1 /*
2 * IBM Accurate Mathematical Library
3 * Copyright (C) 2001-2023 Free Software Foundation, Inc.
4 *
5 * This program is free software; you can redistribute it and/or modify
6 * it under the terms of the GNU Lesser General Public License as published by
7 * the Free Software Foundation; either version 2.1 of the License, or
8 * (at your option) any later version.
9 *
10 * This program is distributed in the hope that it will be useful,
11 * but WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
13 * GNU Lesser General Public License for more details.
14 *
15 * You should have received a copy of the GNU Lesser General Public License
16 * along with this program; if not, see <https://www.gnu.org/licenses/>.
17 */
18
19 #include <math.h>
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. Use it to split a
39 double for better accuracy. */
40 #define CN 134217729.0
41
42
43 /* Exact addition of two single-length floating point numbers, Dekker. */
44 /* The macro produces a double-length number (z,zz) that satisfies */
45 /* z+zz = x+y exactly. */
46
47 #define EADD(x,y,z,zz) \
48 z=(x)+(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
49
50
51 /* Exact subtraction of two single-length floating point numbers, Dekker. */
52 /* The macro produces a double-length number (z,zz) that satisfies */
53 /* z+zz = x-y exactly. */
54
55 #define ESUB(x,y,z,zz) \
56 z=(x)-(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
57
58
59 #ifdef __FP_FAST_FMA
60 # define DLA_FMS(x, y, z) __builtin_fma (x, y, -(z))
61 #endif
62
63 /* Exact multiplication of two single-length floating point numbers, */
64 /* Veltkamp. The macro produces a double-length number (z,zz) that */
65 /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary */
66 /* storage variables of type double. */
67
68 #ifdef DLA_FMS
69 # define EMULV(x, y, z, zz) \
70 z = x * y; zz = DLA_FMS (x, y, z);
71 #else
72 # define EMULV(x, y, z, zz) \
73 ({ __typeof__ (x) __p, hx, tx, hy, ty; \
74 __p = CN * (x); hx = ((x) - __p) + __p; tx = (x) - hx; \
75 __p = CN * (y); hy = ((y) - __p) + __p; ty = (y) - hy; \
76 z = (x) * (y); zz = (((hx * hy - z) + hx * ty) + tx * hy) + tx * ty; \
77 })
78 #endif
79
80
81 /* Exact multiplication of two single-length floating point numbers, Dekker. */
82 /* The macro produces a nearly double-length number (z,zz) (see Dekker) */
83 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */
84 /* storage variables of type double. */
85
86 #ifdef DLA_FMS
87 # define MUL12(x, y, z, zz) \
88 EMULV(x, y, z, zz)
89 #else
90 # define MUL12(x, y, z, zz) \
91 ({ __typeof__ (x) __p, hx, tx, hy, ty, __q; \
92 __p=CN*(x); hx=((x)-__p)+__p; tx=(x)-hx; \
93 __p=CN*(y); hy=((y)-__p)+__p; ty=(y)-hy; \
94 __p=hx*hy; __q=hx*ty+tx*hy; z=__p+__q; zz=((__p-z)+__q)+tx*ty; \
95 })
96 #endif
97
98
99 /* Double-length addition, Dekker. The macro produces a double-length */
100 /* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */
101 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
102 /* are assumed to be double-length numbers. r,s are temporary */
103 /* storage variables of type double. */
104
105 #define ADD2(x, xx, y, yy, z, zz, r, s) \
106 r = (x) + (y); s = (fabs (x) > fabs (y)) ? \
107 (((((x) - r) + (y)) + (yy)) + (xx)) : \
108 (((((y) - r) + (x)) + (xx)) + (yy)); \
109 z = r + s; zz = (r - z) + s;
110
111
112 /* Double-length subtraction, Dekker. The macro produces a double-length */
113 /* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */
114 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
115 /* are assumed to be double-length numbers. r,s are temporary */
116 /* storage variables of type double. */
117
118 #define SUB2(x, xx, y, yy, z, zz, r, s) \
119 r = (x) - (y); s = (fabs (x) > fabs (y)) ? \
120 (((((x) - r) - (y)) - (yy)) + (xx)) : \
121 ((((x) - ((y) + r)) + (xx)) - (yy)); \
122 z = r + s; zz = (r - z) + s;
123
124
125 /* Double-length multiplication, Dekker. The macro produces a double-length */
126 /* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */
127 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */
128 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */
129 /* temporary storage variables of type double. */
130
131 #define MUL2(x, xx, y, yy, z, zz, c, cc) \
132 MUL12 (x, y, c, cc); \
133 cc = ((x) * (yy) + (xx) * (y)) + cc; z = c + cc; zz = (c - z) + cc;
134
135
136 /* Double-length division, Dekker. The macro produces a double-length */
137 /* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */
138 /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */
139 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */
140 /* are temporary storage variables of type double. */
141
142 #define DIV2(x, xx, y, yy, z, zz, c, cc, u, uu) \
143 c=(x)/(y); MUL12(c,y,u,uu); \
144 cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc;
145
146
147 /* Double-length addition, slower but more accurate than ADD2. */
148 /* The macro produces a double-length */
149 /* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */
150 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */
151 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
152 /* are temporary storage variables of type double. */
153
154 #define ADD2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
155 r = (x) + (y); \
156 if (fabs (x) > fabs (y)) { rr = ((x) - r) + (y); s = (rr + (yy)) + (xx); } \
157 else { rr = ((y) - r) + (x); s = (rr + (xx)) + (yy); } \
158 if (rr != 0.0) { \
159 z = r + s; zz = (r - z) + s; } \
160 else { \
161 ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) + (yy)) : (((yy) - s) + (xx));\
162 u = r + s; \
163 uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \
164 w = uu + ss; z = u + w; \
165 zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); }
166
167
168 /* Double-length subtraction, slower but more accurate than SUB2. */
169 /* The macro produces a double-length */
170 /* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */
171 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */
172 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
173 /* are temporary storage variables of type double. */
174
175 #define SUB2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
176 r = (x) - (y); \
177 if (fabs (x) > fabs (y)) { rr = ((x) - r) - (y); s = (rr - (yy)) + (xx); } \
178 else { rr = (x) - ((y) + r); s = (rr + (xx)) - (yy); } \
179 if (rr != 0.0) { \
180 z = r + s; zz = (r - z) + s; } \
181 else { \
182 ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) - (yy)) : ((xx) - ((yy) + s)); \
183 u = r + s; \
184 uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \
185 w = uu + ss; z = u + w; \
186 zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); }