| /* |
| * IBM Accurate Mathematical Library |
| * Written by International Business Machines Corp. |
| * Copyright (C) 2001-2014 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, see <http://www.gnu.org/licenses/>. |
| */ |
| |
| /***********************************************************************/ |
| /*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. Use it to split a |
| double for better accuracy. */ |
| #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. */ |
| |
| #ifdef DLA_FMS |
| # define EMULV(x, y, z, zz, p, hx, tx, hy, ty) \ |
| z = x * y; zz = DLA_FMS (x, y, z); |
| #else |
| # 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; |
| #endif |
| |
| |
| /* 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. */ |
| |
| #ifdef DLA_FMS |
| # define MUL12(x,y,z,zz,p,hx,tx,hy,ty,q) \ |
| EMULV(x,y,z,zz,p,hx,tx,hy,ty) |
| #else |
| # 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; |
| #endif |
| |
| |
| /* 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); } |
