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