/* $NetBSD: n_support.c,v 1.5 2003/08/07 16:44:52 agc Exp $ */ /* * 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. 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" #include "trig.h" #if defined(__vax__)||defined(tahoe) /* VAX D format */ #include 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 ; #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; #endif /* defined(__vax__)||defined(tahoe) */ double scalb(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< -prep1 ) /* gradual underflow */ {*px=(*px&~mexp)|(short)(1<>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) */ } int finite(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(double x, double 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(double x) { double q,s,b,r; double t; 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(s1.0) 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(double x, double 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; 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(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