1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
|
/* IEEE754 floating point arithmetic
* double precision square root
*/
/*
* MIPS floating point support
* Copyright (C) 1994-2000 Algorithmics Ltd.
*
* This program is free software; you can distribute it and/or modify it
* under the terms of the GNU General Public License (Version 2) as
* published by the Free Software Foundation.
*
* This program is distributed in the hope it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*/
#include "ieee754dp.h"
static const unsigned table[] = {
0, 1204, 3062, 5746, 9193, 13348, 18162, 23592,
29598, 36145, 43202, 50740, 58733, 67158, 75992,
85215, 83599, 71378, 60428, 50647, 41945, 34246,
27478, 21581, 16499, 12183, 8588, 5674, 3403,
1742, 661, 130
};
union ieee754dp ieee754dp_sqrt(union ieee754dp x)
{
struct _ieee754_csr oldcsr;
union ieee754dp y, z, t;
unsigned scalx, yh;
COMPXDP;
EXPLODEXDP;
ieee754_clearcx();
FLUSHXDP;
/* x == INF or NAN? */
switch (xc) {
case IEEE754_CLASS_QNAN:
/* sqrt(Nan) = Nan */
return ieee754dp_nanxcpt(x);
case IEEE754_CLASS_SNAN:
ieee754_setcx(IEEE754_INVALID_OPERATION);
return ieee754dp_nanxcpt(ieee754dp_indef());
case IEEE754_CLASS_ZERO:
/* sqrt(0) = 0 */
return x;
case IEEE754_CLASS_INF:
if (xs) {
/* sqrt(-Inf) = Nan */
ieee754_setcx(IEEE754_INVALID_OPERATION);
return ieee754dp_nanxcpt(ieee754dp_indef());
}
/* sqrt(+Inf) = Inf */
return x;
case IEEE754_CLASS_DNORM:
DPDNORMX;
/* fall through */
case IEEE754_CLASS_NORM:
if (xs) {
/* sqrt(-x) = Nan */
ieee754_setcx(IEEE754_INVALID_OPERATION);
return ieee754dp_nanxcpt(ieee754dp_indef());
}
break;
}
/* save old csr; switch off INX enable & flag; set RN rounding */
oldcsr = ieee754_csr;
ieee754_csr.mx &= ~IEEE754_INEXACT;
ieee754_csr.sx &= ~IEEE754_INEXACT;
ieee754_csr.rm = FPU_CSR_RN;
/* adjust exponent to prevent overflow */
scalx = 0;
if (xe > 512) { /* x > 2**-512? */
xe -= 512; /* x = x / 2**512 */
scalx += 256;
} else if (xe < -512) { /* x < 2**-512? */
xe += 512; /* x = x * 2**512 */
scalx -= 256;
}
y = x = builddp(0, xe + DP_EBIAS, xm & ~DP_HIDDEN_BIT);
/* magic initial approximation to almost 8 sig. bits */
yh = y.bits >> 32;
yh = (yh >> 1) + 0x1ff80000;
yh = yh - table[(yh >> 15) & 31];
y.bits = ((u64) yh << 32) | (y.bits & 0xffffffff);
/* Heron's rule once with correction to improve to ~18 sig. bits */
/* t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0; */
t = ieee754dp_div(x, y);
y = ieee754dp_add(y, t);
y.bits -= 0x0010000600000000LL;
y.bits &= 0xffffffff00000000LL;
/* triple to almost 56 sig. bits: y ~= sqrt(x) to within 1 ulp */
/* t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y; */
z = t = ieee754dp_mul(y, y);
t.bexp += 0x001;
t = ieee754dp_add(t, z);
z = ieee754dp_mul(ieee754dp_sub(x, z), y);
/* t=z/(t+x) ; pt[n0]+=0x00100000; y+=t; */
t = ieee754dp_div(z, ieee754dp_add(t, x));
t.bexp += 0x001;
y = ieee754dp_add(y, t);
/* twiddle last bit to force y correctly rounded */
/* set RZ, clear INEX flag */
ieee754_csr.rm = FPU_CSR_RZ;
ieee754_csr.sx &= ~IEEE754_INEXACT;
/* t=x/y; ...chopped quotient, possibly inexact */
t = ieee754dp_div(x, y);
if (ieee754_csr.sx & IEEE754_INEXACT || t.bits != y.bits) {
if (!(ieee754_csr.sx & IEEE754_INEXACT))
/* t = t-ulp */
t.bits -= 1;
/* add inexact to result status */
oldcsr.cx |= IEEE754_INEXACT;
oldcsr.sx |= IEEE754_INEXACT;
switch (oldcsr.rm) {
case FPU_CSR_RU:
y.bits += 1;
/* drop through */
case FPU_CSR_RN:
t.bits += 1;
break;
}
/* y=y+t; ...chopped sum */
y = ieee754dp_add(y, t);
/* adjust scalx for correctly rounded sqrt(x) */
scalx -= 1;
}
/* py[n0]=py[n0]+scalx; ...scale back y */
y.bexp += scalx;
/* restore rounding mode, possibly set inexact */
ieee754_csr = oldcsr;
return y;
}
|
