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//===-- Half-precision acosh(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/acoshf16.h"
#include "explogxf.h"
#include "hdr/errno_macros.h"
#include "hdr/fenv_macros.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/cast.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/sqrt.h"
#include "src/__support/common.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h"
namespace LIBC_NAMESPACE_DECL {
static constexpr size_t N_EXCEPTS = 2;
static constexpr fputil::ExceptValues<float16, N_EXCEPTS> ACOSHF16_EXCEPTS{{
// (input, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.6dcp+1, acoshf16(x) = 0x1.b6p+0 (RZ)
{0x41B7, 0x3ED8, 1, 0, 0},
// x = 0x1.39p+0, acoshf16(x) = 0x1.4f8p-1 (RZ)
{0x3CE4, 0x393E, 1, 0, 1},
}};
LLVM_LIBC_FUNCTION(float16, acoshf16, (float16 x)) {
using FPBits = fputil::FPBits<float16>;
FPBits xbits(x);
uint16_t x_u = xbits.uintval();
// Check for NaN input first.
if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) {
if (xbits.is_signaling_nan()) {
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
if (xbits.is_neg()) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
return x;
}
// Domain error for inputs less than 1.0.
if (LIBC_UNLIKELY(x <= 1.0f)) {
if (x == 1.0f)
return FPBits::zero().get_val();
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
if (auto r = ACOSHF16_EXCEPTS.lookup(xbits.uintval());
LIBC_UNLIKELY(r.has_value()))
return r.value();
float xf = x;
// High-precision polynomial approximation for inputs close to 1.0
// ([1, 1.25)).
//
// Brief derivation:
// 1. Expand acosh(1 + delta) using Taylor series around delta=0:
// acosh(1 + delta) ≈ sqrt(2 * delta) * [1 - delta/12 + 3*delta^2/160
// - 5*delta^3/896 + 35*delta^4/18432 + ...]
// 2. Truncate the series to fit accurately for delta in [0, 0.25].
// 3. Polynomial coefficients (from sollya) used here are:
// P(delta) ≈ 1 - 0x1.555556p-4 * delta + 0x1.333334p-6 * delta^2
// - 0x1.6db6dcp-8 * delta^3 + 0x1.f1c71cp-10 * delta^4
// 4. The Sollya commands used to generate these coefficients were:
// > display = hexadecimal;
// > round(1/12, SG, RN);
// > round(3/160, SG, RN);
// > round(5/896, SG, RN);
// > round(35/18432, SG, RN);
// With hexadecimal display mode enabled, the outputs were:
// 0x1.555556p-4
// 0x1.333334p-6
// 0x1.6db6dcp-8
// 0x1.f1c71cp-10
// 5. The maximum absolute error, estimated using:
// dirtyinfnorm(acosh(1 + x) - sqrt(2*x) * P(x), [0, 0.25])
// is:
// 0x1.d84281p-22
if (LIBC_UNLIKELY(x_u < 0x3D00U)) {
float delta = xf - 1.0f;
float sqrt_2_delta = fputil::sqrt<float>(2.0 * delta);
float pe = fputil::polyeval(delta, 0x1p+0f, -0x1.555556p-4f, 0x1.333334p-6f,
-0x1.6db6dcp-8f, 0x1.f1c71cp-10f);
float approx = sqrt_2_delta * pe;
return fputil::cast<float16>(approx);
}
// acosh(x) = log(x + sqrt(x^2 - 1))
float sqrt_term = fputil::sqrt<float>(fputil::multiply_add(xf, xf, -1.0f));
float result = static_cast<float>(log_eval(xf + sqrt_term));
return fputil::cast<float16>(result);
}
} // namespace LIBC_NAMESPACE_DECL
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