![]() |
Eigen
3.3.3
|
00001 // This file is part of Eigen, a lightweight C++ template library 00002 // for linear algebra. 00003 // 00004 // Copyright (C) 2007 Julien Pommier 00005 // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr> 00006 // 00007 // This Source Code Form is subject to the terms of the Mozilla 00008 // Public License v. 2.0. If a copy of the MPL was not distributed 00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 00010 00011 /* The sin, cos, exp, and log functions of this file come from 00012 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/ 00013 */ 00014 00015 #ifndef EIGEN_MATH_FUNCTIONS_SSE_H 00016 #define EIGEN_MATH_FUNCTIONS_SSE_H 00017 00018 namespace Eigen { 00019 00020 namespace internal { 00021 00022 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00023 Packet4f plog<Packet4f>(const Packet4f& _x) 00024 { 00025 Packet4f x = _x; 00026 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 00027 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 00028 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f); 00029 00030 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inv_mant_mask, ~0x7f800000); 00031 00032 /* the smallest non denormalized float number */ 00033 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(min_norm_pos, 0x00800000); 00034 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(minus_inf, 0xff800000);//-1.f/0.f); 00035 00036 /* natural logarithm computed for 4 simultaneous float 00037 return NaN for x <= 0 00038 */ 00039 _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f); 00040 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292E-2f); 00041 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, - 1.1514610310E-1f); 00042 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740E-1f); 00043 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, - 1.2420140846E-1f); 00044 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, + 1.4249322787E-1f); 00045 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, - 1.6668057665E-1f); 00046 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, + 2.0000714765E-1f); 00047 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, - 2.4999993993E-1f); 00048 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, + 3.3333331174E-1f); 00049 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f); 00050 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f); 00051 00052 00053 Packet4i emm0; 00054 00055 Packet4f invalid_mask = _mm_cmpnge_ps(x, _mm_setzero_ps()); // not greater equal is true if x is NaN 00056 Packet4f iszero_mask = _mm_cmpeq_ps(x, _mm_setzero_ps()); 00057 00058 x = pmax(x, p4f_min_norm_pos); /* cut off denormalized stuff */ 00059 emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23); 00060 00061 /* keep only the fractional part */ 00062 x = _mm_and_ps(x, p4f_inv_mant_mask); 00063 x = _mm_or_ps(x, p4f_half); 00064 00065 emm0 = _mm_sub_epi32(emm0, p4i_0x7f); 00066 Packet4f e = padd(Packet4f(_mm_cvtepi32_ps(emm0)), p4f_1); 00067 00068 /* part2: 00069 if( x < SQRTHF ) { 00070 e -= 1; 00071 x = x + x - 1.0; 00072 } else { x = x - 1.0; } 00073 */ 00074 Packet4f mask = _mm_cmplt_ps(x, p4f_cephes_SQRTHF); 00075 Packet4f tmp = pand(x, mask); 00076 x = psub(x, p4f_1); 00077 e = psub(e, pand(p4f_1, mask)); 00078 x = padd(x, tmp); 00079 00080 Packet4f x2 = pmul(x,x); 00081 Packet4f x3 = pmul(x2,x); 00082 00083 Packet4f y, y1, y2; 00084 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1); 00085 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4); 00086 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7); 00087 y = pmadd(y , x, p4f_cephes_log_p2); 00088 y1 = pmadd(y1, x, p4f_cephes_log_p5); 00089 y2 = pmadd(y2, x, p4f_cephes_log_p8); 00090 y = pmadd(y, x3, y1); 00091 y = pmadd(y, x3, y2); 00092 y = pmul(y, x3); 00093 00094 y1 = pmul(e, p4f_cephes_log_q1); 00095 tmp = pmul(x2, p4f_half); 00096 y = padd(y, y1); 00097 x = psub(x, tmp); 00098 y2 = pmul(e, p4f_cephes_log_q2); 00099 x = padd(x, y); 00100 x = padd(x, y2); 00101 // negative arg will be NAN, 0 will be -INF 00102 return _mm_or_ps(_mm_andnot_ps(iszero_mask, _mm_or_ps(x, invalid_mask)), 00103 _mm_and_ps(iszero_mask, p4f_minus_inf)); 00104 } 00105 00106 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00107 Packet4f pexp<Packet4f>(const Packet4f& _x) 00108 { 00109 Packet4f x = _x; 00110 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 00111 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 00112 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f); 00113 00114 00115 _EIGEN_DECLARE_CONST_Packet4f(exp_hi, 88.3762626647950f); 00116 _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -88.3762626647949f); 00117 00118 _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f); 00119 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f); 00120 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f); 00121 00122 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500E-4f); 00123 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507E-3f); 00124 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073E-3f); 00125 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894E-2f); 00126 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459E-1f); 00127 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201E-1f); 00128 00129 Packet4f tmp, fx; 00130 Packet4i emm0; 00131 00132 // clamp x 00133 x = pmax(pmin(x, p4f_exp_hi), p4f_exp_lo); 00134 00135 /* express exp(x) as exp(g + n*log(2)) */ 00136 fx = pmadd(x, p4f_cephes_LOG2EF, p4f_half); 00137 00138 #ifdef EIGEN_VECTORIZE_SSE4_1 00139 fx = _mm_floor_ps(fx); 00140 #else 00141 emm0 = _mm_cvttps_epi32(fx); 00142 tmp = _mm_cvtepi32_ps(emm0); 00143 /* if greater, substract 1 */ 00144 Packet4f mask = _mm_cmpgt_ps(tmp, fx); 00145 mask = _mm_and_ps(mask, p4f_1); 00146 fx = psub(tmp, mask); 00147 #endif 00148 00149 tmp = pmul(fx, p4f_cephes_exp_C1); 00150 Packet4f z = pmul(fx, p4f_cephes_exp_C2); 00151 x = psub(x, tmp); 00152 x = psub(x, z); 00153 00154 z = pmul(x,x); 00155 00156 Packet4f y = p4f_cephes_exp_p0; 00157 y = pmadd(y, x, p4f_cephes_exp_p1); 00158 y = pmadd(y, x, p4f_cephes_exp_p2); 00159 y = pmadd(y, x, p4f_cephes_exp_p3); 00160 y = pmadd(y, x, p4f_cephes_exp_p4); 00161 y = pmadd(y, x, p4f_cephes_exp_p5); 00162 y = pmadd(y, z, x); 00163 y = padd(y, p4f_1); 00164 00165 // build 2^n 00166 emm0 = _mm_cvttps_epi32(fx); 00167 emm0 = _mm_add_epi32(emm0, p4i_0x7f); 00168 emm0 = _mm_slli_epi32(emm0, 23); 00169 return pmax(pmul(y, Packet4f(_mm_castsi128_ps(emm0))), _x); 00170 } 00171 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00172 Packet2d pexp<Packet2d>(const Packet2d& _x) 00173 { 00174 Packet2d x = _x; 00175 00176 _EIGEN_DECLARE_CONST_Packet2d(1 , 1.0); 00177 _EIGEN_DECLARE_CONST_Packet2d(2 , 2.0); 00178 _EIGEN_DECLARE_CONST_Packet2d(half, 0.5); 00179 00180 _EIGEN_DECLARE_CONST_Packet2d(exp_hi, 709.437); 00181 _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -709.436139303); 00182 00183 _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599); 00184 00185 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4); 00186 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2); 00187 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1); 00188 00189 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6); 00190 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3); 00191 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1); 00192 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0); 00193 00194 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125); 00195 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6); 00196 static const __m128i p4i_1023_0 = _mm_setr_epi32(1023, 1023, 0, 0); 00197 00198 Packet2d tmp, fx; 00199 Packet4i emm0; 00200 00201 // clamp x 00202 x = pmax(pmin(x, p2d_exp_hi), p2d_exp_lo); 00203 /* express exp(x) as exp(g + n*log(2)) */ 00204 fx = pmadd(p2d_cephes_LOG2EF, x, p2d_half); 00205 00206 #ifdef EIGEN_VECTORIZE_SSE4_1 00207 fx = _mm_floor_pd(fx); 00208 #else 00209 emm0 = _mm_cvttpd_epi32(fx); 00210 tmp = _mm_cvtepi32_pd(emm0); 00211 /* if greater, substract 1 */ 00212 Packet2d mask = _mm_cmpgt_pd(tmp, fx); 00213 mask = _mm_and_pd(mask, p2d_1); 00214 fx = psub(tmp, mask); 00215 #endif 00216 00217 tmp = pmul(fx, p2d_cephes_exp_C1); 00218 Packet2d z = pmul(fx, p2d_cephes_exp_C2); 00219 x = psub(x, tmp); 00220 x = psub(x, z); 00221 00222 Packet2d x2 = pmul(x,x); 00223 00224 Packet2d px = p2d_cephes_exp_p0; 00225 px = pmadd(px, x2, p2d_cephes_exp_p1); 00226 px = pmadd(px, x2, p2d_cephes_exp_p2); 00227 px = pmul (px, x); 00228 00229 Packet2d qx = p2d_cephes_exp_q0; 00230 qx = pmadd(qx, x2, p2d_cephes_exp_q1); 00231 qx = pmadd(qx, x2, p2d_cephes_exp_q2); 00232 qx = pmadd(qx, x2, p2d_cephes_exp_q3); 00233 00234 x = pdiv(px,psub(qx,px)); 00235 x = pmadd(p2d_2,x,p2d_1); 00236 00237 // build 2^n 00238 emm0 = _mm_cvttpd_epi32(fx); 00239 emm0 = _mm_add_epi32(emm0, p4i_1023_0); 00240 emm0 = _mm_slli_epi32(emm0, 20); 00241 emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(1,2,0,3)); 00242 return pmax(pmul(x, Packet2d(_mm_castsi128_pd(emm0))), _x); 00243 } 00244 00245 /* evaluation of 4 sines at onces, using SSE2 intrinsics. 00246 00247 The code is the exact rewriting of the cephes sinf function. 00248 Precision is excellent as long as x < 8192 (I did not bother to 00249 take into account the special handling they have for greater values 00250 -- it does not return garbage for arguments over 8192, though, but 00251 the extra precision is missing). 00252 00253 Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the 00254 surprising but correct result. 00255 */ 00256 00257 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00258 Packet4f psin<Packet4f>(const Packet4f& _x) 00259 { 00260 Packet4f x = _x; 00261 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 00262 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 00263 00264 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 00265 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 00266 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 00267 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 00268 00269 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(sign_mask, 0x80000000); 00270 00271 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 00272 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 00273 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 00274 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 00275 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 00276 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 00277 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 00278 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 00279 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 00280 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 00281 00282 Packet4f xmm1, xmm2, xmm3, sign_bit, y; 00283 00284 Packet4i emm0, emm2; 00285 sign_bit = x; 00286 /* take the absolute value */ 00287 x = pabs(x); 00288 00289 /* take the modulo */ 00290 00291 /* extract the sign bit (upper one) */ 00292 sign_bit = _mm_and_ps(sign_bit, p4f_sign_mask); 00293 00294 /* scale by 4/Pi */ 00295 y = pmul(x, p4f_cephes_FOPI); 00296 00297 /* store the integer part of y in mm0 */ 00298 emm2 = _mm_cvttps_epi32(y); 00299 /* j=(j+1) & (~1) (see the cephes sources) */ 00300 emm2 = _mm_add_epi32(emm2, p4i_1); 00301 emm2 = _mm_and_si128(emm2, p4i_not1); 00302 y = _mm_cvtepi32_ps(emm2); 00303 /* get the swap sign flag */ 00304 emm0 = _mm_and_si128(emm2, p4i_4); 00305 emm0 = _mm_slli_epi32(emm0, 29); 00306 /* get the polynom selection mask 00307 there is one polynom for 0 <= x <= Pi/4 00308 and another one for Pi/4<x<=Pi/2 00309 00310 Both branches will be computed. 00311 */ 00312 emm2 = _mm_and_si128(emm2, p4i_2); 00313 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 00314 00315 Packet4f swap_sign_bit = _mm_castsi128_ps(emm0); 00316 Packet4f poly_mask = _mm_castsi128_ps(emm2); 00317 sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit); 00318 00319 /* The magic pass: "Extended precision modular arithmetic" 00320 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 00321 xmm1 = pmul(y, p4f_minus_cephes_DP1); 00322 xmm2 = pmul(y, p4f_minus_cephes_DP2); 00323 xmm3 = pmul(y, p4f_minus_cephes_DP3); 00324 x = padd(x, xmm1); 00325 x = padd(x, xmm2); 00326 x = padd(x, xmm3); 00327 00328 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 00329 y = p4f_coscof_p0; 00330 Packet4f z = _mm_mul_ps(x,x); 00331 00332 y = pmadd(y, z, p4f_coscof_p1); 00333 y = pmadd(y, z, p4f_coscof_p2); 00334 y = pmul(y, z); 00335 y = pmul(y, z); 00336 Packet4f tmp = pmul(z, p4f_half); 00337 y = psub(y, tmp); 00338 y = padd(y, p4f_1); 00339 00340 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 00341 00342 Packet4f y2 = p4f_sincof_p0; 00343 y2 = pmadd(y2, z, p4f_sincof_p1); 00344 y2 = pmadd(y2, z, p4f_sincof_p2); 00345 y2 = pmul(y2, z); 00346 y2 = pmul(y2, x); 00347 y2 = padd(y2, x); 00348 00349 /* select the correct result from the two polynoms */ 00350 y2 = _mm_and_ps(poly_mask, y2); 00351 y = _mm_andnot_ps(poly_mask, y); 00352 y = _mm_or_ps(y,y2); 00353 /* update the sign */ 00354 return _mm_xor_ps(y, sign_bit); 00355 } 00356 00357 /* almost the same as psin */ 00358 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00359 Packet4f pcos<Packet4f>(const Packet4f& _x) 00360 { 00361 Packet4f x = _x; 00362 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 00363 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 00364 00365 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 00366 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 00367 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 00368 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 00369 00370 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 00371 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 00372 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 00373 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 00374 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 00375 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 00376 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 00377 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 00378 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 00379 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 00380 00381 Packet4f xmm1, xmm2, xmm3, y; 00382 Packet4i emm0, emm2; 00383 00384 x = pabs(x); 00385 00386 /* scale by 4/Pi */ 00387 y = pmul(x, p4f_cephes_FOPI); 00388 00389 /* get the integer part of y */ 00390 emm2 = _mm_cvttps_epi32(y); 00391 /* j=(j+1) & (~1) (see the cephes sources) */ 00392 emm2 = _mm_add_epi32(emm2, p4i_1); 00393 emm2 = _mm_and_si128(emm2, p4i_not1); 00394 y = _mm_cvtepi32_ps(emm2); 00395 00396 emm2 = _mm_sub_epi32(emm2, p4i_2); 00397 00398 /* get the swap sign flag */ 00399 emm0 = _mm_andnot_si128(emm2, p4i_4); 00400 emm0 = _mm_slli_epi32(emm0, 29); 00401 /* get the polynom selection mask */ 00402 emm2 = _mm_and_si128(emm2, p4i_2); 00403 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 00404 00405 Packet4f sign_bit = _mm_castsi128_ps(emm0); 00406 Packet4f poly_mask = _mm_castsi128_ps(emm2); 00407 00408 /* The magic pass: "Extended precision modular arithmetic" 00409 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 00410 xmm1 = pmul(y, p4f_minus_cephes_DP1); 00411 xmm2 = pmul(y, p4f_minus_cephes_DP2); 00412 xmm3 = pmul(y, p4f_minus_cephes_DP3); 00413 x = padd(x, xmm1); 00414 x = padd(x, xmm2); 00415 x = padd(x, xmm3); 00416 00417 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 00418 y = p4f_coscof_p0; 00419 Packet4f z = pmul(x,x); 00420 00421 y = pmadd(y,z,p4f_coscof_p1); 00422 y = pmadd(y,z,p4f_coscof_p2); 00423 y = pmul(y, z); 00424 y = pmul(y, z); 00425 Packet4f tmp = _mm_mul_ps(z, p4f_half); 00426 y = psub(y, tmp); 00427 y = padd(y, p4f_1); 00428 00429 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 00430 Packet4f y2 = p4f_sincof_p0; 00431 y2 = pmadd(y2, z, p4f_sincof_p1); 00432 y2 = pmadd(y2, z, p4f_sincof_p2); 00433 y2 = pmul(y2, z); 00434 y2 = pmadd(y2, x, x); 00435 00436 /* select the correct result from the two polynoms */ 00437 y2 = _mm_and_ps(poly_mask, y2); 00438 y = _mm_andnot_ps(poly_mask, y); 00439 y = _mm_or_ps(y,y2); 00440 00441 /* update the sign */ 00442 return _mm_xor_ps(y, sign_bit); 00443 } 00444 00445 #if EIGEN_FAST_MATH 00446 00447 // Functions for sqrt. 00448 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step 00449 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the 00450 // exact solution. It does not handle +inf, or denormalized numbers correctly. 00451 // The main advantage of this approach is not just speed, but also the fact that 00452 // it can be inlined and pipelined with other computations, further reducing its 00453 // effective latency. This is similar to Quake3's fast inverse square root. 00454 // For detail see here: http://www.beyond3d.com/content/articles/8/ 00455 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00456 Packet4f psqrt<Packet4f>(const Packet4f& _x) 00457 { 00458 Packet4f half = pmul(_x, pset1<Packet4f>(.5f)); 00459 Packet4f denormal_mask = _mm_and_ps( 00460 _mm_cmpge_ps(_x, _mm_setzero_ps()), 00461 _mm_cmplt_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)()))); 00462 00463 // Compute approximate reciprocal sqrt. 00464 Packet4f x = _mm_rsqrt_ps(_x); 00465 // Do a single step of Newton's iteration. 00466 x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x)))); 00467 // Flush results for denormals to zero. 00468 return _mm_andnot_ps(denormal_mask, pmul(_x,x)); 00469 } 00470 00471 #else 00472 00473 template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00474 Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); } 00475 00476 #endif 00477 00478 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00479 Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); } 00480 00481 #if EIGEN_FAST_MATH 00482 00483 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00484 Packet4f prsqrt<Packet4f>(const Packet4f& _x) { 00485 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000); 00486 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000); 00487 _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f); 00488 _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f); 00489 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000); 00490 00491 Packet4f neg_half = pmul(_x, p4f_minus_half); 00492 00493 // select only the inverse sqrt of positive normal inputs (denormals are 00494 // flushed to zero and cause infs as well). 00495 Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min); 00496 Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x)); 00497 00498 // Fill in NaNs and Infs for the negative/zero entries. 00499 Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps()); 00500 Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask); 00501 Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan), 00502 _mm_and_ps(zero_mask, p4f_inf)); 00503 00504 // Do a single step of Newton's iteration. 00505 x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five)); 00506 00507 // Insert NaNs and Infs in all the right places. 00508 return _mm_or_ps(x, infs_and_nans); 00509 } 00510 00511 #else 00512 00513 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00514 Packet4f prsqrt<Packet4f>(const Packet4f& x) { 00515 // Unfortunately we can't use the much faster mm_rqsrt_ps since it only provides an approximation. 00516 return _mm_div_ps(pset1<Packet4f>(1.0f), _mm_sqrt_ps(x)); 00517 } 00518 00519 #endif 00520 00521 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 00522 Packet2d prsqrt<Packet2d>(const Packet2d& x) { 00523 // Unfortunately we can't use the much faster mm_rqsrt_pd since it only provides an approximation. 00524 return _mm_div_pd(pset1<Packet2d>(1.0), _mm_sqrt_pd(x)); 00525 } 00526 00527 // Hyperbolic Tangent function. 00528 template <> 00529 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 00530 ptanh<Packet4f>(const Packet4f& x) { 00531 return internal::generic_fast_tanh_float(x); 00532 } 00533 00534 } // end namespace internal 00535 00536 namespace numext { 00537 00538 template<> 00539 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE 00540 float sqrt(const float &x) 00541 { 00542 return internal::pfirst(internal::Packet4f(_mm_sqrt_ss(_mm_set_ss(x)))); 00543 } 00544 00545 template<> 00546 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE 00547 double sqrt(const double &x) 00548 { 00549 #if EIGEN_COMP_GNUC_STRICT 00550 // This works around a GCC bug generating poor code for _mm_sqrt_pd 00551 // See https://bitbucket.org/eigen/eigen/commits/14f468dba4d350d7c19c9b93072e19f7b3df563b 00552 return internal::pfirst(internal::Packet2d(__builtin_ia32_sqrtsd(_mm_set_sd(x)))); 00553 #else 00554 return internal::pfirst(internal::Packet2d(_mm_sqrt_pd(_mm_set_sd(x)))); 00555 #endif 00556 } 00557 00558 } // end namespace numex 00559 00560 } // end namespace Eigen 00561 00562 #endif // EIGEN_MATH_FUNCTIONS_SSE_H