *>(_pi32_256_2));
  imm2 = avx2_mm256_cmpeq_epi32(imm2,
                                *reinterpret_cast<const v8si *>(_pi32_256_0));
#else
  /* we use SSE2 routines to perform the integer ops */
  COPY_IMM_TO_XMM(_mm256_cvttps_epi32(y), imm2_1, imm2_2);

  imm2_1 = _mm_add_epi32(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_1));
  imm2_2 = _mm_add_epi32(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_1));

  imm2_1 =
      _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_inv1));
  imm2_2 =
      _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_inv1));

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
  y = _mm256_cvtepi32_ps(imm2);

  imm0_1 = _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_4));
  imm0_2 = _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_4));

  imm0_1 = _mm_slli_epi32(imm0_1, 29);
  imm0_2 = _mm_slli_epi32(imm0_2, 29);

  COPY_XMM_TO_IMM(imm0_1, imm0_2, imm0);

  imm2_1 = _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_2));
  imm2_2 = _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_2));

  imm2_1 = _mm_cmpeq_epi32(imm2_1, _mm_setzero_si128());
  imm2_2 = _mm_cmpeq_epi32(imm2_2, _mm_setzero_si128());

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
#endif

  v8sf swap_sign_bit = _mm256_castsi256_ps(imm0);
  v8sf poly_mask = _mm256_castsi256_ps(imm2);
  sign_bit = _mm256_xor_ps(sign_bit, swap_sign_bit);

  /* The magic pass: "Extended precision modular arithmetic"
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP1);
  xmm2 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP2);
  xmm3 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP3);
  xmm1 = _mm256_mul_ps(y, xmm1);
  xmm2 = _mm256_mul_ps(y, xmm2);
  xmm3 = _mm256_mul_ps(y, xmm3);
  x = _mm256_add_ps(x, xmm1);
  x = _mm256_add_ps(x, xmm2);
  x = _mm256_add_ps(x, xmm3);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = *reinterpret_cast<const v8sf *>(_ps256_coscof_p0);
  v8sf z = _mm256_mul_ps(x, x);

  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p1));
  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p2));
  y = _mm256_mul_ps(y, z);
  y = _mm256_mul_ps(y, z);
  v8sf tmp = _mm256_mul_ps(z, *reinterpret_cast<const v8sf *>(_ps256_0p5));
  y = _mm256_sub_ps(y, tmp);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_1));

  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v8sf y2 = *reinterpret_cast<const v8sf *>(_ps256_sincof_p0);
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p1));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p2));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_mul_ps(y2, x);
  y2 = _mm256_add_ps(y2, x);

  /* select the correct result from the two polynoms */
  xmm3 = poly_mask;
  y2 = _mm256_and_ps(xmm3, y2);  //, xmm3);
  y = _mm256_andnot_ps(xmm3, y);
  y = _mm256_add_ps(y, y2);
  /* update the sign */
  y = _mm256_xor_ps(y, sign_bit);

  return y;
}

/* almost the same as sin_ps */
v8sf cos256_ps(v8sf x) {  // any x
  v8sf xmm1, xmm2 = _mm256_setzero_ps(), xmm3, y;
  v8si imm0, imm2;

#ifndef __AVX2__
  v4si imm0_1, imm0_2;
  v4si imm2_1, imm2_2;
#endif

  /* take the absolute value */
  x = _mm256_and_ps(x, *reinterpret_cast<const v8sf *>(_ps256_inv_sign_mask));

  /* scale by 4/Pi */
  y = _mm256_mul_ps(x, *reinterpret_cast<const v8sf *>(_ps256_cephes_FOPI));

#ifdef __AVX2__
  /* store the integer part of y in mm0 */
  imm2 = _mm256_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  imm2 =
      avx2_mm256_add_epi32(imm2, *reinterpret_cast<const v8si *>(_pi32_256_1));
  imm2 = avx2_mm256_and_si256(imm2,
                              *reinterpret_cast<const v8si *>(_pi32_256_inv1));
  y = _mm256_cvtepi32_ps(imm2);
  imm2 =
      avx2_mm256_sub_epi32(imm2, *reinterpret_cast<const v8si *>(_pi32_256_2));

  /* get the swap sign flag */
  imm0 = avx2_mm256_andnot_si256(imm2,
                                 *reinterpret_cast<const v8si *>(_pi32_256_4));
  imm0 = avx2_mm256_slli_epi32(imm0, 29);
  /* get the polynom selection mask */
  imm2 =
      avx2_mm256_and_si256(imm2, *reinterpret_cast<const v8si *>(_pi32_256_2));
  imm2 = avx2_mm256_cmpeq_epi32(imm2,
                                *reinterpret_cast<const v8si *>(_pi32_256_0));
#else

  /* we use SSE2 routines to perform the integer ops */
  COPY_IMM_TO_XMM(_mm256_cvttps_epi32(y), imm2_1, imm2_2);

  imm2_1 = _mm_add_epi32(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_1));
  imm2_2 = _mm_add_epi32(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_1));

  imm2_1 =
      _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_inv1));
  imm2_2 =
      _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_inv1));

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
  y = _mm256_cvtepi32_ps(imm2);

  imm2_1 = _mm_sub_epi32(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_2));
  imm2_2 = _mm_sub_epi32(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_2));

  imm0_1 =
      _mm_andnot_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_4));
  imm0_2 =
      _mm_andnot_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_4));

  imm0_1 = _mm_slli_epi32(imm0_1, 29);
  imm0_2 = _mm_slli_epi32(imm0_2, 29);

  COPY_XMM_TO_IMM(imm0_1, imm0_2, imm0);

  imm2_1 = _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_2));
  imm2_2 = _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_2));

  imm2_1 = _mm_cmpeq_epi32(imm2_1, _mm_setzero_si128());
  imm2_2 = _mm_cmpeq_epi32(imm2_2, _mm_setzero_si128());

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
#endif

  v8sf sign_bit = _mm256_castsi256_ps(imm0);
  v8sf poly_mask = _mm256_castsi256_ps(imm2);

  /* The magic pass: "Extended precision modular arithmetic"
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP1);
  xmm2 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP2);
  xmm3 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP3);
  xmm1 = _mm256_mul_ps(y, xmm1);
  xmm2 = _mm256_mul_ps(y, xmm2);
  xmm3 = _mm256_mul_ps(y, xmm3);
  x = _mm256_add_ps(x, xmm1);
  x = _mm256_add_ps(x, xmm2);
  x = _mm256_add_ps(x, xmm3);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = *reinterpret_cast<const v8sf *>(_ps256_coscof_p0);
  v8sf z = _mm256_mul_ps(x, x);

  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p1));
  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p2));
  y = _mm256_mul_ps(y, z);
  y = _mm256_mul_ps(y, z);
  v8sf tmp = _mm256_mul_ps(z, *reinterpret_cast<const v8sf *>(_ps256_0p5));
  y = _mm256_sub_ps(y, tmp);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_1));

  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v8sf y2 = *reinterpret_cast<const v8sf *>(_ps256_sincof_p0);
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p1));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p2));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_mul_ps(y2, x);
  y2 = _mm256_add_ps(y2, x);

  /* select the correct result from the two polynoms */
  xmm3 = poly_mask;
  y2 = _mm256_and_ps(xmm3, y2);  //, xmm3);
  y = _mm256_andnot_ps(xmm3, y);
  y = _mm256_add_ps(y, y2);
  /* update the sign */
  y = _mm256_xor_ps(y, sign_bit);

  return y;
}

/* since sin256_ps and cos256_ps are almost identical, sincos256_ps could
   replace both of them..
   it is almost as fast, and gives you a free cosine with your sine */
void sincos256_ps(v8sf x, v8sf *s, v8sf *c) {
  v8sf xmm1, xmm2, xmm3 = _mm256_setzero_ps(), sign_bit_sin, y;
  v8si imm0, imm2, imm4;

#ifndef __AVX2__
  v4si imm0_1, imm0_2;
  v4si imm2_1, imm2_2;
  v4si imm4_1, imm4_2;
#endif

  sign_bit_sin = x;
  /* take the absolute value */
  x = _mm256_and_ps(x, *reinterpret_cast<const v8sf *>(_ps256_inv_sign_mask));
  /* extract the sign bit (upper one) */
  sign_bit_sin = _mm256_and_ps(
      sign_bit_sin, *reinterpret_cast<const v8sf *>(_ps256_sign_mask));

  /* scale by 4/Pi */
  y = _mm256_mul_ps(x, *reinterpret_cast<const v8sf *>(_ps256_cephes_FOPI));

#ifdef __AVX2__
  /* store the integer part of y in imm2 */
  imm2 = _mm256_cvttps_epi32(y);

  /* j=(j+1) & (~1) (see the cephes sources) */
  imm2 =
      avx2_mm256_add_epi32(imm2, *reinterpret_cast<const v8si *>(_pi32_256_1));
  imm2 = avx2_mm256_and_si256(imm2,
                              *reinterpret_cast<const v8si *>(_pi32_256_inv1));

  y = _mm256_cvtepi32_ps(imm2);
  imm4 = imm2;

  /* get the swap sign flag for the sine */
  imm0 =
      avx2_mm256_and_si256(imm2, *reinterpret_cast<const v8si *>(_pi32_256_4));
  imm0 = avx2_mm256_slli_epi32(imm0, 29);
  // v8sf swap_sign_bit_sin = _mm256_castsi256_ps(imm0);

  /* get the polynom selection mask for the sine*/
  imm2 =
      avx2_mm256_and_si256(imm2, *reinterpret_cast<const v8si *>(_pi32_256_2));
  imm2 = avx2_mm256_cmpeq_epi32(imm2,
                                *reinterpret_cast<const v8si *>(_pi32_256_0));
// v8sf poly_mask = _mm256_castsi256_ps(imm2);
#else
  /* we use SSE2 routines to perform the integer ops */
  COPY_IMM_TO_XMM(_mm256_cvttps_epi32(y), imm2_1, imm2_2);

  imm2_1 = _mm_add_epi32(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_1));
  imm2_2 = _mm_add_epi32(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_1));

  imm2_1 =
      _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_inv1));
  imm2_2 =
      _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_inv1));

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
  y = _mm256_cvtepi32_ps(imm2);

  imm4_1 = imm2_1;
  imm4_2 = imm2_2;

  imm0_1 = _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_4));
  imm0_2 = _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_4));

  imm0_1 = _mm_slli_epi32(imm0_1, 29);
  imm0_2 = _mm_slli_epi32(imm0_2, 29);

  COPY_XMM_TO_IMM(imm0_1, imm0_2, imm0);

  imm2_1 = _mm_and_si128(imm2_1, *reinterpret_cast<const v4si *>(_pi32avx_2));
  imm2_2 = _mm_and_si128(imm2_2, *reinterpret_cast<const v4si *>(_pi32avx_2));

  imm2_1 = _mm_cmpeq_epi32(imm2_1, _mm_setzero_si128());
  imm2_2 = _mm_cmpeq_epi32(imm2_2, _mm_setzero_si128());

  COPY_XMM_TO_IMM(imm2_1, imm2_2, imm2);
#endif
  v8sf swap_sign_bit_sin = _mm256_castsi256_ps(imm0);
  v8sf poly_mask = _mm256_castsi256_ps(imm2);

  /* The magic pass: "Extended precision modular arithmetic"
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP1);
  xmm2 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP2);
  xmm3 = *reinterpret_cast<const v8sf *>(_ps256_minus_cephes_DP3);
  xmm1 = _mm256_mul_ps(y, xmm1);
  xmm2 = _mm256_mul_ps(y, xmm2);
  xmm3 = _mm256_mul_ps(y, xmm3);
  x = _mm256_add_ps(x, xmm1);
  x = _mm256_add_ps(x, xmm2);
  x = _mm256_add_ps(x, xmm3);

#ifdef __AVX2__
  imm4 =
      avx2_mm256_sub_epi32(imm4, *reinterpret_cast<const v8si *>(_pi32_256_2));
  imm4 = avx2_mm256_andnot_si256(imm4,
                                 *reinterpret_cast<const v8si *>(_pi32_256_4));
  imm4 = avx2_mm256_slli_epi32(imm4, 29);
#else
  imm4_1 = _mm_sub_epi32(imm4_1, *reinterpret_cast<const v4si *>(_pi32avx_2));
  imm4_2 = _mm_sub_epi32(imm4_2, *reinterpret_cast<const v4si *>(_pi32avx_2));

  imm4_1 =
      _mm_andnot_si128(imm4_1, *reinterpret_cast<const v4si *>(_pi32avx_4));
  imm4_2 =
      _mm_andnot_si128(imm4_2, *reinterpret_cast<const v4si *>(_pi32avx_4));

  imm4_1 = _mm_slli_epi32(imm4_1, 29);
  imm4_2 = _mm_slli_epi32(imm4_2, 29);

  COPY_XMM_TO_IMM(imm4_1, imm4_2, imm4);
#endif

  v8sf sign_bit_cos = _mm256_castsi256_ps(imm4);

  sign_bit_sin = _mm256_xor_ps(sign_bit_sin, swap_sign_bit_sin);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  v8sf z = _mm256_mul_ps(x, x);
  y = *reinterpret_cast<const v8sf *>(_ps256_coscof_p0);

  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p1));
  y = _mm256_mul_ps(y, z);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_coscof_p2));
  y = _mm256_mul_ps(y, z);
  y = _mm256_mul_ps(y, z);
  v8sf tmp = _mm256_mul_ps(z, *reinterpret_cast<const v8sf *>(_ps256_0p5));
  y = _mm256_sub_ps(y, tmp);
  y = _mm256_add_ps(y, *reinterpret_cast<const v8sf *>(_ps256_1));

  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v8sf y2 = *reinterpret_cast<const v8sf *>(_ps256_sincof_p0);
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p1));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_add_ps(y2, *reinterpret_cast<const v8sf *>(_ps256_sincof_p2));
  y2 = _mm256_mul_ps(y2, z);
  y2 = _mm256_mul_ps(y2, x);
  y2 = _mm256_add_ps(y2, x);

  /* select the correct result from the two polynoms */
  xmm3 = poly_mask;
  v8sf ysin2 = _mm256_and_ps(xmm3, y2);
  v8sf ysin1 = _mm256_andnot_ps(xmm3, y);
  y2 = _mm256_sub_ps(y2, ysin2);
  y = _mm256_sub_ps(y, ysin1);

  xmm1 = _mm256_add_ps(ysin1, ysin2);
  xmm2 = _mm256_add_ps(y, y2);

  /* update the sign */
  *s = _mm256_xor_ps(xmm1, sign_bit_sin);
  *c = _mm256_xor_ps(xmm2, sign_bit_cos);
}