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 /*
15 * __kernel_cos( x, y )
16 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
17 * Input x is assumed to be bounded by ~pi/4 in magnitude.
18 * Input y is the tail of x.
19 *
20 * Algorithm
21 * 1. Since cos(-x) = cos(x), we need only to consider positive x.
22 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
23 * 3. cos(x) is approximated by a polynomial of degree 14 on
24 * [0,pi/4]
25 * 4 14
26 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
27 * where the remez error is
28 *
29 * | 2 4 6 8 10 12 14 | -58
30 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
31 * | |
32 *
33 * 4 6 8 10 12 14
34 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
35 * cos(x) ~ 1 - x*x/2 + r
36 * since cos(x+y) ~ cos(x) - sin(x)*y
37 * ~ cos(x) - x*y,
38 * a correction term is necessary in cos(x) and hence
39 * cos(x+y) = 1 - (x*x/2 - (r - x*y))
40 * For better accuracy, rearrange to
41 * cos(x+y) ~ w + (tmp + (r-x*y))
42 * where w = 1 - x*x/2 and tmp is a tiny correction term
43 * (1 - x*x/2 == w + tmp exactly in infinite precision).
44 * The exactness of w + tmp in infinite precision depends on w
45 * and tmp having the same precision as x. If they have extra
46 * precision due to compiler bugs, then the extra precision is
47 * only good provided it is retained in all terms of the final
48 * expression for cos(). Retention happens in all cases tested
49 * under FreeBSD, so don't pessimize things by forcibly clipping
50 * any extra precision in w.
51 */
52
53 #include "math.h"
54 #include "math_private.h"
55
56 static const double
57 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
58 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
59 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
60 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
61 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
62 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
63 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
64
65 double
__kernel_cos(double x,double y)66 __kernel_cos(double x, double y)
67 {
68 double hz,z,r,w;
69
70 z = x*x;
71 w = z*z;
72 r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
73 hz = 0.5*z;
74 w = one-hz;
75 return w + (((one-w)-hz) + (z*r-x*y));
76 }
77