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      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 #include <sys/cdefs.h>
     15 __FBSDID("$FreeBSD$");
     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, rearrange to
     44  *		cos(x+y) ~ w + (tmp + (r-x*y))
     45  *	   where w = 1 - x*x/2 and tmp is a tiny correction term
     46  *	   (1 - x*x/2 == w + tmp exactly in infinite precision).
     47  *	   The exactness of w + tmp in infinite precision depends on w
     48  *	   and tmp having the same precision as x.  If they have extra
     49  *	   precision due to compiler bugs, then the extra precision is
     50  *	   only good provided it is retained in all terms of the final
     51  *	   expression for cos().  Retention happens in all cases tested
     52  *	   under FreeBSD, so don't pessimize things by forcibly clipping
     53  *	   any extra precision in w.
     54  */
     55 
     56 #include "math.h"
     57 #include "math_private.h"
     58 
     59 static const double
     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 double
     69 __kernel_cos(double x, double y)
     70 {
     71 	double hz,z,r,w;
     72 
     73 	z  = x*x;
     74 	w  = z*z;
     75 	r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
     76 	hz = 0.5*z;
     77 	w  = one-hz;
     78 	return w + (((one-w)-hz) + (z*r-x*y));
     79 }
     80