eigen/Eigen/src/Core/arch/SSE/MathFunctions.h

// 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 and cos and 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

#include "../Default/GenericPacketMathFunctions.h"

namespace Eigen {

namespace internal {

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f plog<Packet4f>(const Packet4f& _x)
{
  return plog_float(_x);
}

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f pexp<Packet4f>(const Packet4f& _x)
{
  return pexp_float(_x);
}

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet2d pexp<Packet2d>(const Packet2d& x)
{
  return pexp_double(x);
}

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f psin<Packet4f>(const Packet4f& _x)
{
  return psin_float(_x);
}

template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4f pcos<Packet4f>(const Packet4f& _x)
{
  return pcos_float(_x);
}

#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, 0x7f800000u);
  _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000u);
  _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, 0x00800000u);

  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