1 //===-- Double-precision e^x - 1 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/expm1.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/except_value_utils.h" 20 #include "src/__support/FPUtil/multiply_add.h" 21 #include "src/__support/FPUtil/nearest_integer.h" 22 #include "src/__support/FPUtil/rounding_mode.h" 23 #include "src/__support/FPUtil/triple_double.h" 24 #include "src/__support/common.h" 25 #include "src/__support/integer_literals.h" 26 #include "src/__support/macros/config.h" 27 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 28 29 #if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0) 30 #define LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS 31 #endif 32 33 // #define DEBUGDEBUG 34 35 #ifdef DEBUGDEBUG 36 #include <iomanip> 37 #include <iostream> 38 #endif 39 40 namespace LIBC_NAMESPACE_DECL { 41 42 using fputil::DoubleDouble; 43 using fputil::TripleDouble; 44 using Float128 = typename fputil::DyadicFloat<128>; 45 46 using LIBC_NAMESPACE::operator""_u128; 47 48 // log2(e) 49 constexpr double LOG2_E = 0x1.71547652b82fep+0; 50 51 // Error bounds: 52 // Errors when using double precision. 53 // 0x1.8p-63; 54 constexpr uint64_t ERR_D = 0x3c08000000000000; 55 // Errors when using double-double precision. 56 // 0x1.0p-99 57 [[maybe_unused]] constexpr uint64_t ERR_DD = 0x39c0000000000000; 58 59 // -2^-12 * log(2) 60 // > a = -2^-12 * log(2); 61 // > b = round(a, 30, RN); 62 // > c = round(a - b, 30, RN); 63 // > d = round(a - b - c, D, RN); 64 // Errors < 1.5 * 2^-133 65 constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13; 66 constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47; 67 constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; 68 constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; 69 70 namespace { 71 72 // Polynomial approximations with double precision: 73 // Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 74 // For |dx| < 2^-13 + 2^-30: 75 // | output - expm1(dx) / dx | < 2^-51. 76 LIBC_INLINE double poly_approx_d(double dx) { 77 // dx^2 78 double dx2 = dx * dx; 79 // c0 = 1 + dx / 2 80 double c0 = fputil::multiply_add(dx, 0.5, 1.0); 81 // c1 = 1/6 + dx / 24 82 double c1 = 83 fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3); 84 // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24 85 double p = fputil::multiply_add(dx2, c1, c0); 86 return p; 87 } 88 89 // Polynomial approximation with double-double precision: 90 // Return expm1(dx) / dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 91 // For |dx| < 2^-13 + 2^-30: 92 // | output - expm1(dx) | < 2^-101 93 DoubleDouble poly_approx_dd(const DoubleDouble &dx) { 94 // Taylor polynomial. 95 constexpr DoubleDouble COEFFS[] = { 96 {0, 0x1p0}, // 1 97 {0, 0x1p-1}, // 1/2 98 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6 99 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24 100 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120 101 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720 102 {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/5040 103 }; 104 105 DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2], 106 COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]); 107 return p; 108 } 109 110 // Polynomial approximation with 128-bit precision: 111 // Return (exp(dx) - 1)/dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 112 // For |dx| < 2^-13 + 2^-30: 113 // | output - exp(dx) | < 2^-126. 114 [[maybe_unused]] Float128 poly_approx_f128(const Float128 &dx) { 115 constexpr Float128 COEFFS_128[]{ 116 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 117 {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5 118 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6 119 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24 120 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120 121 {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720 122 {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040 123 }; 124 125 Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2], 126 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5], 127 COEFFS_128[6]); 128 return p; 129 } 130 131 #ifdef DEBUGDEBUG 132 std::ostream &operator<<(std::ostream &OS, const Float128 &r) { 133 OS << (r.sign == Sign::NEG ? "-(" : "(") << r.mantissa.val[0] << " + " 134 << r.mantissa.val[1] << " * 2^64) * 2^" << r.exponent << "\n"; 135 return OS; 136 } 137 138 std::ostream &operator<<(std::ostream &OS, const DoubleDouble &r) { 139 OS << std::hexfloat << "(" << r.hi << " + " << r.lo << ")" 140 << std::defaultfloat << "\n"; 141 return OS; 142 } 143 #endif 144 145 // Compute exp(x) - 1 using 128-bit precision. 146 // TODO(lntue): investigate triple-double precision implementation for this 147 // step. 148 [[maybe_unused]] Float128 expm1_f128(double x, double kd, int idx1, int idx2) { 149 // Recalculate dx: 150 151 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 152 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 153 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133 154 155 Float128 dx = fputil::quick_add( 156 Float128(t1), fputil::quick_add(Float128(t2), Float128(t3))); 157 158 // TODO: Skip recalculating exp_mid1 and exp_mid2. 159 Float128 exp_mid1 = 160 fputil::quick_add(Float128(EXP2_MID1[idx1].hi), 161 fputil::quick_add(Float128(EXP2_MID1[idx1].mid), 162 Float128(EXP2_MID1[idx1].lo))); 163 164 Float128 exp_mid2 = 165 fputil::quick_add(Float128(EXP2_MID2[idx2].hi), 166 fputil::quick_add(Float128(EXP2_MID2[idx2].mid), 167 Float128(EXP2_MID2[idx2].lo))); 168 169 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); 170 171 int hi = static_cast<int>(kd) >> 12; 172 Float128 minus_one{Sign::NEG, -127 - hi, 173 0x80000000'00000000'00000000'00000000_u128}; 174 175 Float128 exp_mid_m1 = fputil::quick_add(exp_mid, minus_one); 176 177 Float128 p = poly_approx_f128(dx); 178 179 // r = exp_mid * (1 + dx * P) - 1 180 // = (exp_mid - 1) + (dx * exp_mid) * P 181 Float128 r = 182 fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1); 183 184 r.exponent += hi; 185 186 #ifdef DEBUGDEBUG 187 std::cout << "=== VERY SLOW PASS ===\n" 188 << " kd: " << kd << "\n" 189 << " hi: " << hi << "\n" 190 << " minus_one: " << minus_one << " dx: " << dx 191 << "exp_mid_m1: " << exp_mid_m1 << " exp_mid: " << exp_mid 192 << " p: " << p << " r: " << r << std::endl; 193 #endif 194 195 return r; 196 } 197 198 // Compute exp(x) - 1 with double-double precision. 199 DoubleDouble exp_double_double(double x, double kd, const DoubleDouble &exp_mid, 200 const DoubleDouble &hi_part) { 201 // Recalculate dx: 202 // dx = x - k * 2^-12 * log(2) 203 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 204 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 205 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130 206 207 DoubleDouble dx = fputil::exact_add(t1, t2); 208 dx.lo += t3; 209 210 // Degree-6 Taylor polynomial approximation in double-double precision. 211 // | p - exp(x) | < 2^-100. 212 DoubleDouble p = poly_approx_dd(dx); 213 214 // Error bounds: 2^-99. 215 DoubleDouble r = 216 fputil::multiply_add(fputil::quick_mult(exp_mid, dx), p, hi_part); 217 218 #ifdef DEBUGDEBUG 219 std::cout << "=== SLOW PASS ===\n" 220 << " dx: " << dx << " p: " << p << " r: " << r << std::endl; 221 #endif 222 223 return r; 224 } 225 226 // Check for exceptional cases when 227 // |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 228 double set_exceptional(double x) { 229 using FPBits = typename fputil::FPBits<double>; 230 FPBits xbits(x); 231 232 uint64_t x_u = xbits.uintval(); 233 uint64_t x_abs = xbits.abs().uintval(); 234 235 // |x| <= 2^-53. 236 if (x_abs <= 0x3ca0'0000'0000'0000ULL) { 237 // expm1(x) ~ x. 238 239 if (LIBC_UNLIKELY(x_abs <= 0x0370'0000'0000'0000ULL)) { 240 if (LIBC_UNLIKELY(x_abs == 0)) 241 return x; 242 // |x| <= 2^-968, need to scale up a bit before rounding, then scale it 243 // back down. 244 return 0x1.0p-200 * fputil::multiply_add(x, 0x1.0p+200, 0x1.0p-1022); 245 } 246 247 // 2^-968 < |x| <= 2^-53. 248 return fputil::round_result_slightly_up(x); 249 } 250 251 // x < log(2^-54) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. 252 253 // x < log(2^-54) or -inf/nan 254 if (x_u >= 0xc042'b708'8723'20e2ULL) { 255 // expm1(-Inf) = -1 256 if (xbits.is_inf()) 257 return -1.0; 258 259 // exp(nan) = nan 260 if (xbits.is_nan()) 261 return x; 262 263 return fputil::round_result_slightly_up(-1.0); 264 } 265 266 // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan 267 // x is finite 268 if (x_u < 0x7ff0'0000'0000'0000ULL) { 269 int rounding = fputil::quick_get_round(); 270 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 271 return FPBits::max_normal().get_val(); 272 273 fputil::set_errno_if_required(ERANGE); 274 fputil::raise_except_if_required(FE_OVERFLOW); 275 } 276 // x is +inf or nan 277 return x + FPBits::inf().get_val(); 278 } 279 280 } // namespace 281 282 LLVM_LIBC_FUNCTION(double, expm1, (double x)) { 283 using FPBits = typename fputil::FPBits<double>; 284 285 FPBits xbits(x); 286 287 bool x_is_neg = xbits.is_neg(); 288 uint64_t x_u = xbits.uintval(); 289 290 // Upper bound: max normal number = 2^1023 * (2 - 2^-52) 291 // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9 292 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9 293 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9 294 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty 295 296 // Lower bound: log(2^-54) = -0x1.2b708872320e2p5 297 // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5 298 299 // x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 or |x| <= 2^-53. 300 301 if (LIBC_UNLIKELY(x_u >= 0xc042b708872320e2 || 302 (x_u <= 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || 303 x_u <= 0x3ca0000000000000)) { 304 return set_exceptional(x); 305 } 306 307 // Now log(2^-54) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52)) 308 309 // Range reduction: 310 // Let x = log(2) * (hi + mid1 + mid2) + lo 311 // in which: 312 // hi is an integer 313 // mid1 * 2^6 is an integer 314 // mid2 * 2^12 is an integer 315 // then: 316 // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo). 317 // With this formula: 318 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent 319 // field. 320 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables. 321 // - exp(lo) ~ 1 + lo + a0 * lo^2 + ... 322 // 323 // They can be defined by: 324 // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x) 325 // If we store L2E = round(log2(e), D, RN), then: 326 // log2(e) - L2E ~ 1.5 * 2^(-56) 327 // So the errors when computing in double precision is: 328 // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <= 329 // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | + 330 // + | x * 2^12 * L2E - D(x * 2^12 * L2E) | 331 // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN 332 // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes. 333 // So if: 334 // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely 335 // in double precision, the reduced argument: 336 // lo = x - log(2) * (hi + mid1 + mid2) is bounded by: 337 // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x)) 338 // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52)) 339 // < 2^-13 + 2^-41 340 // 341 342 // The following trick computes the round(x * L2E) more efficiently 343 // than using the rounding instructions, with the tradeoff for less accuracy, 344 // and hence a slightly larger range for the reduced argument `lo`. 345 // 346 // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9, 347 // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23, 348 // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t. 349 // Thus, the goal is to be able to use an additional addition and fixed width 350 // shift to get an int32_t representing round(x * 2^12 * L2E). 351 // 352 // Assuming int32_t using 2-complement representation, since the mantissa part 353 // of a double precision is unsigned with the leading bit hidden, if we add an 354 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the 355 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be 356 // considered as a proper 2-complement representations of x*2^12*L2E. 357 // 358 // One small problem with this approach is that the sum (x*2^12*L2E + C) in 359 // double precision is rounded to the least significant bit of the dorminant 360 // factor C. In order to minimize the rounding errors from this addition, we 361 // want to minimize e1. Another constraint that we want is that after 362 // shifting the mantissa so that the least significant bit of int32_t 363 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without 364 // any adjustment. So combining these 2 requirements, we can choose 365 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence 366 // after right shifting the mantissa, the resulting int32_t has correct sign. 367 // With this choice of C, the number of mantissa bits we need to shift to the 368 // right is: 52 - 33 = 19. 369 // 370 // Moreover, since the integer right shifts are equivalent to rounding down, 371 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to- 372 // +infinity. So in particular, we can compute: 373 // hmm = x * 2^12 * L2E + C, 374 // where C = 2^33 + 2^32 + 2^-1, then if 375 // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19), 376 // the reduced argument: 377 // lo = x - log(2) * 2^-12 * k is bounded by: 378 // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19 379 // = 2^-13 + 2^-31 + 2^-41. 380 // 381 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the 382 // exponent 2^12 is not needed. So we can simply define 383 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and 384 // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19). 385 386 // Rounding errors <= 2^-31 + 2^-41. 387 double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21); 388 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19); 389 double kd = static_cast<double>(k); 390 391 uint32_t idx1 = (k >> 6) & 0x3f; 392 uint32_t idx2 = k & 0x3f; 393 int hi = k >> 12; 394 395 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi}; 396 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi}; 397 398 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); 399 400 // -2^(-hi) 401 double one_scaled = 402 FPBits::create_value(Sign::NEG, FPBits::EXP_BIAS - hi, 0).get_val(); 403 404 // 2^(mid1 + mid2) - 2^(-hi) 405 DoubleDouble hi_part = x_is_neg ? fputil::exact_add(one_scaled, exp_mid.hi) 406 : fputil::exact_add(exp_mid.hi, one_scaled); 407 408 hi_part.lo += exp_mid.lo; 409 410 // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo) 411 // = 2^11 * 2^-13 * 2^-52 412 // = 2^-54. 413 // |dx| < 2^-13 + 2^-30. 414 double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 415 double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h); 416 417 // We use the degree-4 Taylor polynomial to approximate exp(lo): 418 // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo) 419 // So that the errors are bounded by: 420 // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58 421 // Let P_ be an evaluation of P where all intermediate computations are in 422 // double precision. Using either Horner's or Estrin's schemes, the evaluated 423 // errors can be bounded by: 424 // |P_(dx) - P(dx)| < 2^-51 425 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64 426 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63. 427 // Since we approximate 428 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo, 429 // We use the expression: 430 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~ 431 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo) 432 // with errors bounded by 1.5 * 2^-63. 433 434 // Finally, we have the following approximation formula: 435 // expm1(x) = 2^hi * 2^(mid1 + mid2) * exp(lo) - 1 436 // = 2^hi * ( 2^(mid1 + mid2) * exp(lo) - 2^(-hi) ) 437 // ~ 2^hi * ( (exp_mid.hi - 2^-hi) + 438 // + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)) 439 440 double mid_lo = dx * exp_mid.hi; 441 442 // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 443 double p = poly_approx_d(dx); 444 445 double lo = fputil::multiply_add(p, mid_lo, hi_part.lo); 446 447 // TODO: The following line leaks encoding abstraction. Use FPBits methods 448 // instead. 449 uint64_t err = x_is_neg ? (static_cast<uint64_t>(-hi) << 52) : 0; 450 451 double err_d = cpp::bit_cast<double>(ERR_D + err); 452 453 double upper = hi_part.hi + (lo + err_d); 454 double lower = hi_part.hi + (lo - err_d); 455 456 #ifdef DEBUGDEBUG 457 std::cout << "=== FAST PASS ===\n" 458 << " x: " << std::hexfloat << x << std::defaultfloat << "\n" 459 << " k: " << k << "\n" 460 << " idx1: " << idx1 << "\n" 461 << " idx2: " << idx2 << "\n" 462 << " hi: " << hi << "\n" 463 << " dx: " << std::hexfloat << dx << std::defaultfloat << "\n" 464 << "exp_mid: " << exp_mid << "hi_part: " << hi_part 465 << " mid_lo: " << std::hexfloat << mid_lo << std::defaultfloat 466 << "\n" 467 << " p: " << std::hexfloat << p << std::defaultfloat << "\n" 468 << " lo: " << std::hexfloat << lo << std::defaultfloat << "\n" 469 << " upper: " << std::hexfloat << upper << std::defaultfloat 470 << "\n" 471 << " lower: " << std::hexfloat << lower << std::defaultfloat 472 << "\n" 473 << std::endl; 474 #endif 475 476 if (LIBC_LIKELY(upper == lower)) { 477 // to multiply by 2^hi, a fast way is to simply add hi to the exponent 478 // field. 479 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 480 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper)); 481 return r; 482 } 483 484 // Use double-double 485 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid, hi_part); 486 487 #ifdef LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS 488 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 489 double r = 490 cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(r_dd.hi + r_dd.lo)); 491 return r; 492 #else 493 double err_dd = cpp::bit_cast<double>(ERR_DD + err); 494 495 double upper_dd = r_dd.hi + (r_dd.lo + err_dd); 496 double lower_dd = r_dd.hi + (r_dd.lo - err_dd); 497 498 if (LIBC_LIKELY(upper_dd == lower_dd)) { 499 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 500 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd)); 501 return r; 502 } 503 504 // Use 128-bit precision 505 Float128 r_f128 = expm1_f128(x, kd, idx1, idx2); 506 507 return static_cast<double>(r_f128); 508 #endif // LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS 509 } 510 511 } // namespace LIBC_NAMESPACE_DECL 512