424 lines
16 KiB
C++
424 lines
16 KiB
C++
//===-- Double-precision e^x function -------------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "src/math/exp.h"
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#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
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#include "explogxf.h" // ziv_test_denorm.
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#include "src/__support/CPP/bit.h"
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#include "src/__support/CPP/optional.h"
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#include "src/__support/FPUtil/FEnvImpl.h"
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#include "src/__support/FPUtil/FPBits.h"
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#include "src/__support/FPUtil/PolyEval.h"
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#include "src/__support/FPUtil/double_double.h"
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#include "src/__support/FPUtil/dyadic_float.h"
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#include "src/__support/FPUtil/multiply_add.h"
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#include "src/__support/FPUtil/nearest_integer.h"
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#include "src/__support/FPUtil/rounding_mode.h"
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#include "src/__support/FPUtil/triple_double.h"
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#include "src/__support/common.h"
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#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
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#include <errno.h>
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namespace LIBC_NAMESPACE {
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using fputil::DoubleDouble;
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using fputil::TripleDouble;
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using Float128 = typename fputil::DyadicFloat<128>;
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using Sign = fputil::Sign;
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// log2(e)
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constexpr double LOG2_E = 0x1.71547652b82fep+0;
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// Error bounds:
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// Errors when using double precision.
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constexpr double ERR_D = 0x1.8p-63;
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// Errors when using double-double precision.
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constexpr double ERR_DD = 0x1.0p-99;
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// -2^-12 * log(2)
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// > a = -2^-12 * log(2);
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// > b = round(a, 30, RN);
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// > c = round(a - b, 30, RN);
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// > d = round(a - b - c, D, RN);
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// Errors < 1.5 * 2^-133
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constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
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constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
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constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
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constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
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namespace {
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// Polynomial approximations with double precision:
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// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
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// For |dx| < 2^-13 + 2^-30:
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// | output - expm1(dx) / dx | < 2^-51.
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LIBC_INLINE double poly_approx_d(double dx) {
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// dx^2
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double dx2 = dx * dx;
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// c0 = 1 + dx / 2
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double c0 = fputil::multiply_add(dx, 0.5, 1.0);
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// c1 = 1/6 + dx / 24
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double c1 =
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fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3);
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// p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
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double p = fputil::multiply_add(dx2, c1, c0);
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return p;
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}
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// Polynomial approximation with double-double precision:
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// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
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// For |dx| < 2^-13 + 2^-30:
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// | output - exp(dx) | < 2^-101
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DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
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// Taylor polynomial.
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constexpr DoubleDouble COEFFS[] = {
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{0, 0x1p0}, // 1
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{0, 0x1p0}, // 1
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{0, 0x1p-1}, // 1/2
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{0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6
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{0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24
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{0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120
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{-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720
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};
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DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
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COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
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return p;
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}
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// Polynomial approximation with 128-bit precision:
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// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040
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// For |dx| < 2^-13 + 2^-30:
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// | output - exp(dx) | < 2^-126.
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Float128 poly_approx_f128(const Float128 &dx) {
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using MType = typename Float128::MantissaType;
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constexpr Float128 COEFFS_128[]{
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{Sign::POS, -127, MType({0, 0x8000000000000000})}, // 1.0
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{Sign::POS, -127, MType({0, 0x8000000000000000})}, // 1.0
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{Sign::POS, -128, MType({0, 0x8000000000000000})}, // 0.5
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{Sign::POS, -130, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/6
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{Sign::POS, -132,
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MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/24
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{Sign::POS, -134,
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MType({0x8888888888888889, 0x8888888888888888})}, // 1/120
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{Sign::POS, -137,
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MType({0x60b60b60b60b60b6, 0xb60b60b60b60b60b})}, // 1/720
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{Sign::POS, -140,
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MType({0x00d00d00d00d00d0, 0xd00d00d00d00d00d})}, // 1/5040
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};
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Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
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COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
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COEFFS_128[6], COEFFS_128[7]);
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return p;
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}
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// Compute exp(x) using 128-bit precision.
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// TODO(lntue): investigate triple-double precision implementation for this
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// step.
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Float128 exp_f128(double x, double kd, int idx1, int idx2) {
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// Recalculate dx:
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double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
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double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
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double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133
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Float128 dx = fputil::quick_add(
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Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
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// TODO: Skip recalculating exp_mid1 and exp_mid2.
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Float128 exp_mid1 =
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fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
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fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
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Float128(EXP2_MID1[idx1].lo)));
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Float128 exp_mid2 =
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fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
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fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
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Float128(EXP2_MID2[idx2].lo)));
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Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
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Float128 p = poly_approx_f128(dx);
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Float128 r = fputil::quick_mul(exp_mid, p);
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r.exponent += static_cast<int>(kd) >> 12;
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return r;
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}
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// Compute exp(x) with double-double precision.
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DoubleDouble exp_double_double(double x, double kd,
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const DoubleDouble &exp_mid) {
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// Recalculate dx:
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// dx = x - k * 2^-12 * log(2)
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double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
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double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
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double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130
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DoubleDouble dx = fputil::exact_add(t1, t2);
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dx.lo += t3;
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// Degree-6 Taylor polynomial approximation in double-double precision.
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// | p - exp(x) | < 2^-100.
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DoubleDouble p = poly_approx_dd(dx);
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// Error bounds: 2^-99.
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DoubleDouble r = fputil::quick_mult(exp_mid, p);
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return r;
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}
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// Check for exceptional cases when
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// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
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double set_exceptional(double x) {
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using FPBits = typename fputil::FPBits<double>;
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FPBits xbits(x);
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uint64_t x_u = xbits.uintval();
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uint64_t x_abs = xbits.abs().uintval();
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// |x| <= 2^-53
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if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
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// exp(x) ~ 1 + x
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return 1 + x;
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}
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// x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
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// x <= log(2^-1075) or -inf/nan
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if (x_u >= 0xc087'4910'd52d'3052ULL) {
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// exp(-Inf) = 0
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if (xbits.is_inf())
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return 0.0;
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// exp(nan) = nan
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if (xbits.is_nan())
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return x;
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if (fputil::quick_get_round() == FE_UPWARD)
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return FPBits::min_denormal();
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fputil::set_errno_if_required(ERANGE);
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fputil::raise_except_if_required(FE_UNDERFLOW);
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return 0.0;
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}
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// x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan
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// x is finite
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if (x_u < 0x7ff0'0000'0000'0000ULL) {
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int rounding = fputil::quick_get_round();
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if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
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return FPBits::max_normal();
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fputil::set_errno_if_required(ERANGE);
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fputil::raise_except_if_required(FE_OVERFLOW);
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}
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// x is +inf or nan
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return x + static_cast<double>(FPBits::inf());
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}
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} // namespace
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LLVM_LIBC_FUNCTION(double, exp, (double x)) {
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using FPBits = typename fputil::FPBits<double>;
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FPBits xbits(x);
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uint64_t x_u = xbits.uintval();
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// Upper bound: max normal number = 2^1023 * (2 - 2^-52)
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// > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
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// > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
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// > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
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// > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
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// Lower bound: min denormal number / 2 = 2^-1075
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// > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
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// Another lower bound: min normal number = 2^-1022
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// > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
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// x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
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if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
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(x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
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x_u < 0x3ca0000000000000)) {
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return set_exceptional(x);
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}
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// Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
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// Range reduction:
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// Let x = log(2) * (hi + mid1 + mid2) + lo
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// in which:
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// hi is an integer
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// mid1 * 2^6 is an integer
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// mid2 * 2^12 is an integer
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// then:
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// exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo).
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// With this formula:
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// - multiplying by 2^hi is exact and cheap, simply by adding the exponent
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// field.
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// - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
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// - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
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//
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// They can be defined by:
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// hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
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// If we store L2E = round(log2(e), D, RN), then:
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// log2(e) - L2E ~ 1.5 * 2^(-56)
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// So the errors when computing in double precision is:
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// | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
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// <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
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// + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
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// <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN
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// 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
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// So if:
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// hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
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// in double precision, the reduced argument:
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// lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
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// |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
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// < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
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// < 2^-13 + 2^-41
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//
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// The following trick computes the round(x * L2E) more efficiently
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// than using the rounding instructions, with the tradeoff for less accuracy,
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// and hence a slightly larger range for the reduced argument `lo`.
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//
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// To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
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// |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
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// So we can fit the rounded result round(x * 2^12 * L2E) in int32_t.
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// Thus, the goal is to be able to use an additional addition and fixed width
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// shift to get an int32_t representing round(x * 2^12 * L2E).
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//
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// Assuming int32_t using 2-complement representation, since the mantissa part
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// of a double precision is unsigned with the leading bit hidden, if we add an
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// extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
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// part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
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// considered as a proper 2-complement representations of x*2^12*L2E.
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//
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// One small problem with this approach is that the sum (x*2^12*L2E + C) in
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// double precision is rounded to the least significant bit of the dorminant
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// factor C. In order to minimize the rounding errors from this addition, we
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// want to minimize e1. Another constraint that we want is that after
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// shifting the mantissa so that the least significant bit of int32_t
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// corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
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// any adjustment. So combining these 2 requirements, we can choose
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// C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
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// after right shifting the mantissa, the resulting int32_t has correct sign.
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// With this choice of C, the number of mantissa bits we need to shift to the
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// right is: 52 - 33 = 19.
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//
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// Moreover, since the integer right shifts are equivalent to rounding down,
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// we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
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// +infinity. So in particular, we can compute:
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// hmm = x * 2^12 * L2E + C,
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// where C = 2^33 + 2^32 + 2^-1, then if
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// k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
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// the reduced argument:
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// lo = x - log(2) * 2^-12 * k is bounded by:
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// |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
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// = 2^-13 + 2^-31 + 2^-41.
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//
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// Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the
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// exponent 2^12 is not needed. So we can simply define
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// C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
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// k = int32_t(lower 51 bits of double(x * L2E + C) >> 19).
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// Rounding errors <= 2^-31 + 2^-41.
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double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21);
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int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
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double kd = static_cast<double>(k);
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uint32_t idx1 = (k >> 6) & 0x3f;
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uint32_t idx2 = k & 0x3f;
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int hi = k >> 12;
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bool denorm = (hi <= -1022);
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DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
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DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
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DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
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// |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
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// = 2^11 * 2^-13 * 2^-52
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// = 2^-54.
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// |dx| < 2^-13 + 2^-30.
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double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
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double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h);
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// We use the degree-4 Taylor polynomial to approximate exp(lo):
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// exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
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// So that the errors are bounded by:
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// |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
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// Let P_ be an evaluation of P where all intermediate computations are in
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// double precision. Using either Horner's or Estrin's schemes, the evaluated
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// errors can be bounded by:
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// |P_(dx) - P(dx)| < 2^-51
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// => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
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// => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.
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// Since we approximate
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// 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
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// We use the expression:
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// (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
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// ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
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// with errors bounded by 1.5 * 2^-63.
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double mid_lo = dx * exp_mid.hi;
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// Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
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double p = poly_approx_d(dx);
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double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
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if (LIBC_UNLIKELY(denorm)) {
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if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
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LIBC_LIKELY(r.has_value()))
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return r.value();
|
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} else {
|
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double upper = exp_mid.hi + (lo + ERR_D);
|
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double lower = exp_mid.hi + (lo - ERR_D);
|
|
|
|
if (LIBC_LIKELY(upper == lower)) {
|
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// to multiply by 2^hi, a fast way is to simply add hi to the exponent
|
|
// field.
|
|
int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
|
|
double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
|
|
return r;
|
|
}
|
|
}
|
|
|
|
// Use double-double
|
|
DoubleDouble r_dd = exp_double_double(x, kd, exp_mid);
|
|
|
|
if (LIBC_UNLIKELY(denorm)) {
|
|
if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
|
|
LIBC_LIKELY(r.has_value()))
|
|
return r.value();
|
|
} else {
|
|
double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
|
|
double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
|
|
|
|
if (LIBC_LIKELY(upper_dd == lower_dd)) {
|
|
int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
|
|
double r =
|
|
cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
|
|
return r;
|
|
}
|
|
}
|
|
|
|
// Use 128-bit precision
|
|
Float128 r_f128 = exp_f128(x, kd, idx1, idx2);
|
|
|
|
return static_cast<double>(r_f128);
|
|
}
|
|
|
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} // namespace LIBC_NAMESPACE
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