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
|
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2025 Free Software Foundation, Inc.
*
* This program 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 2.1 of the License, or
* (at your option) any later version.
*
* This program 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 this program; if not, see <https://www.gnu.org/licenses/>.
*/
/*********************************************************************/
/* MODULE_NAME: uroot.c */
/* */
/* FUNCTION: usqrt */
/* */
/* FILES NEEDED: dla.h endian.h mydefs.h */
/* uroot.tbl */
/* */
/* An ultimate sqrt routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of square */
/* root of x. */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/*********************************************************************/
#include "endian.h"
#include "mydefs.h"
#include <dla.h>
#include "root.tbl"
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
#include <libm-alias-finite.h>
#include <math-use-builtins.h>
/*********************************************************************/
/* An ultimate sqrt routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of square */
/* root of x. */
/*********************************************************************/
double
__ieee754_sqrt (double x)
{
#if USE_SQRT_BUILTIN
return __builtin_sqrt (x);
#else
/* Use generic implementation. */
static const double
rt0 = 9.99999999859990725855365213134618E-01,
rt1 = 4.99999999495955425917856814202739E-01,
rt2 = 3.75017500867345182581453026130850E-01,
rt3 = 3.12523626554518656309172508769531E-01;
static const double big = 134217728.0;
double y, t, del, res, res1, hy, z, zz, s;
mynumber a, c = { { 0, 0 } };
int4 k;
a.x = x;
k = a.i[HIGH_HALF];
a.i[HIGH_HALF] = (k & 0x001fffff) | 0x3fe00000;
t = inroot[(k & 0x001fffff) >> 14];
s = a.x;
/*----------------- 2^-1022 <= | x |< 2^1024 -----------------*/
if (k > 0x000fffff && k < 0x7ff00000)
{
int rm = __fegetround ();
fenv_t env;
libc_feholdexcept_setround (&env, FE_TONEAREST);
double ret;
y = 1.0 - t * (t * s);
t = t * (rt0 + y * (rt1 + y * (rt2 + y * rt3)));
c.i[HIGH_HALF] = 0x20000000 + ((k & 0x7fe00000) >> 1);
y = t * s;
hy = (y + big) - big;
del = 0.5 * t * ((s - hy * hy) - (y - hy) * (y + hy));
res = y + del;
if (res == (res + 1.002 * ((y - res) + del)))
ret = res * c.x;
else
{
res1 = res + 1.5 * ((y - res) + del);
EMULV (res, res1, z, zz); /* (z+zz)=res*res1 */
res = ((((z - s) + zz) < 0) ? max (res, res1) :
min (res, res1));
ret = res * c.x;
}
math_force_eval (ret);
libc_fesetenv (&env);
double dret = x / ret;
if (dret != ret)
{
double force_inexact = 1.0 / 3.0;
math_force_eval (force_inexact);
/* The square root is inexact, ret is the round-to-nearest
value which may need adjusting for other rounding
modes. */
switch (rm)
{
#ifdef FE_UPWARD
case FE_UPWARD:
if (dret > ret)
ret = (res + 0x1p-1022) * c.x;
break;
#endif
#ifdef FE_DOWNWARD
case FE_DOWNWARD:
#endif
#ifdef FE_TOWARDZERO
case FE_TOWARDZERO:
#endif
#if defined FE_DOWNWARD || defined FE_TOWARDZERO
if (dret < ret)
ret = (res - 0x1p-1022) * c.x;
break;
#endif
default:
break;
}
}
/* Otherwise (x / ret == ret), either the square root was exact or
the division was inexact. */
return ret;
}
else
{
if ((k & 0x7ff00000) == 0x7ff00000)
return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
if (x == 0)
return x; /* sqrt(+0)=+0, sqrt(-0)=-0 */
if (k < 0)
return (x - x) / (x - x); /* sqrt(-ve)=sNaN */
return 0x1p-256 * __ieee754_sqrt (x * 0x1p512);
}
#endif /* ! USE_SQRT_BUILTIN */
}
#ifndef __ieee754_sqrt
libm_alias_finite (__ieee754_sqrt, __sqrt)
#endif
|
