cosmopolitan/libc/tinymath/kcos.c

/*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:2;tab-width:8;coding:utf-8 -*-│
│vi: set net ft=c ts=2 sts=2 sw=2 fenc=utf-8                                :vi│
╚──────────────────────────────────────────────────────────────────────────────╝
│                                                                              │
│  Musl Libc                                                                   │
│  Copyright © 2005-2014 Rich Felker, et al.                                   │
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│  Permission is hereby granted, free of charge, to any person obtaining       │
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╚─────────────────────────────────────────────────────────────────────────────*/
#include "libc/math.h"

asm(".ident\t\"\\n\\n\
fdlibm (fdlibm license)\\n\
Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\"");
asm(".ident\t\"\\n\\n\
Musl libc (MIT License)\\n\
Copyright 2005-2014 Rich Felker, et. al.\"");
asm(".include \"libc/disclaimer.inc\"");

/* clang-format off */
/* origin: FreeBSD /usr/src/lib/msun/src/k_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.
 * ====================================================
 */
/*
 * __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.
 */

#define C1  4.16666666666666019037e-02
#define C2 -1.38888888888741095749e-03
#define C3  2.48015872894767294178e-05
#define C4 -2.75573143513906633035e-07
#define C5  2.08757232129817482790e-09
#define C6 -1.13596475577881948265e-11

double __cos(double x, double y)
{
	double_t hz,z,r,w;
	z  = x*x;
	w  = z*z;
	r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
	hz = 0.5*z;
	w  = 1.0-hz;
	return w + (((1.0-w)-hz) + (z*r-x*y));
}