/* * Copyright (c) 1985 Regents of the University of California. * * Use and reproduction of this software are granted in accordance with * the terms and conditions specified in the Berkeley Software License * Agreement (in particular, this entails acknowledgement of the programs' * source, and inclusion of this notice) with the additional understanding * that all recipients should regard themselves as participants in an * ongoing research project and hence should feel obligated to report * their experiences (good or bad) with these elementary function codes, * using "sendbug 4bsd-bugs@BERKELEY", to the authors. */ #ifndef lint static char sccsid[] = "@(#)pow.c 4.5 (Berkeley) 8/21/85"; #endif not lint /* POW(X,Y) * RETURN X**Y * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) * CODED IN C BY K.C. NG, 1/8/85; * REVISED BY K.C. NG on 7/10/85. * * Required system supported functions: * scalb(x,n) * logb(x) * copysign(x,y) * finite(x) * drem(x,y) * * Required kernel functions: * exp__E(a,c) ...return exp(a+c) - 1 - a*a/2 * log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x) * pow_p(x,y) ...return +(anything)**(finite non zero) * * Method * 1. Compute and return log(x) in three pieces: * log(x) = n*ln2 + hi + lo, * where n is an integer. * 2. Perform y*log(x) by simulating muti-precision arithmetic and * return the answer in three pieces: * y*log(x) = m*ln2 + hi + lo, * where m is an integer. * 3. Return x**y = exp(y*log(x)) * = 2^m * ( exp(hi+lo) ). * * Special cases: * (anything) ** 0 is 1 ; * (anything) ** 1 is itself; * (anything) ** NaN is NaN; * NaN ** (anything except 0) is NaN; * +-(anything > 1) ** +INF is +INF; * +-(anything > 1) ** -INF is +0; * +-(anything < 1) ** +INF is +0; * +-(anything < 1) ** -INF is +INF; * +-1 ** +-INF is NaN and signal INVALID; * +0 ** +(anything except 0, NaN) is +0; * -0 ** +(anything except 0, NaN, odd integer) is +0; * +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO; * -0 ** -(anything except 0, NaN, odd integer) is +INF with signal; * -0 ** (odd integer) = -( +0 ** (odd integer) ); * +INF ** +(anything except 0,NaN) is +INF; * +INF ** -(anything except 0,NaN) is +0; * -INF ** (odd integer) = -( +INF ** (odd integer) ); * -INF ** (even integer) = ( +INF ** (even integer) ); * -INF ** -(anything except integer,NaN) is NaN with signal; * -(x=anything) ** (k=integer) is (-1)**k * (x ** k); * -(anything except 0) ** (non-integer) is NaN with signal; * * Accuracy: * pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX, * and a Zilog Z8000, * pow(integer,integer) * always returns the correct integer provided it is representable. * In a test run with 100,000 random arguments with 0 < x, y < 20.0 * on a VAX, the maximum observed error was 1.79 ulps (units in the * last place). * * Constants : * The hexadecimal values are the intended ones for the following constants. * The decimal values may be used, provided that the compiler will convert * from decimal to binary accurately enough to produce the hexadecimal values * shown. */ #ifdef VAX /* VAX D format */ #include extern double infnan(); /* double static */ /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */ /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */ /* invln2 = 1.4426950408889634148E0 , Hex 2^ 1 * .B8AA3B295C17F1 */ /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ static long ln2hix[] = { 0x72174031, 0x0000f7d0}; static long ln2lox[] = { 0xbcd52ce7, 0xd9cce4f1}; static long invln2x[] = { 0xaa3b40b8, 0x17f1295c}; static long sqrt2x[] = { 0x04f340b5, 0xde6533f9}; #define ln2hi (*(double*)ln2hix) #define ln2lo (*(double*)ln2lox) #define invln2 (*(double*)invln2x) #define sqrt2 (*(double*)sqrt2x) #else /* IEEE double */ double static ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */ ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */ invln2 = 1.4426950408889633870E0 , /*Hex 2^ 0 * 1.71547652B82FE */ sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ #endif double static zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0; double pow(x,y) double x,y; { double drem(),pow_p(),copysign(),t; int finite(); if (y==zero) return(one); else if(y==one #ifndef VAX ||x!=x #endif ) return( x ); /* if x is NaN or y=1 */ #ifndef VAX else if(y!=y) return( y ); /* if y is NaN */ #endif else if(!finite(y)) /* if y is INF */ if((t=copysign(x,one))==one) return(zero/zero); else if(t>one) return((y>zero)?y:zero); else return((yzero)?-x:one/(-x)); else { /* return NaN */ #ifdef VAX return (infnan(EDOM)); /* NaN */ #else /* IEEE double */ return(zero/zero); #endif } } /* pow_p(x,y) return x**y for x with sign=1 and finite y */ static double pow_p(x,y) double x,y; { double logb(),scalb(),copysign(),log__L(),exp__E(); double c,s,t,z,tx,ty; float sx,sy; long k=0; int n,m; if(x==zero||!finite(x)) { /* if x is +INF or +0 */ #ifdef VAX return((y>zero)?x:infnan(ERANGE)); /* if yzero)?x:one/x); #endif } if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */ /* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */ z=scalb(x,-(n=logb(x))); #ifndef VAX /* IEEE double */ /* subnormal number */ if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);} #endif if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ; /* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */ s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s)); t= z-(c-tx); tx += (z-t)-c; /* if y*log(x) is neither too big nor too small */ if((s=logb(y)+logb(n+t)) < 12.0) if(s>-60.0) { /* compute y*log(x) ~ mlog2 + t + c */ s=y*(n+invln2*t); m=s+copysign(half,s); /* m := nint(y*log(x)) */ k=y; if((double)k==y) { /* if y is an integer */ k = m-k*n; sx=t; tx+=(t-sx); } else { /* if y is not an integer */ k =m; tx+=n*ln2lo; sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; } /* end of checking whether k==y */ sy=y; ty=y-sy; /* y ~ sy + ty */ s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */ z=(tx*ty-k*ln2lo); tx=tx*sy; ty=sx*ty; t=ty+z; t+=tx; t+=s; c= -((((t-s)-tx)-ty)-z); /* return exp(y*log(x)) */ t += exp__E(t,c); return(scalb(one+t,m)); } /* end of if log(y*log(x)) > -60.0 */ else /* exp(+- tiny) = 1 with inexact flag */ {ln2hi+ln2lo; return(one);} else if(copysign(one,y)*(n+invln2*t)