| // 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 { |
| |
| // Functions for division. |
| // The EIGEN_FAST_MATH version uses the _mm_rcp_ps approximation and one step of |
| // Newton's method, at a cost of 1-2 bits of precision as opposed to the exact |
| // solution. 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. |
| #if EIGEN_FAST_DIV |
| template <> |
| EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f |
| pdiv<Packet4f>(const Packet4f& a, const Packet4f& b) { |
| _EIGEN_DECLARE_CONST_Packet4f(two, 2.0f); |
| |
| /* Start with an estimate of the reciprocal of b. */ |
| Packet4f x = _mm_rcp_ps(b); |
| |
| /* One step of Newton's method on b - x^-1 == 0. */ |
| x = pmul(x, pmadd(-b, x, p4f_two)); |
| |
| // Multiply the inverse of b with a. |
| return pmul(a, x); |
| } |
| #else |
| template <> |
| EIGEN_STRONG_INLINE Packet4f |
| pdiv<Packet4f>(const Packet4f& a, const Packet4f& b) { |
| return _mm_div_ps(a, b); |
| } |
| #endif |
| template <> |
| EIGEN_STRONG_INLINE Packet2d |
| pdiv<Packet2d>(const Packet2d& a, const Packet2d& b) { |
| return _mm_div_pd(a, b); |
| } |
| template <> |
| EIGEN_STRONG_INLINE Packet4i |
| pdiv<Packet4i>(const Packet4i& /*a*/, const Packet4i& /*b*/) { |
| eigen_assert(false && "packet integer division are not supported by SSE"); |
| return pset1<Packet4i>(0); |
| } |
| |
| // Hyperbolic Tangent function. |
| // Doesn't do anything fancy, just a 13/6-degree rational interpolant which |
| // is accurate up to a couple of ulp in the range [-9, 9], outside of which the |
| // fl(tanh(x)) = +/-1. |
| template <> |
| EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f |
| ptanh<Packet4f>(const Packet4f& _x) { |
| // Clamp the inputs to the range [-9, 9] since anything outside |
| // this range is +/-1.0f in single-precision. |
| _EIGEN_DECLARE_CONST_Packet4f(plus_9, 9.0f); |
| _EIGEN_DECLARE_CONST_Packet4f(minus_9, -9.0f); |
| const Packet4f x = pmax(p4f_minus_9, pmin(p4f_plus_9, _x)); |
| |
| // The monomial coefficients of the numerator polynomial (odd). |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_1, 4.89352455891786e-03f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_3, 6.37261928875436e-04f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_5, 1.48572235717979e-05f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_7, 5.12229709037114e-08f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_9, -8.60467152213735e-11f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_11, 2.00018790482477e-13f); |
| _EIGEN_DECLARE_CONST_Packet4f(alpha_13, -2.76076847742355e-16f); |
| |
| // The monomial coefficients of the denominator polynomial (even). |
| _EIGEN_DECLARE_CONST_Packet4f(beta_0, 4.89352518554385e-03f); |
| _EIGEN_DECLARE_CONST_Packet4f(beta_2, 2.26843463243900e-03f); |
| _EIGEN_DECLARE_CONST_Packet4f(beta_4, 1.18534705686654e-04f); |
| _EIGEN_DECLARE_CONST_Packet4f(beta_6, 1.19825839466702e-06f); |
| |
| // Since the polynomials are odd/even, we need x^2. |
| const Packet4f x2 = pmul(x, x); |
| |
| // Evaluate the numerator polynomial p. |
| Packet4f p = pmadd(x2, p4f_alpha_13, p4f_alpha_11); |
| p = pmadd(x2, p, p4f_alpha_9); |
| p = pmadd(x2, p, p4f_alpha_7); |
| p = pmadd(x2, p, p4f_alpha_5); |
| p = pmadd(x2, p, p4f_alpha_3); |
| p = pmadd(x2, p, p4f_alpha_1); |
| p = pmul(x, p); |
| |
| // Evaluate the denominator polynomial p. |
| Packet4f q = pmadd(x2, p4f_beta_6, p4f_beta_4); |
| q = pmadd(x2, q, p4f_beta_2); |
| q = pmadd(x2, q, p4f_beta_0); |
| |
| // Divide the numerator by the denominator. |
| return pdiv(p, q); |
| } |
| |
| 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; |
| |
| // invalid_mask is set to true when x is NaN |
| Packet4f invalid_mask = _mm_cmpnge_ps(x, _mm_setzero_ps()); |
| 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. 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. |
| template <> |
| EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f |
| psqrt<Packet4f>(const Packet4f& _x) { |
| _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); |
| Packet4f denormal_mask = _mm_and_ps(_mm_cmpge_ps(_x, _mm_setzero_ps()), |
| _mm_cmplt_ps(_x, p4f_flt_min)); |
| |
| // Compute approximate reciprocal sqrt. |
| Packet4f x = _mm_rsqrt_ps(_x); |
| |
| // Do a single step of Newton's iteration. |
| x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five)); |
| |
| // Multiply the original _x by it's reciprocal square root to extract the |
| // square root. |
| x = pmul(_x, x); |
| |
| // Flush results for denormals to zero. |
| return _mm_andnot_ps(denormal_mask, x); |
| } |
| |
| #else |
| |
| template<> EIGEN_STRONG_INLINE Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); } |
| |
| #endif |
| |
| template<> EIGEN_STRONG_INLINE Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); } |
| |
| |
| #if EIGEN_FAST_MATH |
| |
| // Functions for rsqrt. |
| // Almost identical to the sqrt routine, just leave out the last multiplication |
| // and fill in NaN/Inf where needed. Note that this function only exists as an |
| // iterative version since there is no instruction for diretly computing the |
| // reciprocal square root in AVX/AVX2 (there will be one in AVX-512). |
| template <> |
| EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f |
| prsqrt<Packet4f>(const Packet4f& _x) { |
| _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f); |
| _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f); |
| _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000); |
| _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000); |
| |
| // Remeber which entries were zero (or almost). |
| Packet4f is_zero = _mm_and_ps(_mm_cmplt_ps(_x, p4f_flt_min), |
| _mm_cmpge_ps(_x, _mm_setzero_ps())); |
| |
| // select only the inverse sqrt of positive normal inputs (denormals are |
| // flushed to zero and cause infs). |
| Packet4f x = _mm_rsqrt_ps(_x); |
| |
| // Do a single step of Newton's iteration. |
| Packet4f neg_half = pmul(_x, p4f_minus_half); |
| x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five)); |
| |
| // Mask-out the zero terms. |
| return _mm_or_ps(_mm_andnot_ps(is_zero, x), _mm_and_ps(is_zero, p4f_inf)); |
| } |
| |
| #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)); |
| } |
| |
| // Identical to the ptanh in GenericPacketMath.h, but for doubles use |
| // a small/medium approximation threshold of 0.001. |
| template<> EIGEN_STRONG_INLINE Packet2d ptanh_approx_threshold() { |
| return pset1<Packet2d>(0.001); |
| } |
| |
| } // end namespace internal |
| |
| |
| namespace numext { |
| |
| template<> |
| EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE |
| float sqrt(const float &x) |
| { |
| return internal::pfirst(_mm_sqrt_ss(_mm_set_ss(x))); |
| } |
| |
| template<> |
| EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE |
| double sqrt(const double &x) |
| { |
| #if EIGEN_COMP_GNUC |
| return internal::pfirst(__builtin_ia32_sqrtsd(_mm_set_sd(x))); |
| #else |
| return internal::pfirst(_mm_sqrt_pd(_mm_set_sd(x))); |
| #endif |
| } |
| |
| } // end namespace numex |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_MATH_FUNCTIONS_SSE_H |