libm/math/
atan.rs

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
/* origin: FreeBSD /usr/src/lib/msun/src/s_atan.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* atan(x)
 * Method
 *   1. Reduce x to positive by atan(x) = -atan(-x).
 *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
 *      is further reduced to one of the following intervals and the
 *      arctangent of t is evaluated by the corresponding formula:
 *
 *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
 *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
 *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
 *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
 *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

use super::fabs;
use core::f64;

const ATANHI: [f64; 4] = [
    4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
    7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
    9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
    1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
];

const ATANLO: [f64; 4] = [
    2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
    3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
    1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
    6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
];

const AT: [f64; 11] = [
    3.33333333333329318027e-01,  /* 0x3FD55555, 0x5555550D */
    -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
    1.42857142725034663711e-01,  /* 0x3FC24924, 0x920083FF */
    -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
    9.09088713343650656196e-02,  /* 0x3FB745CD, 0xC54C206E */
    -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
    6.66107313738753120669e-02,  /* 0x3FB10D66, 0xA0D03D51 */
    -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
    4.97687799461593236017e-02,  /* 0x3FA97B4B, 0x24760DEB */
    -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
    1.62858201153657823623e-02,  /* 0x3F90AD3A, 0xE322DA11 */
];

/// Arctangent (f64)
///
/// Computes the inverse tangent (arc tangent) of the input value.
/// Returns a value in radians, in the range of -pi/2 to pi/2.
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn atan(x: f64) -> f64 {
    let mut x = x;
    let mut ix = (x.to_bits() >> 32) as u32;
    let sign = ix >> 31;
    ix &= 0x7fff_ffff;
    if ix >= 0x4410_0000 {
        if x.is_nan() {
            return x;
        }

        let z = ATANHI[3] + f64::from_bits(0x0380_0000); // 0x1p-120f
        return if sign != 0 { -z } else { z };
    }

    let id = if ix < 0x3fdc_0000 {
        /* |x| < 0.4375 */
        if ix < 0x3e40_0000 {
            /* |x| < 2^-27 */
            if ix < 0x0010_0000 {
                /* raise underflow for subnormal x */
                force_eval!(x as f32);
            }

            return x;
        }

        -1
    } else {
        x = fabs(x);
        if ix < 0x3ff30000 {
            /* |x| < 1.1875 */
            if ix < 0x3fe60000 {
                /* 7/16 <= |x| < 11/16 */
                x = (2. * x - 1.) / (2. + x);
                0
            } else {
                /* 11/16 <= |x| < 19/16 */
                x = (x - 1.) / (x + 1.);
                1
            }
        } else if ix < 0x40038000 {
            /* |x| < 2.4375 */
            x = (x - 1.5) / (1. + 1.5 * x);
            2
        } else {
            /* 2.4375 <= |x| < 2^66 */
            x = -1. / x;
            3
        }
    };

    let z = x * x;
    let w = z * z;
    /* break sum from i=0 to 10 AT[i]z**(i+1) into odd and even poly */
    let s1 = z * (AT[0] + w * (AT[2] + w * (AT[4] + w * (AT[6] + w * (AT[8] + w * AT[10])))));
    let s2 = w * (AT[1] + w * (AT[3] + w * (AT[5] + w * (AT[7] + w * AT[9]))));

    if id < 0 {
        return x - x * (s1 + s2);
    }

    let z = i!(ATANHI, id as usize) - (x * (s1 + s2) - i!(ATANLO, id as usize) - x);

    if sign != 0 {
        -z
    } else {
        z
    }
}

#[cfg(test)]
mod tests {
    use super::atan;
    use core::f64;

    #[test]
    fn sanity_check() {
        for (input, answer) in [
            (3.0_f64.sqrt() / 3.0, f64::consts::FRAC_PI_6),
            (1.0, f64::consts::FRAC_PI_4),
            (3.0_f64.sqrt(), f64::consts::FRAC_PI_3),
            (-3.0_f64.sqrt() / 3.0, -f64::consts::FRAC_PI_6),
            (-1.0, -f64::consts::FRAC_PI_4),
            (-3.0_f64.sqrt(), -f64::consts::FRAC_PI_3),
        ]
        .iter()
        {
            assert!(
                (atan(*input) - answer) / answer < 1e-5,
                "\natan({:.4}/16) = {:.4}, actual: {}",
                input * 16.0,
                answer,
                atan(*input)
            );
        }
    }

    #[test]
    fn zero() {
        assert_eq!(atan(0.0), 0.0);
    }

    #[test]
    fn infinity() {
        assert_eq!(atan(f64::INFINITY), f64::consts::FRAC_PI_2);
    }

    #[test]
    fn minus_infinity() {
        assert_eq!(atan(f64::NEG_INFINITY), -f64::consts::FRAC_PI_2);
    }

    #[test]
    fn nan() {
        assert!(atan(f64::NAN).is_nan());
    }
}