xref: /freebsd/lib/msun/src/k_cos.c (revision 380a989b3223d455375b4fae70fd0b9bdd43bafb)
1 /* @(#)k_cos.c 5.1 93/09/24 */
2 /*
3  * ====================================================
4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5  *
6  * Developed at SunPro, 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 #ifndef lint
14 static char rcsid[] = "$Id$";
15 #endif
16 
17 /*
18  * __kernel_cos( x,  y )
19  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
20  * Input x is assumed to be bounded by ~pi/4 in magnitude.
21  * Input y is the tail of x.
22  *
23  * Algorithm
24  *	1. Since cos(-x) = cos(x), we need only to consider positive x.
25  *	2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
26  *	3. cos(x) is approximated by a polynomial of degree 14 on
27  *	   [0,pi/4]
28  *		  	                 4            14
29  *	   	cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
30  *	   where the remez error is
31  *
32  * 	|              2     4     6     8     10    12     14 |     -58
33  * 	|cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
34  * 	|    					               |
35  *
36  * 	               4     6     8     10    12     14
37  *	4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
38  *	       cos(x) = 1 - x*x/2 + r
39  *	   since cos(x+y) ~ cos(x) - sin(x)*y
40  *			  ~ cos(x) - x*y,
41  *	   a correction term is necessary in cos(x) and hence
42  *		cos(x+y) = 1 - (x*x/2 - (r - x*y))
43  *	   For better accuracy when x > 0.3, let qx = |x|/4 with
44  *	   the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
45  *	   Then
46  *		cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
47  *	   Note that 1-qx and (x*x/2-qx) is EXACT here, and the
48  *	   magnitude of the latter is at least a quarter of x*x/2,
49  *	   thus, reducing the rounding error in the subtraction.
50  */
51 
52 #include "math.h"
53 #include "math_private.h"
54 
55 #ifdef __STDC__
56 static const double
57 #else
58 static double
59 #endif
60 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
61 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
62 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
63 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
64 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
65 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
66 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
67 
68 #ifdef __STDC__
69 	double __kernel_cos(double x, double y)
70 #else
71 	double __kernel_cos(x, y)
72 	double x,y;
73 #endif
74 {
75 	double a,hz,z,r,qx;
76 	int32_t ix;
77 	GET_HIGH_WORD(ix,x);
78 	ix &= 0x7fffffff;			/* ix = |x|'s high word*/
79 	if(ix<0x3e400000) {			/* if x < 2**27 */
80 	    if(((int)x)==0) return one;		/* generate inexact */
81 	}
82 	z  = x*x;
83 	r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
84 	if(ix < 0x3FD33333) 			/* if |x| < 0.3 */
85 	    return one - (0.5*z - (z*r - x*y));
86 	else {
87 	    if(ix > 0x3fe90000) {		/* x > 0.78125 */
88 		qx = 0.28125;
89 	    } else {
90 	        INSERT_WORDS(qx,ix-0x00200000,0);	/* x/4 */
91 	    }
92 	    hz = 0.5*z-qx;
93 	    a  = one-qx;
94 	    return a - (hz - (z*r-x*y));
95 	}
96 }
97