libm/k_cos.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /*
  13  * __kernel_cos( x,  y )
  14  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  15  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  16  * Input y is the tail of x.
  17  *
  18  * Algorithm
  19  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
  20  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
  21  *      3. cos(x) is approximated by a polynomial of degree 14 on
  22  *         [0,pi/4]
  23  *                                       4            14
  24  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  25  *         where the remez error is
  26  *
  27  *      |              2     4     6     8     10    12     14 |     -58
  28  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
  29  *      |                                                      |
  30  *
  31  *                     4     6     8     10    12     14
  32  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
  33  *             cos(x) = 1 - x*x/2 + r
  34  *         since cos(x+y) ~ cos(x) - sin(x)*y
  35  *                        ~ cos(x) - x*y,
  36  *         a correction term is necessary in cos(x) and hence
  37  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
  38  *         For better accuracy when x > 0.3, let qx = |x|/4 with
  39  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  40  *         Then
  41  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
  42  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
  43  *         magnitude of the latter is at least a quarter of x*x/2,
  44  *         thus, reducing the rounding error in the subtraction.
  45  */
  46
  47 #include "math.h"
  48 #include "math_private.h"
  49
  50 static const double
  51 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  52 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
  53 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
  54 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
  55 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
  56 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
  57 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
  58
  59 double __kernel_cos(double x, double y)
  60 {
  61         double a,hz,z,r,qx;
  62         int32_t ix;
  63         GET_HIGH_WORD(ix,x);
  64         ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
  65         if(ix<0x3e400000) {                     /* if x < 2**27 */
  66             if(((int)x)==0) return one;         /* generate inexact */
  67         }
  68         z  = x*x;
  69         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  70         if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
  71             return one - (0.5*z - (z*r - x*y));
  72         else {
  73             if(ix > 0x3fe90000) {               /* x > 0.78125 */
  74                 qx = 0.28125;
  75             } else {
  76                 INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
  77             }
  78             hz = 0.5*z-qx;
  79             a  = one-qx;
  80             return a - (hz - (z*r-x*y));
  81         }
  82 }