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