| 1 |
207 |
jeremybenn |
|
| 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) */
|
