254 lines
8.7 KiB
C
254 lines
8.7 KiB
C
/*
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* Copyright (c) 1985 Regents of the University of California.
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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#ifndef lint
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static char sccsid[] = "@(#)pow.c 5.7 (Berkeley) 10/9/90";
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#endif /* not lint */
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/* POW(X,Y)
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* RETURN X**Y
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* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
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* CODED IN C BY K.C. NG, 1/8/85;
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* REVISED BY K.C. NG on 7/10/85.
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*
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* Required system supported functions:
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* scalb(x,n)
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* logb(x)
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* copysign(x,y)
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* finite(x)
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* drem(x,y)
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*
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* Required kernel functions:
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* exp__E(a,c) ...return exp(a+c) - 1 - a*a/2
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* log__L(x) ...return (log(1+x) - 2s)/s, s=x/(2+x)
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* pow_p(x,y) ...return +(anything)**(finite non zero)
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*
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* Method
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* 1. Compute and return log(x) in three pieces:
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* log(x) = n*ln2 + hi + lo,
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* where n is an integer.
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* 2. Perform y*log(x) by simulating muti-precision arithmetic and
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* return the answer in three pieces:
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* y*log(x) = m*ln2 + hi + lo,
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* where m is an integer.
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* 3. Return x**y = exp(y*log(x))
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* = 2^m * ( exp(hi+lo) ).
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*
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* Special cases:
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* (anything) ** 0 is 1 ;
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* (anything) ** 1 is itself;
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* (anything) ** NaN is NaN;
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* NaN ** (anything except 0) is NaN;
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* +-(anything > 1) ** +INF is +INF;
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* +-(anything > 1) ** -INF is +0;
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* +-(anything < 1) ** +INF is +0;
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* +-(anything < 1) ** -INF is +INF;
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* +-1 ** +-INF is NaN and signal INVALID;
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* +0 ** +(anything except 0, NaN) is +0;
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* -0 ** +(anything except 0, NaN, odd integer) is +0;
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* +0 ** -(anything except 0, NaN) is +INF and signal DIV-BY-ZERO;
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* -0 ** -(anything except 0, NaN, odd integer) is +INF with signal;
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* -0 ** (odd integer) = -( +0 ** (odd integer) );
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* +INF ** +(anything except 0,NaN) is +INF;
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* +INF ** -(anything except 0,NaN) is +0;
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* -INF ** (odd integer) = -( +INF ** (odd integer) );
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* -INF ** (even integer) = ( +INF ** (even integer) );
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* -INF ** -(anything except integer,NaN) is NaN with signal;
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* -(x=anything) ** (k=integer) is (-1)**k * (x ** k);
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* -(anything except 0) ** (non-integer) is NaN with signal;
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*
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* Accuracy:
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* pow(x,y) returns x**y nearly rounded. In particular, on a SUN, a VAX,
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* and a Zilog Z8000,
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* pow(integer,integer)
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* always returns the correct integer provided it is representable.
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* In a test run with 100,000 random arguments with 0 < x, y < 20.0
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* on a VAX, the maximum observed error was 1.79 ulps (units in the
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* last place).
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*
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* Constants :
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* The hexadecimal values are the intended ones for the following constants.
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* The decimal values may be used, provided that the compiler will convert
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* from decimal to binary accurately enough to produce the hexadecimal values
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* shown.
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*/
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#include <errno.h>
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#include <limits.h>
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#include "mathimpl.h"
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vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
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vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
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vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
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vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65)
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ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
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ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)
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ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
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ic(sqrt2, 1.4142135623730951455E0, 0, 1.6A09E667F3BCD)
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#ifdef vccast
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#define ln2hi vccast(ln2hi)
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#define ln2lo vccast(ln2lo)
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#define invln2 vccast(invln2)
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#define sqrt2 vccast(sqrt2)
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#endif
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const static double zero=0.0, half=1.0/2.0, one=1.0, two=2.0, negone= -1.0;
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static double pow_p(double, double);
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double pow(x,y)
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double x,y;
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{
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double t;
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if (y==zero) return(one);
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else if(y==one
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#if !defined(vax)&&!defined(tahoe)
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||x!=x
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#endif /* !defined(vax)&&!defined(tahoe) */
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) return( x ); /* if x is NaN or y=1 */
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#if !defined(vax)&&!defined(tahoe)
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else if(y!=y) return( y ); /* if y is NaN */
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#endif /* !defined(vax)&&!defined(tahoe) */
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else if(!finite(y)) /* if y is INF */
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if((t=copysign(x,one))==one) return(zero/zero);
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else if(t>one) return((y>zero)?y:zero);
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else return((y<zero)?-y:zero);
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else if(y==two) return(x*x);
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else if(y==negone) return(one/x);
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/* sign(x) = 1 */
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else if(copysign(one,x)==one) return(pow_p(x,y));
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/* sign(x)= -1 */
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/* if y is an even integer */
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else if ( (t=drem(y,two)) == zero) return( pow_p(-x,y) );
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/* if y is an odd integer */
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else if (copysign(t,one) == one) return( -pow_p(-x,y) );
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/* Henceforth y is not an integer */
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else if(x==zero) /* x is -0 */
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return((y>zero)?-x:one/(-x));
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else { /* return NaN */
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#if defined(vax)||defined(tahoe)
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return (infnan(EDOM)); /* NaN */
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#else /* defined(vax)||defined(tahoe) */
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return(zero/zero);
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#endif /* defined(vax)||defined(tahoe) */
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}
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}
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#ifndef mc68881
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/* pow_p(x,y) return x**y for x with sign=1 and finite y */
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static double pow_p(x,y)
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double x,y;
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{
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double c,s,t,z,tx,ty;
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#ifdef tahoe
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double tahoe_tmp;
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#endif /* tahoe */
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double errtmp;
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float sx,sy;
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long k=0;
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int n,m;
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if(x==zero||!finite(x)) { /* if x is +INF or +0 */
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#if defined(vax)||defined(tahoe)
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return((y>zero)?x:infnan(ERANGE)); /* if y<zero, return +INF */
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#else /* defined(vax)||defined(tahoe) */
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return((y>zero)?x:one/x);
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#endif /* defined(vax)||defined(tahoe) */
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}
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if(x==1.0) return(x); /* if x=1.0, return 1 since y is finite */
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/* reduce x to z in [sqrt(1/2)-1, sqrt(2)-1] */
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z=scalb(x,-(n=logb(x)));
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#if !defined(vax)&&!defined(tahoe) /* IEEE double; subnormal number */
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if(n <= -1022) {n += (m=logb(z)); z=scalb(z,-m);}
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#endif /* !defined(vax)&&!defined(tahoe) */
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if(z >= sqrt2 ) {n += 1; z *= half;} z -= one ;
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/* log(x) = nlog2+log(1+z) ~ nlog2 + t + tx */
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s=z/(two+z); c=z*z*half; tx=s*(c+log__L(s*s));
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t= z-(c-tx); tx += (z-t)-c;
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/* if y*log(x) is neither too big nor too small */
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if((s=logb(y)+logb(n+t)) < 12.0)
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if(s>-60.0) {
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/* compute y*log(x) ~ mlog2 + t + c */
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s=y*(n+invln2*t);
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m=s+copysign(half,s); /* m := nint(y*log(x)) */
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k=y;
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if(y > (double)LONG_MIN && y < (double)LONG_MAX
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&& (double)(long)y==y) { /* y is an integer */
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k = m-(long)y*n;
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sx=t; tx+=(t-sx); }
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else { /* if y is not an integer */
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k =m;
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tx+=n*ln2lo;
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sx=(c=n*ln2hi)+t; tx+=(c-sx)+t; }
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/* end of checking whether k==y */
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sy=y; ty=y-sy; /* y ~ sy + ty */
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#ifdef tahoe
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s = (tahoe_tmp = sx)*sy-k*ln2hi;
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#else /* tahoe */
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s=(double)sx*sy-k*ln2hi; /* (sy+ty)*(sx+tx)-kln2 */
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#endif /* tahoe */
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z=(tx*ty-k*ln2lo);
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tx=tx*sy; ty=sx*ty;
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t=ty+z; t+=tx; t+=s;
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c= -((((t-s)-tx)-ty)-z);
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/* return exp(y*log(x)) */
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t += exp__E(t,c); return(scalb(one+t,m));
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}
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/* end of if log(y*log(x)) > -60.0 */
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else
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/* exp(+- tiny) = 1 with inexact flag */
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{errtmp=ln2hi+ln2lo; return(one);}
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else if(copysign(one,y)*(n+invln2*t) <zero)
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/* exp(-(big#)) underflows to zero */
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return(scalb(one,-5000));
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else
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/* exp(+(big#)) overflows to INF */
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return(scalb(one, 5000));
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}
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#endif /* mc68881 */
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