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/* 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 core::f64;
use super::fabs;
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 core::f64;
use super::atan;
#[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());
}
}