4.3BSD - /usr/src/usr.lib/libm/asincos.c

   1: /*
   2:  * Copyright (c) 1985 Regents of the University of California.
   3:  *
   4:  * Use and reproduction of this software are granted  in  accordance  with
   5:  * the terms and conditions specified in  the  Berkeley  Software  License
   6:  * Agreement (in particular, this entails acknowledgement of the programs'
   7:  * source, and inclusion of this notice) with the additional understanding
   8:  * that  all  recipients  should regard themselves as participants  in  an
   9:  * ongoing  research  project and hence should  feel  obligated  to report
  10:  * their  experiences (good or bad) with these elementary function  codes,
  11:  * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
  12:  */
  13: 
  14: #ifndef lint
  15: static char sccsid[] = "@(#)asincos.c	1.1 (Berkeley) 8/21/85";
  16: #endif not lint
  17: 
  18: /* ASIN(X)
  19:  * RETURNS ARC SINE OF X
  20:  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
  21:  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
  22:  *
  23:  * Required system supported functions:
  24:  *	copysign(x,y)
  25:  *	sqrt(x)
  26:  *
  27:  * Required kernel function:
  28:  *	atan2(y,x)
  29:  *
  30:  * Method :
  31:  *	asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
  32:  *		  computed as follows
  33:  *			1-x*x                     if x <  0.5,
  34:  *			2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
  35:  *
  36:  * Special cases:
  37:  *	if x is NaN, return x itself;
  38:  *	if |x|>1, return NaN.
  39:  *
  40:  * Accuracy:
  41:  * 1)  If atan2() uses machine PI, then
  42:  *
  43:  *	asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
  44:  *	and PI is the exact pi rounded to machine precision (see atan2 for
  45:  *      details):
  46:  *
  47:  *	in decimal:
  48:  *		pi = 3.141592653589793 23846264338327 .....
  49:  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
  50:  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
  51:  *
  52:  *	in hexadecimal:
  53:  *		pi = 3.243F6A8885A308D313198A2E....
  54:  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
  55:  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
  56:  *
  57:  *	In a test run with more than 200,000 random arguments on a VAX, the
  58:  *	maximum observed error in ulps (units in the last place) was
  59:  *	2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
  60:  *
  61:  * 2)  If atan2() uses true pi, then
  62:  *
  63:  *	asin(x) returns the exact asin(x) with error below about 2 ulps.
  64:  *
  65:  *	In a test run with more than 1,024,000 random arguments on a VAX, the
  66:  *	maximum observed error in ulps (units in the last place) was
  67:  *      1.99 ulps.
  68:  */
  69: 
  70: double asin(x)
  71: double x;
  72: {
  73:     double s,t,copysign(),atan2(),sqrt(),one=1.0;
  74: #ifndef VAX
  75:     if(x!=x) return(x); /* x is NaN */
  76: #endif
  77:     s=copysign(x,one);
  78:     if(s <= 0.5)
  79:         return(atan2(x,sqrt(one-x*x)));
  80:     else
  81:         { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
  82: 
  83: }
  84: 
  85: /* ACOS(X)
  86:  * RETURNS ARC COS OF X
  87:  * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
  88:  * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
  89:  *
  90:  * Required system supported functions:
  91:  *	copysign(x,y)
  92:  *	sqrt(x)
  93:  *
  94:  * Required kernel function:
  95:  *	atan2(y,x)
  96:  *
  97:  * Method :
  98:  *			      ________
  99:  *                           / 1 - x
 100:  *	acos(x) = 2*atan2(  / -------- , 1 ) .
 101:  *                        \/   1 + x
 102:  *
 103:  * Special cases:
 104:  *	if x is NaN, return x itself;
 105:  *	if |x|>1, return NaN.
 106:  *
 107:  * Accuracy:
 108:  * 1)  If atan2() uses machine PI, then
 109:  *
 110:  *	acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
 111:  *	and PI is the exact pi rounded to machine precision (see atan2 for
 112:  *      details):
 113:  *
 114:  *	in decimal:
 115:  *		pi = 3.141592653589793 23846264338327 .....
 116:  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 117:  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 118:  *
 119:  *	in hexadecimal:
 120:  *		pi = 3.243F6A8885A308D313198A2E....
 121:  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
 122:  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 123:  *
 124:  *	In a test run with more than 200,000 random arguments on a VAX, the
 125:  *	maximum observed error in ulps (units in the last place) was
 126:  *	2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
 127:  *
 128:  * 2)  If atan2() uses true pi, then
 129:  *
 130:  *	acos(x) returns the exact acos(x) with error below about 2 ulps.
 131:  *
 132:  *	In a test run with more than 1,024,000 random arguments on a VAX, the
 133:  *	maximum observed error in ulps (units in the last place) was
 134:  *	2.15 ulps.
 135:  */
 136: 
 137: double acos(x)
 138: double x;
 139: {
 140:     double t,copysign(),atan2(),sqrt(),one=1.0;
 141: #ifndef VAX
 142:     if(x!=x) return(x);
 143: #endif
 144:     if( x != -1.0)
 145:         t=atan2(sqrt((one-x)/(one+x)),one);
 146:     else
 147:         t=atan2(one,0.0);   /* t = PI/2 */
 148:     return(t+t);
 149: }

Defined functions

Defined variables