1 | // origin: FreeBSD /usr/src/lib/msun/src/k_cos.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 | const C1: f64 = 4.16666666666666019037e-02; /* 0x3FA55555, 0x5555554C */ |
13 | const C2: f64 = -1.38888888888741095749e-03; /* 0xBF56C16C, 0x16C15177 */ |
14 | const C3: f64 = 2.48015872894767294178e-05; /* 0x3EFA01A0, 0x19CB1590 */ |
15 | const C4: f64 = -2.75573143513906633035e-07; /* 0xBE927E4F, 0x809C52AD */ |
16 | const C5: f64 = 2.08757232129817482790e-09; /* 0x3E21EE9E, 0xBDB4B1C4 */ |
17 | const C6: f64 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
18 | |
19 | // kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
20 | // Input x is assumed to be bounded by ~pi/4 in magnitude. |
21 | // Input y is the tail of x. |
22 | // |
23 | // Algorithm |
24 | // 1. Since cos(-x) = cos(x), we need only to consider positive x. |
25 | // 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
26 | // 3. cos(x) is approximated by a polynomial of degree 14 on |
27 | // [0,pi/4] |
28 | // 4 14 |
29 | // cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
30 | // where the remez error is |
31 | // |
32 | // | 2 4 6 8 10 12 14 | -58 |
33 | // |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
34 | // | | |
35 | // |
36 | // 4 6 8 10 12 14 |
37 | // 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
38 | // cos(x) ~ 1 - x*x/2 + r |
39 | // since cos(x+y) ~ cos(x) - sin(x)*y |
40 | // ~ cos(x) - x*y, |
41 | // a correction term is necessary in cos(x) and hence |
42 | // cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
43 | // For better accuracy, rearrange to |
44 | // cos(x+y) ~ w + (tmp + (r-x*y)) |
45 | // where w = 1 - x*x/2 and tmp is a tiny correction term |
46 | // (1 - x*x/2 == w + tmp exactly in infinite precision). |
47 | // The exactness of w + tmp in infinite precision depends on w |
48 | // and tmp having the same precision as x. If they have extra |
49 | // precision due to compiler bugs, then the extra precision is |
50 | // only good provided it is retained in all terms of the final |
51 | // expression for cos(). Retention happens in all cases tested |
52 | // under FreeBSD, so don't pessimize things by forcibly clipping |
53 | // any extra precision in w. |
54 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
55 | pub(crate) fn k_cos(x: f64, y: f64) -> f64 { |
56 | let z: f64 = x * x; |
57 | let w: f64 = z * z; |
58 | let r: f64 = z * (C1 + z * (C2 + z * C3)) + w * w * (C4 + z * (C5 + z * C6)); |
59 | let hz: f64 = 0.5 * z; |
60 | let w: f64 = 1.0 - hz; |
61 | w + (((1.0 - w) - hz) + (z * r - x * y)) |
62 | } |
63 | |