k_cos.c

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/* @(#)k_cos.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static const char rcsid[] =
    "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
#endif

/*
 * __kernel_cos( x,  y )
 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 *
 * Algorithm
 *  1. Since cos(-x) = cos(x), we need only to consider positive x.
 *  2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
 *  3. cos(x) is approximated by a polynomial of degree 14 on
 *     [0,pi/4]
 *                         4            14
 *      cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
 *     where the remez error is
 *
 *  |              2     4     6     8     10    12     14 |     -58
 *  |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
 *  |                            |
 *
 *                 4     6     8     10    12     14
 *  4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
 *         cos(x) = 1 - x*x/2 + r
 *     since cos(x+y) ~ cos(x) - sin(x)*y
 *        ~ cos(x) - x*y,
 *     a correction term is necessary in cos(x) and hence
 *    cos(x+y) = 1 - (x*x/2 - (r - x*y))
 *     For better accuracy when x > 0.3, let qx = |x|/4 with
 *     the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
 *     Then
 *    cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
 *     Note that 1-qx and (x*x/2-qx) is EXACT here, and the
 *     magnitude of the latter is at least a quarter of x*x/2,
 *     thus, reducing the rounding error in the subtraction.
 */

#include "math_libm.h"
#include "math_private.h"

#ifdef __STDC__
static const double
#else
static double
#endif
  one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
    C1 = 4.16666666666666019037e-02,    /* 0x3FA55555, 0x5555554C */
    C2 = -1.38888888888741095749e-03,   /* 0xBF56C16C, 0x16C15177 */
    C3 = 2.48015872894767294178e-05,    /* 0x3EFA01A0, 0x19CB1590 */
    C4 = -2.75573143513906633035e-07,   /* 0xBE927E4F, 0x809C52AD */
    C5 = 2.08757232129817482790e-09,    /* 0x3E21EE9E, 0xBDB4B1C4 */
    C6 = -1.13596475577881948265e-11;   /* 0xBDA8FAE9, 0xBE8838D4 */

#ifdef __STDC__
double attribute_hidden
__kernel_cos(double x, double y)
#else
double attribute_hidden
__kernel_cos(x, y)
     double x, y;
#endif
{
    double a, hz, z, r, qx;
    int32_t ix;
    GET_HIGH_WORD(ix, x);
    ix &= 0x7fffffff;           /* ix = |x|'s high word */
    if (ix < 0x3e400000) {      /* if x < 2**27 */
        if (((int) x) == 0)
            return one;         /* generate inexact */
    }
    z = x * x;
    r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
    if (ix < 0x3FD33333)        /* if |x| < 0.3 */
        return one - (0.5 * z - (z * r - x * y));
    else {
        if (ix > 0x3fe90000) {  /* x > 0.78125 */
            qx = 0.28125;
        } else {
            INSERT_WORDS(qx, ix - 0x00200000, 0);       /* x/4 */
        }
        hz = 0.5 * z - qx;
        a = one - qx;
        return a - (hz - (z * r - x * y));
    }
}