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
|
//===-- Single-precision atan 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/atanf.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/FPUtil/rounding_mode.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/math/inv_trigf_utils.h"
namespace LIBC_NAMESPACE_DECL {
LLVM_LIBC_FUNCTION(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, const_term, final_sign;
int idx;
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 LIBC_NAMESPACE_DECL
|
