1 /* origin: FreeBSD /usr/src/lib/msun/src/s_atan.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12 /* atan(x)
13 * Method
14 * 1. Reduce x to positive by atan(x) = -atan(-x).
15 * 2. According to the integer k=4t+0.25 chopped, t=x, the argument
16 * is further reduced to one of the following intervals and the
17 * arctangent of t is evaluated by the corresponding formula:
18 *
19 * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
20 * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
21 * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
22 * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
23 * [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
24 *
25 * Constants:
26 * The hexadecimal values are the intended ones for the following
27 * constants. The decimal values may be used, provided that the
28 * compiler will convert from decimal to binary accurately enough
29 * to produce the hexadecimal values shown.
30 */
31
32 use core::f64;
33
34 use super::fabs;
35
36 const ATANHI: [f64; 4] = [
37 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
38 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
39 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
40 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
41 ];
42
43 const ATANLO: [f64; 4] = [
44 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
45 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
46 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
47 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
48 ];
49
50 const AT: [f64; 11] = [
51 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
52 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
53 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
54 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
55 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
56 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
57 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
58 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
59 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
60 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
61 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
62 ];
63
64 /// Arctangent (f64)
65 ///
66 /// Computes the inverse tangent (arc tangent) of the input value.
67 /// Returns a value in radians, in the range of -pi/2 to pi/2.
68 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
atan(x: f64) -> f6469 pub fn atan(x: f64) -> f64 {
70 let mut x = x;
71 let mut ix = (x.to_bits() >> 32) as u32;
72 let sign = ix >> 31;
73 ix &= 0x7fff_ffff;
74 if ix >= 0x4410_0000 {
75 if x.is_nan() {
76 return x;
77 }
78
79 let z = ATANHI[3] + f64::from_bits(0x0380_0000); // 0x1p-120f
80 return if sign != 0 { -z } else { z };
81 }
82
83 let id = if ix < 0x3fdc_0000 {
84 /* |x| < 0.4375 */
85 if ix < 0x3e40_0000 {
86 /* |x| < 2^-27 */
87 if ix < 0x0010_0000 {
88 /* raise underflow for subnormal x */
89 force_eval!(x as f32);
90 }
91
92 return x;
93 }
94
95 -1
96 } else {
97 x = fabs(x);
98 if ix < 0x3ff30000 {
99 /* |x| < 1.1875 */
100 if ix < 0x3fe60000 {
101 /* 7/16 <= |x| < 11/16 */
102 x = (2. * x - 1.) / (2. + x);
103 0
104 } else {
105 /* 11/16 <= |x| < 19/16 */
106 x = (x - 1.) / (x + 1.);
107 1
108 }
109 } else if ix < 0x40038000 {
110 /* |x| < 2.4375 */
111 x = (x - 1.5) / (1. + 1.5 * x);
112 2
113 } else {
114 /* 2.4375 <= |x| < 2^66 */
115 x = -1. / x;
116 3
117 }
118 };
119
120 let z = x * x;
121 let w = z * z;
122 /* break sum from i=0 to 10 AT[i]z**(i+1) into odd and even poly */
123 let s1 = z * (AT[0] + w * (AT[2] + w * (AT[4] + w * (AT[6] + w * (AT[8] + w * AT[10])))));
124 let s2 = w * (AT[1] + w * (AT[3] + w * (AT[5] + w * (AT[7] + w * AT[9]))));
125
126 if id < 0 {
127 return x - x * (s1 + s2);
128 }
129
130 let z = i!(ATANHI, id as usize) - (x * (s1 + s2) - i!(ATANLO, id as usize) - x);
131
132 if sign != 0 { -z } else { z }
133 }
134
135 #[cfg(test)]
136 mod tests {
137 use core::f64;
138
139 use super::atan;
140
141 #[test]
sanity_check()142 fn sanity_check() {
143 for (input, answer) in [
144 (3.0_f64.sqrt() / 3.0, f64::consts::FRAC_PI_6),
145 (1.0, f64::consts::FRAC_PI_4),
146 (3.0_f64.sqrt(), f64::consts::FRAC_PI_3),
147 (-3.0_f64.sqrt() / 3.0, -f64::consts::FRAC_PI_6),
148 (-1.0, -f64::consts::FRAC_PI_4),
149 (-3.0_f64.sqrt(), -f64::consts::FRAC_PI_3),
150 ]
151 .iter()
152 {
153 assert!(
154 (atan(*input) - answer) / answer < 1e-5,
155 "\natan({:.4}/16) = {:.4}, actual: {}",
156 input * 16.0,
157 answer,
158 atan(*input)
159 );
160 }
161 }
162
163 #[test]
zero()164 fn zero() {
165 assert_eq!(atan(0.0), 0.0);
166 }
167
168 #[test]
infinity()169 fn infinity() {
170 assert_eq!(atan(f64::INFINITY), f64::consts::FRAC_PI_2);
171 }
172
173 #[test]
minus_infinity()174 fn minus_infinity() {
175 assert_eq!(atan(f64::NEG_INFINITY), -f64::consts::FRAC_PI_2);
176 }
177
178 #[test]
nan()179 fn nan() {
180 assert!(atan(f64::NAN).is_nan());
181 }
182 }
183