compiler/mlib/k_cos.c

   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 #ifndef lint
  15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_cos.c,v 1.10 2005/10/26 12:36:18 bde 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, rearrange to
  45  *              cos(x+y) ~ w + (tmp + (r-x*y))
  46  *         where w = 1 - x*x/2 and tmp is a tiny correction term
  47  *         (1 - x*x/2 == w + tmp exactly in infinite precision).
  48  *         The exactness of w + tmp in infinite precision depends on w
  49  *         and tmp having the same precision as x.  If they have extra
  50  *         precision due to compiler bugs, then the extra precision is
  51  *         only good provided it is retained in all terms of the final
  52  *         expression for cos().  Retention happens in all cases tested
  53  *         under FreeBSD, so don't pessimize things by forcibly clipping
  54  *         any extra precision in w.
  55  */
  56
  57 #include "math.h"
  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 double
  70 __kernel_cos(double x, double y)
  71 {
  72         double hz,z,r,w;
  73
  74         z  = x*x;
  75         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  76         hz = (float)0.5*z;
  77         w  = one-hz;
  78         return w + (((one-w)-hz) + (z*r-x*y));
  79 }