MathFunctions.h (16159B)
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) 2014 Pedro Gonnet (pedro.gonnet@gmail.com) 6 // Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr> 7 // 8 // Copyright (C) 2018 Wave Computing, Inc. 9 // Written by: 10 // Chris Larsen 11 // Alexey Frunze (afrunze@wavecomp.com) 12 // 13 // This Source Code Form is subject to the terms of the Mozilla 14 // Public License v. 2.0. If a copy of the MPL was not distributed 15 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. 16 17 /* The sin, cos, exp, and log functions of this file come from 18 * Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/ 19 */ 20 21 /* The tanh function of this file is an adaptation of 22 * template<typename T> T generic_fast_tanh_float(const T&) 23 * from MathFunctionsImpl.h. 24 */ 25 26 #ifndef EIGEN_MATH_FUNCTIONS_MSA_H 27 #define EIGEN_MATH_FUNCTIONS_MSA_H 28 29 namespace Eigen { 30 31 namespace internal { 32 33 template <> 34 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 35 plog<Packet4f>(const Packet4f& _x) { 36 static _EIGEN_DECLARE_CONST_Packet4f(cephes_SQRTHF, 0.707106781186547524f); 37 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p0, 7.0376836292e-2f); 38 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p1, -1.1514610310e-1f); 39 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p2, 1.1676998740e-1f); 40 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p3, -1.2420140846e-1f); 41 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p4, +1.4249322787e-1f); 42 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p5, -1.6668057665e-1f); 43 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p6, +2.0000714765e-1f); 44 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p7, -2.4999993993e-1f); 45 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_p8, +3.3333331174e-1f); 46 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q1, -2.12194440e-4f); 47 static _EIGEN_DECLARE_CONST_Packet4f(cephes_log_q2, 0.693359375f); 48 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 49 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f); 50 51 // Convert negative argument into NAN (quiet negative, to be specific). 52 Packet4f zero = (Packet4f)__builtin_msa_ldi_w(0); 53 Packet4i neg_mask = __builtin_msa_fclt_w(_x, zero); 54 Packet4i zero_mask = __builtin_msa_fceq_w(_x, zero); 55 Packet4f non_neg_x_or_nan = padd(_x, (Packet4f)neg_mask); // Add 0.0 or NAN. 56 Packet4f x = non_neg_x_or_nan; 57 58 // Extract exponent from x = mantissa * 2**exponent, where 1.0 <= mantissa < 2.0. 59 // N.B. the exponent is one less of what frexpf() would return. 60 Packet4i e_int = __builtin_msa_ftint_s_w(__builtin_msa_flog2_w(x)); 61 // Multiply x by 2**(-exponent-1) to get 0.5 <= x < 1.0 as from frexpf(). 62 x = __builtin_msa_fexp2_w(x, (Packet4i)__builtin_msa_nori_b((v16u8)e_int, 0)); 63 64 /* 65 if (x < SQRTHF) { 66 x = x + x - 1.0; 67 } else { 68 e += 1; 69 x = x - 1.0; 70 } 71 */ 72 Packet4f xx = padd(x, x); 73 Packet4i ge_mask = __builtin_msa_fcle_w(p4f_cephes_SQRTHF, x); 74 e_int = psub(e_int, ge_mask); 75 x = (Packet4f)__builtin_msa_bsel_v((v16u8)ge_mask, (v16u8)xx, (v16u8)x); 76 x = psub(x, p4f_1); 77 Packet4f e = __builtin_msa_ffint_s_w(e_int); 78 79 Packet4f x2 = pmul(x, x); 80 Packet4f x3 = pmul(x2, x); 81 82 Packet4f y, y1, y2; 83 y = pmadd(p4f_cephes_log_p0, x, p4f_cephes_log_p1); 84 y1 = pmadd(p4f_cephes_log_p3, x, p4f_cephes_log_p4); 85 y2 = pmadd(p4f_cephes_log_p6, x, p4f_cephes_log_p7); 86 y = pmadd(y, x, p4f_cephes_log_p2); 87 y1 = pmadd(y1, x, p4f_cephes_log_p5); 88 y2 = pmadd(y2, x, p4f_cephes_log_p8); 89 y = pmadd(y, x3, y1); 90 y = pmadd(y, x3, y2); 91 y = pmul(y, x3); 92 93 y = pmadd(e, p4f_cephes_log_q1, y); 94 x = __builtin_msa_fmsub_w(x, x2, p4f_half); 95 x = padd(x, y); 96 x = pmadd(e, p4f_cephes_log_q2, x); 97 98 // x is now the logarithm result candidate. We still need to handle the 99 // extreme arguments of zero and positive infinity, though. 100 // N.B. if the argument is +INFINITY, x is NAN because the polynomial terms 101 // contain infinities of both signs (see the coefficients and code above). 102 // INFINITY - INFINITY is NAN. 103 104 // If the argument is +INFINITY, make it the new result candidate. 105 // To achieve that we choose the smaller of the result candidate and the 106 // argument. 107 // This is correct for all finite pairs of values (the logarithm is smaller 108 // than the argument). 109 // This is also correct in the special case when the argument is +INFINITY 110 // and the result candidate is NAN. This is because the fmin.df instruction 111 // prefers non-NANs to NANs. 112 x = __builtin_msa_fmin_w(x, non_neg_x_or_nan); 113 114 // If the argument is zero (including -0.0), the result becomes -INFINITY. 115 Packet4i neg_infs = __builtin_msa_slli_w(zero_mask, 23); 116 x = (Packet4f)__builtin_msa_bsel_v((v16u8)zero_mask, (v16u8)x, (v16u8)neg_infs); 117 118 return x; 119 } 120 121 template <> 122 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 123 pexp<Packet4f>(const Packet4f& _x) { 124 // Limiting single-precision pexp's argument to [-128, +128] lets pexp 125 // reach 0 and INFINITY naturally. 126 static _EIGEN_DECLARE_CONST_Packet4f(exp_lo, -128.0f); 127 static _EIGEN_DECLARE_CONST_Packet4f(exp_hi, +128.0f); 128 static _EIGEN_DECLARE_CONST_Packet4f(cephes_LOG2EF, 1.44269504088896341f); 129 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C1, 0.693359375f); 130 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_C2, -2.12194440e-4f); 131 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p0, 1.9875691500e-4f); 132 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p1, 1.3981999507e-3f); 133 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p2, 8.3334519073e-3f); 134 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p3, 4.1665795894e-2f); 135 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p4, 1.6666665459e-1f); 136 static _EIGEN_DECLARE_CONST_Packet4f(cephes_exp_p5, 5.0000001201e-1f); 137 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 138 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f); 139 140 Packet4f x = _x; 141 142 // Clamp x. 143 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(x, p4f_exp_lo), (v16u8)x, 144 (v16u8)p4f_exp_lo); 145 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(p4f_exp_hi, x), (v16u8)x, 146 (v16u8)p4f_exp_hi); 147 148 // Round to nearest integer by adding 0.5 (with x's sign) and truncating. 149 Packet4f x2_add = (Packet4f)__builtin_msa_binsli_w((v4u32)p4f_half, (v4u32)x, 0); 150 Packet4f x2 = pmadd(x, p4f_cephes_LOG2EF, x2_add); 151 Packet4i x2_int = __builtin_msa_ftrunc_s_w(x2); 152 Packet4f x2_int_f = __builtin_msa_ffint_s_w(x2_int); 153 154 x = __builtin_msa_fmsub_w(x, x2_int_f, p4f_cephes_exp_C1); 155 x = __builtin_msa_fmsub_w(x, x2_int_f, p4f_cephes_exp_C2); 156 157 Packet4f z = pmul(x, x); 158 159 Packet4f y = p4f_cephes_exp_p0; 160 y = pmadd(y, x, p4f_cephes_exp_p1); 161 y = pmadd(y, x, p4f_cephes_exp_p2); 162 y = pmadd(y, x, p4f_cephes_exp_p3); 163 y = pmadd(y, x, p4f_cephes_exp_p4); 164 y = pmadd(y, x, p4f_cephes_exp_p5); 165 y = pmadd(y, z, x); 166 y = padd(y, p4f_1); 167 168 // y *= 2**exponent. 169 y = __builtin_msa_fexp2_w(y, x2_int); 170 171 return y; 172 } 173 174 template <> 175 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 176 ptanh<Packet4f>(const Packet4f& _x) { 177 static _EIGEN_DECLARE_CONST_Packet4f(tanh_tiny, 1e-4f); 178 static _EIGEN_DECLARE_CONST_Packet4f(tanh_hi, 9.0f); 179 // The monomial coefficients of the numerator polynomial (odd). 180 static _EIGEN_DECLARE_CONST_Packet4f(alpha_1, 4.89352455891786e-3f); 181 static _EIGEN_DECLARE_CONST_Packet4f(alpha_3, 6.37261928875436e-4f); 182 static _EIGEN_DECLARE_CONST_Packet4f(alpha_5, 1.48572235717979e-5f); 183 static _EIGEN_DECLARE_CONST_Packet4f(alpha_7, 5.12229709037114e-8f); 184 static _EIGEN_DECLARE_CONST_Packet4f(alpha_9, -8.60467152213735e-11f); 185 static _EIGEN_DECLARE_CONST_Packet4f(alpha_11, 2.00018790482477e-13f); 186 static _EIGEN_DECLARE_CONST_Packet4f(alpha_13, -2.76076847742355e-16f); 187 // The monomial coefficients of the denominator polynomial (even). 188 static _EIGEN_DECLARE_CONST_Packet4f(beta_0, 4.89352518554385e-3f); 189 static _EIGEN_DECLARE_CONST_Packet4f(beta_2, 2.26843463243900e-3f); 190 static _EIGEN_DECLARE_CONST_Packet4f(beta_4, 1.18534705686654e-4f); 191 static _EIGEN_DECLARE_CONST_Packet4f(beta_6, 1.19825839466702e-6f); 192 193 Packet4f x = pabs(_x); 194 Packet4i tiny_mask = __builtin_msa_fclt_w(x, p4f_tanh_tiny); 195 196 // Clamp the inputs to the range [-9, 9] since anything outside 197 // this range is -/+1.0f in single-precision. 198 x = (Packet4f)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_w(p4f_tanh_hi, x), (v16u8)x, 199 (v16u8)p4f_tanh_hi); 200 201 // Since the polynomials are odd/even, we need x**2. 202 Packet4f x2 = pmul(x, x); 203 204 // Evaluate the numerator polynomial p. 205 Packet4f p = pmadd(x2, p4f_alpha_13, p4f_alpha_11); 206 p = pmadd(x2, p, p4f_alpha_9); 207 p = pmadd(x2, p, p4f_alpha_7); 208 p = pmadd(x2, p, p4f_alpha_5); 209 p = pmadd(x2, p, p4f_alpha_3); 210 p = pmadd(x2, p, p4f_alpha_1); 211 p = pmul(x, p); 212 213 // Evaluate the denominator polynomial q. 214 Packet4f q = pmadd(x2, p4f_beta_6, p4f_beta_4); 215 q = pmadd(x2, q, p4f_beta_2); 216 q = pmadd(x2, q, p4f_beta_0); 217 218 // Divide the numerator by the denominator. 219 p = pdiv(p, q); 220 221 // Reinstate the sign. 222 p = (Packet4f)__builtin_msa_binsli_w((v4u32)p, (v4u32)_x, 0); 223 224 // When the argument is very small in magnitude it's more accurate to just return it. 225 p = (Packet4f)__builtin_msa_bsel_v((v16u8)tiny_mask, (v16u8)p, (v16u8)_x); 226 227 return p; 228 } 229 230 template <bool sine> 231 Packet4f psincos_inner_msa_float(const Packet4f& _x) { 232 static _EIGEN_DECLARE_CONST_Packet4f(sincos_max_arg, 13176795.0f); // Approx. (2**24) / (4/Pi). 233 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP1, -0.78515625f); 234 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP2, -2.4187564849853515625e-4f); 235 static _EIGEN_DECLARE_CONST_Packet4f(minus_cephes_DP3, -3.77489497744594108e-8f); 236 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p0, -1.9515295891e-4f); 237 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p1, 8.3321608736e-3f); 238 static _EIGEN_DECLARE_CONST_Packet4f(sincof_p2, -1.6666654611e-1f); 239 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p0, 2.443315711809948e-5f); 240 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p1, -1.388731625493765e-3f); 241 static _EIGEN_DECLARE_CONST_Packet4f(coscof_p2, 4.166664568298827e-2f); 242 static _EIGEN_DECLARE_CONST_Packet4f(cephes_FOPI, 1.27323954473516f); // 4/Pi. 243 static _EIGEN_DECLARE_CONST_Packet4f(half, 0.5f); 244 static _EIGEN_DECLARE_CONST_Packet4f(1, 1.0f); 245 246 Packet4f x = pabs(_x); 247 248 // Translate infinite arguments into NANs. 249 Packet4f zero_or_nan_if_inf = psub(_x, _x); 250 x = padd(x, zero_or_nan_if_inf); 251 // Prevent sin/cos from generating values larger than 1.0 in magnitude 252 // for very large arguments by setting x to 0.0. 253 Packet4i small_or_nan_mask = __builtin_msa_fcult_w(x, p4f_sincos_max_arg); 254 x = pand(x, (Packet4f)small_or_nan_mask); 255 256 // Scale x by 4/Pi to find x's octant. 257 Packet4f y = pmul(x, p4f_cephes_FOPI); 258 // Get the octant. We'll reduce x by this number of octants or by one more than it. 259 Packet4i y_int = __builtin_msa_ftrunc_s_w(y); 260 // x's from even-numbered octants will translate to octant 0: [0, +Pi/4]. 261 // x's from odd-numbered octants will translate to octant -1: [-Pi/4, 0]. 262 // Adjustment for odd-numbered octants: octant = (octant + 1) & (~1). 263 Packet4i y_int1 = __builtin_msa_addvi_w(y_int, 1); 264 Packet4i y_int2 = (Packet4i)__builtin_msa_bclri_w((Packet4ui)y_int1, 0); // bclri = bit-clear 265 y = __builtin_msa_ffint_s_w(y_int2); 266 267 // Compute the sign to apply to the polynomial. 268 Packet4i sign_mask = sine ? pxor(__builtin_msa_slli_w(y_int1, 29), (Packet4i)_x) 269 : __builtin_msa_slli_w(__builtin_msa_addvi_w(y_int, 3), 29); 270 271 // Get the polynomial selection mask. 272 // We'll calculate both (sin and cos) polynomials and then select from the two. 273 Packet4i poly_mask = __builtin_msa_ceqi_w(__builtin_msa_slli_w(y_int2, 30), 0); 274 275 // Reduce x by y octants to get: -Pi/4 <= x <= +Pi/4. 276 // The magic pass: "Extended precision modular arithmetic" 277 // x = ((x - y * DP1) - y * DP2) - y * DP3 278 Packet4f tmp1 = pmul(y, p4f_minus_cephes_DP1); 279 Packet4f tmp2 = pmul(y, p4f_minus_cephes_DP2); 280 Packet4f tmp3 = pmul(y, p4f_minus_cephes_DP3); 281 x = padd(x, tmp1); 282 x = padd(x, tmp2); 283 x = padd(x, tmp3); 284 285 // Evaluate the cos(x) polynomial. 286 y = p4f_coscof_p0; 287 Packet4f z = pmul(x, x); 288 y = pmadd(y, z, p4f_coscof_p1); 289 y = pmadd(y, z, p4f_coscof_p2); 290 y = pmul(y, z); 291 y = pmul(y, z); 292 y = __builtin_msa_fmsub_w(y, z, p4f_half); 293 y = padd(y, p4f_1); 294 295 // Evaluate the sin(x) polynomial. 296 Packet4f y2 = p4f_sincof_p0; 297 y2 = pmadd(y2, z, p4f_sincof_p1); 298 y2 = pmadd(y2, z, p4f_sincof_p2); 299 y2 = pmul(y2, z); 300 y2 = pmadd(y2, x, x); 301 302 // Select the correct result from the two polynomials. 303 y = sine ? (Packet4f)__builtin_msa_bsel_v((v16u8)poly_mask, (v16u8)y, (v16u8)y2) 304 : (Packet4f)__builtin_msa_bsel_v((v16u8)poly_mask, (v16u8)y2, (v16u8)y); 305 306 // Update the sign. 307 sign_mask = pxor(sign_mask, (Packet4i)y); 308 y = (Packet4f)__builtin_msa_binsli_w((v4u32)y, (v4u32)sign_mask, 0); // binsli = bit-insert-left 309 return y; 310 } 311 312 template <> 313 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 314 psin<Packet4f>(const Packet4f& x) { 315 return psincos_inner_msa_float</* sine */ true>(x); 316 } 317 318 template <> 319 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4f 320 pcos<Packet4f>(const Packet4f& x) { 321 return psincos_inner_msa_float</* sine */ false>(x); 322 } 323 324 template <> 325 EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet2d 326 pexp<Packet2d>(const Packet2d& _x) { 327 // Limiting double-precision pexp's argument to [-1024, +1024] lets pexp 328 // reach 0 and INFINITY naturally. 329 static _EIGEN_DECLARE_CONST_Packet2d(exp_lo, -1024.0); 330 static _EIGEN_DECLARE_CONST_Packet2d(exp_hi, +1024.0); 331 static _EIGEN_DECLARE_CONST_Packet2d(cephes_LOG2EF, 1.4426950408889634073599); 332 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C1, 0.693145751953125); 333 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_C2, 1.42860682030941723212e-6); 334 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p0, 1.26177193074810590878e-4); 335 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p1, 3.02994407707441961300e-2); 336 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_p2, 9.99999999999999999910e-1); 337 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q0, 3.00198505138664455042e-6); 338 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q1, 2.52448340349684104192e-3); 339 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q2, 2.27265548208155028766e-1); 340 static _EIGEN_DECLARE_CONST_Packet2d(cephes_exp_q3, 2.00000000000000000009e0); 341 static _EIGEN_DECLARE_CONST_Packet2d(half, 0.5); 342 static _EIGEN_DECLARE_CONST_Packet2d(1, 1.0); 343 static _EIGEN_DECLARE_CONST_Packet2d(2, 2.0); 344 345 Packet2d x = _x; 346 347 // Clamp x. 348 x = (Packet2d)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_d(x, p2d_exp_lo), (v16u8)x, 349 (v16u8)p2d_exp_lo); 350 x = (Packet2d)__builtin_msa_bsel_v((v16u8)__builtin_msa_fclt_d(p2d_exp_hi, x), (v16u8)x, 351 (v16u8)p2d_exp_hi); 352 353 // Round to nearest integer by adding 0.5 (with x's sign) and truncating. 354 Packet2d x2_add = (Packet2d)__builtin_msa_binsli_d((v2u64)p2d_half, (v2u64)x, 0); 355 Packet2d x2 = pmadd(x, p2d_cephes_LOG2EF, x2_add); 356 Packet2l x2_long = __builtin_msa_ftrunc_s_d(x2); 357 Packet2d x2_long_d = __builtin_msa_ffint_s_d(x2_long); 358 359 x = __builtin_msa_fmsub_d(x, x2_long_d, p2d_cephes_exp_C1); 360 x = __builtin_msa_fmsub_d(x, x2_long_d, p2d_cephes_exp_C2); 361 362 x2 = pmul(x, x); 363 364 Packet2d px = p2d_cephes_exp_p0; 365 px = pmadd(px, x2, p2d_cephes_exp_p1); 366 px = pmadd(px, x2, p2d_cephes_exp_p2); 367 px = pmul(px, x); 368 369 Packet2d qx = p2d_cephes_exp_q0; 370 qx = pmadd(qx, x2, p2d_cephes_exp_q1); 371 qx = pmadd(qx, x2, p2d_cephes_exp_q2); 372 qx = pmadd(qx, x2, p2d_cephes_exp_q3); 373 374 x = pdiv(px, psub(qx, px)); 375 x = pmadd(p2d_2, x, p2d_1); 376 377 // x *= 2**exponent. 378 x = __builtin_msa_fexp2_d(x, x2_long); 379 380 return x; 381 } 382 383 } // end namespace internal 384 385 } // end namespace Eigen 386 387 #endif // EIGEN_MATH_FUNCTIONS_MSA_H