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1/* IEEE754 floating point arithmetic
2 * double precision square root
3 */
4/*
5 * MIPS floating point support
6 * Copyright (C) 1994-2000 Algorithmics Ltd.
7 *
8 * ########################################################################
9 *
10 * This program is free software; you can distribute it and/or modify it
11 * under the terms of the GNU General Public License (Version 2) as
12 * published by the Free Software Foundation.
13 *
14 * This program is distributed in the hope it will be useful, but WITHOUT
15 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
16 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
17 * for more details.
18 *
19 * You should have received a copy of the GNU General Public License along
20 * with this program; if not, write to the Free Software Foundation, Inc.,
21 * 59 Temple Place - Suite 330, Boston MA 02111-1307, USA.
22 *
23 * ########################################################################
24 */
25
26
27#include "ieee754dp.h"
28
29static const unsigned table[] = {
30 0, 1204, 3062, 5746, 9193, 13348, 18162, 23592,
31 29598, 36145, 43202, 50740, 58733, 67158, 75992,
32 85215, 83599, 71378, 60428, 50647, 41945, 34246,
33 27478, 21581, 16499, 12183, 8588, 5674, 3403,
34 1742, 661, 130
35};
36
37ieee754dp ieee754dp_sqrt(ieee754dp x)
38{
39 struct _ieee754_csr oldcsr;
40 ieee754dp y, z, t;
41 unsigned scalx, yh;
42 COMPXDP;
43
44 EXPLODEXDP;
45 CLEARCX;
46 FLUSHXDP;
47
48 /* x == INF or NAN? */
49 switch (xc) {
50 case IEEE754_CLASS_QNAN:
51 /* sqrt(Nan) = Nan */
52 return ieee754dp_nanxcpt(x, "sqrt");
53 case IEEE754_CLASS_SNAN:
54 SETCX(IEEE754_INVALID_OPERATION);
55 return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
56 case IEEE754_CLASS_ZERO:
57 /* sqrt(0) = 0 */
58 return x;
59 case IEEE754_CLASS_INF:
60 if (xs) {
61 /* sqrt(-Inf) = Nan */
62 SETCX(IEEE754_INVALID_OPERATION);
63 return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
64 }
65 /* sqrt(+Inf) = Inf */
66 return x;
67 case IEEE754_CLASS_DNORM:
68 DPDNORMX;
69 /* fall through */
70 case IEEE754_CLASS_NORM:
71 if (xs) {
72 /* sqrt(-x) = Nan */
73 SETCX(IEEE754_INVALID_OPERATION);
74 return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
75 }
76 break;
77 }
78
79 /* save old csr; switch off INX enable & flag; set RN rounding */
80 oldcsr = ieee754_csr;
81 ieee754_csr.mx &= ~IEEE754_INEXACT;
82 ieee754_csr.sx &= ~IEEE754_INEXACT;
83 ieee754_csr.rm = IEEE754_RN;
84
85 /* adjust exponent to prevent overflow */
86 scalx = 0;
87 if (xe > 512) { /* x > 2**-512? */
88 xe -= 512; /* x = x / 2**512 */
89 scalx += 256;
90 } else if (xe < -512) { /* x < 2**-512? */
91 xe += 512; /* x = x * 2**512 */
92 scalx -= 256;
93 }
94
95 y = x = builddp(0, xe + DP_EBIAS, xm & ~DP_HIDDEN_BIT);
96
97 /* magic initial approximation to almost 8 sig. bits */
98 yh = y.bits >> 32;
99 yh = (yh >> 1) + 0x1ff80000;
100 yh = yh - table[(yh >> 15) & 31];
101 y.bits = ((u64) yh << 32) | (y.bits & 0xffffffff);
102
103 /* Heron's rule once with correction to improve to ~18 sig. bits */
104 /* t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0; */
105 t = ieee754dp_div(x, y);
106 y = ieee754dp_add(y, t);
107 y.bits -= 0x0010000600000000LL;
108 y.bits &= 0xffffffff00000000LL;
109
110 /* triple to almost 56 sig. bits: y ~= sqrt(x) to within 1 ulp */
111 /* t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y; */
112 z = t = ieee754dp_mul(y, y);
113 t.parts.bexp += 0x001;
114 t = ieee754dp_add(t, z);
115 z = ieee754dp_mul(ieee754dp_sub(x, z), y);
116
117 /* t=z/(t+x) ; pt[n0]+=0x00100000; y+=t; */
118 t = ieee754dp_div(z, ieee754dp_add(t, x));
119 t.parts.bexp += 0x001;
120 y = ieee754dp_add(y, t);
121
122 /* twiddle last bit to force y correctly rounded */
123
124 /* set RZ, clear INEX flag */
125 ieee754_csr.rm = IEEE754_RZ;
126 ieee754_csr.sx &= ~IEEE754_INEXACT;
127
128 /* t=x/y; ...chopped quotient, possibly inexact */
129 t = ieee754dp_div(x, y);
130
131 if (ieee754_csr.sx & IEEE754_INEXACT || t.bits != y.bits) {
132
133 if (!(ieee754_csr.sx & IEEE754_INEXACT))
134 /* t = t-ulp */
135 t.bits -= 1;
136
137 /* add inexact to result status */
138 oldcsr.cx |= IEEE754_INEXACT;
139 oldcsr.sx |= IEEE754_INEXACT;
140
141 switch (oldcsr.rm) {
142 case IEEE754_RP:
143 y.bits += 1;
144 /* drop through */
145 case IEEE754_RN:
146 t.bits += 1;
147 break;
148 }
149
150 /* y=y+t; ...chopped sum */
151 y = ieee754dp_add(y, t);
152
153 /* adjust scalx for correctly rounded sqrt(x) */
154 scalx -= 1;
155 }
156
157 /* py[n0]=py[n0]+scalx; ...scale back y */
158 y.parts.bexp += scalx;
159
160 /* restore rounding mode, possibly set inexact */
161 ieee754_csr = oldcsr;
162
163 return y;
164}
1/* IEEE754 floating point arithmetic
2 * double precision square root
3 */
4/*
5 * MIPS floating point support
6 * Copyright (C) 1994-2000 Algorithmics Ltd.
7 *
8 * This program is free software; you can distribute it and/or modify it
9 * under the terms of the GNU General Public License (Version 2) as
10 * published by the Free Software Foundation.
11 *
12 * This program is distributed in the hope it will be useful, but WITHOUT
13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
15 * for more details.
16 *
17 * You should have received a copy of the GNU General Public License along
18 * with this program; if not, write to the Free Software Foundation, Inc.,
19 * 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 */
21
22#include "ieee754dp.h"
23
24static const unsigned table[] = {
25 0, 1204, 3062, 5746, 9193, 13348, 18162, 23592,
26 29598, 36145, 43202, 50740, 58733, 67158, 75992,
27 85215, 83599, 71378, 60428, 50647, 41945, 34246,
28 27478, 21581, 16499, 12183, 8588, 5674, 3403,
29 1742, 661, 130
30};
31
32union ieee754dp ieee754dp_sqrt(union ieee754dp x)
33{
34 struct _ieee754_csr oldcsr;
35 union ieee754dp y, z, t;
36 unsigned scalx, yh;
37 COMPXDP;
38
39 EXPLODEXDP;
40 ieee754_clearcx();
41 FLUSHXDP;
42
43 /* x == INF or NAN? */
44 switch (xc) {
45 case IEEE754_CLASS_SNAN:
46 return ieee754dp_nanxcpt(x);
47
48 case IEEE754_CLASS_QNAN:
49 /* sqrt(Nan) = Nan */
50 return x;
51
52 case IEEE754_CLASS_ZERO:
53 /* sqrt(0) = 0 */
54 return x;
55
56 case IEEE754_CLASS_INF:
57 if (xs) {
58 /* sqrt(-Inf) = Nan */
59 ieee754_setcx(IEEE754_INVALID_OPERATION);
60 return ieee754dp_indef();
61 }
62 /* sqrt(+Inf) = Inf */
63 return x;
64
65 case IEEE754_CLASS_DNORM:
66 DPDNORMX;
67 /* fall through */
68
69 case IEEE754_CLASS_NORM:
70 if (xs) {
71 /* sqrt(-x) = Nan */
72 ieee754_setcx(IEEE754_INVALID_OPERATION);
73 return ieee754dp_indef();
74 }
75 break;
76 }
77
78 /* save old csr; switch off INX enable & flag; set RN rounding */
79 oldcsr = ieee754_csr;
80 ieee754_csr.mx &= ~IEEE754_INEXACT;
81 ieee754_csr.sx &= ~IEEE754_INEXACT;
82 ieee754_csr.rm = FPU_CSR_RN;
83
84 /* adjust exponent to prevent overflow */
85 scalx = 0;
86 if (xe > 512) { /* x > 2**-512? */
87 xe -= 512; /* x = x / 2**512 */
88 scalx += 256;
89 } else if (xe < -512) { /* x < 2**-512? */
90 xe += 512; /* x = x * 2**512 */
91 scalx -= 256;
92 }
93
94 y = x = builddp(0, xe + DP_EBIAS, xm & ~DP_HIDDEN_BIT);
95
96 /* magic initial approximation to almost 8 sig. bits */
97 yh = y.bits >> 32;
98 yh = (yh >> 1) + 0x1ff80000;
99 yh = yh - table[(yh >> 15) & 31];
100 y.bits = ((u64) yh << 32) | (y.bits & 0xffffffff);
101
102 /* Heron's rule once with correction to improve to ~18 sig. bits */
103 /* t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0; */
104 t = ieee754dp_div(x, y);
105 y = ieee754dp_add(y, t);
106 y.bits -= 0x0010000600000000LL;
107 y.bits &= 0xffffffff00000000LL;
108
109 /* triple to almost 56 sig. bits: y ~= sqrt(x) to within 1 ulp */
110 /* t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y; */
111 z = t = ieee754dp_mul(y, y);
112 t.bexp += 0x001;
113 t = ieee754dp_add(t, z);
114 z = ieee754dp_mul(ieee754dp_sub(x, z), y);
115
116 /* t=z/(t+x) ; pt[n0]+=0x00100000; y+=t; */
117 t = ieee754dp_div(z, ieee754dp_add(t, x));
118 t.bexp += 0x001;
119 y = ieee754dp_add(y, t);
120
121 /* twiddle last bit to force y correctly rounded */
122
123 /* set RZ, clear INEX flag */
124 ieee754_csr.rm = FPU_CSR_RZ;
125 ieee754_csr.sx &= ~IEEE754_INEXACT;
126
127 /* t=x/y; ...chopped quotient, possibly inexact */
128 t = ieee754dp_div(x, y);
129
130 if (ieee754_csr.sx & IEEE754_INEXACT || t.bits != y.bits) {
131
132 if (!(ieee754_csr.sx & IEEE754_INEXACT))
133 /* t = t-ulp */
134 t.bits -= 1;
135
136 /* add inexact to result status */
137 oldcsr.cx |= IEEE754_INEXACT;
138 oldcsr.sx |= IEEE754_INEXACT;
139
140 switch (oldcsr.rm) {
141 case FPU_CSR_RU:
142 y.bits += 1;
143 /* drop through */
144 case FPU_CSR_RN:
145 t.bits += 1;
146 break;
147 }
148
149 /* y=y+t; ...chopped sum */
150 y = ieee754dp_add(y, t);
151
152 /* adjust scalx for correctly rounded sqrt(x) */
153 scalx -= 1;
154 }
155
156 /* py[n0]=py[n0]+scalx; ...scale back y */
157 y.bexp += scalx;
158
159 /* restore rounding mode, possibly set inexact */
160 ieee754_csr = oldcsr;
161
162 return y;
163}