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Annotation of 43BSD/usr.lib/libm/asincos.c, revision 1.1

1.1     ! root        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: }

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