sysdeps/libm-ieee754/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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
  18 #endif
  19
  20 /*
  21  * __kernel_cos( x,  y )
  22  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  23  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  24  * Input y is the tail of x.
  25  *
  26  * Algorithm
  27  *      1. Since cos(-x) = cos(x), we need only to consider positive x.
  28  *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
  29  *      3. cos(x) is approximated by a polynomial of degree 14 on
  30  *         [0,pi/4]
  31  *                                       4            14
  32  *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  33  *         where the remez error is
  34  *
  35  *      |              2     4     6     8     10    12     14 |     -58
  36  *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
  37  *      |                                                      |
  38  *
  39  *                     4     6     8     10    12     14
  40  *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
  41  *             cos(x) = 1 - x*x/2 + r
  42  *         since cos(x+y) ~ cos(x) - sin(x)*y
  43  *                        ~ cos(x) - x*y,
  44  *         a correction term is necessary in cos(x) and hence
  45  *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
  46  *         For better accuracy when x > 0.3, let qx = |x|/4 with
  47  *         the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  48  *         Then
  49  *              cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
  50  *         Note that 1-qx and (x*x/2-qx) is EXACT here, and the
  51  *         magnitude of the latter is at least a quarter of x*x/2,
  52  *         thus, reducing the rounding error in the subtraction.
  53  */
  54
  55 #include "math.h"
  56 #include "math_private.h"
  57
  58 #ifdef __STDC__
  59 static const double
  60 #else
  61 static double
  62 #endif
  63 C[] = {
  64   1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  65   4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
  66  -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
  67   2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
  68  -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
  69   2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
  70  -1.13596475577881948265e-11}; /* 0xBDA8FAE9, 0xBE8838D4 */
  71
  72 #ifdef __STDC__
  73         double __kernel_cos(double x, double y)
  74 #else
  75         double __kernel_cos(x, y)
  76         double x,y;
  77 #endif
  78 {
  79         double a,hz,z,r,qx,r1,r2,r3,z1,z2,z3;
  80         int32_t ix;
  81         z  = x*x;
  82         GET_HIGH_WORD(ix,x);
  83         ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
  84         if(ix<0x3e400000) {                     /* if x < 2**27 */
  85             if(((int)x)==0) return C[0];        /* generate inexact */
  86         }
  87 #ifdef DO_NOT_USE_THIS
  88         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  89 #else
  90         r1=z*C[6];r1=r1+C[5];z1=z*z;
  91         r2=z*C[4];r2=r2+C[3];z2=z1*z;
  92         r3=z*C[2];r3=r3+C[1];z3=z2*z1;
  93         r=z3*r1+z2*r2+z*r3;
  94 #endif
  95         if(ix < 0x3FD33333)                     /* if |x| < 0.3 */
  96             return C[0] - (0.5*z - (z*r - x*y));
  97         else {
  98             if(ix > 0x3fe90000) {               /* x > 0.78125 */
  99                 qx = 0.28125;
 100             } else {
 101                 INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
 102             }
 103             hz = 0.5*z-qx;
 104             a  = C[0]-qx;
 105             return a - (hz - (z*r-x*y));
 106         }
 107 }