Branch data     Line data    Source code

       1                 :             : 
       2                 :             : /* @(#)k_cos.c 1.3 95/01/18 */
       3                 :             : /*
       4                 :             :  * ====================================================
       5                 :             :  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
       6                 :             :  *
       7                 :             :  * Developed at SunSoft, a Sun Microsystems, Inc. business.
       8                 :             :  * Permission to use, copy, modify, and distribute this
       9                 :             :  * software is freely granted, provided that this notice 
      10                 :             :  * is preserved.
      11                 :             :  * ====================================================
      12                 :             :  */
      13                 :             : 
      14                 :             : #include "cdefs-compat.h"
      15                 :             : //__FBSDID("$FreeBSD: src/lib/msun/src/k_cos.c,v 1.12 2008/02/19 12:54:14 bde Exp $");
      16                 :             : 
      17                 :             : /*
      18                 :             :  * __kernel_cos( x,  y )
      19                 :             :  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
      20                 :             :  * Input x is assumed to be bounded by ~pi/4 in magnitude.
      21                 :             :  * Input y is the tail of x. 
      22                 :             :  *
      23                 :             :  * Algorithm
      24                 :             :  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
      25                 :             :  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
      26                 :             :  *      3. cos(x) is approximated by a polynomial of degree 14 on
      27                 :             :  *         [0,pi/4]
      28                 :             :  *                                       4            14
      29                 :             :  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
      30                 :             :  *         where the remez error is
      31                 :             :  *      
      32                 :             :  *      |              2     4     6     8     10    12     14 |     -58
      33                 :             :  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
      34                 :             :  *      |                                                      | 
      35                 :             :  * 
      36                 :             :  *                     4     6     8     10    12     14 
      37                 :             :  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
      38                 :             :  *             cos(x) ~ 1 - x*x/2 + r
      39                 :             :  *         since cos(x+y) ~ cos(x) - sin(x)*y 
      40                 :             :  *                        ~ cos(x) - x*y,
      41                 :             :  *         a correction term is necessary in cos(x) and hence
      42                 :             :  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
      43                 :             :  *         For better accuracy, rearrange to
      44                 :             :  *              cos(x+y) ~ w + (tmp + (r-x*y))
      45                 :             :  *         where w = 1 - x*x/2 and tmp is a tiny correction term
      46                 :             :  *         (1 - x*x/2 == w + tmp exactly in infinite precision).
      47                 :             :  *         The exactness of w + tmp in infinite precision depends on w
      48                 :             :  *         and tmp having the same precision as x.  If they have extra
      49                 :             :  *         precision due to compiler bugs, then the extra precision is
      50                 :             :  *         only good provided it is retained in all terms of the final
      51                 :             :  *         expression for cos().  Retention happens in all cases tested
      52                 :             :  *         under FreeBSD, so don't pessimize things by forcibly clipping
      53                 :             :  *         any extra precision in w.
      54                 :             :  */
      55                 :             : 
      56                 :             : #include <openlibm_math.h>
      57                 :             : 
      58                 :             : #include "math_private.h"
      59                 :             : 
      60                 :             : static const double
      61                 :             : one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
      62                 :             : C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
      63                 :             : C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
      64                 :             : C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
      65                 :             : C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
      66                 :             : C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
      67                 :             : C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
      68                 :             : 
      69                 :             : OLM_DLLEXPORT double
      70                 :          53 : __kernel_cos(double x, double y)
      71                 :             : {
      72                 :             :         double hz,z,r,w;
      73                 :             : 
      74                 :          53 :         z  = x*x;
      75                 :          53 :         w  = z*z;
      76                 :          53 :         r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
      77                 :          53 :         hz = 0.5*z;
      78                 :          53 :         w  = one-hz;
      79                 :          53 :         return w + (((one-w)-hz) + (z*r-x*y));
      80                 :             : }