1/* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, 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/* asin(x)
13 * Method :
14 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
15 * we approximate asin(x) on [0,0.5] by
16 * asin(x) = x + x*x^2*R(x^2)
17 * where
18 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
19 * and its remez error is bounded by
20 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
21 *
22 * For x in [0.5,1]
23 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
24 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
25 * then for x>0.98
26 * asin(x) = pi/2 - 2*(s+s*z*R(z))
27 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
28 * For x<=0.98, let pio4_hi = pio2_hi/2, then
29 * f = hi part of s;
30 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
31 * and
32 * asin(x) = pi/2 - 2*(s+s*z*R(z))
33 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
34 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
35 *
36 * Special cases:
37 * if x is NaN, return x itself;
38 * if |x|>1, return NaN with invalid signal.
39 *
40 */
41
42use super::{fabs, get_high_word, get_low_word, sqrt, with_set_low_word};
43
44const PIO2_HI: f64 = 1.57079632679489655800e+00; /* 0x3FF921FB, 0x54442D18 */
45const PIO2_LO: f64 = 6.12323399573676603587e-17; /* 0x3C91A626, 0x33145C07 */
46/* coefficients for R(x^2) */
47const P_S0: f64 = 1.66666666666666657415e-01; /* 0x3FC55555, 0x55555555 */
48const P_S1: f64 = -3.25565818622400915405e-01; /* 0xBFD4D612, 0x03EB6F7D */
49const P_S2: f64 = 2.01212532134862925881e-01; /* 0x3FC9C155, 0x0E884455 */
50const P_S3: f64 = -4.00555345006794114027e-02; /* 0xBFA48228, 0xB5688F3B */
51const P_S4: f64 = 7.91534994289814532176e-04; /* 0x3F49EFE0, 0x7501B288 */
52const P_S5: f64 = 3.47933107596021167570e-05; /* 0x3F023DE1, 0x0DFDF709 */
53const Q_S1: f64 = -2.40339491173441421878e+00; /* 0xC0033A27, 0x1C8A2D4B */
54const Q_S2: f64 = 2.02094576023350569471e+00; /* 0x40002AE5, 0x9C598AC8 */
55const Q_S3: f64 = -6.88283971605453293030e-01; /* 0xBFE6066C, 0x1B8D0159 */
56const Q_S4: f64 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
57
58fn comp_r(z: f64) -> f64 {
59 let p: f64 = z * (P_S0 + z * (P_S1 + z * (P_S2 + z * (P_S3 + z * (P_S4 + z * P_S5)))));
60 let q: f64 = 1.0 + z * (Q_S1 + z * (Q_S2 + z * (Q_S3 + z * Q_S4)));
61 p / q
62}
63
64/// Arcsine (f64)
65///
66/// Computes the inverse sine (arc sine) of the argument `x`.
67/// Arguments to asin must be in the range -1 to 1.
68/// Returns values in radians, in the range of -pi/2 to pi/2.
69#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
70pub fn asin(mut x: f64) -> f64 {
71 let z: f64;
72 let r: f64;
73 let s: f64;
74 let hx: u32;
75 let ix: u32;
76
77 hx = get_high_word(x);
78 ix = hx & 0x7fffffff;
79 /* |x| >= 1 or nan */
80 if ix >= 0x3ff00000 {
81 let lx: u32;
82 lx = get_low_word(x);
83 if ((ix - 0x3ff00000) | lx) == 0 {
84 /* asin(1) = +-pi/2 with inexact */
85 return x * PIO2_HI + f64::from_bits(0x3870000000000000);
86 } else {
87 return 0.0 / (x - x);
88 }
89 }
90 /* |x| < 0.5 */
91 if ix < 0x3fe00000 {
92 /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
93 if (0x00100000..0x3e500000).contains(&ix) {
94 return x;
95 } else {
96 return x + x * comp_r(x * x);
97 }
98 }
99 /* 1 > |x| >= 0.5 */
100 z = (1.0 - fabs(x)) * 0.5;
101 s = sqrt(z);
102 r = comp_r(z);
103 if ix >= 0x3fef3333 {
104 /* if |x| > 0.975 */
105 x = PIO2_HI - (2. * (s + s * r) - PIO2_LO);
106 } else {
107 let f: f64;
108 let c: f64;
109 /* f+c = sqrt(z) */
110 f = with_set_low_word(s, 0);
111 c = (z - f * f) / (s + f);
112 x = 0.5 * PIO2_HI - (2.0 * s * r - (PIO2_LO - 2.0 * c) - (0.5 * PIO2_HI - 2.0 * f));
113 }
114 if hx >> 31 != 0 { -x } else { x }
115}
116