MathFunctions.h revision c981c48f5bc9aefeffc0bcb0cc3934c2fae179dd
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@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 18namespace Eigen { 19 20namespace internal { 21 22template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 23Packet4f 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 35 /* natural logarithm computed for 4 simultaneous float 36 return NaN for x <= 0 37 */ 38 _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f); 39 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292E-2f); 40 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, - 1.1514610310E-1f); 41 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740E-1f); 42 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, - 1.2420140846E-1f); 43 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, + 1.4249322787E-1f); 44 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, - 1.6668057665E-1f); 45 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, + 2.0000714765E-1f); 46 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, - 2.4999993993E-1f); 47 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, + 3.3333331174E-1f); 48 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f); 49 _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f); 50 51 52 Packet4i emm0; 53 54 Packet4f invalid_mask = _mm_cmple_ps(x, _mm_setzero_ps()); 55 56 x = pmax(x, p4f_min_norm_pos); /* cut off denormalized stuff */ 57 emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23); 58 59 /* keep only the fractional part */ 60 x = _mm_and_ps(x, p4f_inv_mant_mask); 61 x = _mm_or_ps(x, p4f_half); 62 63 emm0 = _mm_sub_epi32(emm0, p4i_0x7f); 64 Packet4f e = padd(_mm_cvtepi32_ps(emm0), p4f_1); 65 66 /* part2: 67 if( x < SQRTHF ) { 68 e -= 1; 69 x = x + x - 1.0; 70 } else { x = x - 1.0; } 71 */ 72 Packet4f mask = _mm_cmplt_ps(x, p4f_cephes_SQRTHF); 73 Packet4f tmp = _mm_and_ps(x, mask); 74 x = psub(x, p4f_1); 75 e = psub(e, _mm_and_ps(p4f_1, mask)); 76 x = padd(x, tmp); 77 78 Packet4f x2 = pmul(x,x); 79 Packet4f x3 = pmul(x2,x); 80 81 Packet4f y, y1, y2; 82 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1); 83 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4); 84 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7); 85 y = pmadd(y , x, p4f_cephes_log_p2); 86 y1 = pmadd(y1, x, p4f_cephes_log_p5); 87 y2 = pmadd(y2, x, p4f_cephes_log_p8); 88 y = pmadd(y, x3, y1); 89 y = pmadd(y, x3, y2); 90 y = pmul(y, x3); 91 92 y1 = pmul(e, p4f_cephes_log_q1); 93 tmp = pmul(x2, p4f_half); 94 y = padd(y, y1); 95 x = psub(x, tmp); 96 y2 = pmul(e, p4f_cephes_log_q2); 97 x = padd(x, y); 98 x = padd(x, y2); 99 return _mm_or_ps(x, invalid_mask); // negative arg will be NAN 100} 101 102template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 103Packet4f pexp<Packet4f>(const Packet4f& _x) 104{ 105 Packet4f x = _x; 106 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 107 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 108 _EIGEN_DECLARE_CONST_Packet4i(0x7f, 0x7f); 109 110 111 _EIGEN_DECLARE_CONST_Packet4f(exp_hi, 88.3762626647950f); 112 _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -88.3762626647949f); 113 114 _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f); 115 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f); 116 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f); 117 118 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500E-4f); 119 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507E-3f); 120 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073E-3f); 121 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894E-2f); 122 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459E-1f); 123 _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201E-1f); 124 125 Packet4f tmp = _mm_setzero_ps(), fx; 126 Packet4i emm0; 127 128 // clamp x 129 x = pmax(pmin(x, p4f_exp_hi), p4f_exp_lo); 130 131 /* express exp(x) as exp(g + n*log(2)) */ 132 fx = pmadd(x, p4f_cephes_LOG2EF, p4f_half); 133 134 /* how to perform a floorf with SSE: just below */ 135 emm0 = _mm_cvttps_epi32(fx); 136 tmp = _mm_cvtepi32_ps(emm0); 137 /* if greater, substract 1 */ 138 Packet4f mask = _mm_cmpgt_ps(tmp, fx); 139 mask = _mm_and_ps(mask, p4f_1); 140 fx = psub(tmp, mask); 141 142 tmp = pmul(fx, p4f_cephes_exp_C1); 143 Packet4f z = pmul(fx, p4f_cephes_exp_C2); 144 x = psub(x, tmp); 145 x = psub(x, z); 146 147 z = pmul(x,x); 148 149 Packet4f y = p4f_cephes_exp_p0; 150 y = pmadd(y, x, p4f_cephes_exp_p1); 151 y = pmadd(y, x, p4f_cephes_exp_p2); 152 y = pmadd(y, x, p4f_cephes_exp_p3); 153 y = pmadd(y, x, p4f_cephes_exp_p4); 154 y = pmadd(y, x, p4f_cephes_exp_p5); 155 y = pmadd(y, z, x); 156 y = padd(y, p4f_1); 157 158 // build 2^n 159 emm0 = _mm_cvttps_epi32(fx); 160 emm0 = _mm_add_epi32(emm0, p4i_0x7f); 161 emm0 = _mm_slli_epi32(emm0, 23); 162 return pmul(y, _mm_castsi128_ps(emm0)); 163} 164 165/* evaluation of 4 sines at onces, using SSE2 intrinsics. 166 167 The code is the exact rewriting of the cephes sinf function. 168 Precision is excellent as long as x < 8192 (I did not bother to 169 take into account the special handling they have for greater values 170 -- it does not return garbage for arguments over 8192, though, but 171 the extra precision is missing). 172 173 Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the 174 surprising but correct result. 175*/ 176 177template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 178Packet4f psin<Packet4f>(const Packet4f& _x) 179{ 180 Packet4f x = _x; 181 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 182 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 183 184 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 185 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 186 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 187 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 188 189 _EIGEN_DECLARE_CONST_Packet4f_FROM_INT(sign_mask, 0x80000000); 190 191 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 192 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 193 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 194 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 195 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 196 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 197 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 198 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 199 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 200 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 201 202 Packet4f xmm1, xmm2 = _mm_setzero_ps(), xmm3, sign_bit, y; 203 204 Packet4i emm0, emm2; 205 sign_bit = x; 206 /* take the absolute value */ 207 x = pabs(x); 208 209 /* take the modulo */ 210 211 /* extract the sign bit (upper one) */ 212 sign_bit = _mm_and_ps(sign_bit, p4f_sign_mask); 213 214 /* scale by 4/Pi */ 215 y = pmul(x, p4f_cephes_FOPI); 216 217 /* store the integer part of y in mm0 */ 218 emm2 = _mm_cvttps_epi32(y); 219 /* j=(j+1) & (~1) (see the cephes sources) */ 220 emm2 = _mm_add_epi32(emm2, p4i_1); 221 emm2 = _mm_and_si128(emm2, p4i_not1); 222 y = _mm_cvtepi32_ps(emm2); 223 /* get the swap sign flag */ 224 emm0 = _mm_and_si128(emm2, p4i_4); 225 emm0 = _mm_slli_epi32(emm0, 29); 226 /* get the polynom selection mask 227 there is one polynom for 0 <= x <= Pi/4 228 and another one for Pi/4<x<=Pi/2 229 230 Both branches will be computed. 231 */ 232 emm2 = _mm_and_si128(emm2, p4i_2); 233 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 234 235 Packet4f swap_sign_bit = _mm_castsi128_ps(emm0); 236 Packet4f poly_mask = _mm_castsi128_ps(emm2); 237 sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit); 238 239 /* The magic pass: "Extended precision modular arithmetic" 240 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 241 xmm1 = pmul(y, p4f_minus_cephes_DP1); 242 xmm2 = pmul(y, p4f_minus_cephes_DP2); 243 xmm3 = pmul(y, p4f_minus_cephes_DP3); 244 x = padd(x, xmm1); 245 x = padd(x, xmm2); 246 x = padd(x, xmm3); 247 248 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 249 y = p4f_coscof_p0; 250 Packet4f z = _mm_mul_ps(x,x); 251 252 y = pmadd(y, z, p4f_coscof_p1); 253 y = pmadd(y, z, p4f_coscof_p2); 254 y = pmul(y, z); 255 y = pmul(y, z); 256 Packet4f tmp = pmul(z, p4f_half); 257 y = psub(y, tmp); 258 y = padd(y, p4f_1); 259 260 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 261 262 Packet4f y2 = p4f_sincof_p0; 263 y2 = pmadd(y2, z, p4f_sincof_p1); 264 y2 = pmadd(y2, z, p4f_sincof_p2); 265 y2 = pmul(y2, z); 266 y2 = pmul(y2, x); 267 y2 = padd(y2, x); 268 269 /* select the correct result from the two polynoms */ 270 y2 = _mm_and_ps(poly_mask, y2); 271 y = _mm_andnot_ps(poly_mask, y); 272 y = _mm_or_ps(y,y2); 273 /* update the sign */ 274 return _mm_xor_ps(y, sign_bit); 275} 276 277/* almost the same as psin */ 278template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 279Packet4f pcos<Packet4f>(const Packet4f& _x) 280{ 281 Packet4f x = _x; 282 _EIGEN_DECLARE_CONST_Packet4f(1 , 1.0f); 283 _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 284 285 _EIGEN_DECLARE_CONST_Packet4i(1, 1); 286 _EIGEN_DECLARE_CONST_Packet4i(not1, ~1); 287 _EIGEN_DECLARE_CONST_Packet4i(2, 2); 288 _EIGEN_DECLARE_CONST_Packet4i(4, 4); 289 290 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1,-0.78515625f); 291 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 292 _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 293 _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891E-4f); 294 _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736E-3f); 295 _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611E-1f); 296 _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948E-005f); 297 _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765E-003f); 298 _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827E-002f); 299 _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4 / M_PI 300 301 Packet4f xmm1, xmm2 = _mm_setzero_ps(), xmm3, y; 302 Packet4i emm0, emm2; 303 304 x = pabs(x); 305 306 /* scale by 4/Pi */ 307 y = pmul(x, p4f_cephes_FOPI); 308 309 /* get the integer part of y */ 310 emm2 = _mm_cvttps_epi32(y); 311 /* j=(j+1) & (~1) (see the cephes sources) */ 312 emm2 = _mm_add_epi32(emm2, p4i_1); 313 emm2 = _mm_and_si128(emm2, p4i_not1); 314 y = _mm_cvtepi32_ps(emm2); 315 316 emm2 = _mm_sub_epi32(emm2, p4i_2); 317 318 /* get the swap sign flag */ 319 emm0 = _mm_andnot_si128(emm2, p4i_4); 320 emm0 = _mm_slli_epi32(emm0, 29); 321 /* get the polynom selection mask */ 322 emm2 = _mm_and_si128(emm2, p4i_2); 323 emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128()); 324 325 Packet4f sign_bit = _mm_castsi128_ps(emm0); 326 Packet4f poly_mask = _mm_castsi128_ps(emm2); 327 328 /* The magic pass: "Extended precision modular arithmetic" 329 x = ((x - y * DP1) - y * DP2) - y * DP3; */ 330 xmm1 = pmul(y, p4f_minus_cephes_DP1); 331 xmm2 = pmul(y, p4f_minus_cephes_DP2); 332 xmm3 = pmul(y, p4f_minus_cephes_DP3); 333 x = padd(x, xmm1); 334 x = padd(x, xmm2); 335 x = padd(x, xmm3); 336 337 /* Evaluate the first polynom (0 <= x <= Pi/4) */ 338 y = p4f_coscof_p0; 339 Packet4f z = pmul(x,x); 340 341 y = pmadd(y,z,p4f_coscof_p1); 342 y = pmadd(y,z,p4f_coscof_p2); 343 y = pmul(y, z); 344 y = pmul(y, z); 345 Packet4f tmp = _mm_mul_ps(z, p4f_half); 346 y = psub(y, tmp); 347 y = padd(y, p4f_1); 348 349 /* Evaluate the second polynom (Pi/4 <= x <= 0) */ 350 Packet4f y2 = p4f_sincof_p0; 351 y2 = pmadd(y2, z, p4f_sincof_p1); 352 y2 = pmadd(y2, z, p4f_sincof_p2); 353 y2 = pmul(y2, z); 354 y2 = pmadd(y2, x, x); 355 356 /* select the correct result from the two polynoms */ 357 y2 = _mm_and_ps(poly_mask, y2); 358 y = _mm_andnot_ps(poly_mask, y); 359 y = _mm_or_ps(y,y2); 360 361 /* update the sign */ 362 return _mm_xor_ps(y, sign_bit); 363} 364 365// This is based on Quake3's fast inverse square root. 366// For detail see here: http://www.beyond3d.com/content/articles/8/ 367template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED 368Packet4f psqrt<Packet4f>(const Packet4f& _x) 369{ 370 Packet4f half = pmul(_x, pset1<Packet4f>(.5f)); 371 372 /* select only the inverse sqrt of non-zero inputs */ 373 Packet4f non_zero_mask = _mm_cmpgt_ps(_x, pset1<Packet4f>(std::numeric_limits<float>::epsilon())); 374 Packet4f x = _mm_and_ps(non_zero_mask, _mm_rsqrt_ps(_x)); 375 376 x = pmul(x, psub(pset1<Packet4f>(1.5f), pmul(half, pmul(x,x)))); 377 return pmul(_x,x); 378} 379 380} // end namespace internal 381 382} // end namespace Eigen 383 384#endif // EIGEN_MATH_FUNCTIONS_SSE_H 385