nuttx/libs/libc/math/__cos.c

/****************************************************************************
 * libs/libc/math/__cos.c
 *
 *   Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 *   Developed at SunSoft, a Sun Microsystems, Inc. business.
 *   Permission to use, copy, modify, and distribute this
 *   software is freely granted, provided that this notice
 *   is preserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 * 3. Neither the name NuttX nor the names of its contributors may be
 *    used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
 * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
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 * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
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 *
 ****************************************************************************/

/* __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, rearrange to
 *              cos(x+y) ~ w + (tmp + (r-x*y))
 *         where w = 1 - x*x/2 and tmp is a tiny correction term
 *         (1 - x*x/2 == w + tmp exactly in infinite precision).
 *         The exactness of w + tmp in infinite precision depends on w
 *         and tmp having the same precision as x.  If they have extra
 *         precision due to compiler bugs, then the extra precision is
 *         only good provided it is retained in all terms of the final
 *         expression for cos().  Retention happens in all cases tested
 *         under FreeBSD, so don't pessimize things by forcibly clipping
 *         any extra precision in w.
 */

/****************************************************************************
 * Included Files
 ****************************************************************************/

#include <nuttx/config.h>
#include <nuttx/compiler.h>

#include <sys/types.h>
#include <math.h>

#ifdef CONFIG_HAVE_DOUBLE

/****************************************************************************
 * Private Data
 ****************************************************************************/

static const double g_c1 =  4.16666666666666019037e-02; /* 0x3fa55555, 0x5555554c */
static const double g_c2 = -1.38888888888741095749e-03; /* 0xbf56C16c, 0x16c15177 */
static const double g_c3 =  2.48015872894767294178e-05; /* 0x3efa01a0, 0x19cb1590 */
static const double g_c4 = -2.75573143513906633035e-07; /* 0xbe927e4e, 0x809c52ad */
static const double g_c5 =  2.08757232129817482790e-09; /* 0x3e21ee9E, 0xbdb4b1c4 */
static const double g_c6 = -1.13596475577881948265e-11; /* 0xbda8fae9, 0xbe8838d4 */

/****************************************************************************
 * Public Functions
 ****************************************************************************/

double __cos(double x, double y)
{
  double hz;
  double z;
  double r;
  double w;

  z = x * x;
  w = z * z;
  r =
    z * (g_c1 + z * (g_c2 + z * g_c3)) +
         w * w * (g_c4 + z * (g_c5 + z * g_c6));
  hz = 0.5 * z;
  w = 1.0 - hz;

  return w + (((1.0 - w) - hz) + (z * r - x * y));
}
#endif