1
2 /* @(#)k_cos.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, 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 when x > 0.3, let qx = |x|/4 with
41 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
42 * Then
43 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
44 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
45 * magnitude of the latter is at least a quarter of x*x/2,
46 * thus, reducing the rounding error in the subtraction.
47 */
48
49 #include "fdlibm.h"
50
51 #ifdef _NEED_FLOAT64
52
53 static const __float64
54 one = _F_64(1.00000000000000000000e+00), /* 0x3FF00000, 0x00000000 */
55 C1 = _F_64(4.16666666666666019037e-02), /* 0x3FA55555, 0x5555554C */
56 C2 = _F_64(-1.38888888888741095749e-03), /* 0xBF56C16C, 0x16C15177 */
57 C3 = _F_64(2.48015872894767294178e-05), /* 0x3EFA01A0, 0x19CB1590 */
58 C4 = _F_64(-2.75573143513906633035e-07), /* 0xBE927E4F, 0x809C52AD */
59 C5 = _F_64(2.08757232129817482790e-09), /* 0x3E21EE9E, 0xBDB4B1C4 */
60 C6 = _F_64(-1.13596475577881948265e-11); /* 0xBDA8FAE9, 0xBE8838D4 */
61
62 __float64
__kernel_cos(__float64 x,__float64 y)63 __kernel_cos(__float64 x, __float64 y)
64 {
65 __float64 a, hz, z, r, qx;
66 __int32_t ix;
67 GET_HIGH_WORD(ix, x);
68 ix &= 0x7fffffff; /* ix = |x|'s high word*/
69 if (ix < 0x3e400000) /* if x < 2**27 */
70 return one;
71 z = x * x;
72 r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
73 if (ix < 0x3FD33333) /* if |x| < 0.3 */
74 return one - (_F_64(0.5) * z - (z * r - x * y));
75 else {
76 if (ix > 0x3fe90000) { /* x > 0.78125 */
77 qx = _F_64(0.28125);
78 } else {
79 INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
80 }
81 hz = _F_64(0.5) * z - qx;
82 a = one - qx;
83 return a - (hz - (z * r - x * y));
84 }
85 }
86
87 #endif /* _NEED_FLOAT64 */
88
