math_func_neon.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. */

/* NEON implementation of sin, cos, exp and log
 *
 *   Inspired by Intel Approximate Math library, and based on the
 *   corresponding algorithms of the cephes math library
 */

/* Copyright (C) 2011  Julien Pommier
 *
 *  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 <arm_neon.h>

#define c_inv_mant_mask ~0x7f800000u
#define c_cephes_SQRTHF 0.707106781186547524
#define c_cephes_log_p0 7.0376836292E-2
#define c_cephes_log_p1 -1.1514610310E-1
#define c_cephes_log_p2 1.1676998740E-1
#define c_cephes_log_p3 -1.2420140846E-1
#define c_cephes_log_p4 +1.4249322787E-1
#define c_cephes_log_p5 -1.6668057665E-1
#define c_cephes_log_p6 +2.0000714765E-1
#define c_cephes_log_p7 -2.4999993993E-1
#define c_cephes_log_p8 +3.3333331174E-1
#define c_cephes_log_q1 -2.12194440e-4
#define c_cephes_log_q2 0.693359375

/* natural logarithm computed for 4 simultaneous float
 *   return NaN for x <= 0
 */
static inline float32x4_t log_ps(float32x4_t x) {
  float32x4_t one = vdupq_n_f32(1);

  x = vmaxq_f32(x, vdupq_n_f32(0)); /* force flush to zero on denormal values */
  uint32x4_t invalid_mask = vcleq_f32(x, vdupq_n_f32(0));

  int32x4_t ux = vreinterpretq_s32_f32(x);

  int32x4_t emm0 = vshrq_n_s32(ux, 23);

  /* keep only the fractional part */
  ux = vandq_s32(ux, vdupq_n_s32(c_inv_mant_mask));
  ux = vorrq_s32(ux, vreinterpretq_s32_f32(vdupq_n_f32(0.5f)));
  x = vreinterpretq_f32_s32(ux);

  emm0 = vsubq_s32(emm0, vdupq_n_s32(0x7f));
  float32x4_t e = vcvtq_f32_s32(emm0);

  e = vaddq_f32(e, one);

  /* part2:
   *     if( x < SQRTHF ) {
   *       e -= 1;
   *       x = x + x - 1.0;
   *     } else { x = x - 1.0; }
   */
  uint32x4_t mask = vcltq_f32(x, vdupq_n_f32(c_cephes_SQRTHF));
  float32x4_t tmp =
      vreinterpretq_f32_u32(vandq_u32(vreinterpretq_u32_f32(x), mask));
  x = vsubq_f32(x, one);
  e = vsubq_f32(
      e, vreinterpretq_f32_u32(vandq_u32(vreinterpretq_u32_f32(one), mask)));
  x = vaddq_f32(x, tmp);

  float32x4_t z = vmulq_f32(x, x);

  float32x4_t y = vdupq_n_f32(c_cephes_log_p0);
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p1));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p2));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p3));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p4));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p5));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p6));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p7));
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, vdupq_n_f32(c_cephes_log_p8));
  y = vmulq_f32(y, x);

  y = vmulq_f32(y, z);

  tmp = vmulq_f32(e, vdupq_n_f32(c_cephes_log_q1));
  y = vaddq_f32(y, tmp);

  tmp = vmulq_f32(z, vdupq_n_f32(0.5f));
  y = vsubq_f32(y, tmp);

  tmp = vmulq_f32(e, vdupq_n_f32(c_cephes_log_q2));
  x = vaddq_f32(x, y);
  x = vaddq_f32(x, tmp);
  x = vreinterpretq_f32_u32(vorrq_u32(
      vreinterpretq_u32_f32(x), invalid_mask));  // negative arg will be NAN
  return x;
}

#define c_exp_hi 88.3762626647949f
#define c_exp_lo -88.3762626647949f

#define c_cephes_LOG2EF 1.44269504088896341
#define c_cephes_exp_C1 0.693359375
#define c_cephes_exp_C2 -2.12194440e-4

#define c_cephes_exp_p0 1.9875691500E-4
#define c_cephes_exp_p1 1.3981999507E-3
#define c_cephes_exp_p2 8.3334519073E-3
#define c_cephes_exp_p3 4.1665795894E-2
#define c_cephes_exp_p4 1.6666665459E-1
#define c_cephes_exp_p5 5.0000001201E-1

/* exp() computed for 4 float at once */
static inline float32x4_t exp_ps(float32x4_t x) {
  float32x4_t tmp, fx;

  float32x4_t one = vdupq_n_f32(1);
  x = vminq_f32(x, vdupq_n_f32(c_exp_hi));
  x = vmaxq_f32(x, vdupq_n_f32(c_exp_lo));

  /* express exp(x) as exp(g + n*log(2)) */
  fx = vmlaq_f32(vdupq_n_f32(0.5f), x, vdupq_n_f32(c_cephes_LOG2EF));

  /* perform a floorf */
  tmp = vcvtq_f32_s32(vcvtq_s32_f32(fx));

  /* if greater, substract 1 */
  uint32x4_t mask = vcgtq_f32(tmp, fx);
  mask = vandq_u32(mask, vreinterpretq_u32_f32(one));

  fx = vsubq_f32(tmp, vreinterpretq_f32_u32(mask));

  tmp = vmulq_f32(fx, vdupq_n_f32(c_cephes_exp_C1));
  float32x4_t z = vmulq_f32(fx, vdupq_n_f32(c_cephes_exp_C2));
  x = vsubq_f32(x, tmp);
  x = vsubq_f32(x, z);

  static const float cephes_exp_p[6] = {c_cephes_exp_p0, c_cephes_exp_p1,
                                        c_cephes_exp_p2, c_cephes_exp_p3,
                                        c_cephes_exp_p4, c_cephes_exp_p5};
  float32x4_t y = vld1q_dup_f32(cephes_exp_p + 0);
  float32x4_t c1 = vld1q_dup_f32(cephes_exp_p + 1);
  float32x4_t c2 = vld1q_dup_f32(cephes_exp_p + 2);
  float32x4_t c3 = vld1q_dup_f32(cephes_exp_p + 3);
  float32x4_t c4 = vld1q_dup_f32(cephes_exp_p + 4);
  float32x4_t c5 = vld1q_dup_f32(cephes_exp_p + 5);

  y = vmulq_f32(y, x);
  z = vmulq_f32(x, x);

  y = vaddq_f32(y, c1);
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, c2);
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, c3);
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, c4);
  y = vmulq_f32(y, x);
  y = vaddq_f32(y, c5);

  y = vmulq_f32(y, z);
  y = vaddq_f32(y, x);
  y = vaddq_f32(y, one);

  /* build 2^n */
  int32x4_t mm;
  mm = vcvtq_s32_f32(fx);
  mm = vaddq_s32(mm, vdupq_n_s32(0x7f));
  mm = vshlq_n_s32(mm, 23);
  float32x4_t pow2n = vreinterpretq_f32_s32(mm);

  y = vmulq_f32(y, pow2n);
  return y;
}

#define c_minus_cephes_DP1 -0.78515625
#define c_minus_cephes_DP2 -2.4187564849853515625e-4
#define c_minus_cephes_DP3 -3.77489497744594108e-8
#define c_sincof_p0 -1.9515295891E-4
#define c_sincof_p1 8.3321608736E-3
#define c_sincof_p2 -1.6666654611E-1
#define c_coscof_p0 2.443315711809948E-005
#define c_coscof_p1 -1.388731625493765E-003
#define c_coscof_p2 4.166664568298827E-002
#define c_cephes_FOPI 1.27323954473516  // 4 / M_PI

/* evaluation of 4 sines & cosines at once.
 *
 *   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.
 *
 *   Note also that when you compute sin(x), cos(x) is available at
 *   almost no extra price so both sin_ps and cos_ps make use of
 *   sincos_ps..
 */
static inline void sincos_ps(float32x4_t x, float32x4_t *ysin,
                             float32x4_t *ycos) {
  // any x
  float32x4_t xmm1, xmm2, xmm3, y;

  uint32x4_t emm2;

  uint32x4_t sign_mask_sin, sign_mask_cos;
  sign_mask_sin = vcltq_f32(x, vdupq_n_f32(0));
  x = vabsq_f32(x);

  /* scale by 4/Pi */
  y = vmulq_f32(x, vdupq_n_f32(c_cephes_FOPI));

  /* store the integer part of y in mm0 */
  emm2 = vcvtq_u32_f32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = vaddq_u32(emm2, vdupq_n_u32(1));
  emm2 = vandq_u32(emm2, vdupq_n_u32(~1));
  y = vcvtq_f32_u32(emm2);

  /* 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.
   */
  uint32x4_t poly_mask = vtstq_u32(emm2, vdupq_n_u32(2));

  /* The magic pass: "Extended precision modular arithmetic"
   *     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = vmulq_n_f32(y, c_minus_cephes_DP1);
  xmm2 = vmulq_n_f32(y, c_minus_cephes_DP2);
  xmm3 = vmulq_n_f32(y, c_minus_cephes_DP3);
  x = vaddq_f32(x, xmm1);
  x = vaddq_f32(x, xmm2);
  x = vaddq_f32(x, xmm3);

  sign_mask_sin = veorq_u32(sign_mask_sin, vtstq_u32(emm2, vdupq_n_u32(4)));
  sign_mask_cos = vtstq_u32(vsubq_u32(emm2, vdupq_n_u32(2)), vdupq_n_u32(4));

  /* Evaluate the first polynom  (0 <= x <= Pi/4) in y1,
   *     and the second polynom      (Pi/4 <= x <= 0) in y2 */
  float32x4_t z = vmulq_f32(x, x);
  float32x4_t y1, y2;

  y1 = vmulq_n_f32(z, c_coscof_p0);
  y2 = vmulq_n_f32(z, c_sincof_p0);
  y1 = vaddq_f32(y1, vdupq_n_f32(c_coscof_p1));
  y2 = vaddq_f32(y2, vdupq_n_f32(c_sincof_p1));
  y1 = vmulq_f32(y1, z);
  y2 = vmulq_f32(y2, z);
  y1 = vaddq_f32(y1, vdupq_n_f32(c_coscof_p2));
  y2 = vaddq_f32(y2, vdupq_n_f32(c_sincof_p2));
  y1 = vmulq_f32(y1, z);
  y2 = vmulq_f32(y2, z);
  y1 = vmulq_f32(y1, z);
  y2 = vmulq_f32(y2, x);
  y1 = vsubq_f32(y1, vmulq_f32(z, vdupq_n_f32(0.5f)));
  y2 = vaddq_f32(y2, x);
  y1 = vaddq_f32(y1, vdupq_n_f32(1));

  /* select the correct result from the two polynoms */
  float32x4_t ys = vbslq_f32(poly_mask, y1, y2);
  float32x4_t yc = vbslq_f32(poly_mask, y2, y1);
  *ysin = vbslq_f32(sign_mask_sin, vnegq_f32(ys), ys);
  *ycos = vbslq_f32(sign_mask_cos, yc, vnegq_f32(yc));
}

static inline float32x4_t sin_ps(float32x4_t x) {
  float32x4_t ysin, ycos;
  sincos_ps(x, &ysin, &ycos);
  return ysin;
}

static inline float32x4_t cos_ps(float32x4_t x) {
  float32x4_t ysin, ycos;
  sincos_ps(x, &ysin, &ycos);
  return ycos;
}

static inline float32x4_t div_ps(float32x4_t a, float32x4_t b) {
  float32x4_t reciprocal = vrecpeq_f32(b);
  reciprocal = vmulq_f32(vrecpsq_f32(b, reciprocal), reciprocal);
  //     reciprocal = vmulq_f32(vrecpsq_f32(b, reciprocal), reciprocal);
  return vmulq_f32(a, reciprocal);
}

static inline float32x4_t pow_ps(float32x4_t a, float32x4_t b) {
  // pow(x, m) = exp(m * log(x))
  return exp_ps(vmulq_f32(b, log_ps(a)));
}