source: trunk/sys/libm/k_cos.c @ 75

/* @(#)k_cos.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/*
 * __kernel_cos( x,  y )
 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x. 
 *
 * Algorithm
 *      1. Since cos(-x) = cos(x), we need only to consider positive x.
 *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
 *      3. cos(x) is approximated by a polynomial of degree 14 on
 *         [0,pi/4]
 *                                       4            14
 *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *         where the remez error is
 *      
 *      |              2     4     6     8     10    12     14 |     -58
 *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *      |                                                      | 
 * 
 *                     4     6     8     10    12     14 
 *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *             cos(x) = 1 - x*x/2 + r
 *         since cos(x+y) ~ cos(x) - sin(x)*y 
 *                        ~ cos(x) - x*y,
 *         a correction term is necessary in cos(x) and hence
 *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *         For better accuracy when x > 0.3, let qx = |x|/4 with
 *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *         Then
 *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *         magnitude of the latter is at least a quarter of x*x/2,
 *         thus, reducing the rounding error in the subtraction.
 */

#include <libm/fdlibm.h>

#ifdef __STDC__
static const double 
#else
static double 
#endif
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */

#ifdef __STDC__
        double __kernel_cos(double x, double y)
#else
        double __kernel_cos(x, y)
        double x,y;
#endif
{
        double a,hz,z,r,qx;
        int n0, ix;
        n0 = ((*(int*)&one)>>29)^1;             /* high word index */
        ix = (*(n0+(int*)&x))&0x7fffffff;       /* ix = |x|'s high word*/
        if(ix<0x3e400000) {                     /* if x < 2**27 */
            if(((int)x)==0) return one;         /* generate inexact */
        }
        z  = x*x;
        r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
        if(ix < 0x3FD33333)                     /* if |x| < 0.3 */ 
            return one - (0.5*z - (z*r - x*y));
        else {
            if(ix > 0x3fe90000) {               /* x > 0.78125 */
                qx = 0.28125;
            } else {
                *(n0+(int*)&qx) = ix-0x00200000;        /* x/4 */
                *(1-n0+(int*)&qx) = 0;
            }
            hz = 0.5*z-qx;
            a  = one-qx;
            return a - (hz - (z*r-x*y));
        }
}