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

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