196 lines
5.9 KiB
C
196 lines
5.9 KiB
C
/*
|
|
(C) Copyright IBM Corp. 2009
|
|
|
|
All rights reserved.
|
|
|
|
Redistribution and use in source and binary forms, with or without
|
|
modification, are permitted provided that the following conditions are met:
|
|
|
|
* Redistributions of source code must retain the above copyright notice,
|
|
this list of conditions and the following disclaimer.
|
|
* 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.
|
|
* Neither the name of IBM nor the names of its contributors may be
|
|
used to endorse or promote products derived from this software without
|
|
specific prior written permission.
|
|
|
|
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "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 COPYRIGHT OWNER OR CONTRIBUTORS 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 <math.h>
|
|
#include "local.h"
|
|
|
|
#ifdef _LDBL_EQ_DBL
|
|
/* On platforms where long double is as wide as double. */
|
|
long double
|
|
sqrtl (long double x)
|
|
{
|
|
return sqrt(x);
|
|
}
|
|
|
|
#else
|
|
|
|
/* This code is based upon the version in the BSD math's library.
|
|
That code is...
|
|
*
|
|
* 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 <float.h>
|
|
#include "ieeefp.h"
|
|
|
|
#ifndef LDBL_NBIT
|
|
#define LDBL_NBIT 0
|
|
#endif
|
|
|
|
#ifndef LDBL_MAX_EXP
|
|
#define LDBL_MAX_EXP DBL_MAX_EXP
|
|
#endif
|
|
|
|
/* Return (x + ulp) for normal positive x. Assumes no overflow. */
|
|
|
|
static inline long double
|
|
inc (long double x)
|
|
{
|
|
union ieee_ext_u ux = { .extu_ld = x, };
|
|
|
|
if (++ux.extu_ext.ext_fracl == 0)
|
|
{
|
|
if (++ux.extu_ext.ext_frach == 0)
|
|
{
|
|
ux.extu_ext.ext_exp++;
|
|
ux.extu_ext.ext_frach |= LDBL_NBIT;
|
|
}
|
|
}
|
|
|
|
return ux.extu_ld;
|
|
}
|
|
|
|
/* Return (x - ulp) for normal positive x. Assumes no underflow. */
|
|
|
|
static inline long double
|
|
dec (long double x)
|
|
{
|
|
union ieee_ext_u ux = { .extu_ld = x, };
|
|
|
|
if (ux.extu_ext.ext_fracl-- == 0)
|
|
{
|
|
if (ux.extu_ext.ext_frach-- == LDBL_NBIT)
|
|
{
|
|
ux.extu_ext.ext_exp--;
|
|
ux.extu_ext.ext_frach |= LDBL_NBIT;
|
|
}
|
|
}
|
|
|
|
return ux.extu_ld;
|
|
}
|
|
|
|
/* This is slow, but simple and portable. */
|
|
|
|
long double
|
|
sqrtl (long double x)
|
|
{
|
|
union ieee_ext_u ux = { .extu_ld = x, };
|
|
int k, r;
|
|
long double lo, xn;
|
|
|
|
/* If x = NaN, then sqrt(x) = NaN. */
|
|
/* If x = Inf, then sqrt(x) = Inf. */
|
|
/* If x = -Inf, then sqrt(x) = NaN. */
|
|
if (ux.extu_ext.ext_exp == LDBL_MAX_EXP * 2 - 1)
|
|
return (x * x + x);
|
|
|
|
/* If x = +-0, then sqrt(x) = +-0. */
|
|
if (x == 0.0L || x == -0.0L)
|
|
return x;
|
|
|
|
/* If x < 0, then raise invalid and return NaN. */
|
|
if (ux.extu_ext.ext_sign)
|
|
return ((x - x) / (x - x));
|
|
|
|
if (ux.extu_ext.ext_exp == 0)
|
|
{
|
|
/* Adjust subnormal numbers. */
|
|
ux.extu_ld *= 0x1.0p514;
|
|
k = -514;
|
|
}
|
|
else
|
|
k = 0;
|
|
|
|
/* ux.extu_ld is a normal number, so break it into ux.extu_ld = e*2^n where
|
|
ux.extu_ld = (2*e)*2^2k for odd n and ux.extu_ld = (4*e)*2^2k for even n. */
|
|
|
|
if ((ux.extu_ext.ext_exp - EXT_EXP_BIAS) & 1)
|
|
{
|
|
/* n is even. */
|
|
k += ux.extu_ext.ext_exp - EXT_EXP_BIAS - 1; /* 2k = n - 2. */
|
|
ux.extu_ext.ext_exp = EXT_EXP_BIAS + 1; /* ux.extu_ld in [2,4). */
|
|
}
|
|
else
|
|
{
|
|
k += ux.extu_ext.ext_exp - EXT_EXP_BIAS; /* 2k = n - 1. */
|
|
ux.extu_ext.ext_exp = EXT_EXP_BIAS; /* ux.extu_ld in [1,2). */
|
|
}
|
|
|
|
/* Newton's iteration.
|
|
Split ux.extu_ld into a high and low part to achieve additional precision. */
|
|
|
|
xn = sqrt ((double) ux.extu_ld); /* 53-bit estimate of sqrtl(x). */
|
|
|
|
#if LDBL_MANT_DIG > 100
|
|
xn = (xn + (ux.extu_ld / xn)) * 0.5; /* 106-bit estimate. */
|
|
#endif
|
|
|
|
lo = ux.extu_ld;
|
|
ux.extu_ext.ext_fracl = 0; /* Zero out lower bits. */
|
|
lo = (lo - ux.extu_ld) / xn; /* Low bits divided by xn. */
|
|
xn = xn + (ux.extu_ld / xn); /* High portion of estimate. */
|
|
ux.extu_ld = xn + lo; /* Combine everything. */
|
|
ux.extu_ext.ext_exp += (k >> 1) - 1;
|
|
|
|
xn = x / ux.extu_ld; /* Chopped quotient (inexact?). */
|
|
|
|
/* For simplicity we round to nearest. */
|
|
xn = inc (xn); /* xn = xn + ulp. */
|
|
|
|
ux.extu_ld = ux.extu_ld + xn; /* Chopped sum. */
|
|
ux.extu_ext.ext_exp--;
|
|
|
|
return ux.extu_ld;
|
|
}
|
|
#endif /* ! _LDBL_EQ_DBL */
|
|
|
|
|