k_cos.c

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/* @(#)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 Remes 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 "math.h"
#include "math_private.h"

static const double 
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 */

double
__kernel_cos(double x, double y)
{
    double a,hz,z,r,qx;
    int32_t ix;
    GET_HIGH_WORD(ix,x);
    ix &= 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 {
            INSERT_WORDS(qx,ix-0x00200000,0);   /* x/4 */
        }
        hz = 0.5*z-qx;
        a  = one-qx;
        return a - (hz - (z*r-x*y));
    }
}