/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak __sqrtl = sqrtl
#include "libm.h"
#include "longdouble.h"
extern int __swapTE(int);
extern int __swapEX(int);
extern enum fp_direction_type __swapRD(enum fp_direction_type);
/*
* in struct longdouble, msw consists of
* unsigned short sgn:1;
* unsigned short exp:15;
* unsigned short frac1:16;
*/
#ifdef __LITTLE_ENDIAN
/* array indices used to access words within a double */
#define HIWORD 1
#define LOWORD 0
/* structure used to access words within a quad */
union longdouble {
struct {
unsigned int frac4;
unsigned int frac3;
unsigned int frac2;
unsigned int msw;
} l;
long double d;
};
/* default NaN returned for sqrt(neg) */
static const union longdouble
qnan = { 0xffffffff, 0xffffffff, 0xffffffff, 0x7fffffff };
/* signalling NaN used to raise invalid */
static const union {
unsigned u[2];
double d;
} snan = { 0, 0x7ff00001 };
#else
/* array indices used to access words within a double */
#define HIWORD 0
#define LOWORD 1
/* structure used to access words within a quad */
union longdouble {
struct {
unsigned int msw;
unsigned int frac2;
unsigned int frac3;
unsigned int frac4;
} l;
long double d;
};
/* default NaN returned for sqrt(neg) */
static const union longdouble
qnan = { 0x7fffffff, 0xffffffff, 0xffffffff, 0xffffffff };
/* signalling NaN used to raise invalid */
static const union {
unsigned u[2];
double d;
} snan = { 0x7ff00001, 0 };
#endif /* __LITTLE_ENDIAN */
static const double
zero = 0.0,
half = 0.5,
one = 1.0,
huge = 1.0e300,
tiny = 1.0e-300,
two36 = 6.87194767360000000000e+10,
two30 = 1.07374182400000000000e+09,
two6 = 6.40000000000000000000e+01,
two4 = 1.60000000000000000000e+01,
twom18 = 3.81469726562500000000e-06,
twom28 = 3.72529029846191406250e-09,
twom42 = 2.27373675443232059479e-13,
twom60 = 8.67361737988403547206e-19,
twom62 = 2.16840434497100886801e-19,
twom66 = 1.35525271560688054251e-20,
twom90 = 8.07793566946316088742e-28,
twom113 = 9.62964972193617926528e-35,
twom124 = 4.70197740328915003187e-38;
/*
* Extract the exponent and normalized significand (represented as
* an array of five doubles) from a finite, nonzero quad.
*/
static int
__q_unpack(const union longdouble *x, double *s)
{
union {
double d;
unsigned int l[2];
} u;
double b;
unsigned int lx, w[3];
int ex;
/* get the normalized significand and exponent */
ex = (int) ((x->l.msw & 0x7fffffff) >> 16);
lx = x->l.msw & 0xffff;
if (ex)
{
lx |= 0x10000;
w[0] = x->l.frac2;
w[1] = x->l.frac3;
w[2] = x->l.frac4;
}
else
{
if (lx | (x->l.frac2 & 0xfffe0000))
{
w[0] = x->l.frac2;
w[1] = x->l.frac3;
w[2] = x->l.frac4;
ex = 1;
}
else if (x->l.frac2 | (x->l.frac3 & 0xfffe0000))
{
lx = x->l.frac2;
w[0] = x->l.frac3;
w[1] = x->l.frac4;
w[2] = 0;
ex = -31;
}
else if (x->l.frac3 | (x->l.frac4 & 0xfffe0000))
{
lx = x->l.frac3;
w[0] = x->l.frac4;
w[1] = w[2] = 0;
ex = -63;
}
else
{
lx = x->l.frac4;
w[0] = w[1] = w[2] = 0;
ex = -95;
}
while ((lx & 0x10000) == 0)
{
lx = (lx << 1) | (w[0] >> 31);
w[0] = (w[0] << 1) | (w[1] >> 31);
w[1] = (w[1] << 1) | (w[2] >> 31);
w[2] <<= 1;
ex--;
}
}
/* extract the significand into five doubles */
u.l[HIWORD] = 0x42300000;
u.l[LOWORD] = 0;
b = u.d;
u.l[LOWORD] = lx;
s[0] = u.d - b;
u.l[HIWORD] = 0x40300000;
u.l[LOWORD] = 0;
b = u.d;
u.l[LOWORD] = w[0] & 0xffffff00;
s[1] = u.d - b;
u.l[HIWORD] = 0x3e300000;
u.l[LOWORD] = 0;
b = u.d;
u.l[HIWORD] |= w[0] & 0xff;
u.l[LOWORD] = w[1] & 0xffff0000;
s[2] = u.d - b;
u.l[HIWORD] = 0x3c300000;
u.l[LOWORD] = 0;
b = u.d;
u.l[HIWORD] |= w[1] & 0xffff;
u.l[LOWORD] = w[2] & 0xff000000;
s[3] = u.d - b;
u.l[HIWORD] = 0x3c300000;
u.l[LOWORD] = 0;
b = u.d;
u.l[LOWORD] = w[2] & 0xffffff;
s[4] = u.d - b;
return ex - 0x3fff;
}
/*
* Pack an exponent and array of three doubles representing a finite,
* nonzero number into a quad. Assume the sign is already there and
* the rounding mode has been fudged accordingly.
*/
static void
__q_pack(const double *z, int exp, enum fp_direction_type rm,
union longdouble *x, int *inexact)
{
union {
double d;
unsigned int l[2];
} u;
double s[3], t, t2;
unsigned int msw, frac2, frac3, frac4;
/* bias exponent and strip off integer bit */
exp += 0x3fff;
s[0] = z[0] - one;
s[1] = z[1];
s[2] = z[2];
/*
* chop the significand to obtain the fraction;
* use round-to-minus-infinity to ensure chopping
*/
(void) __swapRD(fp_negative);
/* extract the first eighty bits of fraction */
t = s[1] + s[2];
u.d = two36 + (s[0] + t);
msw = u.l[LOWORD];
s[0] -= (u.d - two36);
u.d = two4 + (s[0] + t);
frac2 = u.l[LOWORD];
s[0] -= (u.d - two4);
u.d = twom28 + (s[0] + t);
frac3 = u.l[LOWORD];
s[0] -= (u.d - twom28);
/* condense the remaining fraction; errors here won't matter */
t = s[0] + s[1];
s[1] = ((s[0] - t) + s[1]) + s[2];
s[0] = t;
/* get the last word of fraction */
u.d = twom60 + (s[0] + s[1]);
frac4 = u.l[LOWORD];
s[0] -= (u.d - twom60);
/*
* keep track of what's left for rounding; note that
* t2 will be non-negative due to rounding mode
*/
t = s[0] + s[1];
t2 = (s[0] - t) + s[1];
if (t != zero)
{
*inexact = 1;
/* decide whether to round the fraction up */
if (rm == fp_positive || (rm == fp_nearest && (t > twom113 ||
(t == twom113 && (t2 != zero || frac4 & 1)))))
{
/* round up and renormalize if necessary */
if (++frac4 == 0)
if (++frac3 == 0)
if (++frac2 == 0)
if (++msw == 0x10000)
{
msw = 0;
exp++;
}
}
}
/* assemble the result */
x->l.msw |= msw | (exp << 16);
x->l.frac2 = frac2;
x->l.frac3 = frac3;
x->l.frac4 = frac4;
}
/*
* Compute the square root of x and place the TP result in s.
*/
static void
__q_tp_sqrt(const double *x, double *s)
{
double c, rr, r[3], tt[3], t[5];
/* approximate the divisor for the Newton iteration */
c = sqrt((x[0] + x[1]) + x[2]);
rr = half / c;
/* compute the first five "digits" of the square root */
t[0] = (c + two30) - two30;
tt[0] = t[0] + t[0];
r[0] = ((x[0] - t[0] * t[0]) + x[1]) + x[2];
t[1] = (rr * (r[0] + x[3]) + two6) - two6;
tt[1] = t[1] + t[1];
r[0] -= tt[0] * t[1];
r[1] = x[3] - t[1] * t[1];
c = (r[1] + twom18) - twom18;
r[0] += c;
r[1] = (r[1] - c) + x[4];
t[2] = (rr * (r[0] + r[1]) + twom18) - twom18;
tt[2] = t[2] + t[2];
r[0] -= tt[0] * t[2];
r[1] -= tt[1] * t[2];
c = (r[1] + twom42) - twom42;
r[0] += c;
r[1] = (r[1] - c) - t[2] * t[2];
t[3] = (rr * (r[0] + r[1]) + twom42) - twom42;
r[0] = ((r[0] - tt[0] * t[3]) + r[1]) - tt[1] * t[3];
r[1] = -tt[2] * t[3];
c = (r[1] + twom90) - twom90;
r[0] += c;
r[1] = (r[1] - c) - t[3] * t[3];
t[4] = (rr * (r[0] + r[1]) + twom66) - twom66;
/* here we just need to get the sign of the remainder */
c = (((((r[0] - tt[0] * t[4]) - tt[1] * t[4]) + r[1])
- tt[2] * t[4]) - (t[3] + t[3]) * t[4]) - t[4] * t[4];
/* reduce to three doubles */
t[0] += t[1];
t[1] = t[2] + t[3];
t[2] = t[4];
/* if the third term might lie on a rounding boundary, perturb it */
if (c != zero && t[2] == (twom62 + t[2]) - twom62)
{
if (c < zero)
t[2] -= twom124;
else
t[2] += twom124;
}
/* condense the square root */
c = t[1] + t[2];
t[2] += (t[1] - c);
t[1] = c;
c = t[0] + t[1];
s[1] = t[1] + (t[0] - c);
s[0] = c;
if (s[1] == zero)
{
c = s[0] + t[2];
s[1] = t[2] + (s[0] - c);
s[0] = c;
s[2] = zero;
}
else
{
c = s[1] + t[2];
s[2] = t[2] + (s[1] - c);
s[1] = c;
}
}
long double
sqrtl(long double ldx)
{
union longdouble x;
volatile double t;
double xx[5], zz[3];
enum fp_direction_type rm;
int ex, inexact, exc, traps;
/* clear cexc */
t = zero;
t -= zero;
/* check for zero operand */
x.d = ldx;
if (!((x.l.msw & 0x7fffffff) | x.l.frac2 | x.l.frac3 | x.l.frac4))
return ldx;
/* handle nan and inf cases */
if ((x.l.msw & 0x7fffffff) >= 0x7fff0000)
{
if ((x.l.msw & 0xffff) | x.l.frac2 | x.l.frac3 | x.l.frac4)
{
if (!(x.l.msw & 0x8000))
{
/* snan, signal invalid */
t += snan.d;
}
x.l.msw |= 0x8000;
return x.d;
}
if (x.l.msw & 0x80000000)
{
/* sqrt(-inf), signal invalid */
t = -one;
t = sqrt(t);
return qnan.d;
}
/* sqrt(inf), return inf */
return x.d;
}
/* handle negative numbers */
if (x.l.msw & 0x80000000)
{
t = -one;
t = sqrt(t);
return qnan.d;
}
/* now x is finite, positive */
traps = __swapTE(0);
exc = __swapEX(0);
rm = __swapRD(fp_nearest);
ex = __q_unpack(&x, xx);
if (ex & 1)
{
/* make exponent even */
xx[0] += xx[0];
xx[1] += xx[1];
xx[2] += xx[2];
xx[3] += xx[3];
xx[4] += xx[4];
ex--;
}
__q_tp_sqrt(xx, zz);
/* put everything together */
x.l.msw = 0;
inexact = 0;
__q_pack(zz, ex >> 1, rm, &x, &inexact);
(void) __swapRD(rm);
(void) __swapEX(exc);
(void) __swapTE(traps);
if (inexact)
{
t = huge;
t += tiny;
}
return x.d;
}