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 /*
13 * __cos( x, y )
14 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
15 * Input x is assumed to be bounded by ~pi/4 in magnitude.
16 * Input y is the tail of x.
17 *
18 * Algorithm
19 * 1. Since cos(-x) = cos(x), we need only to consider positive x.
20 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
21 * 3. cos(x) is approximated by a polynomial of degree 14 on
22 * [0,pi/4]
23 * 4 14
24 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
25 * where the remez error is
26 *
27 * | 2 4 6 8 10 12 14 | -58
28 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
29 * | |
30 *
31 * 4 6 8 10 12 14
32 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
33 * cos(x) ~ 1 - x*x/2 + r
34 * since cos(x+y) ~ cos(x) - sin(x)*y
35 * ~ cos(x) - x*y,
36 * a correction term is necessary in cos(x) and hence
37 * cos(x+y) = 1 - (x*x/2 - (r - x*y))
38 * For better accuracy, rearrange to
39 * cos(x+y) ~ w + (tmp + (r-x*y))
40 * where w = 1 - x*x/2 and tmp is a tiny correction term
41 * (1 - x*x/2 == w + tmp exactly in infinite precision).
42 * The exactness of w + tmp in infinite precision depends on w
43 * and tmp having the same precision as x. If they have extra
44 * precision due to compiler bugs, then the extra precision is
45 * only good provided it is retained in all terms of the final
46 * expression for cos(). Retention happens in all cases tested
47 * under FreeBSD, so don't pessimize things by forcibly clipping
48 * any extra precision in w.
49 */
50 51 #include "libm.h"
52 53 static const double
54 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
55 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
56 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
57 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
58 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
59 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
60 61 double __cos(double x, double y)
62 {
63 double_t hz,z,r,w;
64 65 z = x*x;
66 w = z*z;
67 r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
68 hz = 0.5*z;
69 w = 1.0-hz;
70 return w + (((1.0-w)-hz) + (z*r-x*y));
71 }
72