1 //===-- Double-precision e^x function -------------------------------------===// 2 // 3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4 // See https://llvm.org/LICENSE.txt for license information. 5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #include "src/math/exp.h" 10 #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. 11 #include "explogxf.h" // ziv_test_denorm. 12 #include "src/__support/CPP/bit.h" 13 #include "src/__support/CPP/optional.h" 14 #include "src/__support/FPUtil/FEnvImpl.h" 15 #include "src/__support/FPUtil/FPBits.h" 16 #include "src/__support/FPUtil/PolyEval.h" 17 #include "src/__support/FPUtil/double_double.h" 18 #include "src/__support/FPUtil/dyadic_float.h" 19 #include "src/__support/FPUtil/multiply_add.h" 20 #include "src/__support/FPUtil/nearest_integer.h" 21 #include "src/__support/FPUtil/rounding_mode.h" 22 #include "src/__support/FPUtil/triple_double.h" 23 #include "src/__support/common.h" 24 #include "src/__support/integer_literals.h" 25 #include "src/__support/macros/config.h" 26 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 27 28 namespace LIBC_NAMESPACE_DECL { 29 30 using fputil::DoubleDouble; 31 using fputil::TripleDouble; 32 using Float128 = typename fputil::DyadicFloat<128>; 33 34 using LIBC_NAMESPACE::operator""_u128; 35 36 // log2(e) 37 constexpr double LOG2_E = 0x1.71547652b82fep+0; 38 39 // Error bounds: 40 // Errors when using double precision. 41 constexpr double ERR_D = 0x1.8p-63; 42 // Errors when using double-double precision. 43 constexpr double ERR_DD = 0x1.0p-99; 44 45 // -2^-12 * log(2) 46 // > a = -2^-12 * log(2); 47 // > b = round(a, 30, RN); 48 // > c = round(a - b, 30, RN); 49 // > d = round(a - b - c, D, RN); 50 // Errors < 1.5 * 2^-133 51 constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13; 52 constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47; 53 constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; 54 constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; 55 56 namespace { 57 58 // Polynomial approximations with double precision: 59 // Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 60 // For |dx| < 2^-13 + 2^-30: 61 // | output - expm1(dx) / dx | < 2^-51. 62 LIBC_INLINE double poly_approx_d(double dx) { 63 // dx^2 64 double dx2 = dx * dx; 65 // c0 = 1 + dx / 2 66 double c0 = fputil::multiply_add(dx, 0.5, 1.0); 67 // c1 = 1/6 + dx / 24 68 double c1 = 69 fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3); 70 // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24 71 double p = fputil::multiply_add(dx2, c1, c0); 72 return p; 73 } 74 75 // Polynomial approximation with double-double precision: 76 // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720 77 // For |dx| < 2^-13 + 2^-30: 78 // | output - exp(dx) | < 2^-101 79 DoubleDouble poly_approx_dd(const DoubleDouble &dx) { 80 // Taylor polynomial. 81 constexpr DoubleDouble COEFFS[] = { 82 {0, 0x1p0}, // 1 83 {0, 0x1p0}, // 1 84 {0, 0x1p-1}, // 1/2 85 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6 86 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24 87 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120 88 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720 89 }; 90 91 DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2], 92 COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]); 93 return p; 94 } 95 96 // Polynomial approximation with 128-bit precision: 97 // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040 98 // For |dx| < 2^-13 + 2^-30: 99 // | output - exp(dx) | < 2^-126. 100 Float128 poly_approx_f128(const Float128 &dx) { 101 constexpr Float128 COEFFS_128[]{ 102 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 103 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 104 {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5 105 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6 106 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24 107 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120 108 {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720 109 {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040 110 }; 111 112 Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2], 113 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5], 114 COEFFS_128[6], COEFFS_128[7]); 115 return p; 116 } 117 118 // Compute exp(x) using 128-bit precision. 119 // TODO(lntue): investigate triple-double precision implementation for this 120 // step. 121 Float128 exp_f128(double x, double kd, int idx1, int idx2) { 122 // Recalculate dx: 123 124 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 125 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 126 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133 127 128 Float128 dx = fputil::quick_add( 129 Float128(t1), fputil::quick_add(Float128(t2), Float128(t3))); 130 131 // TODO: Skip recalculating exp_mid1 and exp_mid2. 132 Float128 exp_mid1 = 133 fputil::quick_add(Float128(EXP2_MID1[idx1].hi), 134 fputil::quick_add(Float128(EXP2_MID1[idx1].mid), 135 Float128(EXP2_MID1[idx1].lo))); 136 137 Float128 exp_mid2 = 138 fputil::quick_add(Float128(EXP2_MID2[idx2].hi), 139 fputil::quick_add(Float128(EXP2_MID2[idx2].mid), 140 Float128(EXP2_MID2[idx2].lo))); 141 142 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); 143 144 Float128 p = poly_approx_f128(dx); 145 146 Float128 r = fputil::quick_mul(exp_mid, p); 147 148 r.exponent += static_cast<int>(kd) >> 12; 149 150 return r; 151 } 152 153 // Compute exp(x) with double-double precision. 154 DoubleDouble exp_double_double(double x, double kd, 155 const DoubleDouble &exp_mid) { 156 // Recalculate dx: 157 // dx = x - k * 2^-12 * log(2) 158 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 159 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 160 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130 161 162 DoubleDouble dx = fputil::exact_add(t1, t2); 163 dx.lo += t3; 164 165 // Degree-6 Taylor polynomial approximation in double-double precision. 166 // | p - exp(x) | < 2^-100. 167 DoubleDouble p = poly_approx_dd(dx); 168 169 // Error bounds: 2^-99. 170 DoubleDouble r = fputil::quick_mult(exp_mid, p); 171 172 return r; 173 } 174 175 // Check for exceptional cases when 176 // |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 177 double set_exceptional(double x) { 178 using FPBits = typename fputil::FPBits<double>; 179 FPBits xbits(x); 180 181 uint64_t x_u = xbits.uintval(); 182 uint64_t x_abs = xbits.abs().uintval(); 183 184 // |x| <= 2^-53 185 if (x_abs <= 0x3ca0'0000'0000'0000ULL) { 186 // exp(x) ~ 1 + x 187 return 1 + x; 188 } 189 190 // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. 191 192 // x <= log(2^-1075) or -inf/nan 193 if (x_u >= 0xc087'4910'd52d'3052ULL) { 194 // exp(-Inf) = 0 195 if (xbits.is_inf()) 196 return 0.0; 197 198 // exp(nan) = nan 199 if (xbits.is_nan()) 200 return x; 201 202 if (fputil::quick_get_round() == FE_UPWARD) 203 return FPBits::min_subnormal().get_val(); 204 fputil::set_errno_if_required(ERANGE); 205 fputil::raise_except_if_required(FE_UNDERFLOW); 206 return 0.0; 207 } 208 209 // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan 210 // x is finite 211 if (x_u < 0x7ff0'0000'0000'0000ULL) { 212 int rounding = fputil::quick_get_round(); 213 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 214 return FPBits::max_normal().get_val(); 215 216 fputil::set_errno_if_required(ERANGE); 217 fputil::raise_except_if_required(FE_OVERFLOW); 218 } 219 // x is +inf or nan 220 return x + FPBits::inf().get_val(); 221 } 222 223 } // namespace 224 225 LLVM_LIBC_FUNCTION(double, exp, (double x)) { 226 using FPBits = typename fputil::FPBits<double>; 227 FPBits xbits(x); 228 229 uint64_t x_u = xbits.uintval(); 230 231 // Upper bound: max normal number = 2^1023 * (2 - 2^-52) 232 // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9 233 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9 234 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9 235 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty 236 237 // Lower bound: min denormal number / 2 = 2^-1075 238 // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9 239 240 // Another lower bound: min normal number = 2^-1022 241 // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9 242 243 // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53. 244 if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 || 245 (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || 246 x_u < 0x3ca0000000000000)) { 247 return set_exceptional(x); 248 } 249 250 // Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52)) 251 252 // Range reduction: 253 // Let x = log(2) * (hi + mid1 + mid2) + lo 254 // in which: 255 // hi is an integer 256 // mid1 * 2^6 is an integer 257 // mid2 * 2^12 is an integer 258 // then: 259 // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo). 260 // With this formula: 261 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent 262 // field. 263 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables. 264 // - exp(lo) ~ 1 + lo + a0 * lo^2 + ... 265 // 266 // They can be defined by: 267 // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x) 268 // If we store L2E = round(log2(e), D, RN), then: 269 // log2(e) - L2E ~ 1.5 * 2^(-56) 270 // So the errors when computing in double precision is: 271 // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <= 272 // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | + 273 // + | x * 2^12 * L2E - D(x * 2^12 * L2E) | 274 // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN 275 // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes. 276 // So if: 277 // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely 278 // in double precision, the reduced argument: 279 // lo = x - log(2) * (hi + mid1 + mid2) is bounded by: 280 // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x)) 281 // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52)) 282 // < 2^-13 + 2^-41 283 // 284 285 // The following trick computes the round(x * L2E) more efficiently 286 // than using the rounding instructions, with the tradeoff for less accuracy, 287 // and hence a slightly larger range for the reduced argument `lo`. 288 // 289 // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9, 290 // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23, 291 // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t. 292 // Thus, the goal is to be able to use an additional addition and fixed width 293 // shift to get an int32_t representing round(x * 2^12 * L2E). 294 // 295 // Assuming int32_t using 2-complement representation, since the mantissa part 296 // of a double precision is unsigned with the leading bit hidden, if we add an 297 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the 298 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be 299 // considered as a proper 2-complement representations of x*2^12*L2E. 300 // 301 // One small problem with this approach is that the sum (x*2^12*L2E + C) in 302 // double precision is rounded to the least significant bit of the dorminant 303 // factor C. In order to minimize the rounding errors from this addition, we 304 // want to minimize e1. Another constraint that we want is that after 305 // shifting the mantissa so that the least significant bit of int32_t 306 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without 307 // any adjustment. So combining these 2 requirements, we can choose 308 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence 309 // after right shifting the mantissa, the resulting int32_t has correct sign. 310 // With this choice of C, the number of mantissa bits we need to shift to the 311 // right is: 52 - 33 = 19. 312 // 313 // Moreover, since the integer right shifts are equivalent to rounding down, 314 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to- 315 // +infinity. So in particular, we can compute: 316 // hmm = x * 2^12 * L2E + C, 317 // where C = 2^33 + 2^32 + 2^-1, then if 318 // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19), 319 // the reduced argument: 320 // lo = x - log(2) * 2^-12 * k is bounded by: 321 // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19 322 // = 2^-13 + 2^-31 + 2^-41. 323 // 324 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the 325 // exponent 2^12 is not needed. So we can simply define 326 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and 327 // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19). 328 329 // Rounding errors <= 2^-31 + 2^-41. 330 double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21); 331 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19); 332 double kd = static_cast<double>(k); 333 334 uint32_t idx1 = (k >> 6) & 0x3f; 335 uint32_t idx2 = k & 0x3f; 336 int hi = k >> 12; 337 338 bool denorm = (hi <= -1022); 339 340 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi}; 341 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi}; 342 343 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); 344 345 // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo) 346 // = 2^11 * 2^-13 * 2^-52 347 // = 2^-54. 348 // |dx| < 2^-13 + 2^-30. 349 double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 350 double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h); 351 352 // We use the degree-4 Taylor polynomial to approximate exp(lo): 353 // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo) 354 // So that the errors are bounded by: 355 // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58 356 // Let P_ be an evaluation of P where all intermediate computations are in 357 // double precision. Using either Horner's or Estrin's schemes, the evaluated 358 // errors can be bounded by: 359 // |P_(dx) - P(dx)| < 2^-51 360 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64 361 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63. 362 // Since we approximate 363 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo, 364 // We use the expression: 365 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~ 366 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo) 367 // with errors bounded by 1.5 * 2^-63. 368 369 double mid_lo = dx * exp_mid.hi; 370 371 // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 372 double p = poly_approx_d(dx); 373 374 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo); 375 376 if (LIBC_UNLIKELY(denorm)) { 377 if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D); 378 LIBC_LIKELY(r.has_value())) 379 return r.value(); 380 } else { 381 double upper = exp_mid.hi + (lo + ERR_D); 382 double lower = exp_mid.hi + (lo - ERR_D); 383 384 if (LIBC_LIKELY(upper == lower)) { 385 // to multiply by 2^hi, a fast way is to simply add hi to the exponent 386 // field. 387 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 388 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper)); 389 return r; 390 } 391 } 392 393 // Use double-double 394 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid); 395 396 if (LIBC_UNLIKELY(denorm)) { 397 if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD); 398 LIBC_LIKELY(r.has_value())) 399 return r.value(); 400 } else { 401 double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD); 402 double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD); 403 404 if (LIBC_LIKELY(upper_dd == lower_dd)) { 405 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 406 double r = 407 cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd)); 408 return r; 409 } 410 } 411 412 // Use 128-bit precision 413 Float128 r_f128 = exp_f128(x, kd, idx1, idx2); 414 415 return static_cast<double>(r_f128); 416 } 417 418 } // namespace LIBC_NAMESPACE_DECL 419