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1 /*
2 * IBM Accurate Mathematical Library
3 * Written by International Business Machines Corp.
4 * Copyright (C) 2001-2014 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, see <http://www.gnu.org/licenses/>.
18 */
20 /***********************************************************************/
21 /*MODULE_NAME: dla.h */
22 /* */
23 /* This file holds C language macros for 'Double Length Floating Point */
24 /* Arithmetic'. The macros are based on the paper: */
25 /* T.J.Dekker, "A floating-point Technique for extending the */
26 /* Available Precision", Number. Math. 18, 224-242 (1971). */
27 /* A Double-Length number is defined by a pair (r,s), of IEEE double */
28 /* precision floating point numbers that satisfy, */
29 /* */
30 /* abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)). */
31 /* */
32 /* The computer arithmetic assumed is IEEE double precision in */
33 /* round to nearest mode. All variables in the macros must be of type */
34 /* IEEE double. */
35 /***********************************************************************/
37 /* CN = 1+2**27 = '41a0000002000000' IEEE double format. Use it to split a
38 double for better accuracy. */
39 #define CN 134217729.0
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. */
46 #define EADD(x,y,z,zz) \
47 z=(x)+(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
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. */
54 #define ESUB(x,y,z,zz) \
55 z=(x)-(y); zz=(ABS(x)>ABS(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
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. */
63 #ifdef DLA_FMS
64 # define EMULV(x, y, z, zz, p, hx, tx, hy, ty) \
65 z = x * y; zz = DLA_FMS (x, y, z);
66 #else
67 # define EMULV(x, y, z, zz, p, hx, tx, hy, ty) \
68 p = CN * (x); hx = ((x) - p) + p; tx = (x) - hx; \
69 p = CN * (y); hy = ((y) - p) + p; ty = (y) - hy; \
70 z = (x) * (y); zz = (((hx * hy - z) + hx * ty) + tx * hy) + tx * ty;
71 #endif
74 /* Exact multiplication of two single-length floating point numbers, Dekker. */
75 /* The macro produces a nearly double-length number (z,zz) (see Dekker) */
76 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */
77 /* storage variables of type double. */
79 #ifdef DLA_FMS
80 # define MUL12(x,y,z,zz,p,hx,tx,hy,ty,q) \
81 EMULV(x,y,z,zz,p,hx,tx,hy,ty)
82 #else
83 # define MUL12(x,y,z,zz,p,hx,tx,hy,ty,q) \
84 p=CN*(x); hx=((x)-p)+p; tx=(x)-hx; \
85 p=CN*(y); hy=((y)-p)+p; ty=(y)-hy; \
86 p=hx*hy; q=hx*ty+tx*hy; z=p+q; zz=((p-z)+q)+tx*ty;
87 #endif
90 /* Double-length addition, Dekker. The macro produces a double-length */
91 /* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */
92 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
93 /* are assumed to be double-length numbers. r,s are temporary */
94 /* storage variables of type double. */
96 #define ADD2(x, xx, y, yy, z, zz, r, s) \
97 r = (x) + (y); s = (ABS (x) > ABS (y)) ? \
98 (((((x) - r) + (y)) + (yy)) + (xx)) : \
99 (((((y) - r) + (x)) + (xx)) + (yy)); \
100 z = r + s; zz = (r - z) + s;
103 /* Double-length subtraction, Dekker. The macro produces a double-length */
104 /* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */
105 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
106 /* are assumed to be double-length numbers. r,s are temporary */
107 /* storage variables of type double. */
109 #define SUB2(x, xx, y, yy, z, zz, r, s) \
110 r = (x) - (y); s = (ABS (x) > ABS (y)) ? \
111 (((((x) - r) - (y)) - (yy)) + (xx)) : \
112 ((((x) - ((y) + r)) + (xx)) - (yy)); \
113 z = r + s; zz = (r - z) + s;
116 /* Double-length multiplication, Dekker. The macro produces a double-length */
117 /* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */
118 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */
119 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */
120 /* temporary storage variables of type double. */
122 #define MUL2(x, xx, y, yy, z, zz, p, hx, tx, hy, ty, q, c, cc) \
123 MUL12 (x, y, c, cc, p, hx, tx, hy, ty, q) \
124 cc = ((x) * (yy) + (xx) * (y)) + cc; z = c + cc; zz = (c - z) + cc;
127 /* Double-length division, Dekker. The macro produces a double-length */
128 /* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */
129 /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */
130 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */
131 /* are temporary storage variables of type double. */
133 #define DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu) \
134 c=(x)/(y); MUL12(c,y,u,uu,p,hx,tx,hy,ty,q) \
135 cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc;
138 /* Double-length addition, slower but more accurate than ADD2. */
139 /* The macro produces a double-length */
140 /* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */
141 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */
142 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
143 /* are temporary storage variables of type double. */
145 #define ADD2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
146 r = (x) + (y); \
147 if (ABS (x) > ABS (y)) { rr = ((x) - r) + (y); s = (rr + (yy)) + (xx); } \
148 else { rr = ((y) - r) + (x); s = (rr + (xx)) + (yy); } \
149 if (rr != 0.0) { \
150 z = r + s; zz = (r - z) + s; } \
151 else { \
152 ss = (ABS (xx) > ABS (yy)) ? (((xx) - s) + (yy)) : (((yy) - s) + (xx));\
153 u = r + s; \
154 uu = (ABS (r) > ABS (s)) ? ((r - u) + s) : ((s - u) + r); \
155 w = uu + ss; z = u + w; \
156 zz = (ABS (u) > ABS (w)) ? ((u - z) + w) : ((w - z) + u); }
159 /* Double-length subtraction, slower but more accurate than SUB2. */
160 /* The macro produces a double-length */
161 /* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */
162 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */
163 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
164 /* are temporary storage variables of type double. */
166 #define SUB2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
167 r = (x) - (y); \
168 if (ABS (x) > ABS (y)) { rr = ((x) - r) - (y); s = (rr - (yy)) + (xx); } \
169 else { rr = (x) - ((y) + r); s = (rr + (xx)) - (yy); } \
170 if (rr != 0.0) { \
171 z = r + s; zz = (r - z) + s; } \
172 else { \
173 ss = (ABS (xx) > ABS (yy)) ? (((xx) - s) - (yy)) : ((xx) - ((yy) + s)); \
174 u = r + s; \
175 uu = (ABS (r) > ABS (s)) ? ((r - u) + s) : ((s - u) + r); \
176 w = uu + ss; z = u + w; \
177 zz = (ABS (u) > ABS (w)) ? ((u - z) + w) : ((w - z) + u); }