gonzui

    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: /*
   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 Remes 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 when x > 0.3, let qx = |x|/4 with
   40:  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
   41:  *         Then
   42:  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
   43:  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
   44:  *         magnitude of the latter is at least a quarter of x*x/2,
   45:  *         thus, reducing the rounding error in the subtraction.
   46:  */
   47: 
   48: #include "math.h"
   49: #include "math_private.h"
   50: 
   51: static const double 
   52: one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
   53: C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
   54: C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
   55: C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
   56: C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
   57: C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
   58: C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
   59: 
   60: double
   61: __kernel_cos(double x, double y)
   62: {
   63:         double a,hz,z,r,qx;
   64:         int32_t ix;
   65:         GET_HIGH_WORD(ix,x);
   66:         ix &= 0x7fffffff;                      /* ix = |x|'s high word*/
   67:         if(ix<0x3e400000) {                    /* if x < 2**27 */
   68:             if(((int)x)==0) return one;                /* generate inexact */
   69:         }
   70:         z  = x*x;
   71:         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
   72:         if(ix < 0x3FD33333)                    /* if |x| < 0.3 */ 
   73:             return one - (0.5*z - (z*r - x*y));
   74:         else {
   75:             if(ix > 0x3fe90000) {              /* x > 0.78125 */
   76:                 qx = 0.28125;
   77:             } else {
   78:                 INSERT_WORDS(qx,ix-0x00200000,0);      /* x/4 */
   79:             }
   80:             hz = 0.5*z-qx;
   81:             a  = one-qx;
   82:             return a - (hz - (z*r-x*y));
   83:         }
   84: }