1
2/* @(#)k_cos.c 1.3 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14#include <sys/cdefs.h>
15__FBSDID("$FreeBSD$");
16
17/*
18 * __kernel_cos( x,  y )
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 */
55
56#include "math.h"
57#include "math_private.h"
58
59static const double
60one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
61C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
62C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
63C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
64C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
65C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
66C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
67
68double
69__kernel_cos(double x, double y)
70{
71	double hz,z,r,w;
72
73	z  = x*x;
74	w  = z*z;
75	r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
76	hz = 0.5*z;
77	w  = one-hz;
78	return w + (((one-w)-hz) + (z*r-x*y));
79}
80