4.3BSD - /usr/src/usr.lib/libm/pow.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[] = "@(#)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: }

Defined functions

Defined variables

Defined macros