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
|
/* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* asin(x)
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
use super::{fabs, get_high_word, get_low_word, sqrt, with_set_low_word};
const PIO2_HI: f64 = 1.57079632679489655800e+00; /* 0x3FF921FB, 0x54442D18 */
const PIO2_LO: f64 = 6.12323399573676603587e-17; /* 0x3C91A626, 0x33145C07 */
/* coefficients for R(x^2) */
const P_S0: f64 = 1.66666666666666657415e-01; /* 0x3FC55555, 0x55555555 */
const P_S1: f64 = -3.25565818622400915405e-01; /* 0xBFD4D612, 0x03EB6F7D */
const P_S2: f64 = 2.01212532134862925881e-01; /* 0x3FC9C155, 0x0E884455 */
const P_S3: f64 = -4.00555345006794114027e-02; /* 0xBFA48228, 0xB5688F3B */
const P_S4: f64 = 7.91534994289814532176e-04; /* 0x3F49EFE0, 0x7501B288 */
const P_S5: f64 = 3.47933107596021167570e-05; /* 0x3F023DE1, 0x0DFDF709 */
const Q_S1: f64 = -2.40339491173441421878e+00; /* 0xC0033A27, 0x1C8A2D4B */
const Q_S2: f64 = 2.02094576023350569471e+00; /* 0x40002AE5, 0x9C598AC8 */
const Q_S3: f64 = -6.88283971605453293030e-01; /* 0xBFE6066C, 0x1B8D0159 */
const Q_S4: f64 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
fn comp_r(z: f64) -> f64 {
let p = z * (P_S0 + z * (P_S1 + z * (P_S2 + z * (P_S3 + z * (P_S4 + z * P_S5)))));
let q = 1.0 + z * (Q_S1 + z * (Q_S2 + z * (Q_S3 + z * Q_S4)));
p / q
}
/// Arcsine (f64)
///
/// Computes the inverse sine (arc sine) of the argument `x`.
/// Arguments to asin must be in the range -1 to 1.
/// Returns values in radians, in the range of -pi/2 to pi/2.
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn asin(mut x: f64) -> f64 {
let z: f64;
let r: f64;
let s: f64;
let hx: u32;
let ix: u32;
hx = get_high_word(x);
ix = hx & 0x7fffffff;
/* |x| >= 1 or nan */
if ix >= 0x3ff00000 {
let lx: u32;
lx = get_low_word(x);
if ((ix - 0x3ff00000) | lx) == 0 {
/* asin(1) = +-pi/2 with inexact */
return x * PIO2_HI + f64::from_bits(0x3870000000000000);
} else {
return 0.0 / (x - x);
}
}
/* |x| < 0.5 */
if ix < 0x3fe00000 {
/* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
if ix < 0x3e500000 && ix >= 0x00100000 {
return x;
} else {
return x + x * comp_r(x * x);
}
}
/* 1 > |x| >= 0.5 */
z = (1.0 - fabs(x)) * 0.5;
s = sqrt(z);
r = comp_r(z);
if ix >= 0x3fef3333 {
/* if |x| > 0.975 */
x = PIO2_HI - (2. * (s + s * r) - PIO2_LO);
} else {
let f: f64;
let c: f64;
/* f+c = sqrt(z) */
f = with_set_low_word(s, 0);
c = (z - f * f) / (s + f);
x = 0.5 * PIO2_HI - (2.0 * s * r - (PIO2_LO - 2.0 * c) - (0.5 * PIO2_HI - 2.0 * f));
}
if hx >> 31 != 0 {
-x
} else {
x
}
}
|
