Skunkware 5/Skunkware 5.iso/src/X11/lib/asincos.c

C/C++ Source or Header | 1995-05-03 | 5.0 KB | 177 lines

/*
 * Copyright (c) 1985 Regents of the University of California.
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms are permitted
 * provided that the above copyright notice and this paragraph are
 * duplicated in all such forms and that any documentation,
 * advertising materials, and other materials related to such
 * distribution and use acknowledge that the software was developed
 * by the University of California, Berkeley.  The name of the
 * University may not be used to endorse or promote products derived
 * from this software without specific prior written permission.
 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 *
 * All recipients should regard themselves as participants in an ongoing
 * research project and hence should feel obligated to report their
 * experiences (good or bad) with these elementary function codes, using
 * the sendbug(8) program, to the authors.
 */

#include "hdr.h"
#include <errno.h>

/* ASIN(X)
 * RETURNS ARC SINE OF X
 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
 *
 * Required system supported functions:
 *    copysign(x,y)
 *    sqrt(x)
 *
 * Required kernel function:
 *    atan2(y,x)
 *
 * Method :
 *    asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
 *          computed as follows
 *            1-x*x                     if x <  0.5,
 *            2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
 *
 * Special cases:
 *    if x is NaN, return x itself;
 *    if |x|>1, return NaN.
 *
 * Accuracy:
 * 1)  If atan2() uses machine PI, then
 *
 *    asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
 *    and PI is the exact pi rounded to machine precision (see atan2 for
 *      details):
 *
 *    in decimal:
 *        pi = 3.141592653589793 23846264338327 .....
 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 *
 *    in hexadecimal:
 *        pi = 3.243F6A8885A308D313198A2E....
 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 *
 *    In a test run with more than 200,000 random arguments on a VAX, the
 *    maximum observed error in ulps (units in the last place) was
 *    2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
 *
 * 2)  If atan2() uses true pi, then
 *
 *    asin(x) returns the exact asin(x) with error below about 2 ulps.
 *
 *    In a test run with more than 1,024,000 random arguments on a VAX, the
 *    maximum observed error in ulps (units in the last place) was
 *      1.99 ulps.
 */

double asin(double x)
{
    double s,t,one=1.0;

    if(NAN(x)) return(x);

    s=copysign(x,one);

    if (s > one ) {
#ifdef mmax
      return (infnan(EDOM));
#else
      double zero = 0.0;
      errno = EDOM;
      return (zero/zero);
#endif
    }

    if(s <= 0.5)
        return(atan2(x,sqrt(one-x*x)));
    else
        { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }

}

/* ACOS(X)
 * RETURNS ARC COS OF X
 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
 *
 * Required system supported functions:
 *    copysign(x,y)
 *    sqrt(x)
 *
 * Required kernel function:
 *    atan2(y,x)
 *
 * Method :
 *                  ________
 *                           / 1 - x
 *    acos(x) = 2*atan2(  / -------- , 1 ) .
 *                        \/   1 + x
 *
 * Special cases:
 *    if x is NaN, return x itself;
 *    if |x|>1, return NaN.
 *
 * Accuracy:
 * 1)  If atan2() uses machine PI, then
 *
 *    acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
 *    and PI is the exact pi rounded to machine precision (see atan2 for
 *      details):
 *
 *    in decimal:
 *        pi = 3.141592653589793 23846264338327 .....
 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 *
 *    in hexadecimal:
 *        pi = 3.243F6A8885A308D313198A2E....
 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 *
 *    In a test run with more than 200,000 random arguments on a VAX, the
 *    maximum observed error in ulps (units in the last place) was
 *    2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
 *
 * 2)  If atan2() uses true pi, then
 *
 *    acos(x) returns the exact acos(x) with error below about 2 ulps.
 *
 *    In a test run with more than 1,024,000 random arguments on a VAX, the
 *    maximum observed error in ulps (units in the last place) was
 *    2.15 ulps.
 */

double acos(double x)
{
    double t,one=1.0;

    if(NAN(x)) return(x);

    if((x > 1.0) || (x < -1.0)) {
#ifdef mmax
      return infnan(EDOM);
#else
      double zero=0.0;
      errno = EDOM;
      return(zero/zero);
#endif
    }

    if( x != -1.0)
        t=atan2(sqrt((one-x)/(one+x)),one);
    else
        t=atan2(one,0.0);    /* t = PI/2 */
    return(t+t);
}