xref: /freebsd/lib/msun/src/k_cos.c (revision 3642298923e528d795e3a30ec165d2b469e28b40)
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 #ifndef lint
15 static char rcsid[] = "$FreeBSD$";
16 #endif
17 
18 /*
19  * __kernel_cos( x,  y )
20  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
21  * Input x is assumed to be bounded by ~pi/4 in magnitude.
22  * Input y is the tail of x.
23  *
24  * Algorithm
25  *	1. Since cos(-x) = cos(x), we need only to consider positive x.
26  *	2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
27  *	3. cos(x) is approximated by a polynomial of degree 14 on
28  *	   [0,pi/4]
29  *		  	                 4            14
30  *	   	cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
31  *	   where the remez error is
32  *
33  * 	|              2     4     6     8     10    12     14 |     -58
34  * 	|cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
35  * 	|    					               |
36  *
37  * 	               4     6     8     10    12     14
38  *	4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
39  *	       cos(x) = 1 - x*x/2 + r
40  *	   since cos(x+y) ~ cos(x) - sin(x)*y
41  *			  ~ cos(x) - x*y,
42  *	   a correction term is necessary in cos(x) and hence
43  *		cos(x+y) = 1 - (x*x/2 - (r - x*y))
44  *	   For better accuracy when x > 0.3, let qx = |x|/4 with
45  *	   the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
46  *	   Then
47  *		cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
48  *	   Note that 1-qx and (x*x/2-qx) is EXACT here, and the
49  *	   magnitude of the latter is at least a quarter of x*x/2,
50  *	   thus, reducing the rounding error in the subtraction.
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
66 __kernel_cos(double x, double y)
67 {
68 	double a,hz,z,r,qx;
69 	int32_t ix;
70 	GET_HIGH_WORD(ix,x);
71 	ix &= 0x7fffffff;			/* ix = |x|'s high word*/
72 	if(ix<0x3e400000) {			/* if x < 2**27 */
73 	    if(((int)x)==0) return one;		/* generate inexact */
74 	}
75 	z  = x*x;
76 	r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
77 	if(ix < 0x3FD33333) 			/* if |x| < 0.3 */
78 	    return one - (0.5*z - (z*r - x*y));
79 	else {
80 	    if(ix > 0x3fe90000) {		/* x > 0.78125 */
81 		qx = 0.28125;
82 	    } else {
83 	        INSERT_WORDS(qx,ix-0x00200000,0);	/* x/4 */
84 	    }
85 	    hz = 0.5*z-qx;
86 	    a  = one-qx;
87 	    return a - (hz - (z*r-x*y));
88 	}
89 }
90