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Annotation of 43BSD/usr.lib/libm/IEEE/cabs.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[] = "@(#)cabs.c     1.2 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* CABS(Z)
        !            19:  * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER  Z = X + iY
        !            20:  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
        !            21:  * CODED IN C BY K.C. NG, 11/28/84.
        !            22:  * REVISED BY K.C. NG, 7/12/85.
        !            23:  *
        !            24:  * Required kernel function :
        !            25:  *     hypot(x,y)
        !            26:  *
        !            27:  * Method :
        !            28:  *     cabs(z) = hypot(x,y) .
        !            29:  */
        !            30: 
        !            31: double cabs(z)
        !            32: struct { double x, y;} z;
        !            33: {
        !            34:        double hypot();
        !            35:        return(hypot(z.x,z.y));
        !            36: }
        !            37: 
        !            38: 
        !            39: /* HYPOT(X,Y)
        !            40:  * RETURN THE SQUARE ROOT OF X^2 + Y^2  WHERE Z=X+iY
        !            41:  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
        !            42:  * CODED IN C BY K.C. NG, 11/28/84; 
        !            43:  * REVISED BY K.C. NG, 7/12/85.
        !            44:  *
        !            45:  * Required system supported functions :
        !            46:  *     copysign(x,y)
        !            47:  *     finite(x)
        !            48:  *     scalb(x,N)
        !            49:  *     sqrt(x)
        !            50:  *
        !            51:  * Method :
        !            52:  *     1. replace x by |x| and y by |y|, and swap x and
        !            53:  *        y if y > x (hence x is never smaller than y).
        !            54:  *     2. Hypot(x,y) is computed by:
        !            55:  *        Case I, x/y > 2
        !            56:  *             
        !            57:  *                                    y
        !            58:  *             hypot = x + -----------------------------
        !            59:  *                                         2
        !            60:  *                         sqrt ( 1 + [x/y]  )  +  x/y
        !            61:  *
        !            62:  *        Case II, x/y <= 2 
        !            63:  *                                                y
        !            64:  *             hypot = x + --------------------------------------------------
        !            65:  *                                                          2 
        !            66:  *                                                     [x/y]   -  2
        !            67:  *                        (sqrt(2)+1) + (x-y)/y + -----------------------------
        !            68:  *                                                               2
        !            69:  *                                               sqrt ( 1 + [x/y]  )  + sqrt(2)
        !            70:  *
        !            71:  *
        !            72:  *
        !            73:  * Special cases:
        !            74:  *     hypot(x,y) is INF if x or y is +INF or -INF; else
        !            75:  *     hypot(x,y) is NAN if x or y is NAN.
        !            76:  *
        !            77:  * Accuracy:
        !            78:  *     hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units
        !            79:  *     in the last place). See Kahan's "Interval Arithmetic Options in the
        !            80:  *     Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics
        !            81:  *      1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate
        !            82:  *     code follows in comments.) In a test run with 500,000 random arguments
        !            83:  *     on a VAX, the maximum observed error was .959 ulps.
        !            84:  *
        !            85:  * Constants:
        !            86:  * The hexadecimal values are the intended ones for the following constants.
        !            87:  * The decimal values may be used, provided that the compiler will convert
        !            88:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            89:  * shown.
        !            90:  */
        !            91: 
        !            92: #ifdef VAX     /* VAX D format */
        !            93: /* static double */
        !            94: /* r2p1hi =  2.4142135623730950345E0     , Hex  2^  2   *  .9A827999FCEF32 */
        !            95: /* r2p1lo =  1.4349369327986523769E-17   , Hex  2^-55   *  .84597D89B3754B */
        !            96: /* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
        !            97: static long    r2p1hix[] = { 0x8279411a, 0xef3299fc};
        !            98: static long    r2p1lox[] = { 0x597d2484, 0x754b89b3};
        !            99: static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
        !           100: #define   r2p1hi    (*(double*)r2p1hix)
        !           101: #define   r2p1lo    (*(double*)r2p1lox)
        !           102: #define    sqrt2    (*(double*)sqrt2x)
        !           103: #else          /* IEEE double format */
        !           104: static double
        !           105: r2p1hi =  2.4142135623730949234E0     , /*Hex  2^1     *  1.3504F333F9DE6 */
        !           106: r2p1lo =  1.2537167179050217666E-16   , /*Hex  2^-53   *  1.21165F626CDD5 */
        !           107: sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
        !           108: #endif
        !           109: 
        !           110: double hypot(x,y)
        !           111: double x, y;
        !           112: {
        !           113:        static double zero=0, one=1, 
        !           114:                      small=1.0E-18;    /* fl(1+small)==1 */
        !           115:        static ibig=30; /* fl(1+2**(2*ibig))==1 */
        !           116:        double copysign(),scalb(),logb(),sqrt(),t,r;
        !           117:        int finite(), exp;
        !           118: 
        !           119:        if(finite(x))
        !           120:            if(finite(y))
        !           121:            {   
        !           122:                x=copysign(x,one);
        !           123:                y=copysign(y,one);
        !           124:                if(y > x) 
        !           125:                    { t=x; x=y; y=t; }
        !           126:                if(x == zero) return(zero);
        !           127:                if(y == zero) return(x);
        !           128:                exp= logb(x);
        !           129:                if(exp-(int)logb(y) > ibig )    
        !           130:                        /* raise inexact flag and return |x| */
        !           131:                   { one+small; return(x); }
        !           132: 
        !           133:            /* start computing sqrt(x^2 + y^2) */
        !           134:                r=x-y;
        !           135:                if(r>y) {       /* x/y > 2 */
        !           136:                    r=x/y;
        !           137:                    r=r+sqrt(one+r*r); }
        !           138:                else {          /* 1 <= x/y <= 2 */
        !           139:                    r/=y; t=r*(r+2.0);
        !           140:                    r+=t/(sqrt2+sqrt(2.0+t));
        !           141:                    r+=r2p1lo; r+=r2p1hi; }
        !           142: 
        !           143:                r=y/r;
        !           144:                return(x+r);
        !           145: 
        !           146:            }
        !           147: 
        !           148:            else if(y==y)          /* y is +-INF */
        !           149:                     return(copysign(y,one));
        !           150:            else 
        !           151:                     return(y);    /* y is NaN and x is finite */
        !           152: 
        !           153:        else if(x==x)              /* x is +-INF */
        !           154:                 return (copysign(x,one));
        !           155:        else if(finite(y))
        !           156:                 return(x);                /* x is NaN, y is finite */
        !           157:        else if(y!=y) return(y);  /* x and y is NaN */
        !           158:        else return(copysign(y,one));   /* y is INF */
        !           159: }
        !           160: 
        !           161: /* A faster but less accurate version of cabs(x,y) */
        !           162: #if 0
        !           163: double hypot(x,y)
        !           164: double x, y;
        !           165: {
        !           166:        static double zero=0, one=1;
        !           167:                      small=1.0E-18;    /* fl(1+small)==1 */
        !           168:        static ibig=30; /* fl(1+2**(2*ibig))==1 */
        !           169:        double copysign(),scalb(),logb(),sqrt(),temp;
        !           170:        int finite(), exp;
        !           171: 
        !           172:        if(finite(x))
        !           173:            if(finite(y))
        !           174:            {   
        !           175:                x=copysign(x,one);
        !           176:                y=copysign(y,one);
        !           177:                if(y > x) 
        !           178:                    { temp=x; x=y; y=temp; }
        !           179:                if(x == zero) return(zero);
        !           180:                if(y == zero) return(x);
        !           181:                exp= logb(x);
        !           182:                x=scalb(x,-exp);
        !           183:                if(exp-(int)logb(y) > ibig ) 
        !           184:                        /* raise inexact flag and return |x| */
        !           185:                   { one+small; return(scalb(x,exp)); }
        !           186:                else y=scalb(y,-exp);
        !           187:                return(scalb(sqrt(x*x+y*y),exp));
        !           188:            }
        !           189: 
        !           190:            else if(y==y)          /* y is +-INF */
        !           191:                     return(copysign(y,one));
        !           192:            else 
        !           193:                     return(y);    /* y is NaN and x is finite */
        !           194: 
        !           195:        else if(x==x)              /* x is +-INF */
        !           196:                 return (copysign(x,one));
        !           197:        else if(finite(y))
        !           198:                 return(x);                /* x is NaN, y is finite */
        !           199:        else if(y!=y) return(y);        /* x and y is NaN */
        !           200:        else return(copysign(y,one));   /* y is INF */
        !           201: }
        !           202: #endif

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