1//===-- Quad-precision atan2 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/atan2f128.h"
10#include "atan_utils.h"
11#include "src/__support/FPUtil/FPBits.h"
12#include "src/__support/FPUtil/dyadic_float.h"
13#include "src/__support/FPUtil/multiply_add.h"
14#include "src/__support/FPUtil/nearest_integer.h"
15#include "src/__support/integer_literals.h"
16#include "src/__support/macros/config.h"
17#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
18#include "src/__support/macros/properties/types.h"
19#include "src/__support/uint128.h"
20
21namespace LIBC_NAMESPACE_DECL {
22
23namespace {
24
25using Float128 = fputil::DyadicFloat<128>;
26
27static constexpr Float128 ZERO = {Sign::POS, 0, 0_u128};
28static constexpr Float128 MZERO = {Sign::NEG, 0, 0_u128};
29static constexpr Float128 PI = {Sign::POS, -126,
30 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
31static constexpr Float128 MPI = {Sign::NEG, -126,
32 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
33static constexpr Float128 PI_OVER_2 = {
34 Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
35static constexpr Float128 MPI_OVER_2 = {
36 Sign::NEG, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
37static constexpr Float128 PI_OVER_4 = {
38 Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
39static constexpr Float128 THREE_PI_OVER_4 = {
40 Sign::POS, -128, 0x96cbe3f9'990e91a7'9394c9e8'a0a5159d_u128};
41
42// Adjustment for constant term:
43// CONST_ADJ[x_sign][y_sign][recip]
44static constexpr Float128 CONST_ADJ[2][2][2] = {
45 {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
46 {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
47
48} // anonymous namespace
49
50// There are several range reduction steps we can take for atan2(y, x) as
51// follow:
52
53// * Range reduction 1: signness
54// atan2(y, x) will return a number between -PI and PI representing the angle
55// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
56// In particular, we have that:
57// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
58// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
59// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
60// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
61// Since atan function is odd, we can use the formula:
62// atan(-u) = -atan(u)
63// to adjust the above conditions a bit further:
64// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
65// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
66// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
67// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
68// Which can be simplified to:
69// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
70// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
71
72// * Range reduction 2: reciprocal
73// Now that the argument inside atan is positive, we can use the formula:
74// atan(1/x) = pi/2 - atan(x)
75// to make the argument inside atan <= 1 as follow:
76// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
77// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
78// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
79// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
80
81// * Range reduction 3: look up table.
82// After the previous two range reduction steps, we reduce the problem to
83// compute atan(u) with 0 <= u <= 1, or to be precise:
84// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
85// An accurate polynomial approximation for the whole [0, 1] input range will
86// require a very large degree. To make it more efficient, we reduce the input
87// range further by finding an integer idx such that:
88// | n/d - idx/64 | <= 1/128.
89// In particular,
90// idx := round(2^6 * n/d)
91// Then for the fast pass, we find a polynomial approximation for:
92// atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
93// For the accurate pass, we use the addition formula:
94// atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
95// = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
96// And for the fast pass, we use degree-13 minimax polynomial to compute the
97// RHS:
98// atan(u) ~ P(u) = u - c_3 * u^3 + c_5 * u^5 - c_7 * u^7 + c_9 *u^9 -
99// - c_11 * u^11 + c_13 * u^13
100// with absolute errors bounded by:
101// |atan(u) - P(u)| < 2^-121
102// and relative errors bounded by:
103// |(atan(u) - P(u)) / P(u)| < 2^-114.
104
105LLVM_LIBC_FUNCTION(float128, atan2f128, (float128 y, float128 x)) {
106 using FPBits = fputil::FPBits<float128>;
107 using Float128 = fputil::DyadicFloat<128>;
108
109 FPBits x_bits(x), y_bits(y);
110 bool x_sign = x_bits.sign().is_neg();
111 bool y_sign = y_bits.sign().is_neg();
112 x_bits = x_bits.abs();
113 y_bits = y_bits.abs();
114 UInt128 x_abs = x_bits.uintval();
115 UInt128 y_abs = y_bits.uintval();
116 bool recip = x_abs < y_abs;
117 UInt128 min_abs = recip ? x_abs : y_abs;
118 UInt128 max_abs = !recip ? x_abs : y_abs;
119 unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
120 unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
121
122 Float128 num(FPBits(min_abs).get_val());
123 Float128 den(FPBits(max_abs).get_val());
124
125 // Check for exceptional cases, whether inputs are 0, inf, nan, or close to
126 // overflow, or close to underflow.
127 if (LIBC_UNLIKELY(max_exp >= 0x7fffU || min_exp == 0U)) {
128 if (x_bits.is_nan() || y_bits.is_nan())
129 return FPBits::quiet_nan().get_val();
130 unsigned x_except = x == 0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
131 unsigned y_except = y == 0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);
132
133 // Exceptional cases:
134 // EXCEPT[y_except][x_except][x_is_neg]
135 // with x_except & y_except:
136 // 0: zero
137 // 1: finite, non-zero
138 // 2: infinity
139 constexpr Float128 EXCEPTS[3][3][2] = {
140 {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
141 {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
142 {{PI_OVER_2, PI_OVER_2},
143 {PI_OVER_2, PI_OVER_2},
144 {PI_OVER_4, THREE_PI_OVER_4}},
145 };
146
147 if ((x_except != 1) || (y_except != 1)) {
148 Float128 r = EXCEPTS[y_except][x_except][x_sign];
149 if (y_sign)
150 r.sign = r.sign.negate();
151 return static_cast<float128>(r);
152 }
153 }
154
155 bool final_sign = ((x_sign != y_sign) != recip);
156 Float128 const_term = CONST_ADJ[x_sign][y_sign][recip];
157 int exp_diff = den.exponent - num.exponent;
158 // We have the following bound for normalized n and d:
159 // 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
160 if (LIBC_UNLIKELY(exp_diff > FPBits::FRACTION_LEN + 2)) {
161 if (final_sign)
162 const_term.sign = const_term.sign.negate();
163 return static_cast<float128>(const_term);
164 }
165
166 // Take 24 leading bits of num and den to convert to float for fast division.
167 // We also multiply the numerator by 64 using integer addition directly to the
168 // exponent field.
169 float num_f =
170 cpp::bit_cast<float>(static_cast<uint32_t>(num.mantissa >> 104) +
171 (6U << fputil::FPBits<float>::FRACTION_LEN));
172 float den_f = cpp::bit_cast<float>(
173 static_cast<uint32_t>(den.mantissa >> 104) +
174 (static_cast<uint32_t>(exp_diff) << fputil::FPBits<float>::FRACTION_LEN));
175
176 float k = fputil::nearest_integer(num_f / den_f);
177 unsigned idx = static_cast<unsigned>(k);
178
179 // k_f128 = idx / 64
180 Float128 k_f128(Sign::POS, -6, Float128::MantissaType(idx));
181
182 // Range reduction:
183 // atan(n/d) - atan(k) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
184 // = atan((n - d * k/64)) / (d + n * k/64))
185 // num_f128 = n - d * k/64
186 Float128 num_f128 = fputil::multiply_add(den, -k_f128, num);
187 // den_f128 = d + n * k/64
188 Float128 den_f128 = fputil::multiply_add(num, k_f128, den);
189
190 // q = (n - d * k) / (d + n * k)
191 Float128 q = fputil::quick_mul(num_f128, fputil::approx_reciprocal(den_f128));
192 // p ~ atan(q)
193 Float128 p = atan_eval(q);
194
195 Float128 r =
196 fputil::quick_add(const_term, fputil::quick_add(ATAN_I_F128[idx], p));
197 if (final_sign)
198 r.sign = r.sign.negate();
199
200 return static_cast<float128>(r);
201}
202
203} // namespace LIBC_NAMESPACE_DECL
204

source code of libc/src/math/generic/atan2f128.cpp