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525 lines
17 KiB
C
525 lines
17 KiB
C
/*
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* Copyright (c) 1985, 1993
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* The Regents of the University of California. 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[] = "@(#)support.c 8.1 (Berkeley) 6/4/93";
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#endif /* not lint */
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/*
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* Some IEEE standard 754 recommended functions and remainder and sqrt for
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* supporting the C elementary functions.
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******************************************************************************
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* WARNING:
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* These codes are developed (in double) to support the C elementary
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* functions temporarily. They are not universal, and some of them are very
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* slow (in particular, drem and sqrt is extremely inefficient). Each
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* computer system should have its implementation of these functions using
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* its own assembler.
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******************************************************************************
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*
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* IEEE 754 required operations:
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* drem(x,p)
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* returns x REM y = x - [x/y]*y , where [x/y] is the integer
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* nearest x/y; in half way case, choose the even one.
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* sqrt(x)
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* returns the square root of x correctly rounded according to
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* the rounding mod.
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*
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* IEEE 754 recommended functions:
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* (a) copysign(x,y)
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* returns x with the sign of y.
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* (b) scalb(x,N)
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* returns x * (2**N), for integer values N.
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* (c) logb(x)
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* returns the unbiased exponent of x, a signed integer in
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* double precision, except that logb(0) is -INF, logb(INF)
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* is +INF, and logb(NAN) is that NAN.
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* (d) finite(x)
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* returns the value TRUE if -INF < x < +INF and returns
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* FALSE otherwise.
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*
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*
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* CODED IN C BY K.C. NG, 11/25/84;
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* REVISED BY K.C. NG on 1/22/85, 2/13/85, 3/24/85.
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*/
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#include "mathimpl.h"
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#if defined(vax)||defined(tahoe) /* VAX D format */
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#include <errno.h>
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static const unsigned short msign=0x7fff , mexp =0x7f80 ;
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static const short prep1=57, gap=7, bias=129 ;
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static const double novf=1.7E38, nunf=3.0E-39, zero=0.0 ;
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#else /* defined(vax)||defined(tahoe) */
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static const unsigned short msign=0x7fff, mexp =0x7ff0 ;
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static const short prep1=54, gap=4, bias=1023 ;
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static const double novf=1.7E308, nunf=3.0E-308,zero=0.0;
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#endif /* defined(vax)||defined(tahoe) */
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double scalb(x,N)
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double x; int N;
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{
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int k;
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#ifdef national
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unsigned short *px=(unsigned short *) &x + 3;
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#else /* national */
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unsigned short *px=(unsigned short *) &x;
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#endif /* national */
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if( x == zero ) return(x);
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#if defined(vax)||defined(tahoe)
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if( (k= *px & mexp ) != ~msign ) {
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if (N < -260)
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return(nunf*nunf);
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else if (N > 260) {
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return(copysign(infnan(ERANGE),x));
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}
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#else /* defined(vax)||defined(tahoe) */
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if( (k= *px & mexp ) != mexp ) {
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if( N<-2100) return(nunf*nunf); else if(N>2100) return(novf+novf);
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if( k == 0 ) {
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x *= scalb(1.0,(int)prep1); N -= prep1; return(scalb(x,N));}
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#endif /* defined(vax)||defined(tahoe) */
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if((k = (k>>gap)+ N) > 0 )
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if( k < (mexp>>gap) ) *px = (*px&~mexp) | (k<<gap);
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else x=novf+novf; /* overflow */
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else
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if( k > -prep1 )
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/* gradual underflow */
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{*px=(*px&~mexp)|(short)(1<<gap); x *= scalb(1.0,k-1);}
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else
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return(nunf*nunf);
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}
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return(x);
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}
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double copysign(x,y)
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double x,y;
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{
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#ifdef national
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unsigned short *px=(unsigned short *) &x+3,
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*py=(unsigned short *) &y+3;
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#else /* national */
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unsigned short *px=(unsigned short *) &x,
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*py=(unsigned short *) &y;
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#endif /* national */
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#if defined(vax)||defined(tahoe)
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if ( (*px & mexp) == 0 ) return(x);
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#endif /* defined(vax)||defined(tahoe) */
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*px = ( *px & msign ) | ( *py & ~msign );
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return(x);
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}
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double logb(x)
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double x;
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{
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#ifdef national
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short *px=(short *) &x+3, k;
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#else /* national */
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short *px=(short *) &x, k;
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#endif /* national */
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#if defined(vax)||defined(tahoe)
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return (int)(((*px&mexp)>>gap)-bias);
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#else /* defined(vax)||defined(tahoe) */
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if( (k= *px & mexp ) != mexp )
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if ( k != 0 )
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return ( (k>>gap) - bias );
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else if( x != zero)
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return ( -1022.0 );
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else
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return(-(1.0/zero));
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else if(x != x)
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return(x);
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else
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{*px &= msign; return(x);}
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#endif /* defined(vax)||defined(tahoe) */
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}
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finite(x)
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double x;
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{
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#if defined(vax)||defined(tahoe)
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return(1);
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#else /* defined(vax)||defined(tahoe) */
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#ifdef national
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return( (*((short *) &x+3 ) & mexp ) != mexp );
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#else /* national */
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return( (*((short *) &x ) & mexp ) != mexp );
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#endif /* national */
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#endif /* defined(vax)||defined(tahoe) */
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}
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double drem(x,p)
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double x,p;
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{
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short sign;
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double hp,dp,tmp;
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unsigned short k;
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#ifdef national
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unsigned short
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*px=(unsigned short *) &x +3,
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*pp=(unsigned short *) &p +3,
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*pd=(unsigned short *) &dp +3,
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*pt=(unsigned short *) &tmp+3;
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#else /* national */
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unsigned short
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*px=(unsigned short *) &x ,
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*pp=(unsigned short *) &p ,
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*pd=(unsigned short *) &dp ,
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*pt=(unsigned short *) &tmp;
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#endif /* national */
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*pp &= msign ;
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#if defined(vax)||defined(tahoe)
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if( ( *px & mexp ) == ~msign ) /* is x a reserved operand? */
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#else /* defined(vax)||defined(tahoe) */
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if( ( *px & mexp ) == mexp )
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#endif /* defined(vax)||defined(tahoe) */
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return (x-p)-(x-p); /* create nan if x is inf */
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if (p == zero) {
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#if defined(vax)||defined(tahoe)
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return(infnan(EDOM));
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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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#if defined(vax)||defined(tahoe)
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if( ( *pp & mexp ) == ~msign ) /* is p a reserved operand? */
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#else /* defined(vax)||defined(tahoe) */
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if( ( *pp & mexp ) == mexp )
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#endif /* defined(vax)||defined(tahoe) */
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{ if (p != p) return p; else return x;}
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else if ( ((*pp & mexp)>>gap) <= 1 )
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/* subnormal p, or almost subnormal p */
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{ double b; b=scalb(1.0,(int)prep1);
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p *= b; x = drem(x,p); x *= b; return(drem(x,p)/b);}
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else if ( p >= novf/2)
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{ p /= 2 ; x /= 2; return(drem(x,p)*2);}
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else
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{
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dp=p+p; hp=p/2;
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sign= *px & ~msign ;
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*px &= msign ;
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while ( x > dp )
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{
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k=(*px & mexp) - (*pd & mexp) ;
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tmp = dp ;
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*pt += k ;
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#if defined(vax)||defined(tahoe)
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if( x < tmp ) *pt -= 128 ;
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#else /* defined(vax)||defined(tahoe) */
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if( x < tmp ) *pt -= 16 ;
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#endif /* defined(vax)||defined(tahoe) */
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x -= tmp ;
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}
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if ( x > hp )
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{ x -= p ; if ( x >= hp ) x -= p ; }
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#if defined(vax)||defined(tahoe)
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if (x)
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#endif /* defined(vax)||defined(tahoe) */
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*px ^= sign;
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return( x);
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}
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}
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double sqrt(x)
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double x;
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{
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double q,s,b,r;
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double t;
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double const zero=0.0;
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int m,n,i;
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#if defined(vax)||defined(tahoe)
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int k=54;
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#else /* defined(vax)||defined(tahoe) */
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int k=51;
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#endif /* defined(vax)||defined(tahoe) */
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/* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
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if(x!=x||x==zero) return(x);
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/* sqrt(negative) is invalid */
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if(x<zero) {
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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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/* sqrt(INF) is INF */
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if(!finite(x)) return(x);
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/* scale x to [1,4) */
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n=logb(x);
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x=scalb(x,-n);
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if((m=logb(x))!=0) x=scalb(x,-m); /* subnormal number */
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m += n;
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n = m/2;
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if((n+n)!=m) {x *= 2; m -=1; n=m/2;}
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/* generate sqrt(x) bit by bit (accumulating in q) */
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q=1.0; s=4.0; x -= 1.0; r=1;
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for(i=1;i<=k;i++) {
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t=s+1; x *= 4; r /= 2;
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if(t<=x) {
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s=t+t+2, x -= t; q += r;}
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else
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s *= 2;
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}
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/* generate the last bit and determine the final rounding */
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r/=2; x *= 4;
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if(x==zero) goto end; 100+r; /* trigger inexact flag */
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if(s<x) {
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q+=r; x -=s; s += 2; s *= 2; x *= 4;
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t = (x-s)-5;
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b=1.0+3*r/4; if(b==1.0) goto end; /* b==1 : Round-to-zero */
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b=1.0+r/4; if(b>1.0) t=1; /* b>1 : Round-to-(+INF) */
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if(t>=0) q+=r; } /* else: Round-to-nearest */
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else {
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s *= 2; x *= 4;
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t = (x-s)-1;
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b=1.0+3*r/4; if(b==1.0) goto end;
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b=1.0+r/4; if(b>1.0) t=1;
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if(t>=0) q+=r; }
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end: return(scalb(q,n));
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}
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#if 0
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/* DREM(X,Y)
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* RETURN X REM Y =X-N*Y, N=[X/Y] ROUNDED (ROUNDED TO EVEN IN THE HALF WAY CASE)
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* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
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* INTENDED FOR ASSEMBLY LANGUAGE
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* CODED IN C BY K.C. NG, 3/23/85, 4/8/85.
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*
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* Warning: this code should not get compiled in unless ALL of
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* the following machine-dependent routines are supplied.
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*
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* Required machine dependent functions (not on a VAX):
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* swapINX(i): save inexact flag and reset it to "i"
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* swapENI(e): save inexact enable and reset it to "e"
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*/
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double drem(x,y)
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double x,y;
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{
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#ifdef national /* order of words in floating point number */
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static const n0=3,n1=2,n2=1,n3=0;
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#else /* VAX, SUN, ZILOG, TAHOE */
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static const n0=0,n1=1,n2=2,n3=3;
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#endif
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static const unsigned short mexp =0x7ff0, m25 =0x0190, m57 =0x0390;
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static const double zero=0.0;
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double hy,y1,t,t1;
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short k;
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long n;
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int i,e;
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unsigned short xexp,yexp, *px =(unsigned short *) &x ,
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nx,nf, *py =(unsigned short *) &y ,
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sign, *pt =(unsigned short *) &t ,
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*pt1 =(unsigned short *) &t1 ;
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xexp = px[n0] & mexp ; /* exponent of x */
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yexp = py[n0] & mexp ; /* exponent of y */
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sign = px[n0] &0x8000; /* sign of x */
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/* return NaN if x is NaN, or y is NaN, or x is INF, or y is zero */
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if(x!=x) return(x); if(y!=y) return(y); /* x or y is NaN */
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if( xexp == mexp ) return(zero/zero); /* x is INF */
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if(y==zero) return(y/y);
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/* save the inexact flag and inexact enable in i and e respectively
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* and reset them to zero
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*/
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i=swapINX(0); e=swapENI(0);
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/* subnormal number */
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nx=0;
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if(yexp==0) {t=1.0,pt[n0]+=m57; y*=t; nx=m57;}
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/* if y is tiny (biased exponent <= 57), scale up y to y*2**57 */
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if( yexp <= m57 ) {py[n0]+=m57; nx+=m57; yexp+=m57;}
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nf=nx;
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py[n0] &= 0x7fff;
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px[n0] &= 0x7fff;
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/* mask off the least significant 27 bits of y */
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t=y; pt[n3]=0; pt[n2]&=0xf800; y1=t;
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/* LOOP: argument reduction on x whenever x > y */
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loop:
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while ( x > y )
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{
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t=y;
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t1=y1;
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xexp=px[n0]&mexp; /* exponent of x */
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k=xexp-yexp-m25;
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if(k>0) /* if x/y >= 2**26, scale up y so that x/y < 2**26 */
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{pt[n0]+=k;pt1[n0]+=k;}
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n=x/t; x=(x-n*t1)-n*(t-t1);
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}
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/* end while (x > y) */
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if(nx!=0) {t=1.0; pt[n0]+=nx; x*=t; nx=0; goto loop;}
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/* final adjustment */
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hy=y/2.0;
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if(x>hy||((x==hy)&&n%2==1)) x-=y;
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px[n0] ^= sign;
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if(nf!=0) { t=1.0; pt[n0]-=nf; x*=t;}
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/* restore inexact flag and inexact enable */
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swapINX(i); swapENI(e);
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return(x);
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}
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#endif
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#if 0
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/* SQRT
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* RETURN CORRECTLY ROUNDED (ACCORDING TO THE ROUNDING MODE) SQRT
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* FOR IEEE DOUBLE PRECISION ONLY, INTENDED FOR ASSEMBLY LANGUAGE
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* CODED IN C BY K.C. NG, 3/22/85.
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*
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* Warning: this code should not get compiled in unless ALL of
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* the following machine-dependent routines are supplied.
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*
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* Required machine dependent functions:
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* swapINX(i) ...return the status of INEXACT flag and reset it to "i"
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* swapRM(r) ...return the current Rounding Mode and reset it to "r"
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* swapENI(e) ...return the status of inexact enable and reset it to "e"
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* addc(t) ...perform t=t+1 regarding t as a 64 bit unsigned integer
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* subc(t) ...perform t=t-1 regarding t as a 64 bit unsigned integer
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*/
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static const unsigned long table[] = {
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0, 1204, 3062, 5746, 9193, 13348, 18162, 23592, 29598, 36145, 43202, 50740,
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58733, 67158, 75992, 85215, 83599, 71378, 60428, 50647, 41945, 34246, 27478,
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21581, 16499, 12183, 8588, 5674, 3403, 1742, 661, 130, };
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double newsqrt(x)
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double x;
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{
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double y,z,t,addc(),subc()
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double const b54=134217728.*134217728.; /* b54=2**54 */
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long mx,scalx;
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long const mexp=0x7ff00000;
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int i,j,r,e,swapINX(),swapRM(),swapENI();
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unsigned long *py=(unsigned long *) &y ,
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*pt=(unsigned long *) &t ,
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*px=(unsigned long *) &x ;
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#ifdef national /* ordering of word in a floating point number */
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const int n0=1, n1=0;
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#else
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const int n0=0, n1=1;
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#endif
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/* Rounding Mode: RN ...round-to-nearest
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* RZ ...round-towards 0
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* RP ...round-towards +INF
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* RM ...round-towards -INF
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*/
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const int RN=0,RZ=1,RP=2,RM=3;
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/* machine dependent: work on a Zilog Z8070
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* and a National 32081 & 16081
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*/
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/* exceptions */
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if(x!=x||x==0.0) return(x); /* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
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if(x<0) return((x-x)/(x-x)); /* sqrt(negative) is invalid */
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if((mx=px[n0]&mexp)==mexp) return(x); /* sqrt(+INF) is +INF */
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/* save, reset, initialize */
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e=swapENI(0); /* ...save and reset the inexact enable */
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i=swapINX(0); /* ...save INEXACT flag */
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r=swapRM(RN); /* ...save and reset the Rounding Mode to RN */
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scalx=0;
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/* subnormal number, scale up x to x*2**54 */
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if(mx==0) {x *= b54 ; scalx-=0x01b00000;}
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/* scale x to avoid intermediate over/underflow:
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* if (x > 2**512) x=x/2**512; if (x < 2**-512) x=x*2**512 */
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if(mx>0x5ff00000) {px[n0] -= 0x20000000; scalx+= 0x10000000;}
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if(mx<0x1ff00000) {px[n0] += 0x20000000; scalx-= 0x10000000;}
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/* magic initial approximation to almost 8 sig. bits */
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py[n0]=(px[n0]>>1)+0x1ff80000;
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py[n0]=py[n0]-table[(py[n0]>>15)&31];
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/* Heron's rule once with correction to improve y to almost 18 sig. bits */
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t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0;
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/* triple to almost 56 sig. bits; now y approx. sqrt(x) to within 1 ulp */
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t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y;
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t=z/(t+x) ; pt[n0]+=0x00100000; y+=t;
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/* twiddle last bit to force y correctly rounded */
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swapRM(RZ); /* ...set Rounding Mode to round-toward-zero */
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swapINX(0); /* ...clear INEXACT flag */
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swapENI(e); /* ...restore inexact enable status */
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t=x/y; /* ...chopped quotient, possibly inexact */
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j=swapINX(i); /* ...read and restore inexact flag */
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if(j==0) { if(t==y) goto end; else t=subc(t); } /* ...t=t-ulp */
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b54+0.1; /* ..trigger inexact flag, sqrt(x) is inexact */
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if(r==RN) t=addc(t); /* ...t=t+ulp */
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else if(r==RP) { t=addc(t);y=addc(y);}/* ...t=t+ulp;y=y+ulp; */
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y=y+t; /* ...chopped sum */
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py[n0]=py[n0]-0x00100000; /* ...correctly rounded sqrt(x) */
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end: py[n0]=py[n0]+scalx; /* ...scale back y */
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swapRM(r); /* ...restore Rounding Mode */
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return(y);
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}
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#endif
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