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