1*2fe8fb19SBen Gras /* @(#)k_cos.c 5.1 93/09/24 */
2*2fe8fb19SBen Gras /*
3*2fe8fb19SBen Gras * ====================================================
4*2fe8fb19SBen Gras * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*2fe8fb19SBen Gras *
6*2fe8fb19SBen Gras * Developed at SunPro, a Sun Microsystems, Inc. business.
7*2fe8fb19SBen Gras * Permission to use, copy, modify, and distribute this
8*2fe8fb19SBen Gras * software is freely granted, provided that this notice
9*2fe8fb19SBen Gras * is preserved.
10*2fe8fb19SBen Gras * ====================================================
11*2fe8fb19SBen Gras */
12*2fe8fb19SBen Gras
13*2fe8fb19SBen Gras #include <sys/cdefs.h>
14*2fe8fb19SBen Gras #if defined(LIBM_SCCS) && !defined(lint)
15*2fe8fb19SBen Gras __RCSID("$NetBSD: k_cos.c,v 1.11 2002/05/26 22:01:53 wiz Exp $");
16*2fe8fb19SBen Gras #endif
17*2fe8fb19SBen Gras
18*2fe8fb19SBen Gras /*
19*2fe8fb19SBen Gras * __kernel_cos( x, y )
20*2fe8fb19SBen Gras * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
21*2fe8fb19SBen Gras * Input x is assumed to be bounded by ~pi/4 in magnitude.
22*2fe8fb19SBen Gras * Input y is the tail of x.
23*2fe8fb19SBen Gras *
24*2fe8fb19SBen Gras * Algorithm
25*2fe8fb19SBen Gras * 1. Since cos(-x) = cos(x), we need only to consider positive x.
26*2fe8fb19SBen Gras * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
27*2fe8fb19SBen Gras * 3. cos(x) is approximated by a polynomial of degree 14 on
28*2fe8fb19SBen Gras * [0,pi/4]
29*2fe8fb19SBen Gras * 4 14
30*2fe8fb19SBen Gras * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
31*2fe8fb19SBen Gras * where the remez error is
32*2fe8fb19SBen Gras *
33*2fe8fb19SBen Gras * | 2 4 6 8 10 12 14 | -58
34*2fe8fb19SBen Gras * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
35*2fe8fb19SBen Gras * | |
36*2fe8fb19SBen Gras *
37*2fe8fb19SBen Gras * 4 6 8 10 12 14
38*2fe8fb19SBen Gras * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
39*2fe8fb19SBen Gras * cos(x) = 1 - x*x/2 + r
40*2fe8fb19SBen Gras * since cos(x+y) ~ cos(x) - sin(x)*y
41*2fe8fb19SBen Gras * ~ cos(x) - x*y,
42*2fe8fb19SBen Gras * a correction term is necessary in cos(x) and hence
43*2fe8fb19SBen Gras * cos(x+y) = 1 - (x*x/2 - (r - x*y))
44*2fe8fb19SBen Gras * For better accuracy when x > 0.3, let qx = |x|/4 with
45*2fe8fb19SBen Gras * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
46*2fe8fb19SBen Gras * Then
47*2fe8fb19SBen Gras * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
48*2fe8fb19SBen Gras * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
49*2fe8fb19SBen Gras * magnitude of the latter is at least a quarter of x*x/2,
50*2fe8fb19SBen Gras * thus, reducing the rounding error in the subtraction.
51*2fe8fb19SBen Gras */
52*2fe8fb19SBen Gras
53*2fe8fb19SBen Gras #include "math.h"
54*2fe8fb19SBen Gras #include "math_private.h"
55*2fe8fb19SBen Gras
56*2fe8fb19SBen Gras static const double
57*2fe8fb19SBen Gras one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
58*2fe8fb19SBen Gras C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
59*2fe8fb19SBen Gras C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
60*2fe8fb19SBen Gras C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
61*2fe8fb19SBen Gras C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
62*2fe8fb19SBen Gras C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
63*2fe8fb19SBen Gras C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
64*2fe8fb19SBen Gras
65*2fe8fb19SBen Gras double
__kernel_cos(double x,double y)66*2fe8fb19SBen Gras __kernel_cos(double x, double y)
67*2fe8fb19SBen Gras {
68*2fe8fb19SBen Gras double a,hz,z,r,qx;
69*2fe8fb19SBen Gras int32_t ix;
70*2fe8fb19SBen Gras GET_HIGH_WORD(ix,x);
71*2fe8fb19SBen Gras ix &= 0x7fffffff; /* ix = |x|'s high word*/
72*2fe8fb19SBen Gras if(ix<0x3e400000) { /* if x < 2**27 */
73*2fe8fb19SBen Gras if(((int)x)==0) return one; /* generate inexact */
74*2fe8fb19SBen Gras }
75*2fe8fb19SBen Gras z = x*x;
76*2fe8fb19SBen Gras r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
77*2fe8fb19SBen Gras if(ix < 0x3FD33333) /* if |x| < 0.3 */
78*2fe8fb19SBen Gras return one - (0.5*z - (z*r - x*y));
79*2fe8fb19SBen Gras else {
80*2fe8fb19SBen Gras if(ix > 0x3fe90000) { /* x > 0.78125 */
81*2fe8fb19SBen Gras qx = 0.28125;
82*2fe8fb19SBen Gras } else {
83*2fe8fb19SBen Gras INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
84*2fe8fb19SBen Gras }
85*2fe8fb19SBen Gras hz = 0.5*z-qx;
86*2fe8fb19SBen Gras a = one-qx;
87*2fe8fb19SBen Gras return a - (hz - (z*r-x*y));
88*2fe8fb19SBen Gras }
89*2fe8fb19SBen Gras }
90