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