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//===-- Implementation of exp2m1f function --------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/exp2m1f.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h"
#include "src/__support/macros/properties/cpu_features.h"
#include "src/errno/libc_errno.h"
#include "explogxf.h"
namespace LIBC_NAMESPACE {
static constexpr size_t N_EXCEPTS_LO = 8;
static constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO =
{{
// (input, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ)
{0x2d9b'6e47U, 0x2d57'7909U, 1U, 0U, 0U},
// x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ)
{0x3611'249bU, 0x35c9'3607U, 1U, 0U, 1U},
// x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ)
{0x35e8'b690U, 0x35a1'4df6U, 1U, 0U, 1U},
// x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ)
{0x388b'ca4fU, 0x3841'cb80U, 1U, 0U, 1U},
// x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ)
{0xacce'1f0fU, 0xac8e'df56U, 0U, 1U, 0U},
// x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ)
{0xafa6'c4daU, 0xaf67'30dbU, 0U, 1U, 1U},
// x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ)
{0xbad3'7562U, 0xba92'7d6dU, 0U, 1U, 1U},
// x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ)
{0xbcf3'a937U, 0xbca7'29efU, 0U, 1U, 1U},
}};
static constexpr size_t N_EXCEPTS_HI = 3;
static constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI =
{{
// (input, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ)
{0x3f0b'54b9U, 0x3eea'a2d9U, 1U, 0U, 0U},
// x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ)
{0xbd4f'8956U, 0xbd0d'5b46U, 0U, 1U, 0U},
// x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ)
{0xbd6f'3dceU, 0xbd22'847aU, 0U, 1U, 1U},
}};
LLVM_LIBC_FUNCTION(float, exp2m1f, (float x)) {
using FPBits = fputil::FPBits<float>;
FPBits xbits(x);
uint32_t x_u = xbits.uintval();
uint32_t x_abs = x_u & 0x7fff'ffffU;
// When |x| >= 128, or x is nan, or |x| <= 2^-5
if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
// |x| <= 2^-5
if (x_abs <= 0x3d00'0000U) {
if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_u); LIBC_UNLIKELY(r.has_value()))
return r.value();
// Minimax polynomial generated by Sollya with:
// > display = hexadecimal;
// > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
constexpr double COEFFS[] = {
0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
double xd = x;
double xsq = xd * xd;
double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);
double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);
double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);
double p = fputil::polyeval(xsq, c0, c1, c2);
return static_cast<float>(p * xd);
}
// x >= 128, or x is nan
if (xbits.is_pos()) {
if (xbits.is_finite()) {
int rounding = fputil::quick_get_round();
if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
return FPBits::max_normal().get_val();
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_OVERFLOW);
}
// x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan
return x + FPBits::inf().get_val();
}
}
if (LIBC_UNLIKELY(x <= -25.0f)) {
// 2^(-inf) - 1 = -1
if (xbits.is_inf())
return -1.0f;
// 2^nan - 1 = nan
if (xbits.is_nan())
return x;
int rounding = fputil::quick_get_round();
if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO)
return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_UNDERFLOW);
return -1.0f;
}
if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_u); LIBC_UNLIKELY(r.has_value()))
return r.value();
// For -25 < x < 128, to compute 2^x, we perform the following range
// reduction: find hi, mid, lo such that:
// x = hi + mid + lo, in which:
// hi is an integer,
// 0 <= mid * 2^5 < 32 is an integer,
// -2^(-6) <= lo <= 2^(-6).
// In particular,
// hi + mid = round(x * 2^5) * 2^(-5).
// Then,
// 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
// 2^mid is stored in the lookup table of 32 elements.
// 2^lo is computed using a degree-4 minimax polynomial generated by Sollya.
// We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid.
// kf = (hi + mid) * 2^5 = round(x * 2^5)
float kf;
int k;
#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
kf = fputil::nearest_integer(x * 32.0f);
k = static_cast<int>(kf);
#else
constexpr float HALF[2] = {0.5f, -0.5f};
k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));
kf = static_cast<float>(k);
#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
// lo = x - (hi + mid) = x - kf * 2^(-5)
double lo = fputil::multiply_add(-0x1.0p-5f, kf, x);
// hi = floor(kf * 2^(-4))
// exp2_hi = shift hi to the exponent field of double precision.
int64_t exp2_hi =
static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
<< fputil::FPBits<double>::FRACTION_LEN);
// mh = 2^hi * 2^mid
// mh_bits = bit field of mh
int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi;
double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val();
// Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with:
// > display = hexadecimal;
// > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]);
constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
0x1.5d88091198529p-10};
double lo_sq = lo * lo;
double c1 = fputil::multiply_add(lo, COEFFS[0], 1.0);
double c2 = fputil::multiply_add(lo, COEFFS[2], COEFFS[1]);
double c3 = fputil::multiply_add(lo, COEFFS[4], COEFFS[3]);
double exp2_lo = fputil::polyeval(lo_sq, c1, c2, c3);
// 2^x - 1 = 2^(hi + mid + lo) - 1
// = 2^(hi + mid) * 2^lo - 1
// ~ mh * (1 + lo * P(lo)) - 1
// = mh * exp2_lo - 1
return static_cast<float>(fputil::multiply_add(exp2_lo, mh, -1.0));
}
} // namespace LIBC_NAMESPACE
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