// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com) // // 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/. #ifndef EIGEN_MATH_FUNCTIONS_AVX_H #define EIGEN_MATH_FUNCTIONS_AVX_H /* The sin, cos, and exp functions of this file are loosely derived from * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/ */ namespace Eigen { namespace internal { inline Packet8i pshiftleft(Packet8i v, int n) { #ifdef EIGEN_VECTORIZE_AVX2 return _mm256_slli_epi32(v, n); #else __m128i lo = _mm_slli_epi32(_mm256_extractf128_si256(v, 0), n); __m128i hi = _mm_slli_epi32(_mm256_extractf128_si256(v, 1), n); return _mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1); #endif } // Sine function // Computes sin(x) by wrapping x to the interval [-Pi/4,3*Pi/4] and // evaluating interpolants in [-Pi/4,Pi/4] or [Pi/4,3*Pi/4]. The interpolants // are (anti-)symmetric and thus have only odd/even coefficients template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f psin(const Packet8f& _x) { Packet8f x = _x; // Some useful values. _EIGEN_DECLARE_CONST_Packet8i(one, 1); _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f); _EIGEN_DECLARE_CONST_Packet8f(two, 2.0f); _EIGEN_DECLARE_CONST_Packet8f(one_over_four, 0.25f); _EIGEN_DECLARE_CONST_Packet8f(one_over_pi, 3.183098861837907e-01f); _EIGEN_DECLARE_CONST_Packet8f(neg_pi_first, -3.140625000000000e+00f); _EIGEN_DECLARE_CONST_Packet8f(neg_pi_second, -9.670257568359375e-04f); _EIGEN_DECLARE_CONST_Packet8f(neg_pi_third, -6.278329571784980e-07f); _EIGEN_DECLARE_CONST_Packet8f(four_over_pi, 1.273239544735163e+00f); // Map x from [-Pi/4,3*Pi/4] to z in [-1,3] and subtract the shifted period. Packet8f z = pmul(x, p8f_one_over_pi); Packet8f shift = _mm256_floor_ps(padd(z, p8f_one_over_four)); x = pmadd(shift, p8f_neg_pi_first, x); x = pmadd(shift, p8f_neg_pi_second, x); x = pmadd(shift, p8f_neg_pi_third, x); z = pmul(x, p8f_four_over_pi); // Make a mask for the entries that need flipping, i.e. wherever the shift // is odd. Packet8i shift_ints = _mm256_cvtps_epi32(shift); Packet8i shift_isodd = _mm256_castps_si256(_mm256_and_ps(_mm256_castsi256_ps(shift_ints), _mm256_castsi256_ps(p8i_one))); Packet8i sign_flip_mask = pshiftleft(shift_isodd, 31); // Create a mask for which interpolant to use, i.e. if z > 1, then the mask // is set to ones for that entry. Packet8f ival_mask = _mm256_cmp_ps(z, p8f_one, _CMP_GT_OQ); // Evaluate the polynomial for the interval [1,3] in z. _EIGEN_DECLARE_CONST_Packet8f(coeff_right_0, 9.999999724233232e-01f); _EIGEN_DECLARE_CONST_Packet8f(coeff_right_2, -3.084242535619928e-01f); _EIGEN_DECLARE_CONST_Packet8f(coeff_right_4, 1.584991525700324e-02f); _EIGEN_DECLARE_CONST_Packet8f(coeff_right_6, -3.188805084631342e-04f); Packet8f z_minus_two = psub(z, p8f_two); Packet8f z_minus_two2 = pmul(z_minus_two, z_minus_two); Packet8f right = pmadd(p8f_coeff_right_6, z_minus_two2, p8f_coeff_right_4); right = pmadd(right, z_minus_two2, p8f_coeff_right_2); right = pmadd(right, z_minus_two2, p8f_coeff_right_0); // Evaluate the polynomial for the interval [-1,1] in z. _EIGEN_DECLARE_CONST_Packet8f(coeff_left_1, 7.853981525427295e-01f); _EIGEN_DECLARE_CONST_Packet8f(coeff_left_3, -8.074536727092352e-02f); _EIGEN_DECLARE_CONST_Packet8f(coeff_left_5, 2.489871967827018e-03f); _EIGEN_DECLARE_CONST_Packet8f(coeff_left_7, -3.587725841214251e-05f); Packet8f z2 = pmul(z, z); Packet8f left = pmadd(p8f_coeff_left_7, z2, p8f_coeff_left_5); left = pmadd(left, z2, p8f_coeff_left_3); left = pmadd(left, z2, p8f_coeff_left_1); left = pmul(left, z); // Assemble the results, i.e. select the left and right polynomials. left = _mm256_andnot_ps(ival_mask, left); right = _mm256_and_ps(ival_mask, right); Packet8f res = _mm256_or_ps(left, right); // Flip the sign on the odd intervals and return the result. res = _mm256_xor_ps(res, _mm256_castsi256_ps(sign_flip_mask)); return res; } template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f plog(const Packet8f& _x) { return plog_float(_x); } // Exponential function. Works by writing "x = m*log(2) + r" where // "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then // "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1). template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f pexp(const Packet8f& _x) { return pexp_float(_x); } // Hyperbolic Tangent function. template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f ptanh(const Packet8f& x) { return internal::generic_fast_tanh_float(x); } template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d pexp(const Packet4d& x) { return pexp_double(x); } // 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/ #if EIGEN_FAST_MATH template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f psqrt(const Packet8f& _x) { Packet8f half = pmul(_x, pset1(.5f)); Packet8f denormal_mask = _mm256_and_ps( _mm256_cmp_ps(_x, pset1((std::numeric_limits::min)()), _CMP_LT_OQ), _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_GE_OQ)); // Compute approximate reciprocal sqrt. Packet8f x = _mm256_rsqrt_ps(_x); // Do a single step of Newton's iteration. x = pmul(x, psub(pset1(1.5f), pmul(half, pmul(x,x)))); // Flush results for denormals to zero. return _mm256_andnot_ps(denormal_mask, pmul(_x,x)); } #else template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f psqrt(const Packet8f& x) { return _mm256_sqrt_ps(x); } #endif template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d psqrt(const Packet4d& x) { return _mm256_sqrt_pd(x); } #if EIGEN_FAST_MATH template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f prsqrt(const Packet8f& _x) { _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inf, 0x7f800000); _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(nan, 0x7fc00000); _EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f); _EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f); _EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000); Packet8f neg_half = pmul(_x, p8f_minus_half); // select only the inverse sqrt of positive normal inputs (denormals are // flushed to zero and cause infs as well). Packet8f le_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_LT_OQ); Packet8f x = _mm256_andnot_ps(le_zero_mask, _mm256_rsqrt_ps(_x)); // Fill in NaNs and Infs for the negative/zero entries. Packet8f neg_mask = _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_LT_OQ); Packet8f zero_mask = _mm256_andnot_ps(neg_mask, le_zero_mask); Packet8f infs_and_nans = _mm256_or_ps(_mm256_and_ps(neg_mask, p8f_nan), _mm256_and_ps(zero_mask, p8f_inf)); // Do a single step of Newton's iteration. x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five)); // Insert NaNs and Infs in all the right places. return _mm256_or_ps(x, infs_and_nans); } #else template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f prsqrt(const Packet8f& x) { _EIGEN_DECLARE_CONST_Packet8f(one, 1.0f); return _mm256_div_ps(p8f_one, _mm256_sqrt_ps(x)); } #endif template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d prsqrt(const Packet4d& x) { _EIGEN_DECLARE_CONST_Packet4d(one, 1.0); return _mm256_div_pd(p4d_one, _mm256_sqrt_pd(x)); } } // end namespace internal } // end namespace Eigen #endif // EIGEN_MATH_FUNCTIONS_AVX_H