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
|
/* Double-precision floating point square root.
Copyright (C) 1997-2013 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C 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 2.1 of the License, or (at your option) any later version.
The GNU C 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 C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
#include <fenv_libc.h>
#include <inttypes.h>
#include <stdint.h>
#include <sysdep.h>
#include <ldsodefs.h>
static const double almost_half = 0.5000000000000001; /* 0.5 + 2^-53 */
static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 };
static const ieee_float_shape_type a_inf = {.word = 0x7f800000 };
static const float two108 = 3.245185536584267269e+32;
static const float twom54 = 5.551115123125782702e-17;
extern const float __t_sqrt[1024];
/* The method is based on a description in
Computation of elementary functions on the IBM RISC System/6000 processor,
P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
Basically, it consists of two interleaved Newton-Raphson approximations,
one to find the actual square root, and one to find its reciprocal
without the expense of a division operation. The tricky bit here
is the use of the POWER/PowerPC multiply-add operation to get the
required accuracy with high speed.
The argument reduction works by a combination of table lookup to
obtain the initial guesses, and some careful modification of the
generated guesses (which mostly runs on the integer unit, while the
Newton-Raphson is running on the FPU). */
double
__slow_ieee754_sqrt (double x)
{
const float inf = a_inf.value;
if (x > 0)
{
/* schedule the EXTRACT_WORDS to get separation between the store
and the load. */
ieee_double_shape_type ew_u;
ieee_double_shape_type iw_u;
ew_u.value = (x);
if (x != inf)
{
/* Variables named starting with 's' exist in the
argument-reduced space, so that 2 > sx >= 0.5,
1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
Variables named ending with 'i' are integer versions of
floating-point values. */
double sx; /* The value of which we're trying to find the
square root. */
double sg, g; /* Guess of the square root of x. */
double sd, d; /* Difference between the square of the guess and x. */
double sy; /* Estimate of 1/2g (overestimated by 1ulp). */
double sy2; /* 2*sy */
double e; /* Difference between y*g and 1/2 (se = e * fsy). */
double shx; /* == sx * fsg */
double fsg; /* sg*fsg == g. */
fenv_t fe; /* Saved floating-point environment (stores rounding
mode and whether the inexact exception is
enabled). */
uint32_t xi0, xi1, sxi, fsgi;
const float *t_sqrt;
fe = fegetenv_register ();
/* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */
xi0 = ew_u.parts.msw;
xi1 = ew_u.parts.lsw;
relax_fenv_state ();
sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
/* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
between the store and the load. */
iw_u.parts.msw = sxi;
iw_u.parts.lsw = xi1;
t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
sg = t_sqrt[0];
sy = t_sqrt[1];
/* complete the INSERT_WORDS (sx, sxi, xi1) operation. */
sx = iw_u.value;
/* Here we have three Newton-Raphson iterations each of a
division and a square root and the remainder of the
argument reduction, all interleaved. */
sd = -(sg * sg - sx);
fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
sy2 = sy + sy;
sg = sy * sd + sg; /* 16-bit approximation to sqrt(sx). */
/* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
between the store and the load. */
INSERT_WORDS (fsg, fsgi, 0);
iw_u.parts.msw = fsgi;
iw_u.parts.lsw = (0);
e = -(sy * sg - almost_half);
sd = -(sg * sg - sx);
if ((xi0 & 0x7ff00000) == 0)
goto denorm;
sy = sy + e * sy2;
sg = sg + sy * sd; /* 32-bit approximation to sqrt(sx). */
sy2 = sy + sy;
/* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */
fsg = iw_u.value;
e = -(sy * sg - almost_half);
sd = -(sg * sg - sx);
sy = sy + e * sy2;
shx = sx * fsg;
sg = sg + sy * sd; /* 64-bit approximation to sqrt(sx),
but perhaps rounded incorrectly. */
sy2 = sy + sy;
g = sg * fsg;
e = -(sy * sg - almost_half);
d = -(g * sg - shx);
sy = sy + e * sy2;
fesetenv_register (fe);
return g + sy * d;
denorm:
/* For denormalised numbers, we normalise, calculate the
square root, and return an adjusted result. */
fesetenv_register (fe);
return __slow_ieee754_sqrt (x * two108) * twom54;
}
}
else if (x < 0)
{
/* For some reason, some PowerPC32 processors don't implement
FE_INVALID_SQRT. */
#ifdef FE_INVALID_SQRT
feraiseexcept (FE_INVALID_SQRT);
fenv_union_t u = { .fenv = fegetenv_register () };
if ((u.l & FE_INVALID) == 0)
#endif
feraiseexcept (FE_INVALID);
x = a_nan.value;
}
return f_wash (x);
}
#undef __ieee754_sqrt
double
__ieee754_sqrt (double x)
{
double z;
/* If the CPU is 64-bit we can use the optional FP instructions. */
if (__CPU_HAS_FSQRT)
{
/* Volatile is required to prevent the compiler from moving the
fsqrt instruction above the branch. */
__asm __volatile (" fsqrt %0,%1\n"
:"=f" (z):"f" (x));
}
else
z = __slow_ieee754_sqrt (x);
return z;
}
strong_alias (__ieee754_sqrt, __sqrt_finite)
|
