1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
|
//===-- Implementation header for atanf -------------------------*- C++ -*-===//
//
// 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
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ATANF_H
#define LLVM_LIBC_SRC___SUPPORT_MATH_ATANF_H
#include "inv_trigf_utils.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/nearest_integer.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
// We use float-float implementation to reduce size.
#include "atanf_float.h"
#else
namespace LIBC_NAMESPACE_DECL {
namespace math {
LIBC_INLINE static constexpr float atanf(float x) {
using namespace inv_trigf_utils_internal;
using FPBits = typename fputil::FPBits<float>;
constexpr double FINAL_SIGN[2] = {1.0, -1.0};
constexpr double SIGNED_PI_OVER_2[2] = {0x1.921fb54442d18p0,
-0x1.921fb54442d18p0};
FPBits x_bits(x);
Sign sign = x_bits.sign();
x_bits.set_sign(Sign::POS);
uint32_t x_abs = x_bits.uintval();
// x is inf or nan, |x| < 2^-4 or |x|= > 16.
if (LIBC_UNLIKELY(x_abs <= 0x3d80'0000U || x_abs >= 0x4180'0000U)) {
double x_d = static_cast<double>(x);
double const_term = 0.0;
if (LIBC_UNLIKELY(x_abs >= 0x4180'0000)) {
// atan(+-Inf) = +-pi/2.
if (x_bits.is_inf()) {
volatile double sign_pi_over_2 = SIGNED_PI_OVER_2[sign.is_neg()];
return static_cast<float>(sign_pi_over_2);
}
if (x_bits.is_nan())
return x;
// x >= 16
x_d = -1.0 / x_d;
const_term = SIGNED_PI_OVER_2[sign.is_neg()];
}
// 0 <= x < 1/16;
if (LIBC_UNLIKELY(x_bits.is_zero()))
return x;
// x <= 2^-12;
if (LIBC_UNLIKELY(x_abs < 0x3980'0000)) {
#if defined(LIBC_TARGET_CPU_HAS_FMA_FLOAT)
return fputil::multiply_add(x, -0x1.0p-25f, x);
#else
double x_d = static_cast<double>(x);
return static_cast<float>(fputil::multiply_add(x_d, -0x1.0p-25, x_d));
#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
}
// Use Taylor polynomial:
// atan(x) ~ x * (1 - x^2 / 3 + x^4 / 5 - x^6 / 7 + x^8 / 9 - x^10 / 11).
constexpr double ATAN_TAYLOR[6] = {
0x1.0000000000000p+0, -0x1.5555555555555p-2, 0x1.999999999999ap-3,
-0x1.2492492492492p-3, 0x1.c71c71c71c71cp-4, -0x1.745d1745d1746p-4,
};
double x2 = x_d * x_d;
double x4 = x2 * x2;
double c0 = fputil::multiply_add(x2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]);
double c1 = fputil::multiply_add(x2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]);
double c2 = fputil::multiply_add(x2, ATAN_TAYLOR[5], ATAN_TAYLOR[4]);
double p = fputil::polyeval(x4, c0, c1, c2);
double r = fputil::multiply_add(x_d, p, const_term);
return static_cast<float>(r);
}
// Range reduction steps:
// 1) atan(x) = sign(x) * atan(|x|)
// 2) If |x| > 1, atan(|x|) = pi/2 - atan(1/|x|)
// 3) For 1/16 < x <= 1, we find k such that: |x - k/16| <= 1/32.
// 4) Then we use polynomial approximation:
// atan(x) ~ atan((k/16) + (x - (k/16)) * Q(x - k/16)
// = P(x - k/16)
double x_d = 0, const_term = 0, final_sign = 0;
int idx = 0;
if (x_abs > 0x3f80'0000U) {
// |x| > 1, we need to invert x, so we will perform range reduction in
// double precision.
x_d = 1.0 / static_cast<double>(x_bits.get_val());
double k_d = fputil::nearest_integer(x_d * 0x1.0p4);
x_d = fputil::multiply_add(k_d, -0x1.0p-4, x_d);
idx = static_cast<int>(k_d);
final_sign = FINAL_SIGN[sign.is_pos()];
// Adjust constant term of the polynomial by +- pi/2.
const_term = fputil::multiply_add(final_sign, ATAN_COEFFS[idx][0],
SIGNED_PI_OVER_2[sign.is_neg()]);
} else {
// Exceptional value:
if (LIBC_UNLIKELY(x_abs == 0x3d8d'6b23U)) { // |x| = 0x1.1ad646p-4
return sign.is_pos() ? fputil::round_result_slightly_down(0x1.1a6386p-4f)
: fputil::round_result_slightly_up(-0x1.1a6386p-4f);
}
// Perform range reduction in single precision.
float x_f = x_bits.get_val();
float k_f = fputil::nearest_integer(x_f * 0x1.0p4f);
x_f = fputil::multiply_add(k_f, -0x1.0p-4f, x_f);
x_d = static_cast<double>(x_f);
idx = static_cast<int>(k_f);
final_sign = FINAL_SIGN[sign.is_neg()];
const_term = final_sign * ATAN_COEFFS[idx][0];
}
double p = atan_eval(x_d, idx);
double r = fputil::multiply_add(final_sign * x_d, p, const_term);
return static_cast<float>(r);
}
} // namespace math
} // namespace LIBC_NAMESPACE_DECL
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATANF_H
|
