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
59 static const double
60 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
61 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
62 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
63 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
64 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
65 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
66 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
67
68 double
__kernel_cos(double x,double y)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