4.4BSD/usr/src/old/libm/libm/pow.c
/*
* 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 <errno.h>
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((y<zero)?-y:zero);
else if(y==two) return(x*x);
else if(y==negone) return(one/x);
/* sign(x) = 1 */
else if(copysign(one,x)==one) return(pow_p(x,y));
/* sign(x)= -1 */
/* if y is an even integer */
else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) );
/* if y is an odd integer */
else if (copysign(t,one) == one) return( -pow_p(-x,y) );
/* Henceforth y is not an integer */
else if(x==zero) /* x is -0 */
return((y>zero)?-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 y<zero, return +INF */
#else
return((y>zero)?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) <zero)
/* exp(-(big#)) underflows to zero */
return(scalb(one,-5000));
else
/* exp(+(big#)) overflows to INF */
return(scalb(one, 5000));
}