* 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.
"@(#)pow.c 4.5 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) %G%";
* 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:
* 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)
* 1. Compute and return log(x) in three pieces:
* log(x) = n*ln2 + hi + lo,
* 2. Perform y*log(x) by simulating muti-precision arithmetic and
* return the answer in three pieces:
* y*log(x) = m*ln2 + hi + lo,
* 3. Return x**y = exp(y*log(x))
* = 2^m * ( exp(hi+lo) ).
* (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;
* pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX,
* 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
* 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
#ifdef VAX /* VAX D format */
/* 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)
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 */double static zero
=0.0, half
=1.0/2.0, one
=1.0, two
=2.0, negone
= -1.0;
double drem(),pow_p(),copysign(),t
;
if (y
==zero
) return(one
);
) return( x
); /* if x is NaN or y=1 */
else if(y
!=y
) return( y
); /* if y is NaN */
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
);
else if(copysign(one
,x
)==one
) return(pow_p(x
,y
));
/* 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
));
return (infnan(EDOM
)); /* NaN */
/* pow_p(x,y) return x**y for x with sign=1 and finite y */
double logb(),scalb(),copysign(),log__L(),exp__E();
if(x
==zero
||!finite(x
)) { /* if x is +INF or +0 */
return((y
>zero
)?x
:infnan(ERANGE
)); /* if y<zero, return +INF */
return((y
>zero
)?x
:one
/x
);
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] */
#ifndef VAX /* IEEE double */ /* subnormal number */
if(n
<= -1022) {n
+= (m
=logb(z
)); z
=scalb(z
,-m
);}
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)
/* compute y*log(x) ~ mlog2 + t + c */
m
=s
+copysign(half
,s
); /* m := nint(y*log(x)) */ if((double)k
==y
) { /* if y is an integer */
else { /* if y is not an integer */
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 */ /* return exp(y*log(x)) */
t
+= exp__E(t
,c
); return(scalb(one
+t
,m
)); /* end of if log(y*log(x)) > -60.0 */
/* 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));
/* exp(+(big#)) overflows to INF */
return(scalb(one
, 5000));
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