sysdeps/generic/asincos.c

   1 /*
   2  * Copyright (c) 1985, 1993
   3  *      The Regents of the University of California.  All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  * 3. All advertising materials mentioning features or use of this software
  14  *    must display the following acknowledgement:
  15  *      This product includes software developed by the University of
  16  *      California, Berkeley and its contributors.
  17  * 4. Neither the name of the University nor the names of its contributors
  18  *    may be used to endorse or promote products derived from this software
  19  *    without specific prior written permission.
  20  *
  21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  31  * SUCH DAMAGE.
  32  */
  33
  34 #ifndef lint
  35 static char sccsid[] = "@(#)asincos.c   8.1 (Berkeley) 6/4/93";
  36 #endif /* not lint */
  37
  38 /* ASIN(X)
  39  * RETURNS ARC SINE OF X
  40  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
  41  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
  42  *
  43  * Required system supported functions:
  44  *      copysign(x,y)
  45  *      sqrt(x)
  46  *
  47  * Required kernel function:
  48  *      atan2(y,x)
  49  *
  50  * Method :
  51  *      asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
  52  *                computed as follows
  53  *                      1-x*x                     if x <  0.5,
  54  *                      2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
  55  *
  56  * Special cases:
  57  *      if x is NaN, return x itself;
  58  *      if |x|>1, return NaN.
  59  *
  60  * Accuracy:
  61  * 1)  If atan2() uses machine PI, then
  62  *
  63  *      asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
  64  *      and PI is the exact pi rounded to machine precision (see atan2 for
  65  *      details):
  66  *
  67  *      in decimal:
  68  *              pi = 3.141592653589793 23846264338327 .....
  69  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
  70  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
  71  *
  72  *      in hexadecimal:
  73  *              pi = 3.243F6A8885A308D313198A2E....
  74  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
  75  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
  76  *
  77  *      In a test run with more than 200,000 random arguments on a VAX, the
  78  *      maximum observed error in ulps (units in the last place) was
  79  *      2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
  80  *
  81  * 2)  If atan2() uses true pi, then
  82  *
  83  *      asin(x) returns the exact asin(x) with error below about 2 ulps.
  84  *
  85  *      In a test run with more than 1,024,000 random arguments on a VAX, the
  86  *      maximum observed error in ulps (units in the last place) was
  87  *      1.99 ulps.
  88  */
  89
  90 double asin(x)
  91 double x;
  92 {
  93         double s,t,copysign(),atan2(),sqrt(),one=1.0;
  94 #if !defined(vax)&&!defined(tahoe)
  95         if(x!=x) return(x);     /* x is NaN */
  96 #endif  /* !defined(vax)&&!defined(tahoe) */
  97         s=copysign(x,one);
  98         if(s <= 0.5)
  99             return(atan2(x,sqrt(one-x*x)));
 100         else
 101             { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
 102
 103 }
 104
 105 /* ACOS(X)
 106  * RETURNS ARC COS OF X
 107  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
 108  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
 109  *
 110  * Required system supported functions:
 111  *      copysign(x,y)
 112  *      sqrt(x)
 113  *
 114  * Required kernel function:
 115  *      atan2(y,x)
 116  *
 117  * Method :
 118  *                            ________
 119  *                           / 1 - x
 120  *      acos(x) = 2*atan2(  / -------- , 1 ) .
 121  *                        \/   1 + x
 122  *
 123  * Special cases:
 124  *      if x is NaN, return x itself;
 125  *      if |x|>1, return NaN.
 126  *
 127  * Accuracy:
 128  * 1)  If atan2() uses machine PI, then
 129  *
 130  *      acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
 131  *      and PI is the exact pi rounded to machine precision (see atan2 for
 132  *      details):
 133  *
 134  *      in decimal:
 135  *              pi = 3.141592653589793 23846264338327 .....
 136  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 137  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 138  *
 139  *      in hexadecimal:
 140  *              pi = 3.243F6A8885A308D313198A2E....
 141  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
 142  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 143  *
 144  *      In a test run with more than 200,000 random arguments on a VAX, the
 145  *      maximum observed error in ulps (units in the last place) was
 146  *      2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
 147  *
 148  * 2)  If atan2() uses true pi, then
 149  *
 150  *      acos(x) returns the exact acos(x) with error below about 2 ulps.
 151  *
 152  *      In a test run with more than 1,024,000 random arguments on a VAX, the
 153  *      maximum observed error in ulps (units in the last place) was
 154  *      2.15 ulps.
 155  */
 156
 157 double acos(x)
 158 double x;
 159 {
 160         double t,copysign(),atan2(),sqrt(),one=1.0;
 161 #if !defined(vax)&&!defined(tahoe)
 162         if(x!=x) return(x);
 163 #endif  /* !defined(vax)&&!defined(tahoe) */
 164         if( x != -1.0)
 165             t=atan2(sqrt((one-x)/(one+x)),one);
 166         else
 167             t=atan2(one,0.0);   /* t = PI/2 */
 168         return(t+t);
 169 }