xref: /freebsd/lib/msun/src/k_cos.c (revision a03411e84728e9b267056fd31c7d1d9d1dc1b01e)
1 
2 /*
3  * ====================================================
4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5  *
6  * Developed at SunSoft, 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 /*
14  * __kernel_cos( x,  y )
15  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
16  * Input x is assumed to be bounded by ~pi/4 in magnitude.
17  * Input y is the tail of x.
18  *
19  * Algorithm
20  *	1. Since cos(-x) = cos(x), we need only to consider positive x.
21  *	2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
22  *	3. cos(x) is approximated by a polynomial of degree 14 on
23  *	   [0,pi/4]
24  *		  	                 4            14
25  *	   	cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
26  *	   where the remez error is
27  *
28  * 	|              2     4     6     8     10    12     14 |     -58
29  * 	|cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
30  * 	|    					               |
31  *
32  * 	               4     6     8     10    12     14
33  *	4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
34  *	       cos(x) ~ 1 - x*x/2 + r
35  *	   since cos(x+y) ~ cos(x) - sin(x)*y
36  *			  ~ cos(x) - x*y,
37  *	   a correction term is necessary in cos(x) and hence
38  *		cos(x+y) = 1 - (x*x/2 - (r - x*y))
39  *	   For better accuracy, rearrange to
40  *		cos(x+y) ~ w + (tmp + (r-x*y))
41  *	   where w = 1 - x*x/2 and tmp is a tiny correction term
42  *	   (1 - x*x/2 == w + tmp exactly in infinite precision).
43  *	   The exactness of w + tmp in infinite precision depends on w
44  *	   and tmp having the same precision as x.  If they have extra
45  *	   precision due to compiler bugs, then the extra precision is
46  *	   only good provided it is retained in all terms of the final
47  *	   expression for cos().  Retention happens in all cases tested
48  *	   under FreeBSD, so don't pessimize things by forcibly clipping
49  *	   any extra precision in w.
50  */
51 
52 #include "math.h"
53 #include "math_private.h"
54 
55 static const double
56 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
57 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
58 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
59 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
60 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
61 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
62 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
63 
64 double
65 __kernel_cos(double x, double y)
66 {
67 	double hz,z,r,w;
68 
69 	z  = x*x;
70 	w  = z*z;
71 	r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
72 	hz = 0.5*z;
73 	w  = one-hz;
74 	return w + (((one-w)-hz) + (z*r-x*y));
75 }
76