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)]
k_cos(x: f64, y: f64) -> f6455 pub(crate) fn k_cos(x: f64, y: f64) -> f64 {
56 let z = x * x;
57 let w = z * z;
58 let r = z * (C1 + z * (C2 + z * C3)) + w * w * (C4 + z * (C5 + z * C6));
59 let hz = 0.5 * z;
60 let w = 1.0 - hz;
61 w + (((1.0 - w) - hz) + (z * r - x * y))
62 }
63