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
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
|
/* mpfr_sqrt -- square root of a floating-point number
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library 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 3 of the License, or (at your
option) any later version.
The GNU MPFR Library 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 the GNU MPFR Library; see the file COPYING.LIB. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#include "mpfr-impl.h"
int
mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mp_rnd_t rnd_mode)
{
mp_size_t rsize; /* number of limbs of r */
mp_size_t rrsize;
mp_size_t usize; /* number of limbs of u */
mp_size_t tsize; /* number of limbs of the sqrtrem remainder */
mp_size_t k;
mp_size_t l;
mp_ptr rp;
mp_ptr up;
mp_ptr sp;
mp_ptr tp;
mp_limb_t sticky0; /* truncated part of input */
mp_limb_t sticky1; /* truncated part of rp[0] */
mp_limb_t sticky;
int odd_exp;
int sh; /* number of extra bits in rp[0] */
int inexact; /* return ternary flag */
mp_exp_t expr;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", u, u, rnd_mode),
("y[%#R]=%R inexact=%d", r, r, inexact));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(u)))
{
if (MPFR_IS_NAN(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
else if (MPFR_IS_ZERO(u))
{
/* 0+ or 0- */
MPFR_SET_SAME_SIGN(r, u);
MPFR_SET_ZERO(r);
MPFR_RET(0); /* zero is exact */
}
else
{
MPFR_ASSERTD(MPFR_IS_INF(u));
/* sqrt(-Inf) = NAN */
if (MPFR_IS_NEG(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_SET_POS(r);
MPFR_SET_INF(r);
MPFR_RET(0);
}
}
if (MPFR_UNLIKELY(MPFR_IS_NEG(u)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_CLEAR_FLAGS(r);
MPFR_SET_POS(r);
rsize = MPFR_LIMB_SIZE(r); /* number of limbs of r */
rrsize = rsize + rsize;
usize = MPFR_LIMB_SIZE(u); /* number of limbs of u */
rp = MPFR_MANT(r);
up = MPFR_MANT(u);
sticky0 = MPFR_LIMB_ZERO; /* truncated part of input */
sticky1 = MPFR_LIMB_ZERO; /* truncated part of rp[0] */
odd_exp = (unsigned int) MPFR_GET_EXP (u) & 1;
inexact = -1; /* return ternary flag */
MPFR_TMP_MARK (marker);
sp = (mp_limb_t *) MPFR_TMP_ALLOC (rrsize * sizeof (mp_limb_t));
/* copy the most significant limbs of u to {sp, rrsize} */
if (MPFR_LIKELY(usize <= rrsize)) /* in case r and u have the same precision,
we have indeed rrsize = 2 * usize */
{
k = rrsize - usize;
if (MPFR_LIKELY(k))
MPN_ZERO (sp, k);
if (odd_exp)
{
if (MPFR_LIKELY(k))
sp[k - 1] = mpn_rshift (sp + k, up, usize, 1);
else
sticky0 = mpn_rshift (sp, up, usize, 1);
}
else
MPN_COPY (sp + rrsize - usize, up, usize);
}
else /* usize > rrsize: truncate the input */
{
k = usize - rrsize;
if (odd_exp)
sticky0 = mpn_rshift (sp, up + k, rrsize, 1);
else
MPN_COPY (sp, up + k, rrsize);
l = k;
while (sticky0 == MPFR_LIMB_ZERO && l != 0)
sticky0 = up[--l];
}
/* sticky0 is non-zero iff the truncated part of the input is non-zero */
tsize = mpn_sqrtrem (rp, tp = sp, sp, rrsize);
l = tsize;
sticky = sticky0;
while (sticky == MPFR_LIMB_ZERO && l != 0)
sticky = tp[--l];
/* truncated low bits of rp[0] */
MPFR_UNSIGNED_MINUS_MODULO(sh,MPFR_PREC(r));
sticky1 = rp[0] & MPFR_LIMB_MASK(sh);
rp[0] -= sticky1;
sticky = sticky || sticky1;
expr = (MPFR_GET_EXP(u) + odd_exp) / 2; /* exact */
if (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD || sticky == MPFR_LIMB_ZERO)
{
inexact = (sticky == MPFR_LIMB_ZERO) ? 0 : -1;
goto truncate;
}
else if (rnd_mode == MPFR_RNDN)
{
/* if sh>0, the round bit is bit (sh-1) of sticky1
and the sticky bit is formed by the low sh-1 bits from
sticky1, together with {tp, tsize} and sticky0. */
if (sh)
{
if (sticky1 & (MPFR_LIMB_ONE << (sh - 1)))
{ /* round bit is set */
if (sticky1 == (MPFR_LIMB_ONE << (sh - 1)) && tsize == 0
&& sticky0 == 0)
goto even_rule;
else
goto add_one_ulp;
}
else /* round bit is zero */
goto truncate; /* with the default inexact=-1 */
}
else
{
/* if sh=0, we have to compare {tp, tsize} with {rp, rsize}:
if {tp, tsize} < {rp, rsize}: truncate
if {tp, tsize} > {rp, rsize}: round up
if {tp, tsize} = {rp, rsize}: compare the truncated part of the
input to 1/4
if < 1/4: truncate
if > 1/4: round up
if = 1/4: even rounding rule
Set inexact = -1 if truncate
inexact = 1 if add one ulp
inexact = 0 if even rounding rule
*/
if (tsize < rsize)
inexact = -1;
else if (tsize > rsize) /* FIXME: may happen? */
inexact = 1;
else /* tsize = rsize */
{
int cmp;
cmp = mpn_cmp (tp, rp, rsize);
if (cmp > 0)
inexact = 1;
else if (cmp < 0 || sticky0 == MPFR_LIMB_ZERO)
inexact = -1;
/* now tricky case {tp, tsize} = {rp, rsize} */
/* in case usize <= rrsize, the only case where sticky0 <> 0
is when the exponent of u is odd and usize = rrsize (k=0),
but in that case the truncated part is exactly 1/2, thus
we have to round up.
If the exponent of u is odd, and up[k] is odd, the truncated
part is >= 1/2, so we round up too. */
else if (usize <= rrsize || (odd_exp && (up[k] & MPFR_LIMB_ONE)))
inexact = 1;
else
{
/* now usize > rrsize:
(a) if the exponent of u is even, the 1/4 bit is the
2nd most significant bit of up[k-1];
(b) if the exponent of u is odd, the 1/4 bit is the
1st most significant bit of up[k-1]; */
sticky1 = MPFR_LIMB_ONE << (BITS_PER_MP_LIMB - 2 + odd_exp);
if (up[k - 1] < sticky1)
inexact = -1;
else if (up[k - 1] > sticky1)
inexact = 1;
else
{
/* up[k - 1] == sticky1: consider low k-1 limbs */
while (--k > 0 && up[k - 1] == MPFR_LIMB_ZERO)
;
inexact = (k != 0);
}
} /* end of case {tp, tsize} = {rp, rsize} */
} /* end of case tsize = rsize */
if (inexact == -1)
goto truncate;
else if (inexact == 1)
goto add_one_ulp;
/* else go through even_rule */
}
}
else /* rnd_mode=GMP_RDNU, necessarily sticky <> 0, thus add 1 ulp */
goto add_one_ulp;
even_rule: /* has to set inexact */
inexact = (rp[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
if (inexact == -1)
goto truncate;
/* else go through add_one_ulp */
add_one_ulp:
inexact = 1; /* always here */
if (mpn_add_1 (rp, rp, rsize, MPFR_LIMB_ONE << sh))
{
expr ++;
rp[rsize - 1] = MPFR_LIMB_HIGHBIT;
}
truncate: /* inexact = 0 or -1 */
MPFR_ASSERTN (expr >= MPFR_EMIN_MIN && expr <= MPFR_EMAX_MAX);
MPFR_EXP (r) = expr;
MPFR_TMP_FREE(marker);
return mpfr_check_range (r, inexact, rnd_mode);
}
|
