/* * IBM Accurate Mathematical Library * Written by International Business Machines Corp. * Copyright (C) 2001 Free Software Foundation, Inc. * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU Lesser General Public License as published by * the Free Software Foundation; either version 2.1 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU Lesser General Public License for more details. * * You should have received a copy of the GNU Lesser General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */ /***********************************************************************/ /*MODULE_NAME: dla.h */ /* */ /* This file holds C language macros for 'Double Length Floating Point */ /* Arithmetic'. The macros are based on the paper: */ /* T.J.Dekker, "A floating-point Technique for extending the */ /* Available Precision", Number. Math. 18, 224-242 (1971). */ /* A Double-Length number is defined by a pair (r,s), of IEEE double */ /* precision floating point numbers that satisfy, */ /* */ /* abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)). */ /* */ /* The computer arithmetic assumed is IEEE double precision in */ /* round to nearest mode. All variables in the macros must be of type */ /* IEEE double. */ /***********************************************************************/ /* CN = 1+2**27 = '41a0000002000000' IEEE double format */ #define CN 134217729.0 /* Exact addition of two single-length floating point numbers, Dekker. */ /* The macro produces a double-length number (z,zz) that satisfies */ /* z+zz = x+y exactly. */ #define EADD(x,y,z,zz) \ z=(x)+(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x)); /* Exact subtraction of two single-length floating point numbers, Dekker. */ /* The macro produces a double-length number (z,zz) that satisfies */ /* z+zz = x-y exactly. */ #define ESUB(x,y,z,zz) \ z=(x)-(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z))); /* Exact multiplication of two single-length floating point numbers, */ /* Veltkamp. The macro produces a double-length number (z,zz) that */ /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary */ /* storage variables of type double. */ #define EMULV(x,y,z,zz,p,hx,tx,hy,ty) \ p=CN*(x); hx=((x)-p)+p; tx=(x)-hx; \ p=CN*(y); hy=((y)-p)+p; ty=(y)-hy; \ z=(x)*(y); zz=(((hx*hy-z)+hx*ty)+tx*hy)+tx*ty; /* Exact multiplication of two single-length floating point numbers, Dekker. */ /* The macro produces a nearly double-length number (z,zz) (see Dekker) */ /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */ /* storage variables of type double. */ #define MUL12(x,y,z,zz,p,hx,tx,hy,ty,q) \ p=CN*(x); hx=((x)-p)+p; tx=(x)-hx; \ p=CN*(y); hy=((y)-p)+p; ty=(y)-hy; \ p=hx*hy; q=hx*ty+tx*hy; z=p+q; zz=((p-z)+q)+tx*ty; /* Double-length addition, Dekker. The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */ /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. r,s are temporary */ /* storage variables of type double. */ #define ADD2(x,xx,y,yy,z,zz,r,s) \ r=(x)+(y); s=(ABS(x)>ABS(y)) ? \ (((((x)-r)+(y))+(yy))+(xx)) : \ (((((y)-r)+(x))+(xx))+(yy)); \ z=r+s; zz=(r-z)+s; /* Double-length subtraction, Dekker. The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */ /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. r,s are temporary */ /* storage variables of type double. */ #define SUB2(x,xx,y,yy,z,zz,r,s) \ r=(x)-(y); s=(ABS(x)>ABS(y)) ? \ (((((x)-r)-(y))-(yy))+(xx)) : \ ((((x)-((y)+r))+(xx))-(yy)); \ z=r+s; zz=(r-z)+s; /* Double-length multiplication, Dekker. The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */ /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */ /* temporary storage variables of type double. */ #define MUL2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc) \ MUL12(x,y,c,cc,p,hx,tx,hy,ty,q) \ cc=((x)*(yy)+(xx)*(y))+cc; z=c+cc; zz=(c-z)+cc; /* Double-length division, Dekker. The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */ /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */ /* are temporary storage variables of type double. */ #define DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu) \ c=(x)/(y); MUL12(c,y,u,uu,p,hx,tx,hy,ty,q) \ cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc; /* Double-length addition, slower but more accurate than ADD2. */ /* The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */ /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ /* are temporary storage variables of type double. */ #define ADD2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w) \ r=(x)+(y); \ if (ABS(x)>ABS(y)) { rr=((x)-r)+(y); s=(rr+(yy))+(xx); } \ else { rr=((y)-r)+(x); s=(rr+(xx))+(yy); } \ if (rr!=0.0) { \ z=r+s; zz=(r-z)+s; } \ else { \ ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)+(yy)) : (((yy)-s)+(xx)); \ u=r+s; \ uu=(ABS(r)>ABS(s)) ? ((r-u)+s) : ((s-u)+r) ; \ w=uu+ss; z=u+w; \ zz=(ABS(u)>ABS(w)) ? ((u-z)+w) : ((w-z)+u) ; } /* Double-length subtraction, slower but more accurate than SUB2. */ /* The macro produces a double-length */ /* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */ /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */ /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ /* are temporary storage variables of type double. */ #define SUB2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w) \ r=(x)-(y); \ if (ABS(x)>ABS(y)) { rr=((x)-r)-(y); s=(rr-(yy))+(xx); } \ else { rr=(x)-((y)+r); s=(rr+(xx))-(yy); } \ if (rr!=0.0) { \ z=r+s; zz=(r-z)+s; } \ else { \ ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)-(yy)) : ((xx)-((yy)+s)); \ u=r+s; \ uu=(ABS(r)>ABS(s)) ? ((r-u)+s) : ((s-u)+r) ; \ w=uu+ss; z=u+w; \ zz=(ABS(u)>ABS(w)) ? ((u-z)+w) : ((w-z)+u) ; }