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//===-- Single-precision e^x 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/expf.h"
#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
#include "src/__support/FPUtil/BasicOperations.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include <errno.h>
namespace __llvm_libc {
LLVM_LIBC_FUNCTION(float, expf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
uint32_t x_u = xbits.uintval();
uint32_t x_abs = x_u & 0x7fff'ffffU;
// Exceptional values
if (LIBC_UNLIKELY(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f
return 0x1.108a58p-66f - x * 0x1.0p-95f;
}
// When |x| >= 89, |x| < 2^-25, or x is nan
if (LIBC_UNLIKELY(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) {
// |x| < 2^-25
if (xbits.get_unbiased_exponent() <= 101) {
return 1.0f + x;
}
// When x < log(2^-150) or nan
if (xbits.uintval() >= 0xc2cf'f1b5U) {
// exp(-Inf) = 0
if (xbits.is_inf())
return 0.0f;
// exp(nan) = nan
if (xbits.is_nan())
return x;
if (fputil::get_round() == FE_UPWARD)
return static_cast<float>(FPBits(FPBits::MIN_SUBNORMAL));
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_UNDERFLOW);
return 0.0f;
}
// x >= 89 or nan
if (!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000)) {
// x is finite
if (xbits.uintval() < 0x7f80'0000U) {
int rounding = fputil::get_round();
if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_OVERFLOW);
}
// x is +inf or nan
return x + static_cast<float>(FPBits::inf());
}
}
// For -104 < x < 89, to compute exp(x), we perform the following range
// reduction: find hi, mid, lo such that:
// x = hi + mid + lo, in which
// hi is an integer,
// mid * 2^7 is an integer
// -2^(-8) <= lo < 2^-8.
// In particular,
// hi + mid = round(x * 2^7) * 2^(-7).
// Then,
// exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo).
// We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2
// respectively. exp(lo) is computed using a degree-4 minimax polynomial
// generated by Sollya.
// x_hi = (hi + mid) * 2^7 = round(x * 2^7).
float kf = fputil::nearest_integer(x * 0x1.0p7f);
// Subtract (hi + mid) from x to get lo.
double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x));
int x_hi = static_cast<int>(kf);
x_hi += 104 << 7;
// hi = x_hi >> 7
double exp_hi = EXP_M1[x_hi >> 7];
// mid * 2^7 = x_hi & 0x0000'007fU;
double exp_mid = EXP_M2[x_hi & 0x7f];
// Degree-4 minimax polynomial generated by Sollya with the following
// commands:
// > display = hexadecimal;
// > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]);
// > Q;
double exp_lo =
fputil::polyeval(xd, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,
0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);
return static_cast<float>(exp_hi * exp_mid * exp_lo);
}
} // namespace __llvm_libc
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