1/* 2 * IBM Accurate Mathematical Library 3 * Written by International Business Machines Corp. 4 * Copyright (C) 2001 Free Software Foundation, Inc. 5 * 6 * This program is free software; you can redistribute it and/or modify 7 * it under the terms of the GNU Lesser General Public License as published by 8 * the Free Software Foundation; either version 2.1 of the License, or 9 * (at your option) any later version. 10 * 11 * This program is distributed in the hope that it will be useful, 12 * but WITHOUT ANY WARRANTY; without even the implied warranty of 13 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 14 * GNU Lesser General Public License for more details. 15 * 16 * You should have received a copy of the GNU Lesser General Public License 17 * along with this program; if not, write to the Free Software 18 * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. 19 */ 20 21/***********************************************************************/ 22/*MODULE_NAME: dla.h */ 23/* */ 24/* This file holds C language macros for 'Double Length Floating Point */ 25/* Arithmetic'. The macros are based on the paper: */ 26/* T.J.Dekker, "A floating-point Technique for extending the */ 27/* Available Precision", Number. Math. 18, 224-242 (1971). */ 28/* A Double-Length number is defined by a pair (r,s), of IEEE double */ 29/* precision floating point numbers that satisfy, */ 30/* */ 31/* abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)). */ 32/* */ 33/* The computer arithmetic assumed is IEEE double precision in */ 34/* round to nearest mode. All variables in the macros must be of type */ 35/* IEEE double. */ 36/***********************************************************************/ 37 38/* CN = 1+2**27 = '41a0000002000000' IEEE double format */ 39#define CN 134217729.0 40 41 42/* Exact addition of two single-length floating point numbers, Dekker. */ 43/* The macro produces a double-length number (z,zz) that satisfies */ 44/* z+zz = x+y exactly. */ 45 46#define EADD(x,y,z,zz) \ 47 z=(x)+(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x)); 48 49 50/* Exact subtraction of two single-length floating point numbers, Dekker. */ 51/* The macro produces a double-length number (z,zz) that satisfies */ 52/* z+zz = x-y exactly. */ 53 54#define ESUB(x,y,z,zz) \ 55 z=(x)-(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z))); 56 57 58/* Exact multiplication of two single-length floating point numbers, */ 59/* Veltkamp. The macro produces a double-length number (z,zz) that */ 60/* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary */ 61/* storage variables of type double. */ 62 63#define EMULV(x,y,z,zz,p,hx,tx,hy,ty) \ 64 p=CN*(x); hx=((x)-p)+p; tx=(x)-hx; \ 65 p=CN*(y); hy=((y)-p)+p; ty=(y)-hy; \ 66 z=(x)*(y); zz=(((hx*hy-z)+hx*ty)+tx*hy)+tx*ty; 67 68 69/* Exact multiplication of two single-length floating point numbers, Dekker. */ 70/* The macro produces a nearly double-length number (z,zz) (see Dekker) */ 71/* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */ 72/* storage variables of type double. */ 73 74#define MUL12(x,y,z,zz,p,hx,tx,hy,ty,q) \ 75 p=CN*(x); hx=((x)-p)+p; tx=(x)-hx; \ 76 p=CN*(y); hy=((y)-p)+p; ty=(y)-hy; \ 77 p=hx*hy; q=hx*ty+tx*hy; z=p+q; zz=((p-z)+q)+tx*ty; 78 79 80/* Double-length addition, Dekker. The macro produces a double-length */ 81/* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */ 82/* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ 83/* are assumed to be double-length numbers. r,s are temporary */ 84/* storage variables of type double. */ 85 86#define ADD2(x,xx,y,yy,z,zz,r,s) \ 87 r=(x)+(y); s=(ABS(x)>ABS(y)) ? \ 88 (((((x)-r)+(y))+(yy))+(xx)) : \ 89 (((((y)-r)+(x))+(xx))+(yy)); \ 90 z=r+s; zz=(r-z)+s; 91 92 93/* Double-length subtraction, Dekker. The macro produces a double-length */ 94/* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */ 95/* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ 96/* are assumed to be double-length numbers. r,s are temporary */ 97/* storage variables of type double. */ 98 99#define SUB2(x,xx,y,yy,z,zz,r,s) \ 100 r=(x)-(y); s=(ABS(x)>ABS(y)) ? \ 101 (((((x)-r)-(y))-(yy))+(xx)) : \ 102 ((((x)-((y)+r))+(xx))-(yy)); \ 103 z=r+s; zz=(r-z)+s; 104 105 106/* Double-length multiplication, Dekker. The macro produces a double-length */ 107/* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */ 108/* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */ 109/* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */ 110/* temporary storage variables of type double. */ 111 112#define MUL2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc) \ 113 MUL12(x,y,c,cc,p,hx,tx,hy,ty,q) \ 114 cc=((x)*(yy)+(xx)*(y))+cc; z=c+cc; zz=(c-z)+cc; 115 116 117/* Double-length division, Dekker. The macro produces a double-length */ 118/* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */ 119/* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */ 120/* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */ 121/* are temporary storage variables of type double. */ 122 123#define DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu) \ 124 c=(x)/(y); MUL12(c,y,u,uu,p,hx,tx,hy,ty,q) \ 125 cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc; 126 127 128/* Double-length addition, slower but more accurate than ADD2. */ 129/* The macro produces a double-length */ 130/* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */ 131/* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */ 132/* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ 133/* are temporary storage variables of type double. */ 134 135#define ADD2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w) \ 136 r=(x)+(y); \ 137 if (ABS(x)>ABS(y)) { rr=((x)-r)+(y); s=(rr+(yy))+(xx); } \ 138 else { rr=((y)-r)+(x); s=(rr+(xx))+(yy); } \ 139 if (rr!=0.0) { \ 140 z=r+s; zz=(r-z)+s; } \ 141 else { \ 142 ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)+(yy)) : (((yy)-s)+(xx)); \ 143 u=r+s; \ 144 uu=(ABS(r)>ABS(s)) ? ((r-u)+s) : ((s-u)+r) ; \ 145 w=uu+ss; z=u+w; \ 146 zz=(ABS(u)>ABS(w)) ? ((u-z)+w) : ((w-z)+u) ; } 147 148 149/* Double-length subtraction, slower but more accurate than SUB2. */ 150/* The macro produces a double-length */ 151/* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */ 152/* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */ 153/* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ 154/* are temporary storage variables of type double. */ 155 156#define SUB2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w) \ 157 r=(x)-(y); \ 158 if (ABS(x)>ABS(y)) { rr=((x)-r)-(y); s=(rr-(yy))+(xx); } \ 159 else { rr=(x)-((y)+r); s=(rr+(xx))-(yy); } \ 160 if (rr!=0.0) { \ 161 z=r+s; zz=(r-z)+s; } \ 162 else { \ 163 ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)-(yy)) : ((xx)-((yy)+s)); \ 164 u=r+s; \ 165 uu=(ABS(r)>ABS(s)) ? ((r-u)+s) : ((s-u)+r) ; \ 166 w=uu+ss; z=u+w; \ 167 zz=(ABS(u)>ABS(w)) ? ((u-z)+w) : ((w-z)+u) ; } 168 169 170 171 172 173 174 175