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
|
/*-
* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
*
* Copyright (c) 2007 Steven G. Kargl
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <fenv.h>
#include <float.h>
#include "fpmath.h"
#include "math.h"
/* Return (x + ulp) for normal positive x. Assumes no overflow. */
static inline long double
inc(long double x)
{
union IEEEl2bits u;
u.e = x;
if (++u.bits.manl == 0) {
if (++u.bits.manh == 0) {
u.bits.exp++;
u.bits.manh |= LDBL_NBIT;
}
}
return (u.e);
}
/* Return (x - ulp) for normal positive x. Assumes no underflow. */
static inline long double
dec(long double x)
{
union IEEEl2bits u;
u.e = x;
if (u.bits.manl-- == 0) {
if (u.bits.manh-- == LDBL_NBIT) {
u.bits.exp--;
u.bits.manh |= LDBL_NBIT;
}
}
return (u.e);
}
#pragma STDC FENV_ACCESS ON
/*
* This is slow, but simple and portable. You should use hardware sqrt
* if possible.
*/
long double
sqrtl(long double x)
{
union IEEEl2bits u;
int k, r;
long double lo, xn;
fenv_t env;
u.e = x;
/* If x = NaN, then sqrt(x) = NaN. */
/* If x = Inf, then sqrt(x) = Inf. */
/* If x = -Inf, then sqrt(x) = NaN. */
if (u.bits.exp == LDBL_MAX_EXP * 2 - 1)
return (x * x + x);
/* If x = +-0, then sqrt(x) = +-0. */
if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
return (x);
/* If x < 0, then raise invalid and return NaN */
if (u.bits.sign)
return ((x - x) / (x - x));
feholdexcept(&env);
if (u.bits.exp == 0) {
/* Adjust subnormal numbers. */
u.e *= 0x1.0p514;
k = -514;
} else {
k = 0;
}
/*
* u.e is a normal number, so break it into u.e = e*2^n where
* u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
*/
if ((u.bits.exp - 0x3ffe) & 1) { /* n is odd. */
k += u.bits.exp - 0x3fff; /* 2k = n - 1. */
u.bits.exp = 0x3fff; /* u.e in [1,2). */
} else {
k += u.bits.exp - 0x4000; /* 2k = n - 2. */
u.bits.exp = 0x4000; /* u.e in [2,4). */
}
/*
* Newton's iteration.
* Split u.e into a high and low part to achieve additional precision.
*/
xn = sqrt(u.e); /* 53-bit estimate of sqrtl(x). */
#if LDBL_MANT_DIG > 100
xn = (xn + (u.e / xn)) * 0.5; /* 106-bit estimate. */
#endif
lo = u.e;
u.bits.manl = 0; /* Zero out lower bits. */
lo = (lo - u.e) / xn; /* Low bits divided by xn. */
xn = xn + (u.e / xn); /* High portion of estimate. */
u.e = xn + lo; /* Combine everything. */
u.bits.exp += (k >> 1) - 1;
feclearexcept(FE_INEXACT);
r = fegetround();
fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
xn = x / u.e; /* Chopped quotient (inexact?). */
if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
if (xn == u.e) {
fesetenv(&env);
return (u.e);
}
/* Round correctly for inputs like x = y**2 - ulp. */
xn = dec(xn); /* xn = xn - ulp. */
}
if (r == FE_TONEAREST) {
xn = inc(xn); /* xn = xn + ulp. */
} else if (r == FE_UPWARD) {
u.e = inc(u.e); /* u.e = u.e + ulp. */
xn = inc(xn); /* xn = xn + ulp. */
}
u.e = u.e + xn; /* Chopped sum. */
feupdateenv(&env); /* Restore env and raise inexact */
u.bits.exp--;
return (u.e);
}
|
