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
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
/*
* // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
use crate::acospi::INV_PI_DD;
use crate::atan2::{ATAN_I, atan_eval};
use crate::common::f_fmla;
use crate::dekker::Dekker;
/// Computes atan(x)/PI
///
/// Max found ULP 0.5
#[inline]
pub fn f_atan2pi(y: f64, x: f64) -> f64 {
const IS_NEG: [f64; 2] = [1.0, -1.0];
const ZERO: Dekker = Dekker::new(0.0, 0.0);
const MZERO: Dekker = Dekker::new(-0.0, -0.0);
const PI: Dekker = Dekker::new(
f64::from_bits(0x3ca1a62633145c07),
f64::from_bits(0x400921fb54442d18),
);
const MPI: Dekker = Dekker::new(
f64::from_bits(0xbca1a62633145c07),
f64::from_bits(0xc00921fb54442d18),
);
const PI_OVER_2: Dekker = Dekker::new(
f64::from_bits(0x3c91a62633145c07),
f64::from_bits(0x3ff921fb54442d18),
);
const MPI_OVER_2: Dekker = Dekker::new(
f64::from_bits(0xbc91a62633145c07),
f64::from_bits(0xbff921fb54442d18),
);
const PI_OVER_4: Dekker = Dekker::new(
f64::from_bits(0x3c81a62633145c07),
f64::from_bits(0x3fe921fb54442d18),
);
const THREE_PI_OVER_4: Dekker = Dekker::new(
f64::from_bits(0x3c9a79394c9e8a0a),
f64::from_bits(0x4002d97c7f3321d2),
);
// Adjustment for constant term:
// CONST_ADJ[x_sign][y_sign][recip]
const CONST_ADJ: [[[Dekker; 2]; 2]; 2] = [
[[ZERO, MPI_OVER_2], [MZERO, MPI_OVER_2]],
[[MPI, PI_OVER_2], [MPI, PI_OVER_2]],
];
let x_sign = if x.is_sign_negative() { 1 } else { 0 };
let y_sign = if y.is_sign_negative() { 1 } else { 0 };
let x_bits = x.to_bits() & 0x7fff_ffff_ffff_ffff;
let y_bits = y.to_bits() & 0x7fff_ffff_ffff_ffff;
let x_abs = x_bits;
let y_abs = y_bits;
let recip = x_abs < y_abs;
let mut min_abs = if recip { x_abs } else { y_abs };
let mut max_abs = if !recip { x_abs } else { y_abs };
let mut min_exp = min_abs.wrapping_shr(52);
let mut max_exp = max_abs.wrapping_shr(52);
let mut num = f64::from_bits(min_abs);
let mut den = f64::from_bits(max_abs);
// Check for exceptional cases, whether inputs are 0, inf, nan, or close to
// overflow, or close to underflow.
if max_exp > 0x7ffu64 - 128u64 || min_exp < 128u64 {
if x.is_nan() || y.is_nan() {
return f64::NAN;
}
let x_except = if x == 0.0 {
0
} else if x.is_infinite() {
2
} else {
1
};
let y_except = if y == 0.0 {
0
} else if y.is_infinite() {
2
} else {
1
};
// Exceptional cases:
// EXCEPT[y_except][x_except][x_is_neg]
// with x_except & y_except:
// 0: zero
// 1: finite, non-zero
// 2: infinity
const EXCEPTS: [[[Dekker; 2]; 3]; 3] = [
[[ZERO, PI], [ZERO, PI], [ZERO, PI]],
[[PI_OVER_2, PI_OVER_2], [ZERO, ZERO], [ZERO, PI]],
[
[PI_OVER_2, PI_OVER_2],
[PI_OVER_2, PI_OVER_2],
[PI_OVER_4, THREE_PI_OVER_4],
],
];
if (x_except != 1) || (y_except != 1) {
let mut r = EXCEPTS[y_except][x_except][x_sign];
r = Dekker::quick_mult(r, INV_PI_DD);
return f_fmla(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);
}
let scale_up = min_exp < 128u64;
let scale_down = max_exp > 0x7ffu64 - 128u64;
// At least one input is denormal, multiply both numerator and denominator
// by some large enough power of 2 to normalize denormal inputs.
if scale_up {
num *= f64::from_bits(0x43f0000000000000);
if !scale_down {
den *= f64::from_bits(0x43f0000000000000)
}
} else if scale_down {
den *= f64::from_bits(0x3bf0000000000000);
if !scale_up {
num *= f64::from_bits(0x3bf0000000000000);
}
}
min_abs = num.to_bits();
max_abs = den.to_bits();
min_exp = min_abs.wrapping_shr(52);
max_exp = max_abs.wrapping_shr(52);
}
let final_sign = IS_NEG[if (x_sign != y_sign) != recip { 1 } else { 0 }];
let const_term = CONST_ADJ[x_sign][y_sign][if recip { 1 } else { 0 }];
let exp_diff = max_exp - min_exp;
// We have the following bound for normalized n and d:
// 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
if exp_diff > 54 {
let r = f_fmla(
final_sign,
const_term.hi,
final_sign * (const_term.lo + num / den),
);
let p = Dekker::f64_mult(r, INV_PI_DD);
return p.to_f64();
}
let mut k = (64.0 * num / den).round();
let idx = k as u64;
// k = idx / 64
k *= f64::from_bits(0x3f90000000000000);
// Range reduction:
// atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
// = atan((n - d * k/64)) / (d + n * k/64))
let num_k = Dekker::from_exact_mult(num, k);
let den_k = Dekker::from_exact_mult(den, k);
// num_dd = n - d * k
let num_dd = Dekker::from_exact_add(num - den_k.hi, -den_k.lo);
// den_dd = d + n * k
let mut den_dd = Dekker::from_exact_add(den, num_k.hi);
den_dd.lo += num_k.lo;
// q = (n - d * k) / (d + n * k)
let q = Dekker::div(num_dd, den_dd);
// p ~ atan(q)
let p = atan_eval(q);
let vl = ATAN_I[idx as usize];
let vlo = Dekker::new(f64::from_bits(vl.0), f64::from_bits(vl.1));
let mut r = Dekker::add(const_term, Dekker::add(vlo, p));
r = Dekker::quick_mult(r, INV_PI_DD);
r.hi *= final_sign;
r.lo *= final_sign;
r.hi + r.lo
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_atan2pi() {
assert_eq!(f_atan2pi(-5., 2.), -0.3788810584091566);
assert_eq!(f_atan2pi(2., -5.), 0.8788810584091566);
}
}