mirror of
https://sourceware.org/git/glibc.git
synced 2024-11-23 05:20:06 +00:00
521 lines
17 KiB
C
521 lines
17 KiB
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.
|
|
* 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
|