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
|
/* Single-precision floating point square root.
Copyright (C) 1997 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, write to the Free
Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
02111-1307 USA. */
#include <math.h>
#include <math_private.h>
#include <fenv_libc.h>
#include <inttypes.h>
static const float almost_half = 0.50000006; /* 0.5 + 2^-24 */
static const uint32_t a_nan = 0x7fc00000;
static const uint32_t a_inf = 0x7f800000;
static const float two48 = 281474976710656.0;
static const float twom24 = 5.9604644775390625e-8;
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-Rhapson 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-Rhapson is running on the FPU). */
float
__sqrtf(float x)
{
const float inf = *(const float *)&a_inf;
/* x = f_washf(x); *//* This ensures only one exception for SNaN. */
if (x > 0)
{
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. */
float sx; /* The value of which we're trying to find the square
root. */
float sg,g; /* Guess of the square root of x. */
float sd,d; /* Difference between the square of the guess and x. */
float sy; /* Estimate of 1/2g (overestimated by 1ulp). */
float sy2; /* 2*sy */
float e; /* Difference between y*g and 1/2 (note that e==se). */
float shx; /* == sx * fsg */
float fsg; /* sg*fsg == g. */
fenv_t fe; /* Saved floating-point environment (stores rounding
mode and whether the inexact exception is
enabled). */
uint32_t xi, sxi, fsgi;
const float *t_sqrt;
GET_FLOAT_WORD (xi, x);
fe = fegetenv_register ();
relax_fenv_state ();
sxi = (xi & 0x3fffffff) | 0x3f000000;
SET_FLOAT_WORD (sx, sxi);
t_sqrt = __t_sqrt + (xi >> (23-8-1) & 0x3fe);
sg = t_sqrt[0];
sy = t_sqrt[1];
/* Here we have three Newton-Rhapson iterations each of a
division and a square root and the remainder of the
argument reduction, all interleaved. */
sd = -(sg*sg - sx);
fsgi = (xi + 0x40000000) >> 1 & 0x7f800000;
sy2 = sy + sy;
sg = sy*sd + sg; /* 16-bit approximation to sqrt(sx). */
e = -(sy*sg - almost_half);
SET_FLOAT_WORD (fsg, fsgi);
sd = -(sg*sg - sx);
sy = sy + e*sy2;
if ((xi & 0x7f800000) == 0)
goto denorm;
shx = sx * fsg;
sg = sg + sy*sd; /* 32-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 __sqrtf(x * two48) * twom24;
}
}
else if (x < 0)
{
#ifdef FE_INVALID_SQRT
feraiseexcept (FE_INVALID_SQRT);
/* For some reason, some PowerPC processors don't implement
FE_INVALID_SQRT. I guess no-one ever thought they'd be
used for square roots... :-) */
if (!fetestexcept (FE_INVALID))
#endif
feraiseexcept (FE_INVALID);
#ifndef _IEEE_LIBM
if (_LIB_VERSION != _IEEE_)
x = __kernel_standard(x,x,126);
else
#endif
x = *(const float*)&a_nan;
}
return f_washf(x);
}
weak_alias (__sqrtf, sqrtf)
/* Strictly, this is wrong, but the only places where _ieee754_sqrt is
used will not pass in a negative result. */
strong_alias(__sqrtf,__ieee754_sqrtf)
|
