1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2016 Pedro Gonnet (pedro.gonnet@gmail.com) 5 // 6 // This Source Code Form is subject to the terms of the Mozilla 7 // Public License v. 2.0. If a copy of the MPL was not distributed 8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 9 10 #ifndef THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_ 11 #define THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_ 12 13 namespace Eigen { 14 15 namespace internal { 16 17 // Disable the code for older versions of gcc that don't support many of the required avx512 instrinsics. 18 #if EIGEN_GNUC_AT_LEAST(5, 3) 19 20 #define _EIGEN_DECLARE_CONST_Packet16f(NAME, X) \ 21 const Packet16f p16f_##NAME = pset1<Packet16f>(X) 22 23 #define _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(NAME, X) \ 24 const Packet16f p16f_##NAME = (__m512)pset1<Packet16i>(X) 25 26 #define _EIGEN_DECLARE_CONST_Packet8d(NAME, X) \ 27 const Packet8d p8d_##NAME = pset1<Packet8d>(X) 28 29 #define _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(NAME, X) \ 30 const Packet8d p8d_##NAME = _mm512_castsi512_pd(_mm512_set1_epi64(X)) 31 32 // Natural logarithm 33 // Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2) 34 // and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can 35 // be easily approximated by a polynomial centered on m=1 for stability. 36 #if defined(EIGEN_VECTORIZE_AVX512DQ) 37 template <> 38 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f 39 plog<Packet16f>(const Packet16f& _x) { 40 Packet16f x = _x; 41 _EIGEN_DECLARE_CONST_Packet16f(1, 1.0f); 42 _EIGEN_DECLARE_CONST_Packet16f(half, 0.5f); 43 _EIGEN_DECLARE_CONST_Packet16f(126f, 126.0f); 44 45 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(inv_mant_mask, ~0x7f800000); 46 47 // The smallest non denormalized float number. 48 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(min_norm_pos, 0x00800000); 49 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(minus_inf, 0xff800000); 50 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(nan, 0x7fc00000); 51 52 // Polynomial coefficients. 53 _EIGEN_DECLARE_CONST_Packet16f(cephes_SQRTHF, 0.707106781186547524f); 54 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p0, 7.0376836292E-2f); 55 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p1, -1.1514610310E-1f); 56 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p2, 1.1676998740E-1f); 57 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p3, -1.2420140846E-1f); 58 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p4, +1.4249322787E-1f); 59 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p5, -1.6668057665E-1f); 60 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p6, +2.0000714765E-1f); 61 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p7, -2.4999993993E-1f); 62 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_p8, +3.3333331174E-1f); 63 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_q1, -2.12194440e-4f); 64 _EIGEN_DECLARE_CONST_Packet16f(cephes_log_q2, 0.693359375f); 65 66 // invalid_mask is set to true when x is NaN 67 __mmask16 invalid_mask = 68 _mm512_cmp_ps_mask(x, _mm512_setzero_ps(), _CMP_NGE_UQ); 69 __mmask16 iszero_mask = 70 _mm512_cmp_ps_mask(x, _mm512_setzero_ps(), _CMP_EQ_UQ); 71 72 // Truncate input values to the minimum positive normal. 73 x = pmax(x, p16f_min_norm_pos); 74 75 // Extract the shifted exponents. 76 Packet16f emm0 = _mm512_cvtepi32_ps(_mm512_srli_epi32((__m512i)x, 23)); 77 Packet16f e = _mm512_sub_ps(emm0, p16f_126f); 78 79 // Set the exponents to -1, i.e. x are in the range [0.5,1). 80 x = _mm512_and_ps(x, p16f_inv_mant_mask); 81 x = _mm512_or_ps(x, p16f_half); 82 83 // part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2)) 84 // and shift by -1. The values are then centered around 0, which improves 85 // the stability of the polynomial evaluation. 86 // if( x < SQRTHF ) { 87 // e -= 1; 88 // x = x + x - 1.0; 89 // } else { x = x - 1.0; } 90 __mmask16 mask = _mm512_cmp_ps_mask(x, p16f_cephes_SQRTHF, _CMP_LT_OQ); 91 Packet16f tmp = _mm512_mask_blend_ps(mask, _mm512_setzero_ps(), x); 92 x = psub(x, p16f_1); 93 e = psub(e, _mm512_mask_blend_ps(mask, _mm512_setzero_ps(), p16f_1)); 94 x = padd(x, tmp); 95 96 Packet16f x2 = pmul(x, x); 97 Packet16f x3 = pmul(x2, x); 98 99 // Evaluate the polynomial approximant of degree 8 in three parts, probably 100 // to improve instruction-level parallelism. 101 Packet16f y, y1, y2; 102 y = pmadd(p16f_cephes_log_p0, x, p16f_cephes_log_p1); 103 y1 = pmadd(p16f_cephes_log_p3, x, p16f_cephes_log_p4); 104 y2 = pmadd(p16f_cephes_log_p6, x, p16f_cephes_log_p7); 105 y = pmadd(y, x, p16f_cephes_log_p2); 106 y1 = pmadd(y1, x, p16f_cephes_log_p5); 107 y2 = pmadd(y2, x, p16f_cephes_log_p8); 108 y = pmadd(y, x3, y1); 109 y = pmadd(y, x3, y2); 110 y = pmul(y, x3); 111 112 // Add the logarithm of the exponent back to the result of the interpolation. 113 y1 = pmul(e, p16f_cephes_log_q1); 114 tmp = pmul(x2, p16f_half); 115 y = padd(y, y1); 116 x = psub(x, tmp); 117 y2 = pmul(e, p16f_cephes_log_q2); 118 x = padd(x, y); 119 x = padd(x, y2); 120 121 // Filter out invalid inputs, i.e. negative arg will be NAN, 0 will be -INF. 122 return _mm512_mask_blend_ps(iszero_mask, 123 _mm512_mask_blend_ps(invalid_mask, x, p16f_nan), 124 p16f_minus_inf); 125 } 126 #endif 127 128 // Exponential function. Works by writing "x = m*log(2) + r" where 129 // "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then 130 // "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1). 131 template <> 132 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f 133 pexp<Packet16f>(const Packet16f& _x) { 134 _EIGEN_DECLARE_CONST_Packet16f(1, 1.0f); 135 _EIGEN_DECLARE_CONST_Packet16f(half, 0.5f); 136 _EIGEN_DECLARE_CONST_Packet16f(127, 127.0f); 137 138 _EIGEN_DECLARE_CONST_Packet16f(exp_hi, 88.3762626647950f); 139 _EIGEN_DECLARE_CONST_Packet16f(exp_lo, -88.3762626647949f); 140 141 _EIGEN_DECLARE_CONST_Packet16f(cephes_LOG2EF, 1.44269504088896341f); 142 143 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p0, 1.9875691500E-4f); 144 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p1, 1.3981999507E-3f); 145 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p2, 8.3334519073E-3f); 146 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p3, 4.1665795894E-2f); 147 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p4, 1.6666665459E-1f); 148 _EIGEN_DECLARE_CONST_Packet16f(cephes_exp_p5, 5.0000001201E-1f); 149 150 // Clamp x. 151 Packet16f x = pmax(pmin(_x, p16f_exp_hi), p16f_exp_lo); 152 153 // Express exp(x) as exp(m*ln(2) + r), start by extracting 154 // m = floor(x/ln(2) + 0.5). 155 Packet16f m = _mm512_floor_ps(pmadd(x, p16f_cephes_LOG2EF, p16f_half)); 156 157 // Get r = x - m*ln(2). Note that we can do this without losing more than one 158 // ulp precision due to the FMA instruction. 159 _EIGEN_DECLARE_CONST_Packet16f(nln2, -0.6931471805599453f); 160 Packet16f r = _mm512_fmadd_ps(m, p16f_nln2, x); 161 Packet16f r2 = pmul(r, r); 162 163 // TODO(gonnet): Split into odd/even polynomials and try to exploit 164 // instruction-level parallelism. 165 Packet16f y = p16f_cephes_exp_p0; 166 y = pmadd(y, r, p16f_cephes_exp_p1); 167 y = pmadd(y, r, p16f_cephes_exp_p2); 168 y = pmadd(y, r, p16f_cephes_exp_p3); 169 y = pmadd(y, r, p16f_cephes_exp_p4); 170 y = pmadd(y, r, p16f_cephes_exp_p5); 171 y = pmadd(y, r2, r); 172 y = padd(y, p16f_1); 173 174 // Build emm0 = 2^m. 175 Packet16i emm0 = _mm512_cvttps_epi32(padd(m, p16f_127)); 176 emm0 = _mm512_slli_epi32(emm0, 23); 177 178 // Return 2^m * exp(r). 179 return pmax(pmul(y, _mm512_castsi512_ps(emm0)), _x); 180 } 181 182 /*template <> 183 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d 184 pexp<Packet8d>(const Packet8d& _x) { 185 Packet8d x = _x; 186 187 _EIGEN_DECLARE_CONST_Packet8d(1, 1.0); 188 _EIGEN_DECLARE_CONST_Packet8d(2, 2.0); 189 190 _EIGEN_DECLARE_CONST_Packet8d(exp_hi, 709.437); 191 _EIGEN_DECLARE_CONST_Packet8d(exp_lo, -709.436139303); 192 193 _EIGEN_DECLARE_CONST_Packet8d(cephes_LOG2EF, 1.4426950408889634073599); 194 195 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p0, 1.26177193074810590878e-4); 196 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p1, 3.02994407707441961300e-2); 197 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_p2, 9.99999999999999999910e-1); 198 199 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q0, 3.00198505138664455042e-6); 200 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q1, 2.52448340349684104192e-3); 201 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q2, 2.27265548208155028766e-1); 202 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_q3, 2.00000000000000000009e0); 203 204 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_C1, 0.693145751953125); 205 _EIGEN_DECLARE_CONST_Packet8d(cephes_exp_C2, 1.42860682030941723212e-6); 206 207 // clamp x 208 x = pmax(pmin(x, p8d_exp_hi), p8d_exp_lo); 209 210 // Express exp(x) as exp(g + n*log(2)). 211 const Packet8d n = 212 _mm512_mul_round_pd(p8d_cephes_LOG2EF, x, _MM_FROUND_TO_NEAREST_INT); 213 214 // Get the remainder modulo log(2), i.e. the "g" described above. Subtract 215 // n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last 216 // digits right. 217 const Packet8d nC1 = pmul(n, p8d_cephes_exp_C1); 218 const Packet8d nC2 = pmul(n, p8d_cephes_exp_C2); 219 x = psub(x, nC1); 220 x = psub(x, nC2); 221 222 const Packet8d x2 = pmul(x, x); 223 224 // Evaluate the numerator polynomial of the rational interpolant. 225 Packet8d px = p8d_cephes_exp_p0; 226 px = pmadd(px, x2, p8d_cephes_exp_p1); 227 px = pmadd(px, x2, p8d_cephes_exp_p2); 228 px = pmul(px, x); 229 230 // Evaluate the denominator polynomial of the rational interpolant. 231 Packet8d qx = p8d_cephes_exp_q0; 232 qx = pmadd(qx, x2, p8d_cephes_exp_q1); 233 qx = pmadd(qx, x2, p8d_cephes_exp_q2); 234 qx = pmadd(qx, x2, p8d_cephes_exp_q3); 235 236 // I don't really get this bit, copied from the SSE2 routines, so... 237 // TODO(gonnet): Figure out what is going on here, perhaps find a better 238 // rational interpolant? 239 x = _mm512_div_pd(px, psub(qx, px)); 240 x = pmadd(p8d_2, x, p8d_1); 241 242 // Build e=2^n. 243 const Packet8d e = _mm512_castsi512_pd(_mm512_slli_epi64( 244 _mm512_add_epi64(_mm512_cvtpd_epi64(n), _mm512_set1_epi64(1023)), 52)); 245 246 // Construct the result 2^n * exp(g) = e * x. The max is used to catch 247 // non-finite values in the input. 248 return pmax(pmul(x, e), _x); 249 }*/ 250 251 // Functions for sqrt. 252 // The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step 253 // of Newton's method, at a cost of 1-2 bits of precision as opposed to the 254 // exact solution. The main advantage of this approach is not just speed, but 255 // also the fact that it can be inlined and pipelined with other computations, 256 // further reducing its effective latency. 257 #if EIGEN_FAST_MATH 258 template <> 259 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f 260 psqrt<Packet16f>(const Packet16f& _x) { 261 _EIGEN_DECLARE_CONST_Packet16f(one_point_five, 1.5f); 262 _EIGEN_DECLARE_CONST_Packet16f(minus_half, -0.5f); 263 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(flt_min, 0x00800000); 264 265 Packet16f neg_half = pmul(_x, p16f_minus_half); 266 267 // select only the inverse sqrt of positive normal inputs (denormals are 268 // flushed to zero and cause infs as well). 269 __mmask16 non_zero_mask = _mm512_cmp_ps_mask(_x, p16f_flt_min, _CMP_GE_OQ); 270 Packet16f x = _mm512_mask_blend_ps(non_zero_mask, _mm512_setzero_ps(), _mm512_rsqrt14_ps(_x)); 271 272 // Do a single step of Newton's iteration. 273 x = pmul(x, pmadd(neg_half, pmul(x, x), p16f_one_point_five)); 274 275 // Multiply the original _x by it's reciprocal square root to extract the 276 // square root. 277 return pmul(_x, x); 278 } 279 280 template <> 281 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d 282 psqrt<Packet8d>(const Packet8d& _x) { 283 _EIGEN_DECLARE_CONST_Packet8d(one_point_five, 1.5); 284 _EIGEN_DECLARE_CONST_Packet8d(minus_half, -0.5); 285 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(dbl_min, 0x0010000000000000LL); 286 287 Packet8d neg_half = pmul(_x, p8d_minus_half); 288 289 // select only the inverse sqrt of positive normal inputs (denormals are 290 // flushed to zero and cause infs as well). 291 __mmask8 non_zero_mask = _mm512_cmp_pd_mask(_x, p8d_dbl_min, _CMP_GE_OQ); 292 Packet8d x = _mm512_mask_blend_pd(non_zero_mask, _mm512_setzero_pd(), _mm512_rsqrt14_pd(_x)); 293 294 // Do a first step of Newton's iteration. 295 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five)); 296 297 // Do a second step of Newton's iteration. 298 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five)); 299 300 // Multiply the original _x by it's reciprocal square root to extract the 301 // square root. 302 return pmul(_x, x); 303 } 304 #else 305 template <> 306 EIGEN_STRONG_INLINE Packet16f psqrt<Packet16f>(const Packet16f& x) { 307 return _mm512_sqrt_ps(x); 308 } 309 template <> 310 EIGEN_STRONG_INLINE Packet8d psqrt<Packet8d>(const Packet8d& x) { 311 return _mm512_sqrt_pd(x); 312 } 313 #endif 314 315 // Functions for rsqrt. 316 // Almost identical to the sqrt routine, just leave out the last multiplication 317 // and fill in NaN/Inf where needed. Note that this function only exists as an 318 // iterative version for doubles since there is no instruction for diretly 319 // computing the reciprocal square root in AVX-512. 320 #ifdef EIGEN_FAST_MATH 321 template <> 322 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet16f 323 prsqrt<Packet16f>(const Packet16f& _x) { 324 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(inf, 0x7f800000); 325 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(nan, 0x7fc00000); 326 _EIGEN_DECLARE_CONST_Packet16f(one_point_five, 1.5f); 327 _EIGEN_DECLARE_CONST_Packet16f(minus_half, -0.5f); 328 _EIGEN_DECLARE_CONST_Packet16f_FROM_INT(flt_min, 0x00800000); 329 330 Packet16f neg_half = pmul(_x, p16f_minus_half); 331 332 // select only the inverse sqrt of positive normal inputs (denormals are 333 // flushed to zero and cause infs as well). 334 __mmask16 le_zero_mask = _mm512_cmp_ps_mask(_x, p16f_flt_min, _CMP_LT_OQ); 335 Packet16f x = _mm512_mask_blend_ps(le_zero_mask, _mm512_rsqrt14_ps(_x), _mm512_setzero_ps()); 336 337 // Fill in NaNs and Infs for the negative/zero entries. 338 __mmask16 neg_mask = _mm512_cmp_ps_mask(_x, _mm512_setzero_ps(), _CMP_LT_OQ); 339 Packet16f infs_and_nans = _mm512_mask_blend_ps( 340 neg_mask, _mm512_mask_blend_ps(le_zero_mask, _mm512_setzero_ps(), p16f_inf), p16f_nan); 341 342 // Do a single step of Newton's iteration. 343 x = pmul(x, pmadd(neg_half, pmul(x, x), p16f_one_point_five)); 344 345 // Insert NaNs and Infs in all the right places. 346 return _mm512_mask_blend_ps(le_zero_mask, x, infs_and_nans); 347 } 348 349 template <> 350 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8d 351 prsqrt<Packet8d>(const Packet8d& _x) { 352 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(inf, 0x7ff0000000000000LL); 353 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(nan, 0x7ff1000000000000LL); 354 _EIGEN_DECLARE_CONST_Packet8d(one_point_five, 1.5); 355 _EIGEN_DECLARE_CONST_Packet8d(minus_half, -0.5); 356 _EIGEN_DECLARE_CONST_Packet8d_FROM_INT64(dbl_min, 0x0010000000000000LL); 357 358 Packet8d neg_half = pmul(_x, p8d_minus_half); 359 360 // select only the inverse sqrt of positive normal inputs (denormals are 361 // flushed to zero and cause infs as well). 362 __mmask8 le_zero_mask = _mm512_cmp_pd_mask(_x, p8d_dbl_min, _CMP_LT_OQ); 363 Packet8d x = _mm512_mask_blend_pd(le_zero_mask, _mm512_rsqrt14_pd(_x), _mm512_setzero_pd()); 364 365 // Fill in NaNs and Infs for the negative/zero entries. 366 __mmask8 neg_mask = _mm512_cmp_pd_mask(_x, _mm512_setzero_pd(), _CMP_LT_OQ); 367 Packet8d infs_and_nans = _mm512_mask_blend_pd( 368 neg_mask, _mm512_mask_blend_pd(le_zero_mask, _mm512_setzero_pd(), p8d_inf), p8d_nan); 369 370 // Do a first step of Newton's iteration. 371 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five)); 372 373 // Do a second step of Newton's iteration. 374 x = pmul(x, pmadd(neg_half, pmul(x, x), p8d_one_point_five)); 375 376 // Insert NaNs and Infs in all the right places. 377 return _mm512_mask_blend_pd(le_zero_mask, x, infs_and_nans); 378 } 379 #elif defined(EIGEN_VECTORIZE_AVX512ER) 380 template <> 381 EIGEN_STRONG_INLINE Packet16f prsqrt<Packet16f>(const Packet16f& x) { 382 return _mm512_rsqrt28_ps(x); 383 } 384 #endif 385 #endif 386 387 } // end namespace internal 388 389 } // end namespace Eigen 390 391 #endif // THIRD_PARTY_EIGEN3_EIGEN_SRC_CORE_ARCH_AVX512_MATHFUNCTIONS_H_ 392