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/* Double-precision floating point square root.
Copyright (C) 1997-2015 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)
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