lib/libm/src/k_cos.c

   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 #include <sys/cdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: k_cos.c,v 1.11 2002/05/26 22:01:53 wiz Exp $");
  16 #endif
  17
  18 /*
  19  * __kernel_cos( x,  y )
  20  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  21  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  22  * Input y is the tail of x.
  23  *
  24  * Algorithm
  25  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
  26  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
  27  *      3. cos(x) is approximated by a polynomial of degree 14 on
  28  *         [0,pi/4]
  29  *                                       4            14
  30  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  31  *         where the remez error is
  32  *
  33  *      |              2     4     6     8     10    12     14 |     -58
  34  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
  35  *      |                                                      |
  36  *
  37  *                     4     6     8     10    12     14
  38  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
  39  *             cos(x) = 1 - x*x/2 + r
  40  *         since cos(x+y) ~ cos(x) - sin(x)*y
  41  *                        ~ cos(x) - x*y,
  42  *         a correction term is necessary in cos(x) and hence
  43  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
  44  *         For better accuracy when x > 0.3, let qx = |x|/4 with
  45  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  46  *         Then
  47  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
  48  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
  49  *         magnitude of the latter is at least a quarter of x*x/2,
  50  *         thus, reducing the rounding error in the subtraction.
  51  */
  52
  53 #include "math.h"
  54 #include "math_private.h"
  55
  56 static const double
  57 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  58 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
  59 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
  60 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
  61 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
  62 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
  63 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
  64
  65 double
  66 __kernel_cos(double x, double y)
  67 {
  68         double a,hz,z,r,qx;
  69         int32_t ix;
  70         GET_HIGH_WORD(ix,x);
  71         ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
  72         if(ix<0x3e400000) {                     /* if x < 2**27 */
  73             if(((int)x)==0) return one;         /* generate inexact */
  74         }
  75         z  = x*x;
  76         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  77         if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
  78             return one - (0.5*z - (z*r - x*y));
  79         else {
  80             if(ix > 0x3fe90000) {               /* x > 0.78125 */
  81                 qx = 0.28125;
  82             } else {
  83                 INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
  84             }
  85             hz = 0.5*z-qx;
  86             a  = one-qx;
  87             return a - (hz - (z*r-x*y));
  88         }
  89 }