1 /* mpf_sqrt -- Compute the square root of a float.
2
3 Copyright 1993, 1994, 1996, 2000, 2001, 2004, 2005, 2012 Free Software
4 Foundation, Inc.
5
6 This file is part of the GNU MP Library.
7
8 The GNU MP Library is free software; you can redistribute it and/or modify
9 it under the terms of either:
10
11 * the GNU Lesser General Public License as published by the Free
12 Software Foundation; either version 3 of the License, or (at your
13 option) any later version.
14
15 or
16
17 * the GNU General Public License as published by the Free Software
18 Foundation; either version 2 of the License, or (at your option) any
19 later version.
20
21 or both in parallel, as here.
22
23 The GNU MP Library is distributed in the hope that it will be useful, but
24 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
25 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
26 for more details.
27
28 You should have received copies of the GNU General Public License and the
29 GNU Lesser General Public License along with the GNU MP Library. If not,
30 see https://www.gnu.org/licenses/. */
31
32 #include <stdio.h> /* for NULL */
33 #include "gmp-impl.h"
34
35
36 /* As usual, the aim is to produce PREC(r) limbs of result, with the high
37 limb non-zero. This is accomplished by applying mpn_sqrtrem to either
38 2*prec or 2*prec-1 limbs, both such sizes resulting in prec limbs.
39
40 The choice between 2*prec or 2*prec-1 limbs is based on the input
41 exponent. With b=2^GMP_NUMB_BITS the limb base then we can think of
42 effectively taking out a factor b^(2k), for suitable k, to get to an
43 integer input of the desired size ready for mpn_sqrtrem. It must be an
44 even power taken out, ie. an even number of limbs, so the square root
45 gives factor b^k and the radix point is still on a limb boundary. So if
46 EXP(r) is even we'll get an even number of input limbs 2*prec, or if
47 EXP(r) is odd we get an odd number 2*prec-1.
48
49 Further limbs below the 2*prec or 2*prec-1 used don't affect the result
50 and are simply truncated. This can be seen by considering an integer x,
51 with s=floor(sqrt(x)). s is the unique integer satisfying s^2 <= x <
52 (s+1)^2. Notice that adding a fraction part to x (ie. some further bits)
53 doesn't change the inequality, s remains the unique solution. Working
54 suitable factors of 2 into this argument lets it apply to an intended
55 precision at any position for any x, not just the integer binary point.
56
57 If the input is smaller than 2*prec or 2*prec-1, then we just pad with
58 zeros, that of course being our usual interpretation of short inputs.
59 The effect is to extend the root beyond the size of the input (for
60 instance into fractional limbs if u is an integer). */
61
62 void
mpf_sqrt(mpf_ptr r,mpf_srcptr u)63 mpf_sqrt (mpf_ptr r, mpf_srcptr u)
64 {
65 mp_size_t usize;
66 mp_ptr up, tp;
67 mp_size_t prec, tsize;
68 mp_exp_t uexp, expodd;
69 TMP_DECL;
70
71 usize = u->_mp_size;
72 if (UNLIKELY (usize <= 0))
73 {
74 if (usize < 0)
75 SQRT_OF_NEGATIVE;
76 r->_mp_size = 0;
77 r->_mp_exp = 0;
78 return;
79 }
80
81 TMP_MARK;
82
83 uexp = u->_mp_exp;
84 prec = r->_mp_prec;
85 up = u->_mp_d;
86
87 expodd = (uexp & 1);
88 tsize = 2 * prec - expodd;
89 r->_mp_size = prec;
90 r->_mp_exp = (uexp + expodd) / 2; /* ceil(uexp/2) */
91
92 /* root size is ceil(tsize/2), this will be our desired "prec" limbs */
93 ASSERT ((tsize + 1) / 2 == prec);
94
95 tp = TMP_ALLOC_LIMBS (tsize);
96
97 if (usize > tsize)
98 {
99 up += usize - tsize;
100 usize = tsize;
101 MPN_COPY (tp, up, tsize);
102 }
103 else
104 {
105 MPN_ZERO (tp, tsize - usize);
106 MPN_COPY (tp + (tsize - usize), up, usize);
107 }
108
109 mpn_sqrtrem (r->_mp_d, NULL, tp, tsize);
110
111 TMP_FREE;
112 }
113