505 lines · cpp
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-exception6//7//===----------------------------------------------------------------------===//8 9#include "src/math/expm1.h"10#include "src/__support/CPP/bit.h"11#include "src/__support/FPUtil/FEnvImpl.h"12#include "src/__support/FPUtil/FPBits.h"13#include "src/__support/FPUtil/PolyEval.h"14#include "src/__support/FPUtil/double_double.h"15#include "src/__support/FPUtil/dyadic_float.h"16#include "src/__support/FPUtil/except_value_utils.h"17#include "src/__support/FPUtil/multiply_add.h"18#include "src/__support/FPUtil/rounding_mode.h"19#include "src/__support/FPUtil/triple_double.h"20#include "src/__support/common.h"21#include "src/__support/integer_literals.h"22#include "src/__support/macros/config.h"23#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY24#include "src/__support/math/common_constants.h" // Lookup tables EXP_M1 and EXP_M2.25#include "src/__support/math/exp_constants.h"26 27#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0)28#define LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS29#endif30 31namespace LIBC_NAMESPACE_DECL {32 33using fputil::DoubleDouble;34using fputil::TripleDouble;35using Float128 = typename fputil::DyadicFloat<128>;36 37using LIBC_NAMESPACE::operator""_u128;38 39// log2(e)40constexpr double LOG2_E = 0x1.71547652b82fep+0;41 42// Error bounds:43// Errors when using double precision.44// 0x1.8p-63;45constexpr uint64_t ERR_D = 0x3c08000000000000;46// Errors when using double-double precision.47// 0x1.0p-9948[[maybe_unused]] constexpr uint64_t ERR_DD = 0x39c0000000000000;49 50// -2^-12 * log(2)51// > a = -2^-12 * log(2);52// > b = round(a, 30, RN);53// > c = round(a - b, 30, RN);54// > d = round(a - b - c, D, RN);55// Errors < 1.5 * 2^-13356constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;57constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;58constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;59constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;60 61namespace {62 63using namespace common_constants_internal;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^-51.69LIBC_INLINE double poly_approx_d(double dx) {70 // dx^271 double dx2 = dx * dx;72 // c0 = 1 + dx / 273 double c0 = fputil::multiply_add(dx, 0.5, 1.0);74 // c1 = 1/6 + dx / 2475 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 / 2478 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 / 504084// For |dx| < 2^-13 + 2^-30:85// | output - expm1(dx) | < 2^-10186DoubleDouble poly_approx_dd(const DoubleDouble &dx) {87 // Taylor polynomial.88 constexpr DoubleDouble COEFFS[] = {89 {0, 0x1p0}, // 190 {0, 0x1p-1}, // 1/291 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/692 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/2493 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/12094 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/72095 {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/504096 };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 / 5040105// 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}, // 1.0110 {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5111 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6112 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24113 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120114 {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720115 {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040116 };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 DEBUGDEBUG125std::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 131std::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#endif137 138// Compute exp(x) - 1 using 128-bit precision.139// TODO(lntue): investigate triple-double precision implementation for this140// 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); // exact145 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact146 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133147 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) - 1173 // = (exp_mid - 1) + (dx * exp_mid) * P174 Float128 r =175 fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1);176 177 r.exponent += hi;178 179#ifdef DEBUGDEBUG180 std::cout << "=== VERY SLOW PASS ===\n"181 << " kd: " << kd << "\n"182 << " hi: " << hi << "\n"183 << " minus_one: " << minus_one << " dx: " << dx184 << "exp_mid_m1: " << exp_mid_m1 << " exp_mid: " << exp_mid185 << " p: " << p << " r: " << r << std::endl;186#endif187 188 return r;189}190 191// Compute exp(x) - 1 with double-double precision.192DoubleDouble 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); // exact197 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact198 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130199 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 DEBUGDEBUG212 std::cout << "=== SLOW PASS ===\n"213 << " dx: " << dx << " p: " << p << " r: " << r << std::endl;214#endif215 216 return r;217}218 219// Check for exceptional cases when220// |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9221double 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 it236 // 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/nan247 if (x_u >= 0xc042'b708'8723'20e2ULL) {248 // expm1(-Inf) = -1249 if (xbits.is_inf())250 return -1.0;251 252 // exp(nan) = nan253 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/nan260 // x is finite261 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 nan270 return x + FPBits::inf().get_val();271}272 273} // namespace274 275LLVM_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+9285 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9286 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9287 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty288 289 // Lower bound: log(2^-54) = -0x1.2b708872320e2p5290 // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5291 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) + lo304 // in which:305 // hi is an integer306 // mid1 * 2^6 is an integer307 // mid2 * 2^12 is an integer308 // 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 exponent312 // 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 RN325 // 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 entirely328 // 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^-41333 //334 335 // The following trick computes the round(x * L2E) more efficiently336 // 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 width343 // shift to get an int32_t representing round(x * 2^12 * L2E).344 //345 // Assuming int32_t using 2-complement representation, since the mantissa part346 // of a double precision is unsigned with the leading bit hidden, if we add an347 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the348 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be349 // 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) in352 // double precision is rounded to the least significant bit of the dorminant353 // factor C. In order to minimize the rounding errors from this addition, we354 // want to minimize e1. Another constraint that we want is that after355 // shifting the mantissa so that the least significant bit of int32_t356 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without357 // any adjustment. So combining these 2 requirements, we can choose358 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence359 // 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 the361 // 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 if368 // 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^-19372 // = 2^-13 + 2^-31 + 2^-41.373 //374 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the375 // exponent 2^12 is not needed. So we can simply define376 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and377 // 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^-52405 // = 2^-54.406 // |dx| < 2^-13 + 2^-30.407 double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact408 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^-58414 // Let P_ be an evaluation of P where all intermediate computations are in415 // double precision. Using either Horner's or Estrin's schemes, the evaluated416 // errors can be bounded by:417 // |P_(dx) - P(dx)| < 2^-51418 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64419 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.420 // Since we approximate421 // 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) - 1429 // = 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 methods441 // 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 DEBUGDEBUG450 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_part458 << " mid_lo: " << std::hexfloat << mid_lo << std::defaultfloat459 << "\n"460 << " p: " << std::hexfloat << p << std::defaultfloat << "\n"461 << " lo: " << std::hexfloat << lo << std::defaultfloat << "\n"462 << " upper: " << std::hexfloat << upper << std::defaultfloat463 << "\n"464 << " lower: " << std::hexfloat << lower << std::defaultfloat465 << "\n"466 << std::endl;467#endif468 469 if (LIBC_LIKELY(upper == lower)) {470 // to multiply by 2^hi, a fast way is to simply add hi to the exponent471 // 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-double478 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid, hi_part);479 480#ifdef LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS481 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#else486 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 precision498 Float128 r_f128 = expm1_f128(x, kd, idx1, idx2);499 500 return static_cast<double>(r_f128);501#endif // LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS502}503 504} // namespace LIBC_NAMESPACE_DECL505