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
53static const double
54C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
55C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
56C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
57C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
58C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
59C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
60
61double __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