4.4BSD/usr/src/lib/libm/ieee/support.c
/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)support.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
/*
* Some IEEE standard 754 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 754 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 754 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.
*/
#include "mathimpl.h"
#if defined(vax)||defined(tahoe) /* VAX D format */
#include <errno.h>
static const unsigned short msign=0x7fff , mexp =0x7f80 ;
static const short prep1=57, gap=7, bias=129 ;
static const double novf=1.7E38, nunf=3.0E-39, zero=0.0 ;
#else /* defined(vax)||defined(tahoe) */
static const unsigned short msign=0x7fff, mexp =0x7ff0 ;
static const short prep1=54, gap=4, bias=1023 ;
static const double novf=1.7E308, nunf=3.0E-308,zero=0.0;
#endif /* defined(vax)||defined(tahoe) */
double scalb(x,N)
double x; int N;
{
int k;
#ifdef national
unsigned short *px=(unsigned short *) &x + 3;
#else /* national */
unsigned short *px=(unsigned short *) &x;
#endif /* national */
if( x == zero ) return(x);
#if defined(vax)||defined(tahoe)
if( (k= *px & mexp ) != ~msign ) {
if (N < -260)
return(nunf*nunf);
else if (N > 260) {
return(copysign(infnan(ERANGE),x));
}
#else /* defined(vax)||defined(tahoe) */
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 /* defined(vax)||defined(tahoe) */
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 /* national */
unsigned short *px=(unsigned short *) &x,
*py=(unsigned short *) &y;
#endif /* national */
#if defined(vax)||defined(tahoe)
if ( (*px & mexp) == 0 ) return(x);
#endif /* defined(vax)||defined(tahoe) */
*px = ( *px & msign ) | ( *py & ~msign );
return(x);
}
double logb(x)
double x;
{
#ifdef national
short *px=(short *) &x+3, k;
#else /* national */
short *px=(short *) &x, k;
#endif /* national */
#if defined(vax)||defined(tahoe)
return (int)(((*px&mexp)>>gap)-bias);
#else /* defined(vax)||defined(tahoe) */
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 /* defined(vax)||defined(tahoe) */
}
finite(x)
double x;
{
#if defined(vax)||defined(tahoe)
return(1);
#else /* defined(vax)||defined(tahoe) */
#ifdef national
return( (*((short *) &x+3 ) & mexp ) != mexp );
#else /* national */
return( (*((short *) &x ) & mexp ) != mexp );
#endif /* national */
#endif /* defined(vax)||defined(tahoe) */
}
double drem(x,p)
double x,p;
{
short sign;
double hp,dp,tmp;
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 /* national */
unsigned short
*px=(unsigned short *) &x ,
*pp=(unsigned short *) &p ,
*pd=(unsigned short *) &dp ,
*pt=(unsigned short *) &tmp;
#endif /* national */
*pp &= msign ;
#if defined(vax)||defined(tahoe)
if( ( *px & mexp ) == ~msign ) /* is x a reserved operand? */
#else /* defined(vax)||defined(tahoe) */
if( ( *px & mexp ) == mexp )
#endif /* defined(vax)||defined(tahoe) */
return (x-p)-(x-p); /* create nan if x is inf */
if (p == zero) {
#if defined(vax)||defined(tahoe)
return(infnan(EDOM));
#else /* defined(vax)||defined(tahoe) */
return zero/zero;
#endif /* defined(vax)||defined(tahoe) */
}
#if defined(vax)||defined(tahoe)
if( ( *pp & mexp ) == ~msign ) /* is p a reserved operand? */
#else /* defined(vax)||defined(tahoe) */
if( ( *pp & mexp ) == mexp )
#endif /* defined(vax)||defined(tahoe) */
{ if (p != p) return p; else return x;}
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 ;
#if defined(vax)||defined(tahoe)
if( x < tmp ) *pt -= 128 ;
#else /* defined(vax)||defined(tahoe) */
if( x < tmp ) *pt -= 16 ;
#endif /* defined(vax)||defined(tahoe) */
x -= tmp ;
}
if ( x > hp )
{ x -= p ; if ( x >= hp ) x -= p ; }
#if defined(vax)||defined(tahoe)
if (x)
#endif /* defined(vax)||defined(tahoe) */
*px ^= sign;
return( x);
}
}
double sqrt(x)
double x;
{
double q,s,b,r;
double t;
double const zero=0.0;
int m,n,i;
#if defined(vax)||defined(tahoe)
int k=54;
#else /* defined(vax)||defined(tahoe) */
int k=51;
#endif /* defined(vax)||defined(tahoe) */
/* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
if(x!=x||x==zero) return(x);
/* sqrt(negative) is invalid */
if(x<zero) {
#if defined(vax)||defined(tahoe)
return (infnan(EDOM)); /* NaN */
#else /* defined(vax)||defined(tahoe) */
return(zero/zero);
#endif /* defined(vax)||defined(tahoe) */
}
/* 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 const n0=3,n1=2,n2=1,n3=0;
#else /* VAX, SUN, ZILOG, TAHOE */
static const n0=0,n1=1,n2=2,n3=3;
#endif
static const unsigned short mexp =0x7ff0, m25 =0x0190, m57 =0x0390;
static const 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 const 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()
double const b54=134217728.*134217728.; /* b54=2**54 */
long mx,scalx;
long const 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 */
const int n0=1, n1=0;
#else
const int n0=0, n1=1;
#endif
/* Rounding Mode: RN ...round-to-nearest
* RZ ...round-towards 0
* RP ...round-towards +INF
* RM ...round-towards -INF
*/
const 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