1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2007 Julien Pommier 5 // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud (at) inria.fr> 6 // 7 // This Source Code Form is subject to the terms of the Mozilla 8 // Public License v. 2.0. If a copy of the MPL was not distributed 9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 10 11 /* The sin, cos, exp, and log functions of this file come from 12 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/ 13 */ 14 15 #ifndef EIGEN_MATH_FUNCTIONS_SSE_H 16 #define EIGEN_MATH_FUNCTIONS_SSE_H 17 18 namespace Eigen { 19 20 namespace internal { 21 22 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 23 Packet4f plog<Packet4f>(const Packet4f& _x) 24 { 25 Packet4f x = _x; 26 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 27 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 28 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f); 29 30 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inv_mant_mask, ~0x7f800000); 31 32 /* the smallest non denormalized float number */ 33 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(min_norm_pos, 0x00800000); 34 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(minus_inf, 0xff800000);//-1.f/0.f); 35 36 /* natural logarithm computed for 4 simultaneous float 37 return NaN for x <= 0 38 */ 39 _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f); 40 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292E-2f); 41 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, - 1.1514610310E-1f); 42 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740E-1f); 43 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, - 1.2420140846E-1f); 44 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, + 1.4249322787E-1f); 45 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, - 1.6668057665E-1f); 46 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, + 2.0000714765E-1f); 47 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, - 2.4999993993E-1f); 48 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, + 3.3333331174E-1f); 49 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f); 50 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f); 51 52 53 Packet4i emm0; 54 55 Packet4f invalid_mask = _mm_cmpnge_ps(x, _mm_setzero_ps()); // not greater equal is true if x is NaN 56 Packet4f iszero_mask = _mm_cmpeq_ps(x, _mm_setzero_ps()); 57 58 x = pmax(x, p4f_min_norm_pos); /* cut off denormalized stuff */ 59 emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23); 60 61 /* keep only the fractional part */ 62 x = _mm_and_ps(x, p4f_inv_mant_mask); 63 x = _mm_or_ps(x, p4f_half); 64 65 emm0 = _mm_sub_epi32(emm0, p4i_0x7f); 66 Packet4f e = padd(Packet4f(_mm_cvtepi32_ps(emm0)), p4f_1); 67 68 /* part2: 69 if( x < SQRTHF ) { 70 e -= 1; 71 x = x + x - 1.0; 72 } else { x = x - 1.0; } 73 */ 74 Packet4f mask = _mm_cmplt_ps(x, p4f_cephes_SQRTHF); 75 Packet4f tmp = pand(x, mask); 76 x = psub(x, p4f_1); 77 e = psub(e, pand(p4f_1, mask)); 78 x = padd(x, tmp); 79 80 Packet4f x2 = pmul(x,x); 81 Packet4f x3 = pmul(x2,x); 82 83 Packet4f y, y1, y2; 84 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1); 85 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4); 86 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7); 87 y = pmadd(y , x, p4f_cephes_log_p2); 88 y1 = pmadd(y1, x, p4f_cephes_log_p5); 89 y2 = pmadd(y2, x, p4f_cephes_log_p8); 90 y = pmadd(y, x3, y1); 91 y = pmadd(y, x3, y2); 92 y = pmul(y, x3); 93 94 y1 = pmul(e, p4f_cephes_log_q1); 95 tmp = pmul(x2, p4f_half); 96 y = padd(y, y1); 97 x = psub(x, tmp); 98 y2 = pmul(e, p4f_cephes_log_q2); 99 x = padd(x, y); 100 x = padd(x, y2); 101 // negative arg will be NAN, 0 will be -INF 102 return _mm_or_ps(_mm_andnot_ps(iszero_mask, _mm_or_ps(x, invalid_mask)), 103 _mm_and_ps(iszero_mask, p4f_minus_inf)); 104 } 105 106 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 107 Packet4f pexp<Packet4f>(const Packet4f& _x) 108 { 109 Packet4f x = _x; 110 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 111 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 112 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f); 113 114 115 _EIGEN_DECLARE_CONST_Packet4f(exp_hi, 88.3762626647950f); 116 _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -88.3762626647949f); 117 118 _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f); 119 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f); 120 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f); 121 122 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500E-4f); 123 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507E-3f); 124 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073E-3f); 125 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894E-2f); 126 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459E-1f); 127 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201E-1f); 128 129 Packet4f tmp, fx; 130 Packet4i emm0; 131 132 // clamp x 133 x = pmax(pmin(x, p4f_exp_hi), p4f_exp_lo); 134 135 /* express exp(x) as exp(g + n*log(2)) */ 136 fx = pmadd(x, p4f_cephes_LOG2EF, p4f_half); 137 138 #ifdef EIGEN_VECTORIZE_SSE4_1 139 fx = _mm_floor_ps(fx); 140 #else 141 emm0 = _mm_cvttps_epi32(fx); 142 tmp = _mm_cvtepi32_ps(emm0); 143 /* if greater, substract 1 */ 144 Packet4f mask = _mm_cmpgt_ps(tmp, fx); 145 mask = _mm_and_ps(mask, p4f_1); 146 fx = psub(tmp, mask); 147 #endif 148 149 tmp = pmul(fx, p4f_cephes_exp_C1); 150 Packet4f z = pmul(fx, p4f_cephes_exp_C2); 151 x = psub(x, tmp); 152 x = psub(x, z); 153 154 z = pmul(x,x); 155 156 Packet4f y = p4f_cephes_exp_p0; 157 y = pmadd(y, x, p4f_cephes_exp_p1); 158 y = pmadd(y, x, p4f_cephes_exp_p2); 159 y = pmadd(y, x, p4f_cephes_exp_p3); 160 y = pmadd(y, x, p4f_cephes_exp_p4); 161 y = pmadd(y, x, p4f_cephes_exp_p5); 162 y = pmadd(y, z, x); 163 y = padd(y, p4f_1); 164 165 // build 2^n 166 emm0 = _mm_cvttps_epi32(fx); 167 emm0 = _mm_add_epi32(emm0, p4i_0x7f); 168 emm0 = _mm_slli_epi32(emm0, 23); 169 return pmax(pmul(y, Packet4f(_mm_castsi128_ps(emm0))), _x); 170 } 171 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 172 Packet2d pexp<Packet2d>(const Packet2d& _x) 173 { 174 Packet2d x = _x; 175 176 _EIGEN_DECLARE_CONST_Packet2d(1 , 1.0); 177 _EIGEN_DECLARE_CONST_Packet2d(2 , 2.0); 178 _EIGEN_DECLARE_CONST_Packet2d(half, 0.5); 179 180 _EIGEN_DECLARE_CONST_Packet2d(exp_hi, 709.437); 181 _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -709.436139303); 182 183 _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599); 184 185 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4); 186 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2); 187 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1); 188 189 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6); 190 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3); 191 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1); 192 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0); 193 194 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125); 195 _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6); 196 static const __m128i p4i_1023_0 = _mm_setr_epi32(1023, 1023, 0, 0); 197 198 Packet2d tmp, fx; 199 Packet4i emm0; 200 201 // clamp x 202 x = pmax(pmin(x, p2d_exp_hi), p2d_exp_lo); 203 /* express exp(x) as exp(g + n*log(2)) */ 204 fx = pmadd(p2d_cephes_LOG2EF, x, p2d_half); 205 206 #ifdef EIGEN_VECTORIZE_SSE4_1 207 fx = _mm_floor_pd(fx); 208 #else 209 emm0 = _mm_cvttpd_epi32(fx); 210 tmp = _mm_cvtepi32_pd(emm0); 211 /* if greater, substract 1 */ 212 Packet2d mask = _mm_cmpgt_pd(tmp, fx); 213 mask = _mm_and_pd(mask, p2d_1); 214 fx = psub(tmp, mask); 215 #endif 216 217 tmp = pmul(fx, p2d_cephes_exp_C1); 218 Packet2d z = pmul(fx, p2d_cephes_exp_C2); 219 x = psub(x, tmp); 220 x = psub(x, z); 221 222 Packet2d x2 = pmul(x,x); 223 224 Packet2d px = p2d_cephes_exp_p0; 225 px = pmadd(px, x2, p2d_cephes_exp_p1); 226 px = pmadd(px, x2, p2d_cephes_exp_p2); 227 px = pmul (px, x); 228 229 Packet2d qx = p2d_cephes_exp_q0; 230 qx = pmadd(qx, x2, p2d_cephes_exp_q1); 231 qx = pmadd(qx, x2, p2d_cephes_exp_q2); 232 qx = pmadd(qx, x2, p2d_cephes_exp_q3); 233 234 x = pdiv(px,psub(qx,px)); 235 x = pmadd(p2d_2,x,p2d_1); 236 237 // build 2^n 238 emm0 = _mm_cvttpd_epi32(fx); 239 emm0 = _mm_add_epi32(emm0, p4i_1023_0); 240 emm0 = _mm_slli_epi32(emm0, 20); 241 emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(1,2,0,3)); 242 return pmax(pmul(x, Packet2d(_mm_castsi128_pd(emm0))), _x); 243 } 244 245 /* evaluation of 4 sines at onces, using SSE2 intrinsics. 246 247 The code is the exact rewriting of the cephes sinf function. 248 Precision is excellent as long as x < 8192 (I did not bother to 249 take into account the special handling they have for greater values 250 -- it does not return garbage for arguments over 8192, though, but 251 the extra precision is missing). 252 253 Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the 254 surprising but correct result. 255 */ 256 257 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 258 Packet4f psin<Packet4f>(const Packet4f& _x) 259 { 260 Packet4f x = _x; 261 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 262 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 263 264 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 265 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 266 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 267 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 268 269 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(sign_mask, 0x80000000); 270 271 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 272 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 273 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 274 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 275 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 276 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 277 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 278 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 279 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 280 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 281 282 Packet4f xmm1, xmm2, xmm3, sign_bit, y; 283 284 Packet4i emm0, emm2; 285 sign_bit = x; 286 /* take the absolute value */ 287 x = pabs(x); 288 289 /* take the modulo */ 290 291 /* extract the sign bit (upper one) */ 292 sign_bit = _mm_and_ps(sign_bit, p4f_sign_mask); 293 294 /* scale by 4/Pi */ 295 y = pmul(x, p4f_cephes_FOPI); 296 297 /* store the integer part of y in mm0 */ 298 emm2 = _mm_cvttps_epi32(y); 299 /* j=(j+1) & (~1) (see the cephes sources) */ 300 emm2 = _mm_add_epi32(emm2, p4i_1); 301 emm2 = _mm_and_si128(emm2, p4i_not1); 302 y = _mm_cvtepi32_ps(emm2); 303 /* get the swap sign flag */ 304 emm0 = _mm_and_si128(emm2, p4i_4); 305 emm0 = _mm_slli_epi32(emm0, 29); 306 /* get the polynom selection mask 307 there is one polynom for 0 <= x <= Pi/4 308 and another one for Pi/4<x<=Pi/2 309 310 Both branches will be computed. 311 */ 312 emm2 = _mm_and_si128(emm2, p4i_2); 313 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 314 315 Packet4f swap_sign_bit = _mm_castsi128_ps(emm0); 316 Packet4f poly_mask = _mm_castsi128_ps(emm2); 317 sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit); 318 319 /* The magic pass: "Extended precision modular arithmetic" 320 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 321 xmm1 = pmul(y, p4f_minus_cephes_DP1); 322 xmm2 = pmul(y, p4f_minus_cephes_DP2); 323 xmm3 = pmul(y, p4f_minus_cephes_DP3); 324 x = padd(x, xmm1); 325 x = padd(x, xmm2); 326 x = padd(x, xmm3); 327 328 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 329 y = p4f_coscof_p0; 330 Packet4f z = _mm_mul_ps(x,x); 331 332 y = pmadd(y, z, p4f_coscof_p1); 333 y = pmadd(y, z, p4f_coscof_p2); 334 y = pmul(y, z); 335 y = pmul(y, z); 336 Packet4f tmp = pmul(z, p4f_half); 337 y = psub(y, tmp); 338 y = padd(y, p4f_1); 339 340 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 341 342 Packet4f y2 = p4f_sincof_p0; 343 y2 = pmadd(y2, z, p4f_sincof_p1); 344 y2 = pmadd(y2, z, p4f_sincof_p2); 345 y2 = pmul(y2, z); 346 y2 = pmul(y2, x); 347 y2 = padd(y2, x); 348 349 /* select the correct result from the two polynoms */ 350 y2 = _mm_and_ps(poly_mask, y2); 351 y = _mm_andnot_ps(poly_mask, y); 352 y = _mm_or_ps(y,y2); 353 /* update the sign */ 354 return _mm_xor_ps(y, sign_bit); 355 } 356 357 /* almost the same as psin */ 358 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 359 Packet4f pcos<Packet4f>(const Packet4f& _x) 360 { 361 Packet4f x = _x; 362 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 363 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 364 365 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 366 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 367 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 368 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 369 370 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 371 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 372 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 373 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 374 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 375 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 376 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 377 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 378 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 379 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 380 381 Packet4f xmm1, xmm2, xmm3, y; 382 Packet4i emm0, emm2; 383 384 x = pabs(x); 385 386 /* scale by 4/Pi */ 387 y = pmul(x, p4f_cephes_FOPI); 388 389 /* get the integer part of y */ 390 emm2 = _mm_cvttps_epi32(y); 391 /* j=(j+1) & (~1) (see the cephes sources) */ 392 emm2 = _mm_add_epi32(emm2, p4i_1); 393 emm2 = _mm_and_si128(emm2, p4i_not1); 394 y = _mm_cvtepi32_ps(emm2); 395 396 emm2 = _mm_sub_epi32(emm2, p4i_2); 397 398 /* get the swap sign flag */ 399 emm0 = _mm_andnot_si128(emm2, p4i_4); 400 emm0 = _mm_slli_epi32(emm0, 29); 401 /* get the polynom selection mask */ 402 emm2 = _mm_and_si128(emm2, p4i_2); 403 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 404 405 Packet4f sign_bit = _mm_castsi128_ps(emm0); 406 Packet4f poly_mask = _mm_castsi128_ps(emm2); 407 408 /* The magic pass: "Extended precision modular arithmetic" 409 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 410 xmm1 = pmul(y, p4f_minus_cephes_DP1); 411 xmm2 = pmul(y, p4f_minus_cephes_DP2); 412 xmm3 = pmul(y, p4f_minus_cephes_DP3); 413 x = padd(x, xmm1); 414 x = padd(x, xmm2); 415 x = padd(x, xmm3); 416 417 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 418 y = p4f_coscof_p0; 419 Packet4f z = pmul(x,x); 420 421 y = pmadd(y,z,p4f_coscof_p1); 422 y = pmadd(y,z,p4f_coscof_p2); 423 y = pmul(y, z); 424 y = pmul(y, z); 425 Packet4f tmp = _mm_mul_ps(z, p4f_half); 426 y = psub(y, tmp); 427 y = padd(y, p4f_1); 428 429 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 430 Packet4f y2 = p4f_sincof_p0; 431 y2 = pmadd(y2, z, p4f_sincof_p1); 432 y2 = pmadd(y2, z, p4f_sincof_p2); 433 y2 = pmul(y2, z); 434 y2 = pmadd(y2, x, x); 435 436 /* select the correct result from the two polynoms */ 437 y2 = _mm_and_ps(poly_mask, y2); 438 y = _mm_andnot_ps(poly_mask, y); 439 y = _mm_or_ps(y,y2); 440 441 /* update the sign */ 442 return _mm_xor_ps(y, sign_bit); 443 } 444 445 #if EIGEN_FAST_MATH 446 447 // Functions for sqrt. 448 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step 449 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the 450 // exact solution. It does not handle +inf, or denormalized numbers correctly. 451 // The main advantage of this approach is not just speed, but also the fact that 452 // it can be inlined and pipelined with other computations, further reducing its 453 // effective latency. This is similar to Quake3's fast inverse square root. 454 // For detail see here: http://www.beyond3d.com/content/articles/8/ 455 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 456 Packet4f psqrt<Packet4f>(const Packet4f& _x) 457 { 458 Packet4f half = pmul(_x, pset1<Packet4f>(.5f)); 459 Packet4f denormal_mask = _mm_and_ps( 460 _mm_cmpge_ps(_x, _mm_setzero_ps()), 461 _mm_cmplt_ps(_x, pset1<Packet4f>((std::numeric_limits<float>::min)()))); 462 463 // Compute approximate reciprocal sqrt. 464 Packet4f x = _mm_rsqrt_ps(_x); 465 // Do a single step of Newton's iteration. 466 x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x)))); 467 // Flush results for denormals to zero. 468 return _mm_andnot_ps(denormal_mask, pmul(_x,x)); 469 } 470 471 #else 472 473 template<>EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 474 Packet4f psqrt<Packet4f>(const Packet4f& x) { return _mm_sqrt_ps(x); } 475 476 #endif 477 478 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 479 Packet2d psqrt<Packet2d>(const Packet2d& x) { return _mm_sqrt_pd(x); } 480 481 #if EIGEN_FAST_MATH 482 483 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 484 Packet4f prsqrt<Packet4f>(const Packet4f& _x) { 485 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(inf, 0x7f800000); 486 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(nan, 0x7fc00000); 487 _EIGEN_DECLARE_CONST_Packet4f(one_point_five, 1.5f); 488 _EIGEN_DECLARE_CONST_Packet4f(minus_half, -0.5f); 489 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(flt_min, 0x00800000); 490 491 Packet4f neg_half = pmul(_x, p4f_minus_half); 492 493 // select only the inverse sqrt of positive normal inputs (denormals are 494 // flushed to zero and cause infs as well). 495 Packet4f le_zero_mask = _mm_cmple_ps(_x, p4f_flt_min); 496 Packet4f x = _mm_andnot_ps(le_zero_mask, _mm_rsqrt_ps(_x)); 497 498 // Fill in NaNs and Infs for the negative/zero entries. 499 Packet4f neg_mask = _mm_cmplt_ps(_x, _mm_setzero_ps()); 500 Packet4f zero_mask = _mm_andnot_ps(neg_mask, le_zero_mask); 501 Packet4f infs_and_nans = _mm_or_ps(_mm_and_ps(neg_mask, p4f_nan), 502 _mm_and_ps(zero_mask, p4f_inf)); 503 504 // Do a single step of Newton's iteration. 505 x = pmul(x, pmadd(neg_half, pmul(x, x), p4f_one_point_five)); 506 507 // Insert NaNs and Infs in all the right places. 508 return _mm_or_ps(x, infs_and_nans); 509 } 510 511 #else 512 513 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 514 Packet4f prsqrt<Packet4f>(const Packet4f& x) { 515 // Unfortunately we can't use the much faster mm_rqsrt_ps since it only provides an approximation. 516 return _mm_div_ps(pset1<Packet4f>(1.0f), _mm_sqrt_ps(x)); 517 } 518 519 #endif 520 521 template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 522 Packet2d prsqrt<Packet2d>(const Packet2d& x) { 523 // Unfortunately we can't use the much faster mm_rqsrt_pd since it only provides an approximation. 524 return _mm_div_pd(pset1<Packet2d>(1.0), _mm_sqrt_pd(x)); 525 } 526 527 // Hyperbolic Tangent function. 528 template <> 529 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 530 ptanh<Packet4f>(const Packet4f& x) { 531 return internal::generic_fast_tanh_float(x); 532 } 533 534 } // end namespace internal 535 536 namespace numext { 537 538 template<> 539 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE 540 float sqrt(const float &x) 541 { 542 return internal::pfirst(internal::Packet4f(_mm_sqrt_ss(_mm_set_ss(x)))); 543 } 544 545 template<> 546 EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE 547 double sqrt(const double &x) 548 { 549 #if EIGEN_COMP_GNUC_STRICT 550 // This works around a GCC bug generating poor code for _mm_sqrt_pd 551 // See https://bitbucket.org/eigen/eigen/commits/14f468dba4d350d7c19c9b93072e19f7b3df563b 552 return internal::pfirst(internal::Packet2d(__builtin_ia32_sqrtsd(_mm_set_sd(x)))); 553 #else 554 return internal::pfirst(internal::Packet2d(_mm_sqrt_pd(_mm_set_sd(x)))); 555 #endif 556 } 557 558 } // end namespace numex 559 560 } // end namespace Eigen 561 562 #endif // EIGEN_MATH_FUNCTIONS_SSE_H 563
