arch/SSE/MathFunctions.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2007 Julien Pommier
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

/* The sin, cos, exp, and log functions of this file come from
 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
 */

#ifndef EIGEN_MATH_FUNCTIONS_SSE_H
#define EIGEN_MATH_FUNCTIONS_SSE_H

namespace Eigen {

namespace internal {

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f plog<Packet4f>(const Packet4f& _x)
{
  Packet4f x = _x;
  _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
  _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
  _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f);

  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inv_mant_mask, ~0x7f800000);

  /* the smallest non denormalized float number */
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(min_norm_pos,  0x00800000);
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(minus_inf,     0xff800000);//-1.f/0.f);

  /* natural logarithm computed for 4 simultaneous float
    return NaN for x <= 0
  */
  _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292E-2f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, - 1.1514610310E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, - 1.2420140846E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, + 1.4249322787E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, - 1.6668057665E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, + 2.0000714765E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, - 2.4999993993E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, + 3.3333331174E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f);


  Packet4i emm0;

  Packet4f invalid_mask = _mm_cmpnge_ps(x, _mm_setzero_ps()); // not greater equal is true if x is NaN
  Packet4f iszero_mask = _mm_cmpeq_ps(x, _mm_setzero_ps());

  x = pmax(x, p4f_min_norm_pos);  /* cut off denormalized stuff */
  emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23);

  /* keep only the fractional part */
  x = _mm_and_ps(x, p4f_inv_mant_mask);
  x = _mm_or_ps(x, p4f_half);

  emm0 = _mm_sub_epi32(emm0, p4i_0x7f);
  Packet4f e = padd(Packet4f(_mm_cvtepi32_ps(emm0)), p4f_1);

  /* part2:
     if( x < SQRTHF ) {
       e -= 1;
       x = x + x - 1.0;
     } else { x = x - 1.0; }
  */
  Packet4f mask = _mm_cmplt_ps(x, p4f_cephes_SQRTHF);
  Packet4f tmp = pand(x, mask);
  x = psub(x, p4f_1);
  e = psub(e, pand(p4f_1, mask));
  x = padd(x, tmp);

  Packet4f x2 = pmul(x,x);
  Packet4f x3 = pmul(x2,x);

  Packet4f y, y1, y2;
  y  = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1);
  y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4);
  y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7);
  y  = pmadd(y , x, p4f_cephes_log_p2);
  y1 = pmadd(y1, x, p4f_cephes_log_p5);
  y2 = pmadd(y2, x, p4f_cephes_log_p8);
  y = pmadd(y, x3, y1);
  y = pmadd(y, x3, y2);
  y = pmul(y, x3);

  y1 = pmul(e, p4f_cephes_log_q1);
  tmp = pmul(x2, p4f_half);
  y = padd(y, y1);
  x = psub(x, tmp);
  y2 = pmul(e, p4f_cephes_log_q2);
  x = padd(x, y);
  x = padd(x, y2);
  // negative arg will be NAN, 0 will be -INF
  return _mm_or_ps(_mm_andnot_ps(iszero_mask, _mm_or_ps(x, invalid_mask)),
                   _mm_and_ps(iszero_mask, p4f_minus_inf));
}

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f pexp<Packet4f>(const Packet4f& _x)
{
  Packet4f x = _x;
  _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
  _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);
  _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f);


  _EIGEN_DECLARE_CONST_Packet4f(exp_hi,  88.3762626647950f);
  _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -88.3762626647949f);

  _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f);

  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500E-4f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507E-3f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073E-3f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894E-2f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201E-1f);

  Packet4f tmp, fx;
  Packet4i emm0;

  // clamp x
  x = pmax(pmin(x, p4f_exp_hi), p4f_exp_lo);

  /* express exp(x) as exp(g + n*log(2)) */
  fx = pmadd(x, p4f_cephes_LOG2EF, p4f_half);

#ifdef EIGEN_VECTORIZE_SSE4_1
  fx = _mm_floor_ps(fx);
#else
  emm0 = _mm_cvttps_epi32(fx);
  tmp  = _mm_cvtepi32_ps(emm0);
  /* if greater, substract 1 */
  Packet4f mask = _mm_cmpgt_ps(tmp, fx);
  mask = _mm_and_ps(mask, p4f_1);
  fx = psub(tmp, mask);
#endif

  tmp = pmul(fx, p4f_cephes_exp_C1);
  Packet4f z = pmul(fx, p4f_cephes_exp_C2);
  x = psub(x, tmp);
  x = psub(x, z);

  z = pmul(x,x);

  Packet4f y = p4f_cephes_exp_p0;
  y = pmadd(y, x, p4f_cephes_exp_p1);
  y = pmadd(y, x, p4f_cephes_exp_p2);
  y = pmadd(y, x, p4f_cephes_exp_p3);
  y = pmadd(y, x, p4f_cephes_exp_p4);
  y = pmadd(y, x, p4f_cephes_exp_p5);
  y = pmadd(y, z, x);
  y = padd(y, p4f_1);

  // build 2^n
  emm0 = _mm_cvttps_epi32(fx);
  emm0 = _mm_add_epi32(emm0, p4i_0x7f);
  emm0 = _mm_slli_epi32(emm0, 23);
  return pmax(pmul(y, Packet4f(_mm_castsi128_ps(emm0))), _x);
}
template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet2d pexp<Packet2d>(const Packet2d& _x)
{
  Packet2d x = _x;

  _EIGEN_DECLARE_CONST_Packet2d(1 , 1.0);
  _EIGEN_DECLARE_CONST_Packet2d(2 , 2.0);
  _EIGEN_DECLARE_CONST_Packet2d(half, 0.5);

  _EIGEN_DECLARE_CONST_Packet2d(exp_hi,  709.437);
  _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -709.436139303);

  _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599);

  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1);

  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0);

  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125);
  _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6);
  static const __m128i p4i_1023_0 = _mm_setr_epi32(1023, 1023, 0, 0);

  Packet2d tmp, fx;
  Packet4i emm0;

  // clamp x
  x = pmax(pmin(x, p2d_exp_hi), p2d_exp_lo);
  /* express exp(x) as exp(g + n*log(2)) */
  fx = pmadd(p2d_cephes_LOG2EF, x, p2d_half);

#ifdef EIGEN_VECTORIZE_SSE4_1
  fx = _mm_floor_pd(fx);
#else
  emm0 = _mm_cvttpd_epi32(fx);
  tmp  = _mm_cvtepi32_pd(emm0);
  /* if greater, substract 1 */
  Packet2d mask = _mm_cmpgt_pd(tmp, fx);
  mask = _mm_and_pd(mask, p2d_1);
  fx = psub(tmp, mask);
#endif

  tmp = pmul(fx, p2d_cephes_exp_C1);
  Packet2d z = pmul(fx, p2d_cephes_exp_C2);
  x = psub(x, tmp);
  x = psub(x, z);

  Packet2d x2 = pmul(x,x);

  Packet2d px = p2d_cephes_exp_p0;
  px = pmadd(px, x2, p2d_cephes_exp_p1);
  px = pmadd(px, x2, p2d_cephes_exp_p2);
  px = pmul (px, x);

  Packet2d qx = p2d_cephes_exp_q0;
  qx = pmadd(qx, x2, p2d_cephes_exp_q1);
  qx = pmadd(qx, x2, p2d_cephes_exp_q2);
  qx = pmadd(qx, x2, p2d_cephes_exp_q3);

  x = pdiv(px,psub(qx,px));
  x = pmadd(p2d_2,x,p2d_1);

  // build 2^n
  emm0 = _mm_cvttpd_epi32(fx);
  emm0 = _mm_add_epi32(emm0, p4i_1023_0);
  emm0 = _mm_slli_epi32(emm0, 20);
  emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(1,2,0,3));
  return pmax(pmul(x, Packet2d(_mm_castsi128_pd(emm0))), _x);
}

/* evaluation of 4 sines at onces, using SSE2 intrinsics.

   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.
*/

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psin<Packet4f>(const Packet4f& _x)
{
  Packet4f x = _x;
  _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
  _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);

  _EIGEN_DECLARE_CONST_Packet4i(1, 1);
  _EIGEN_DECLARE_CONST_Packet4i(not1, ~1);
  _EIGEN_DECLARE_CONST_Packet4i(2, 2);
  _EIGEN_DECLARE_CONST_Packet4i(4, 4);

  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(sign_mask, 0x80000000);

  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f);
  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f);
  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p1,  8.3321608736E-3f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p0,  2.443315711809948E-005f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p2,  4.166664568298827E-002f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI

  Packet4f xmm1, xmm2, xmm3, sign_bit, y;

  Packet4i emm0, emm2;
  sign_bit = x;
  /* take the absolute value */
  x = pabs(x);

  /* take the modulo */

  /* extract the sign bit (upper one) */
  sign_bit = _mm_and_ps(sign_bit, p4f_sign_mask);

  /* scale by 4/Pi */
  y = pmul(x, p4f_cephes_FOPI);

  /* store the integer part of y in mm0 */
  emm2 = _mm_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = _mm_add_epi32(emm2, p4i_1);
  emm2 = _mm_and_si128(emm2, p4i_not1);
  y = _mm_cvtepi32_ps(emm2);
  /* get the swap sign flag */
  emm0 = _mm_and_si128(emm2, p4i_4);
  emm0 = _mm_slli_epi32(emm0, 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.
  */
  emm2 = _mm_and_si128(emm2, p4i_2);
  emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());

  Packet4f swap_sign_bit = _mm_castsi128_ps(emm0);
  Packet4f poly_mask = _mm_castsi128_ps(emm2);
  sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit);

  /* The magic pass: "Extended precision modular arithmetic"
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = pmul(y, p4f_minus_cephes_DP1);
  xmm2 = pmul(y, p4f_minus_cephes_DP2);
  xmm3 = pmul(y, p4f_minus_cephes_DP3);
  x = padd(x, xmm1);
  x = padd(x, xmm2);
  x = padd(x, xmm3);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = p4f_coscof_p0;
  Packet4f z = _mm_mul_ps(x,x);

  y = pmadd(y, z, p4f_coscof_p1);
  y = pmadd(y, z, p4f_coscof_p2);
  y = pmul(y, z);
  y = pmul(y, z);
  Packet4f tmp = pmul(z, p4f_half);
  y = psub(y, tmp);
  y = padd(y, p4f_1);

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

  Packet4f y2 = p4f_sincof_p0;
  y2 = pmadd(y2, z, p4f_sincof_p1);
  y2 = pmadd(y2, z, p4f_sincof_p2);
  y2 = pmul(y2, z);
  y2 = pmul(y2, x);
  y2 = padd(y2, x);

  /* select the correct result from the two polynoms */
  y2 = _mm_and_ps(poly_mask, y2);
  y = _mm_andnot_ps(poly_mask, y);
  y = _mm_or_ps(y,y2);
  /* update the sign */
  return _mm_xor_ps(y, sign_bit);
}

/* almost the same as psin */
template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f pcos<Packet4f>(const Packet4f& _x)
{
  Packet4f x = _x;
  _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f);
  _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f);

  _EIGEN_DECLARE_CONST_Packet4i(1, 1);
  _EIGEN_DECLARE_CONST_Packet4i(not1, ~1);
  _EIGEN_DECLARE_CONST_Packet4i(2, 2);
  _EIGEN_DECLARE_CONST_Packet4i(4, 4);

  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f);
  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f);
  _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p1,  8.3321608736E-3f);
  _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p0,  2.443315711809948E-005f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f);
  _EIGEN_DECLARE_CONST_Packet4f(coscof_p2,  4.166664568298827E-002f);
  _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI

  Packet4f xmm1, xmm2, xmm3, y;
  Packet4i emm0, emm2;

  x = pabs(x);

  /* scale by 4/Pi */
  y = pmul(x, p4f_cephes_FOPI);

  /* get the integer part of y */
  emm2 = _mm_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = _mm_add_epi32(emm2, p4i_1);
  emm2 = _mm_and_si128(emm2, p4i_not1);
  y = _mm_cvtepi32_ps(emm2);

  emm2 = _mm_sub_epi32(emm2, p4i_2);

  /* get the swap sign flag */
  emm0 = _mm_andnot_si128(emm2, p4i_4);
  emm0 = _mm_slli_epi32(emm0, 29);
  /* get the polynom selection mask */
  emm2 = _mm_and_si128(emm2, p4i_2);
  emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());

  Packet4f sign_bit = _mm_castsi128_ps(emm0);
  Packet4f poly_mask = _mm_castsi128_ps(emm2);

  /* The magic pass: "Extended precision modular arithmetic"
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = pmul(y, p4f_minus_cephes_DP1);
  xmm2 = pmul(y, p4f_minus_cephes_DP2);
  xmm3 = pmul(y, p4f_minus_cephes_DP3);
  x = padd(x, xmm1);
  x = padd(x, xmm2);
  x = padd(x, xmm3);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = p4f_coscof_p0;
  Packet4f z = pmul(x,x);

  y = pmadd(y,z,p4f_coscof_p1);
  y = pmadd(y,z,p4f_coscof_p2);
  y = pmul(y, z);
  y = pmul(y, z);
  Packet4f tmp = _mm_mul_ps(z, p4f_half);
  y = psub(y, tmp);
  y = padd(y, p4f_1);

  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */
  Packet4f y2 = p4f_sincof_p0;
  y2 = pmadd(y2, z, p4f_sincof_p1);
  y2 = pmadd(y2, z, p4f_sincof_p2);
  y2 = pmul(y2, z);
  y2 = pmadd(y2, x, x);

  /* select the correct result from the two polynoms */
  y2 = _mm_and_ps(poly_mask, y2);
  y  = _mm_andnot_ps(poly_mask, y);
  y  = _mm_or_ps(y,y2);

  /* update the sign */
  return _mm_xor_ps(y, sign_bit);
}

#if EIGEN_FAST_MATH

// Functions for sqrt.
// The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
// of Newton's method, at a cost of 1-2 bits of precision as opposed to the
// exact solution. It does not handle +inf, or denormalized numbers correctly.
// The main advantage of this approach is not just speed, but also the fact that
// it can be inlined and pipelined with other computations, further reducing its
// effective latency. This is similar to Quake3's fast inverse square root.
// For detail see here: http://www.beyond3d.com/content/articles/8/
template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psqrt<Packet4f>(const Packet4f& _x)
{
  Packet4f half = pmul(_x, pset1<Packet4f>(.5f));
  Packet4f denormal_mask = _mm_and_ps(
      _mm_cmpge_ps(_x, _mm_setzero_ps()),
      _mm_cmplt_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)())));

  // Compute approximate reciprocal sqrt.
  Packet4f x = _mm_rsqrt_ps(_x);
  // Do a single step of Newton's iteration.
  x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x))));
  // Flush results for denormals to zero.
  return _mm_andnot_ps(denormal_mask, pmul(_x,x));
}

#else

template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); }

#endif

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); }

#if EIGEN_FAST_MATH

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f prsqrt<Packet4f>(const Packet4f& _x) {
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000);
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000);
  _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f);
  _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f);
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000);

  Packet4f neg_half = pmul(_x, p4f_minus_half);

  // select only the inverse sqrt of positive normal inputs (denormals are
  // flushed to zero and cause infs as well).
  Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min);
  Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x));

  // Fill in NaNs and Infs for the negative/zero entries.
  Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps());
  Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask);
  Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan),
                                     _mm_and_ps(zero_mask, p4f_inf));

  // Do a single step of Newton's iteration.
  x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five));

  // Insert NaNs and Infs in all the right places.
  return _mm_or_ps(x, infs_and_nans);
}

#else

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f prsqrt<Packet4f>(const Packet4f& x) {
  // Unfortunately we can't use the much faster mm_rqsrt_ps since it only provides an approximation.
  return _mm_div_ps(pset1<Packet4f>(1.0f), _mm_sqrt_ps(x));
}

#endif

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet2d prsqrt<Packet2d>(const Packet2d& x) {
  // Unfortunately we can't use the much faster mm_rqsrt_pd since it only provides an approximation.
  return _mm_div_pd(pset1<Packet2d>(1.0), _mm_sqrt_pd(x));
}

// Hyperbolic Tangent function.
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f
ptanh<Packet4f>(const Packet4f& x) {
  return internal::generic_fast_tanh_float(x);
}

} // end namespace internal

namespace numext {

template<>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
float sqrt(const float &x)
{
  return internal::pfirst(internal::Packet4f(_mm_sqrt_ss(_mm_set_ss(x))));
}

template<>
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE
double sqrt(const double &x)
{
#if EIGEN_COMP_GNUC_STRICT
  // This works around a GCC bug generating poor code for _mm_sqrt_pd
  // See https://bitbucket.org/eigen/eigen/commits/14f468dba4d350d7c19c9b93072e19f7b3df563b
  return internal::pfirst(internal::Packet2d(__builtin_ia32_sqrtsd(_mm_set_sd(x))));
#else
  return internal::pfirst(internal::Packet2d(_mm_sqrt_pd(_mm_set_sd(x))));
#endif
}

} // end namespace numex

} // end namespace Eigen

#endif // EIGEN_MATH_FUNCTIONS_SSE_H