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/macros/optimization.h" // LIBC_UNLIKELY 26 27 #include <errno.h> 28 29 // #define DEBUGDEBUG 30 31 #ifdef DEBUGDEBUG 32 #include <iomanip> 33 #include <iostream> 34 #endif 35 36 namespace LIBC_NAMESPACE { 37 38 using fputil::DoubleDouble; 39 using fputil::TripleDouble; 40 using Float128 = typename fputil::DyadicFloat<128>; 41 42 // log2(e) 43 constexpr double LOG2_E = 0x1.71547652b82fep+0; 44 45 // Error bounds: 46 // Errors when using double precision. 47 // 0x1.8p-63; 48 constexpr uint64_t ERR_D = 0x3c08000000000000; 49 // Errors when using double-double precision. 50 // 0x1.0p-99 51 constexpr uint64_t ERR_DD = 0x39c0000000000000; 52 53 // -2^-12 * log(2) 54 // > a = -2^-12 * log(2); 55 // > b = round(a, 30, RN); 56 // > c = round(a - b, 30, RN); 57 // > d = round(a - b - c, D, RN); 58 // Errors < 1.5 * 2^-133 59 constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13; 60 constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47; 61 constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; 62 constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; 63 64 namespace { 65 66 // Polynomial approximations with double precision: 67 // Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 68 // For |dx| < 2^-13 + 2^-30: 69 // | output - expm1(dx) / dx | < 2^-51. 70 LIBC_INLINE double poly_approx_d(double dx) { 71 // dx^2 72 double dx2 = dx * dx; 73 // c0 = 1 + dx / 2 74 double c0 = fputil::multiply_add(dx, 0.5, 1.0); 75 // c1 = 1/6 + dx / 24 76 double c1 = 77 fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3); 78 // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24 79 double p = fputil::multiply_add(dx2, c1, c0); 80 return p; 81 } 82 83 // Polynomial approximation with double-double precision: 84 // Return expm1(dx) / dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 85 // For |dx| < 2^-13 + 2^-30: 86 // | output - expm1(dx) | < 2^-101 87 DoubleDouble poly_approx_dd(const DoubleDouble &dx) { 88 // Taylor polynomial. 89 constexpr DoubleDouble COEFFS[] = { 90 {0, 0x1p0}, // 1 91 {0, 0x1p-1}, // 1/2 92 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6 93 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24 94 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120 95 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720 96 {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/5040 97 }; 98 99 DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2], 100 COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]); 101 return p; 102 } 103 104 // Polynomial approximation with 128-bit precision: 105 // Return (exp(dx) - 1)/dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 106 // For |dx| < 2^-13 + 2^-30: 107 // | output - exp(dx) | < 2^-126. 108 Float128 poly_approx_f128(const Float128 &dx) { 109 using MType = typename Float128::MantissaType; 110 111 constexpr Float128 COEFFS_128[]{ 112 {false, -127, MType({0, 0x8000000000000000})}, // 1.0 113 {false, -128, MType({0, 0x8000000000000000})}, // 0.5 114 {false, -130, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/6 115 {false, -132, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/24 116 {false, -134, MType({0x8888888888888889, 0x8888888888888888})}, // 1/120 117 {false, -137, MType({0x60b60b60b60b60b6, 0xb60b60b60b60b60b})}, // 1/720 118 {false, -140, MType({0x00d00d00d00d00d0, 0xd00d00d00d00d00d})}, // 1/5040 119 }; 120 121 Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2], 122 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5], 123 COEFFS_128[6]); 124 return p; 125 } 126 127 #ifdef DEBUGDEBUG 128 std::ostream &operator<<(std::ostream &OS, const Float128 &r) { 129 OS << (r.sign ? "-(" : "(") << r.mantissa.val[0] << " + " << r.mantissa.val[1] 130 << " * 2^64) * 2^" << r.exponent << "\n"; 131 return OS; 132 } 133 134 std::ostream &operator<<(std::ostream &OS, const DoubleDouble &r) { 135 OS << std::hexfloat << r.hi << " + " << r.lo << std::defaultfloat << "\n"; 136 return OS; 137 } 138 #endif 139 140 // Compute exp(x) - 1 using 128-bit precision. 141 // TODO(lntue): investigate triple-double precision implementation for this 142 // step. 143 Float128 expm1_f128(double x, double kd, int idx1, int idx2) { 144 using MType = typename Float128::MantissaType; 145 // Recalculate dx: 146 147 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 148 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 149 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133 150 151 Float128 dx = fputil::quick_add( 152 Float128(t1), fputil::quick_add(Float128(t2), Float128(t3))); 153 154 // TODO: Skip recalculating exp_mid1 and exp_mid2. 155 Float128 exp_mid1 = 156 fputil::quick_add(Float128(EXP2_MID1[idx1].hi), 157 fputil::quick_add(Float128(EXP2_MID1[idx1].mid), 158 Float128(EXP2_MID1[idx1].lo))); 159 160 Float128 exp_mid2 = 161 fputil::quick_add(Float128(EXP2_MID2[idx2].hi), 162 fputil::quick_add(Float128(EXP2_MID2[idx2].mid), 163 Float128(EXP2_MID2[idx2].lo))); 164 165 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); 166 167 int hi = static_cast<int>(kd) >> 12; 168 Float128 minus_one{true, -127 - hi, MType({0, 0x8000000000000000})}; 169 170 Float128 exp_mid_m1 = fputil::quick_add(exp_mid, minus_one); 171 172 Float128 p = poly_approx_f128(dx); 173 174 // r = exp_mid * (1 + dx * P) - 1 175 // = (exp_mid - 1) + (dx * exp_mid) * P 176 Float128 r = 177 fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1); 178 179 r.exponent += hi; 180 181 #ifdef DEBUGDEBUG 182 std::cout << "=== VERY SLOW PASS ===\n" 183 << " kd: " << kd << "\n" 184 << " dx: " << dx << "exp_mid_m1: " << exp_mid_m1 185 << " exp_mid: " << exp_mid << " p: " << p 186 << " r: " << r << std::endl; 187 #endif 188 189 return r; 190 } 191 192 // Compute exp(x) - 1 with double-double precision. 193 DoubleDouble exp_double_double(double x, double kd, const DoubleDouble &exp_mid, 194 const DoubleDouble &hi_part) { 195 // Recalculate dx: 196 // dx = x - k * 2^-12 * log(2) 197 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 198 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact 199 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130 200 201 DoubleDouble dx = fputil::exact_add(t1, t2); 202 dx.lo += t3; 203 204 // Degree-6 Taylor polynomial approximation in double-double precision. 205 // | p - exp(x) | < 2^-100. 206 DoubleDouble p = poly_approx_dd(dx); 207 208 // Error bounds: 2^-99. 209 DoubleDouble r = 210 fputil::multiply_add(fputil::quick_mult(exp_mid, dx), p, hi_part); 211 212 #ifdef DEBUGDEBUG 213 std::cout << "=== SLOW PASS ===\n" 214 << " dx: " << dx << " p: " << p << " r: " << r << std::endl; 215 #endif 216 217 return r; 218 } 219 220 // Check for exceptional cases when 221 // |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 222 double set_exceptional(double x) { 223 using FPBits = typename fputil::FPBits<double>; 224 FPBits xbits(x); 225 226 uint64_t x_u = xbits.uintval(); 227 uint64_t x_abs = xbits.abs().uintval(); 228 229 // |x| <= 2^-53. 230 if (x_abs <= 0x3ca0'0000'0000'0000ULL) { 231 // expm1(x) ~ x. 232 233 if (LIBC_UNLIKELY(x_abs <= 0x0370'0000'0000'0000ULL)) { 234 if (LIBC_UNLIKELY(x_abs == 0)) 235 return x; 236 // |x| <= 2^-968, need to scale up a bit before rounding, then scale it 237 // back down. 238 return 0x1.0p-200 * fputil::multiply_add(x, 0x1.0p+200, 0x1.0p-1022); 239 } 240 241 // 2^-968 < |x| <= 2^-53. 242 return fputil::round_result_slightly_up(x); 243 } 244 245 // x < log(2^-54) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. 246 247 // x < log(2^-54) or -inf/nan 248 if (x_u >= 0xc042'b708'8723'20e2ULL) { 249 // expm1(-Inf) = -1 250 if (xbits.is_inf()) 251 return -1.0; 252 253 // exp(nan) = nan 254 if (xbits.is_nan()) 255 return x; 256 257 return fputil::round_result_slightly_up(-1.0); 258 } 259 260 // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan 261 // x is finite 262 if (x_u < 0x7ff0'0000'0000'0000ULL) { 263 int rounding = fputil::quick_get_round(); 264 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 265 return FPBits::max_normal(); 266 267 fputil::set_errno_if_required(ERANGE); 268 fputil::raise_except_if_required(FE_OVERFLOW); 269 } 270 // x is +inf or nan 271 return x + static_cast<double>(FPBits::inf()); 272 } 273 274 } // namespace 275 276 LLVM_LIBC_FUNCTION(double, expm1, (double x)) { 277 using FPBits = typename fputil::FPBits<double>; 278 using Sign = fputil::Sign; 279 FPBits xbits(x); 280 281 bool x_is_neg = xbits.is_neg(); 282 uint64_t x_u = xbits.uintval(); 283 284 // Upper bound: max normal number = 2^1023 * (2 - 2^-52) 285 // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9 286 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9 287 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9 288 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty 289 290 // Lower bound: log(2^-54) = -0x1.2b708872320e2p5 291 // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5 292 293 // x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 or |x| <= 2^-53. 294 295 if (LIBC_UNLIKELY(x_u >= 0xc042b708872320e2 || 296 (x_u <= 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || 297 x_u <= 0x3ca0000000000000)) { 298 return set_exceptional(x); 299 } 300 301 // Now log(2^-54) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52)) 302 303 // Range reduction: 304 // Let x = log(2) * (hi + mid1 + mid2) + lo 305 // in which: 306 // hi is an integer 307 // mid1 * 2^6 is an integer 308 // mid2 * 2^12 is an integer 309 // then: 310 // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo). 311 // With this formula: 312 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent 313 // field. 314 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables. 315 // - exp(lo) ~ 1 + lo + a0 * lo^2 + ... 316 // 317 // They can be defined by: 318 // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x) 319 // If we store L2E = round(log2(e), D, RN), then: 320 // log2(e) - L2E ~ 1.5 * 2^(-56) 321 // So the errors when computing in double precision is: 322 // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <= 323 // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | + 324 // + | x * 2^12 * L2E - D(x * 2^12 * L2E) | 325 // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN 326 // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes. 327 // So if: 328 // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely 329 // in double precision, the reduced argument: 330 // lo = x - log(2) * (hi + mid1 + mid2) is bounded by: 331 // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x)) 332 // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52)) 333 // < 2^-13 + 2^-41 334 // 335 336 // The following trick computes the round(x * L2E) more efficiently 337 // than using the rounding instructions, with the tradeoff for less accuracy, 338 // and hence a slightly larger range for the reduced argument `lo`. 339 // 340 // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9, 341 // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23, 342 // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t. 343 // Thus, the goal is to be able to use an additional addition and fixed width 344 // shift to get an int32_t representing round(x * 2^12 * L2E). 345 // 346 // Assuming int32_t using 2-complement representation, since the mantissa part 347 // of a double precision is unsigned with the leading bit hidden, if we add an 348 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the 349 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be 350 // considered as a proper 2-complement representations of x*2^12*L2E. 351 // 352 // One small problem with this approach is that the sum (x*2^12*L2E + C) in 353 // double precision is rounded to the least significant bit of the dorminant 354 // factor C. In order to minimize the rounding errors from this addition, we 355 // want to minimize e1. Another constraint that we want is that after 356 // shifting the mantissa so that the least significant bit of int32_t 357 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without 358 // any adjustment. So combining these 2 requirements, we can choose 359 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence 360 // after right shifting the mantissa, the resulting int32_t has correct sign. 361 // With this choice of C, the number of mantissa bits we need to shift to the 362 // right is: 52 - 33 = 19. 363 // 364 // Moreover, since the integer right shifts are equivalent to rounding down, 365 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to- 366 // +infinity. So in particular, we can compute: 367 // hmm = x * 2^12 * L2E + C, 368 // where C = 2^33 + 2^32 + 2^-1, then if 369 // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19), 370 // the reduced argument: 371 // lo = x - log(2) * 2^-12 * k is bounded by: 372 // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19 373 // = 2^-13 + 2^-31 + 2^-41. 374 // 375 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the 376 // exponent 2^12 is not needed. So we can simply define 377 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and 378 // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19). 379 380 // Rounding errors <= 2^-31 + 2^-41. 381 double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21); 382 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19); 383 double kd = static_cast<double>(k); 384 385 uint32_t idx1 = (k >> 6) & 0x3f; 386 uint32_t idx2 = k & 0x3f; 387 int hi = k >> 12; 388 389 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi}; 390 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi}; 391 392 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); 393 394 // -2^(-hi) 395 double one_scaled = 396 FPBits::create_value(Sign::NEG, FPBits::EXP_BIAS - hi, 0).get_val(); 397 398 // 2^(mid1 + mid2) - 2^(-hi) 399 DoubleDouble hi_part = x_is_neg ? fputil::exact_add(one_scaled, exp_mid.hi) 400 : fputil::exact_add(exp_mid.hi, one_scaled); 401 402 hi_part.lo += exp_mid.lo; 403 404 // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo) 405 // = 2^11 * 2^-13 * 2^-52 406 // = 2^-54. 407 // |dx| < 2^-13 + 2^-30. 408 double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact 409 double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h); 410 411 // We use the degree-4 Taylor polynomial to approximate exp(lo): 412 // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo) 413 // So that the errors are bounded by: 414 // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58 415 // Let P_ be an evaluation of P where all intermediate computations are in 416 // double precision. Using either Horner's or Estrin's schemes, the evaluated 417 // errors can be bounded by: 418 // |P_(dx) - P(dx)| < 2^-51 419 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64 420 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63. 421 // Since we approximate 422 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo, 423 // We use the expression: 424 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~ 425 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo) 426 // with errors bounded by 1.5 * 2^-63. 427 428 // Finally, we have the following approximation formula: 429 // expm1(x) = 2^hi * 2^(mid1 + mid2) * exp(lo) - 1 430 // = 2^hi * ( 2^(mid1 + mid2) * exp(lo) - 2^(-hi) ) 431 // ~ 2^hi * ( (exp_mid.hi - 2^-hi) + 432 // + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)) 433 434 double mid_lo = dx * exp_mid.hi; 435 436 // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. 437 double p = poly_approx_d(dx); 438 439 double lo = fputil::multiply_add(p, mid_lo, hi_part.lo); 440 441 // TODO: The following line leaks encoding abstraction. Use FPBits methods 442 // instead. 443 uint64_t err = x_is_neg ? (static_cast<uint64_t>(-hi) << 52) : 0; 444 445 double err_d = cpp::bit_cast<double>(ERR_D + err); 446 447 double upper = hi_part.hi + (lo + err_d); 448 double lower = hi_part.hi + (lo - err_d); 449 450 #ifdef DEBUGDEBUG 451 std::cout << "=== FAST PASS ===\n" 452 << " x: " << std::hexfloat << x << std::defaultfloat << "\n" 453 << " k: " << k << "\n" 454 << " idx1: " << idx1 << "\n" 455 << " idx2: " << idx2 << "\n" 456 << " hi: " << hi << "\n" 457 << " dx: " << std::hexfloat << dx << std::defaultfloat << "\n" 458 << "exp_mid: " << exp_mid << "hi_part: " << hi_part 459 << " mid_lo: " << std::hexfloat << mid_lo << std::defaultfloat 460 << "\n" 461 << " p: " << std::hexfloat << p << std::defaultfloat << "\n" 462 << " lo: " << std::hexfloat << lo << std::defaultfloat << "\n" 463 << " upper: " << std::hexfloat << upper << std::defaultfloat 464 << "\n" 465 << " lower: " << std::hexfloat << lower << std::defaultfloat 466 << "\n" 467 << std::endl; 468 #endif 469 470 if (LIBC_LIKELY(upper == lower)) { 471 // to multiply by 2^hi, a fast way is to simply add hi to the exponent 472 // field. 473 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 474 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper)); 475 return r; 476 } 477 478 // Use double-double 479 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid, hi_part); 480 481 double err_dd = cpp::bit_cast<double>(ERR_DD + err); 482 483 double upper_dd = r_dd.hi + (r_dd.lo + err_dd); 484 double lower_dd = r_dd.hi + (r_dd.lo - err_dd); 485 486 if (LIBC_LIKELY(upper_dd == lower_dd)) { 487 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; 488 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd)); 489 return r; 490 } 491 492 // Use 128-bit precision 493 Float128 r_f128 = expm1_f128(x, kd, idx1, idx2); 494 495 return static_cast<double>(r_f128); 496 } 497 498 } // namespace LIBC_NAMESPACE 499