github.com/afumu/libc@v0.0.6/musl/src/math/__cos.c (about)

     1  /* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */
     2  /*
     3   * ====================================================
     4   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     5   *
     6   * Developed at SunSoft, 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  /*
    13   * __cos( x,  y )
    14   * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
    15   * Input x is assumed to be bounded by ~pi/4 in magnitude.
    16   * Input y is the tail of x.
    17   *
    18   * Algorithm
    19   *      1. Since cos(-x) = cos(x), we need only to consider positive x.
    20   *      2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
    21   *      3. cos(x) is approximated by a polynomial of degree 14 on
    22   *         [0,pi/4]
    23   *                                       4            14
    24   *              cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
    25   *         where the remez error is
    26   *
    27   *      |              2     4     6     8     10    12     14 |     -58
    28   *      |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
    29   *      |                                                      |
    30   *
    31   *                     4     6     8     10    12     14
    32   *      4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
    33   *             cos(x) ~ 1 - x*x/2 + r
    34   *         since cos(x+y) ~ cos(x) - sin(x)*y
    35   *                        ~ cos(x) - x*y,
    36   *         a correction term is necessary in cos(x) and hence
    37   *              cos(x+y) = 1 - (x*x/2 - (r - x*y))
    38   *         For better accuracy, rearrange to
    39   *              cos(x+y) ~ w + (tmp + (r-x*y))
    40   *         where w = 1 - x*x/2 and tmp is a tiny correction term
    41   *         (1 - x*x/2 == w + tmp exactly in infinite precision).
    42   *         The exactness of w + tmp in infinite precision depends on w
    43   *         and tmp having the same precision as x.  If they have extra
    44   *         precision due to compiler bugs, then the extra precision is
    45   *         only good provided it is retained in all terms of the final
    46   *         expression for cos().  Retention happens in all cases tested
    47   *         under FreeBSD, so don't pessimize things by forcibly clipping
    48   *         any extra precision in w.
    49   */
    50  
    51  #include "libm.h"
    52  
    53  static const double
    54  C1  =  4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
    55  C2  = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
    56  C3  =  2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
    57  C4  = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
    58  C5  =  2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
    59  C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
    60  
    61  double __cos(double x, double y)
    62  {
    63  	double_t hz,z,r,w;
    64  
    65  	z  = x*x;
    66  	w  = z*z;
    67  	r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
    68  	hz = 0.5*z;
    69  	w  = 1.0-hz;
    70  	return w + (((1.0-w)-hz) + (z*r-x*y));
    71  }