toolchain/gcc/newlib/libm/math/k_cos.c

   1
   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 /*
  15  * __kernel_cos( x,  y )
  16  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  17  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  18  * Input y is the tail of x.
  19  *
  20  * Algorithm
  21  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
  22  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
  23  *      3. cos(x) is approximated by a polynomial of degree 14 on
  24  *         [0,pi/4]
  25  *                                       4            14
  26  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  27  *         where the remez error is
  28  *
  29  *      |              2     4     6     8     10    12     14 |     -58
  30  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
  31  *      |                                                      |
  32  *
  33  *                     4     6     8     10    12     14
  34  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
  35  *             cos(x) = 1 - x*x/2 + r
  36  *         since cos(x+y) ~ cos(x) - sin(x)*y
  37  *                        ~ cos(x) - x*y,
  38  *         a correction term is necessary in cos(x) and hence
  39  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
  40  *         For better accuracy when x > 0.3, let qx = |x|/4 with
  41  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  42  *         Then
  43  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
  44  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
  45  *         magnitude of the latter is at least a quarter of x*x/2,
  46  *         thus, reducing the rounding error in the subtraction.
  47  */
  48
  49 #include "fdlibm.h"
  50
  51 #ifndef _DOUBLE_IS_32BITS
  52
  53 #ifdef __STDC__
  54 static const double
  55 #else
  56 static double
  57 #endif
  58 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  59 C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
  60 C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
  61 C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
  62 C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
  63 C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
  64 C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
  65
  66 #ifdef __STDC__
  67         double __kernel_cos(double x, double y)
  68 #else
  69         double __kernel_cos(x, y)
  70         double x,y;
  71 #endif
  72 {
  73         double a,hz,z,r,qx;
  74         __int32_t ix;
  75         GET_HIGH_WORD(ix,x);
  76         ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
  77         if(ix<0x3e400000) {                     /* if x < 2**27 */
  78             if(((int)x)==0) return one;         /* generate inexact */
  79         }
  80         z  = x*x;
  81         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  82         if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
  83             return one - (0.5*z - (z*r - x*y));
  84         else {
  85             if(ix > 0x3fe90000) {               /* x > 0.78125 */
  86                 qx = 0.28125;
  87             } else {
  88                 INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
  89             }
  90             hz = 0.5*z-qx;
  91             a  = one-qx;
  92             return a - (hz - (z*r-x*y));
  93         }
  94 }
  95
  96 #endif /* defined(_DOUBLE_IS_32BITS) */