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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* @(#)k_cos.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, 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. #if defined(LIBM_SCCS) && !defined(lint)
  15. static char rcsid[] = "$Id: k_cos.c,v 1.6 1994/08/18 23:06:08 jtc 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. #ifdef __STDC__
  57. static const double
  58. #else
  59. static double
  60. #endif
  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. #ifdef __STDC__
  70.         double __kernel_cos(double x, double y)
  71. #else
  72.         double __kernel_cos(x, y)
  73.         double x,y;
  74. #endif
  75. {
  76.         double a,hz,z,r,qx;
  77.         int32_t ix;
  78.         GET_HIGH_WORD(ix,x);
  79.         ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
  80.         if(ix<0x3e400000) {                     /* if x < 2**27 */
  81.             if(((int)x)==0) return one;         /* generate inexact */
  82.         }
  83.         z  = x*x;
  84.         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  85.         if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
  86.             return one - (0.5*z - (z*r - x*y));
  87.         else {
  88.             if(ix > 0x3fe90000) {               /* x > 0.78125 */
  89.                 qx = 0.28125;
  90.             } else {
  91.                 INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
  92.             }
  93.             hz = 0.5*z-qx;
  94.             a  = one-qx;
  95.             return a - (hz - (z*r-x*y));
  96.         }
  97. }
  98.