1//===-- Half-precision acosh(x) 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-exception
6//
7//===----------------------------------------------------------------------===//
8
9#include "src/math/acoshf16.h"
10#include "explogxf.h"
11#include "hdr/errno_macros.h"
12#include "hdr/fenv_macros.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/PolyEval.h"
16#include "src/__support/FPUtil/cast.h"
17#include "src/__support/FPUtil/except_value_utils.h"
18#include "src/__support/FPUtil/multiply_add.h"
19#include "src/__support/FPUtil/sqrt.h"
20#include "src/__support/common.h"
21#include "src/__support/macros/config.h"
22#include "src/__support/macros/optimization.h"
23
24namespace LIBC_NAMESPACE_DECL {
25
26static constexpr size_t N_EXCEPTS = 2;
27static constexpr fputil::ExceptValues<float16, N_EXCEPTS> ACOSHF16_EXCEPTS{{
28 // (input, RZ output, RU offset, RD offset, RN offset)
29 // x = 0x1.6dcp+1, acoshf16(x) = 0x1.b6p+0 (RZ)
30 {0x41B7, 0x3ED8, 1, 0, 0},
31 // x = 0x1.39p+0, acoshf16(x) = 0x1.4f8p-1 (RZ)
32 {0x3CE4, 0x393E, 1, 0, 1},
33}};
34
35LLVM_LIBC_FUNCTION(float16, acoshf16, (float16 x)) {
36 using FPBits = fputil::FPBits<float16>;
37 FPBits xbits(x);
38 uint16_t x_u = xbits.uintval();
39
40 // Check for NaN input first.
41 if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) {
42 if (xbits.is_signaling_nan()) {
43 fputil::raise_except_if_required(FE_INVALID);
44 return FPBits::quiet_nan().get_val();
45 }
46 if (xbits.is_neg()) {
47 fputil::set_errno_if_required(EDOM);
48 fputil::raise_except_if_required(FE_INVALID);
49 return FPBits::quiet_nan().get_val();
50 }
51 return x;
52 }
53
54 // Domain error for inputs less than 1.0.
55 if (LIBC_UNLIKELY(x <= 1.0f)) {
56 if (x == 1.0f)
57 return FPBits::zero().get_val();
58 fputil::set_errno_if_required(EDOM);
59 fputil::raise_except_if_required(FE_INVALID);
60 return FPBits::quiet_nan().get_val();
61 }
62
63 if (auto r = ACOSHF16_EXCEPTS.lookup(xbits.uintval());
64 LIBC_UNLIKELY(r.has_value()))
65 return r.value();
66
67 float xf = x;
68 // High-precision polynomial approximation for inputs close to 1.0
69 // ([1, 1.25)).
70 //
71 // Brief derivation:
72 // 1. Expand acosh(1 + delta) using Taylor series around delta=0:
73 // acosh(1 + delta) ≈ sqrt(2 * delta) * [1 - delta/12 + 3*delta^2/160
74 // - 5*delta^3/896 + 35*delta^4/18432 + ...]
75 // 2. Truncate the series to fit accurately for delta in [0, 0.25].
76 // 3. Polynomial coefficients (from sollya) used here are:
77 // P(delta) ≈ 1 - 0x1.555556p-4 * delta + 0x1.333334p-6 * delta^2
78 // - 0x1.6db6dcp-8 * delta^3 + 0x1.f1c71cp-10 * delta^4
79 // 4. The Sollya commands used to generate these coefficients were:
80 // > display = hexadecimal;
81 // > round(1/12, SG, RN);
82 // > round(3/160, SG, RN);
83 // > round(5/896, SG, RN);
84 // > round(35/18432, SG, RN);
85 // With hexadecimal display mode enabled, the outputs were:
86 // 0x1.555556p-4
87 // 0x1.333334p-6
88 // 0x1.6db6dcp-8
89 // 0x1.f1c71cp-10
90 // 5. The maximum absolute error, estimated using:
91 // dirtyinfnorm(acosh(1 + x) - sqrt(2*x) * P(x), [0, 0.25])
92 // is:
93 // 0x1.d84281p-22
94 if (LIBC_UNLIKELY(x_u < 0x3D00U)) {
95 float delta = xf - 1.0f;
96 float sqrt_2_delta = fputil::sqrt<float>(2.0 * delta);
97 float pe = fputil::polyeval(delta, 0x1p+0f, -0x1.555556p-4f, 0x1.333334p-6f,
98 -0x1.6db6dcp-8f, 0x1.f1c71cp-10f);
99 float approx = sqrt_2_delta * pe;
100 return fputil::cast<float16>(approx);
101 }
102
103 // acosh(x) = log(x + sqrt(x^2 - 1))
104 float sqrt_term = fputil::sqrt<float>(fputil::multiply_add(xf, xf, -1.0f));
105 float result = static_cast<float>(log_eval(xf + sqrt_term));
106
107 return fputil::cast<float16>(result);
108}
109
110} // namespace LIBC_NAMESPACE_DECL
111

source code of libc/src/math/generic/acoshf16.cpp