+From Prof. Kahan at UC at Berkeley
+/*
+ * 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[] = "@(#)support.c 1.1 (Berkeley) %G%";
+#endif not lint
+
+/*
+ * Some IEEE standard p754 recommended functions and remainder and sqrt for
+ * supporting the C elementary functions.
+ ******************************************************************************
+ * WARNING:
+ * These codes are developed (in double) to support the C elementary
+ * functions temporarily. They are not universal, and some of them are very
+ * slow (in particular, drem and sqrt is extremely inefficient). Each
+ * computer system should have its implementation of these functions using
+ * its own assembler.
+ ******************************************************************************
+ *
+ * IEEE p754 required operations:
+ * drem(x,p)
+ * returns x REM y = x - [x/y]*y , where [x/y] is the integer
+ * nearest x/y; in half way case, choose the even one.
+ * sqrt(x)
+ * returns the square root of x correctly rounded according to
+ * the rounding mod.
+ *
+ * IEEE p754 recommended functions:
+ * (a) copysign(x,y)
+ * returns x with the sign of y.
+ * (b) scalb(x,N)
+ * returns x * (2**N), for integer values N.
+ * (c) logb(x)
+ * returns the unbiased exponent of x, a signed integer in
+ * double precision, except that logb(0) is -INF, logb(INF)
+ * is +INF, and logb(NAN) is that NAN.
+ * (d) finite(x)
+ * returns the value TRUE if -INF < x < +INF and returns
+ * FALSE otherwise.
+ *
+ *
+ * CODED IN C BY K.C. NG, 11/25/84;
+ * REVISED BY K.C. NG on 1/22/85, 2/13/85, 3/24/85.
+ */
+
+
+#ifdef VAX /* VAX D format */
+ static unsigned short msign=0x7fff , mexp =0x7f80 ;
+ static short prep1=57, gap=7, bias=129 ;
+ static double novf=1.7E38, nunf=3.0E-39, zero=0.0 ;
+#else /*IEEE double format */
+ static unsigned short msign=0x7fff, mexp =0x7ff0 ;
+ static short prep1=54, gap=4, bias=1023 ;
+ static double novf=1.7E308, nunf=3.0E-308,zero=0.0;
+#endif
+
+double scalb(x,N)
+double x; int N;
+{
+ int k;
+ double scalb();
+
+#ifdef NATIONAL
+ unsigned short *px=(unsigned short *) &x + 3;
+#else /* VAX, SUN, ZILOG */
+ unsigned short *px=(unsigned short *) &x;
+#endif
+
+ if( x == zero ) return(x);
+
+#ifdef VAX
+ if( (k= *px & mexp ) != ~msign ) {
+ if( N<-260) return(nunf*nunf); else if(N>260) return(novf+novf);
+#else /* IEEE */
+ if( (k= *px & mexp ) != mexp ) {
+ if( N<-2100) return(nunf*nunf); else if(N>2100) return(novf+novf);
+ if( k == 0 ) {
+ x *= scalb(1.0,(int)prep1); N -= prep1; return(scalb(x,N));}
+#endif
+
+ if((k = (k>>gap)+ N) > 0 )
+ if( k < (mexp>>gap) ) *px = (*px&~mexp) | (k<<gap);
+ else x=novf+novf; /* overflow */
+ else
+ if( k > -prep1 )
+ /* gradual underflow */
+ {*px=(*px&~mexp)|(short)(1<<gap); x *= scalb(1.0,k-1);}
+ else
+ return(nunf*nunf);
+ }
+ return(x);
+}
+
+
+double copysign(x,y)
+double x,y;
+{
+#ifdef NATIONAL
+ unsigned short *px=(unsigned short *) &x+3,
+ *py=(unsigned short *) &y+3;
+#else /* VAX, SUN, ZILOG */
+ unsigned short *px=(unsigned short *) &x,
+ *py=(unsigned short *) &y;
+#endif
+
+#ifdef VAX
+ if ( (*px & mexp) == 0 ) return(x);
+#endif
+
+ *px = ( *px & msign ) | ( *py & ~msign );
+ return(x);
+}
+
+double logb(x)
+double x;
+{
+
+#ifdef NATIONAL
+ short *px=(short *) &x+3, k;
+#else /* VAX, SUN, ZILOG */
+ short *px=(short *) &x, k;
+#endif
+
+#ifdef VAX
+ return( ((*px & mexp)>>gap) - bias);
+#else /* IEEE */
+ if( (k= *px & mexp ) != mexp )
+ if ( k != 0 )
+ return ( (k>>gap) - bias );
+ else if( x != zero)
+ return ( -1022.0 );
+ else
+ return(-(1.0/zero));
+ else if(x != x)
+ return(x);
+ else
+ {*px &= msign; return(x);}
+#endif
+}
+
+finite(x)
+double x;
+{
+#ifdef VAX
+ return(1.0);
+#else /* IEEE */
+#ifdef NATIONAL
+ return( (*((short *) &x+3 ) & mexp ) != mexp );
+#else /* SUN, ZILOG */
+ return( (*((short *) &x ) & mexp ) != mexp );
+#endif
+#endif
+}
+
+double drem(x,p)
+double x,p;
+{
+ short sign;
+ double hp,dp,tmp,drem(),scalb();
+ unsigned short k;
+#ifdef NATIONAL
+ unsigned short
+ *px=(unsigned short *) &x +3,
+ *pp=(unsigned short *) &p +3,
+ *pd=(unsigned short *) &dp +3,
+ *pt=(unsigned short *) &tmp+3;
+#else /* VAX, SUN, ZILOG */
+ unsigned short
+ *px=(unsigned short *) &x ,
+ *pp=(unsigned short *) &p ,
+ *pd=(unsigned short *) &dp ,
+ *pt=(unsigned short *) &tmp;
+#endif
+
+ *pp &= msign ;
+
+#ifdef VAX
+ if( ( *px & mexp ) == ~msign || p == zero )
+#else /* IEEE */
+ if( ( *px & mexp ) == mexp || p == zero )
+#endif
+
+ return( (x != x)? x:zero/zero );
+
+ else if ( ((*pp & mexp)>>gap) <= 1 )
+ /* subnormal p, or almost subnormal p */
+ { double b; b=scalb(1.0,(int)prep1);
+ p *= b; x = drem(x,p); x *= b; return(drem(x,p)/b);}
+ else if ( p >= novf/2)
+ { p /= 2 ; x /= 2; return(drem(x,p)*2);}
+ else
+ {
+ dp=p+p; hp=p/2;
+ sign= *px & ~msign ;
+ *px &= msign ;
+ while ( x > dp )
+ {
+ k=(*px & mexp) - (*pd & mexp) ;
+ tmp = dp ;
+ *pt += k ;
+
+#ifdef VAX
+ if( x < tmp ) *pt -= 128 ;
+#else /* IEEE */
+ if( x < tmp ) *pt -= 16 ;
+#endif
+
+ x -= tmp ;
+ }
+ if ( x > hp )
+ { x -= p ; if ( x >= hp ) x -= p ; }
+
+ *px = *px ^ sign;
+ return( x);
+
+ }
+}
+double sqrt(x)
+double x;
+{
+ double q,s,b,r;
+ double logb(),scalb();
+ double t,zero=0.0;
+ int m,n,i,finite();
+#ifdef VAX
+ int k=54;
+#else /* IEEE */
+ int k=51;
+#endif
+
+ /* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
+ if(x!=x||x==zero) return(x);
+
+ /* sqrt(negative) is invalid */
+ if(x<zero) return(zero/zero);
+
+ /* sqrt(INF) is INF */
+ if(!finite(x)) return(x);
+
+ /* scale x to [1,4) */
+ n=logb(x);
+ x=scalb(x,-n);
+ if((m=logb(x))!=0) x=scalb(x,-m); /* subnormal number */
+ m += n;
+ n = m/2;
+ if((n+n)!=m) {x *= 2; m -=1; n=m/2;}
+
+ /* generate sqrt(x) bit by bit (accumulating in q) */
+ q=1.0; s=4.0; x -= 1.0; r=1;
+ for(i=1;i<=k;i++) {
+ t=s+1; x *= 4; r /= 2;
+ if(t<=x) {
+ s=t+t+2, x -= t; q += r;}
+ else
+ s *= 2;
+ }
+
+ /* generate the last bit and determine the final rounding */
+ r/=2; x *= 4;
+ if(x==zero) goto end; 100+r; /* trigger inexact flag */
+ if(s<x) {
+ q+=r; x -=s; s += 2; s *= 2; x *= 4;
+ t = (x-s)-5;
+ b=1.0+3*r/4; if(b==1.0) goto end; /* b==1 : Round-to-zero */
+ b=1.0+r/4; if(b>1.0) t=1; /* b>1 : Round-to-(+INF) */
+ if(t>=0) q+=r; } /* else: Round-to-nearest */
+ else {
+ s *= 2; x *= 4;
+ t = (x-s)-1;
+ b=1.0+3*r/4; if(b==1.0) goto end;
+ b=1.0+r/4; if(b>1.0) t=1;
+ if(t>=0) q+=r; }
+
+end: return(scalb(q,n));
+}
+
+#if 0
+/* DREM(X,Y)
+ * RETURN X REM Y =X-N*Y, N=[X/Y] ROUNDED (ROUNDED TO EVEN IN THE HALF WAY CASE)
+ * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
+ * INTENDED FOR ASSEMBLY LANGUAGE
+ * CODED IN C BY K.C. NG, 3/23/85, 4/8/85.
+ *
+ * Warning: this code should not get compiled in unless ALL of
+ * the following machine-dependent routines are supplied.
+ *
+ * Required machine dependent functions (not on a VAX):
+ * swapINX(i): save inexact flag and reset it to "i"
+ * swapENI(e): save inexact enable and reset it to "e"
+ */
+
+double drem(x,y)
+double x,y;
+{
+
+#ifdef NATIONAL /* order of words in floating point number */
+ static n0=3,n1=2,n2=1,n3=0;
+#else /* VAX, SUN, ZILOG */
+ static n0=0,n1=1,n2=2,n3=3;
+#endif
+
+ static unsigned short mexp =0x7ff0, m25 =0x0190, m57 =0x0390;
+ static double zero=0.0;
+ double hy,y1,t,t1;
+ short k;
+ long n;
+ int i,e;
+ unsigned short xexp,yexp, *px =(unsigned short *) &x ,
+ nx,nf, *py =(unsigned short *) &y ,
+ sign, *pt =(unsigned short *) &t ,
+ *pt1 =(unsigned short *) &t1 ;
+
+ xexp = px[n0] & mexp ; /* exponent of x */
+ yexp = py[n0] & mexp ; /* exponent of y */
+ sign = px[n0] &0x8000; /* sign of x */
+
+/* return NaN if x is NaN, or y is NaN, or x is INF, or y is zero */
+ if(x!=x) return(x); if(y!=y) return(y); /* x or y is NaN */
+ if( xexp == mexp ) return(zero/zero); /* x is INF */
+ if(y==zero) return(y/y);
+
+/* save the inexact flag and inexact enable in i and e respectively
+ * and reset them to zero
+ */
+ i=swapINX(0); e=swapENI(0);
+
+/* subnormal number */
+ nx=0;
+ if(yexp==0) {t=1.0,pt[n0]+=m57; y*=t; nx=m57;}
+
+/* if y is tiny (biased exponent <= 57), scale up y to y*2**57 */
+ if( yexp <= m57 ) {py[n0]+=m57; nx+=m57; yexp+=m57;}
+
+ nf=nx;
+ py[n0] &= 0x7fff;
+ px[n0] &= 0x7fff;
+
+/* mask off the least significant 27 bits of y */
+ t=y; pt[n3]=0; pt[n2]&=0xf800; y1=t;
+
+/* LOOP: argument reduction on x whenever x > y */
+loop:
+ while ( x > y )
+ {
+ t=y;
+ t1=y1;
+ xexp=px[n0]&mexp; /* exponent of x */
+ k=xexp-yexp-m25;
+ if(k>0) /* if x/y >= 2**26, scale up y so that x/y < 2**26 */
+ {pt[n0]+=k;pt1[n0]+=k;}
+ n=x/t; x=(x-n*t1)-n*(t-t1);
+ }
+ /* end while (x > y) */
+
+ if(nx!=0) {t=1.0; pt[n0]+=nx; x*=t; nx=0; goto loop;}
+
+/* final adjustment */
+
+ hy=y/2.0;
+ if(x>hy||((x==hy)&&n%2==1)) x-=y;
+ px[n0] ^= sign;
+ if(nf!=0) { t=1.0; pt[n0]-=nf; x*=t;}
+
+/* restore inexact flag and inexact enable */
+ swapINX(i); swapENI(e);
+
+ return(x);
+}
+#endif
+
+#if 0
+/* SQRT
+ * RETURN CORRECTLY ROUNDED (ACCORDING TO THE ROUNDING MODE) SQRT
+ * FOR IEEE DOUBLE PRECISION ONLY, INTENDED FOR ASSEMBLY LANGUAGE
+ * CODED IN C BY K.C. NG, 3/22/85.
+ *
+ * Warning: this code should not get compiled in unless ALL of
+ * the following machine-dependent routines are supplied.
+ *
+ * Required machine dependent functions:
+ * swapINX(i) ...return the status of INEXACT flag and reset it to "i"
+ * swapRM(r) ...return the current Rounding Mode and reset it to "r"
+ * swapENI(e) ...return the status of inexact enable and reset it to "e"
+ * addc(t) ...perform t=t+1 regarding t as a 64 bit unsigned integer
+ * subc(t) ...perform t=t-1 regarding t as a 64 bit unsigned integer
+ */
+
+static unsigned long table[] = {
+0, 1204, 3062, 5746, 9193, 13348, 18162, 23592, 29598, 36145, 43202, 50740,
+58733, 67158, 75992, 85215, 83599, 71378, 60428, 50647, 41945, 34246, 27478,
+21581, 16499, 12183, 8588, 5674, 3403, 1742, 661, 130, };
+
+double newsqrt(x)
+double x;
+{
+ double y,z,t,addc(),subc(),b54=134217728.*134217728.; /* b54=2**54 */
+ long mx,scalx,mexp=0x7ff00000;
+ int i,j,r,e,swapINX(),swapRM(),swapENI();
+ unsigned long *py=(unsigned long *) &y ,
+ *pt=(unsigned long *) &t ,
+ *px=(unsigned long *) &x ;
+#ifdef NATIONAL /* ordering of word in a floating point number */
+ int n0=1, n1=0;
+#else
+ int n0=0, n1=1;
+#endif
+/* Rounding Mode: RN ...round-to-nearest
+ * RZ ...round-towards 0
+ * RP ...round-towards +INF
+ * RM ...round-towards -INF
+ */
+ int RN=0,RZ=1,RP=2,RM=3;/* machine dependent: work on a Zilog Z8070
+ * and a National 32081 & 16081
+ */
+
+/* exceptions */
+ if(x!=x||x==0.0) return(x); /* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
+ if(x<0) return((x-x)/(x-x)); /* sqrt(negative) is invalid */
+ if((mx=px[n0]&mexp)==mexp) return(x); /* sqrt(+INF) is +INF */
+
+/* save, reset, initialize */
+ e=swapENI(0); /* ...save and reset the inexact enable */
+ i=swapINX(0); /* ...save INEXACT flag */
+ r=swapRM(RN); /* ...save and reset the Rounding Mode to RN */
+ scalx=0;
+
+/* subnormal number, scale up x to x*2**54 */
+ if(mx==0) {x *= b54 ; scalx-=0x01b00000;}
+
+/* scale x to avoid intermediate over/underflow:
+ * if (x > 2**512) x=x/2**512; if (x < 2**-512) x=x*2**512 */
+ if(mx>0x5ff00000) {px[n0] -= 0x20000000; scalx+= 0x10000000;}
+ if(mx<0x1ff00000) {px[n0] += 0x20000000; scalx-= 0x10000000;}
+
+/* magic initial approximation to almost 8 sig. bits */
+ py[n0]=(px[n0]>>1)+0x1ff80000;
+ py[n0]=py[n0]-table[(py[n0]>>15)&31];
+
+/* Heron's rule once with correction to improve y to almost 18 sig. bits */
+ t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0;
+
+/* triple to almost 56 sig. bits; now y approx. sqrt(x) to within 1 ulp */
+ t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y;
+ t=z/(t+x) ; pt[n0]+=0x00100000; y+=t;
+
+/* twiddle last bit to force y correctly rounded */
+ swapRM(RZ); /* ...set Rounding Mode to round-toward-zero */
+ swapINX(0); /* ...clear INEXACT flag */
+ swapENI(e); /* ...restore inexact enable status */
+ t=x/y; /* ...chopped quotient, possibly inexact */
+ j=swapINX(i); /* ...read and restore inexact flag */
+ if(j==0) { if(t==y) goto end; else t=subc(t); } /* ...t=t-ulp */
+ b54+0.1; /* ..trigger inexact flag, sqrt(x) is inexact */
+ if(r==RN) t=addc(t); /* ...t=t+ulp */
+ else if(r==RP) { t=addc(t);y=addc(y);}/* ...t=t+ulp;y=y+ulp; */
+ y=y+t; /* ...chopped sum */
+ py[n0]=py[n0]-0x00100000; /* ...correctly rounded sqrt(x) */
+end: py[n0]=py[n0]+scalx; /* ...scale back y */
+ swapRM(r); /* ...restore Rounding Mode */
+ return(y);
+}
+#endif
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