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Annotation of 43BSD/usr.lib/libm/pow.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[] = "@(#)pow.c      4.5 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* POW(X,Y)  
        !            19:  * RETURN X**Y 
        !            20:  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
        !            21:  * CODED IN C BY K.C. NG, 1/8/85; 
        !            22:  * REVISED BY K.C. NG on 7/10/85.
        !            23:  *
        !            24:  * Required system supported functions:
        !            25:  *      scalb(x,n)      
        !            26:  *      logb(x)         
        !            27:  *     copysign(x,y)   
        !            28:  *     finite(x)       
        !            29:  *     drem(x,y)
        !            30:  *
        !            31:  * Required kernel functions:
        !            32:  *     exp__E(a,c)     ...return  exp(a+c) - 1 - a*a/2
        !            33:  *     log__L(x)       ...return  (log(1+x) - 2s)/s, s=x/(2+x) 
        !            34:  *     pow_p(x,y)      ...return  +(anything)**(finite non zero)
        !            35:  *
        !            36:  * Method
        !            37:  *     1. Compute and return log(x) in three pieces:
        !            38:  *             log(x) = n*ln2 + hi + lo,
        !            39:  *        where n is an integer.
        !            40:  *     2. Perform y*log(x) by simulating muti-precision arithmetic and 
        !            41:  *        return the answer in three pieces:
        !            42:  *             y*log(x) = m*ln2 + hi + lo,
        !            43:  *        where m is an integer.
        !            44:  *     3. Return x**y = exp(y*log(x))
        !            45:  *             = 2^m * ( exp(hi+lo) ).
        !            46:  *
        !            47:  * Special cases:
        !            48:  *     (anything) ** 0  is 1 ;
        !            49:  *     (anything) ** 1  is itself;
        !            50:  *     (anything) ** NaN is NaN;
        !            51:  *     NaN ** (anything except 0) is NaN;
        !            52:  *     +-(anything > 1) ** +INF is +INF;
        !            53:  *     +-(anything > 1) ** -INF is +0;
        !            54:  *     +-(anything < 1) ** +INF is +0;
        !            55:  *     +-(anything < 1) ** -INF is +INF;
        !            56:  *     +-1 ** +-INF is NaN and signal INVALID;
        !            57:  *     +0 ** +(anything except 0, NaN)  is +0;
        !            58:  *     -0 ** +(anything except 0, NaN, odd integer)  is +0;
        !            59:  *     +0 ** -(anything except 0, NaN)  is +INF and signal DIV-BY-ZERO;
        !            60:  *     -0 ** -(anything except 0, NaN, odd integer)  is +INF with signal;
        !            61:  *     -0 ** (odd integer) = -( +0 ** (odd integer) );
        !            62:  *     +INF ** +(anything except 0,NaN) is +INF;
        !            63:  *     +INF ** -(anything except 0,NaN) is +0;
        !            64:  *     -INF ** (odd integer) = -( +INF ** (odd integer) );
        !            65:  *     -INF ** (even integer) = ( +INF ** (even integer) );
        !            66:  *     -INF ** -(anything except integer,NaN) is NaN with signal;
        !            67:  *     -(x=anything) ** (k=integer) is (-1)**k * (x ** k);
        !            68:  *     -(anything except 0) ** (non-integer) is NaN with signal;
        !            69:  *
        !            70:  * Accuracy:
        !            71:  *     pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX,
        !            72:  *     and a Zilog Z8000,
        !            73:  *                     pow(integer,integer)
        !            74:  *     always returns the correct integer provided it is representable.
        !            75:  *     In a test run with 100,000 random arguments with 0 < x, y < 20.0
        !            76:  *     on a VAX, the maximum observed error was 1.79 ulps (units in the 
        !            77:  *     last place).
        !            78:  *
        !            79:  * Constants :
        !            80:  * The hexadecimal values are the intended ones for the following constants.
        !            81:  * The decimal values may be used, provided that the compiler will convert
        !            82:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            83:  * shown.
        !            84:  */
        !            85: 
        !            86: #ifdef VAX     /* VAX D format */
        !            87: #include <errno.h>
        !            88: extern double infnan();
        !            89: 
        !            90: /* double static */
        !            91: /* ln2hi  =  6.9314718055829871446E-1    , Hex  2^  0   *  .B17217F7D00000 */
        !            92: /* ln2lo  =  1.6465949582897081279E-12   , Hex  2^-39   *  .E7BCD5E4F1D9CC */
        !            93: /* invln2 =  1.4426950408889634148E0     , Hex  2^  1   *  .B8AA3B295C17F1 */
        !            94: /* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
        !            95: static long     ln2hix[] = { 0x72174031, 0x0000f7d0};
        !            96: static long     ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1};
        !            97: static long    invln2x[] = { 0xaa3b40b8, 0x17f1295c};
        !            98: static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
        !            99: #define    ln2hi    (*(double*)ln2hix)
        !           100: #define    ln2lo    (*(double*)ln2lox)
        !           101: #define   invln2    (*(double*)invln2x)
        !           102: #define    sqrt2    (*(double*)sqrt2x)
        !           103: #else  /* IEEE double */
        !           104: double static
        !           105: ln2hi  =  6.9314718036912381649E-1    , /*Hex  2^ -1   *  1.62E42FEE00000 */
        !           106: ln2lo  =  1.9082149292705877000E-10   , /*Hex  2^-33   *  1.A39EF35793C76 */
        !           107: invln2 =  1.4426950408889633870E0     , /*Hex  2^  0   *  1.71547652B82FE */
        !           108: sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
        !           109: #endif
        !           110: 
        !           111: double static zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0;
        !           112: 
        !           113: double pow(x,y)        
        !           114: double x,y;
        !           115: {
        !           116:        double drem(),pow_p(),copysign(),t;
        !           117:        int finite();
        !           118: 
        !           119:        if     (y==zero)      return(one);
        !           120:        else if(y==one
        !           121: #ifndef VAX
        !           122:                ||x!=x
        !           123: #endif
        !           124:                ) return( x );      /* if x is NaN or y=1 */
        !           125: #ifndef VAX
        !           126:        else if(y!=y)         return( y );      /* if y is NaN */
        !           127: #endif
        !           128:        else if(!finite(y))                     /* if y is INF */
        !           129:             if((t=copysign(x,one))==one) return(zero/zero);
        !           130:             else if(t>one) return((y>zero)?y:zero);
        !           131:             else return((y<zero)?-y:zero);
        !           132:        else if(y==two)       return(x*x);
        !           133:        else if(y==negone)    return(one/x);
        !           134: 
        !           135:     /* sign(x) = 1 */
        !           136:        else if(copysign(one,x)==one) return(pow_p(x,y));
        !           137: 
        !           138:     /* sign(x)= -1 */
        !           139:        /* if y is an even integer */
        !           140:        else if ( (t=drem(y,two)) == zero)      return( pow_p(-x,y) );
        !           141: 
        !           142:        /* if y is an odd integer */
        !           143:        else if (copysign(t,one) == one) return( -pow_p(-x,y) );
        !           144: 
        !           145:        /* Henceforth y is not an integer */
        !           146:        else if(x==zero)        /* x is -0 */
        !           147:            return((y>zero)?-x:one/(-x));
        !           148:        else {                  /* return NaN */
        !           149: #ifdef VAX
        !           150:            return (infnan(EDOM));      /* NaN */
        !           151: #else  /* IEEE double */
        !           152:            return(zero/zero);
        !           153: #endif
        !           154:        }
        !           155: }
        !           156: 
        !           157: /* pow_p(x,y) return x**y for x with sign=1 and finite y */
        !           158: static double pow_p(x,y)       
        !           159: double x,y;
        !           160: {
        !           161:         double logb(),scalb(),copysign(),log__L(),exp__E();
        !           162:         double c,s,t,z,tx,ty;
        !           163:         float sx,sy;
        !           164:        long k=0;
        !           165:         int n,m;
        !           166: 
        !           167:        if(x==zero||!finite(x)) {           /* if x is +INF or +0 */
        !           168: #ifdef VAX
        !           169:             return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */
        !           170: #else
        !           171:             return((y>zero)?x:one/x);
        !           172: #endif
        !           173:        }
        !           174:        if(x==1.0) return(x);   /* if x=1.0, return 1 since y is finite */
        !           175: 
        !           176:     /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */
        !           177:         z=scalb(x,-(n=logb(x)));  
        !           178: #ifndef VAX    /* IEEE double */       /* subnormal number */
        !           179:         if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} 
        !           180: #endif
        !           181:         if(z >= sqrt2 ) {n += 1; z *= half;}  z -= one ;
        !           182: 
        !           183:     /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */
        !           184:        s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); 
        !           185:        t= z-(c-tx); tx += (z-t)-c;
        !           186: 
        !           187:    /* if y*log(x) is neither too big nor too small */
        !           188:        if((s=logb(y)+logb(n+t)) < 12.0) 
        !           189:            if(s>-60.0) {
        !           190: 
        !           191:        /* compute y*log(x) ~ mlog2 + t + c */
        !           192:                s=y*(n+invln2*t);
        !           193:                 m=s+copysign(half,s);   /* m := nint(y*log(x)) */ 
        !           194:                k=y; 
        !           195:                if((double)k==y) {      /* if y is an integer */
        !           196:                    k = m-k*n;
        !           197:                    sx=t; tx+=(t-sx); }
        !           198:                else    {               /* if y is not an integer */    
        !           199:                    k =m;
        !           200:                    tx+=n*ln2lo;
        !           201:                    sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; }
        !           202:           /* end of checking whether k==y */
        !           203: 
        !           204:                 sy=y; ty=y-sy;          /* y ~ sy + ty */
        !           205:                s=(double)sx*sy-k*ln2hi;        /* (sy+ty)*(sx+tx)-kln2 */
        !           206:                z=(tx*ty-k*ln2lo);
        !           207:                tx=tx*sy; ty=sx*ty;
        !           208:                t=ty+z; t+=tx; t+=s;
        !           209:                c= -((((t-s)-tx)-ty)-z);
        !           210: 
        !           211:            /* return exp(y*log(x)) */
        !           212:                t += exp__E(t,c); return(scalb(one+t,m));
        !           213:             }
        !           214:        /* end of if log(y*log(x)) > -60.0 */
        !           215:            
        !           216:            else
        !           217:                /* exp(+- tiny) = 1 with inexact flag */
        !           218:                        {ln2hi+ln2lo; return(one);}
        !           219:            else if(copysign(one,y)*(n+invln2*t) <zero)
        !           220:                /* exp(-(big#)) underflows to zero */
        !           221:                        return(scalb(one,-5000)); 
        !           222:            else
        !           223:                /* exp(+(big#)) overflows to INF */
        !           224:                        return(scalb(one, 5000)); 
        !           225: 
        !           226: }

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