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/* $OpenBSD: e_sqrtl.c,v 1.3 2016/09/12 19:47:02 guenther Exp $ */
/*-
* 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 <sys/types.h>
#include <machine/ieee.h> /* for struct ieee_ext */
#include <fenv.h>
#include <float.h>
#include <math.h>
#ifdef EXT_IMPLICIT_NBIT
#define LDBL_NBIT 0
#else /* EXT_IMPLICIT_NBIT */
#define LDBL_NBIT 0x80000000
#endif /* EXT_IMPLICIT_NBIT */
/* Return (x + ulp) for normal positive x. Assumes no overflow. */
static inline long double
inc(long double x)
{
struct ieee_ext *p = (struct ieee_ext *)&x;
#ifdef EXT_FRACHMBITS
uint64_t frach;
frach = ((uint64_t)p->ext_frach << EXT_FRACHMBITS) | p->ext_frachm;
frach++;
p->ext_frach = frach >> EXT_FRACHMBITS;
p->ext_frachm = frach & 0x00000000ffffffff;
#else /* EXT_FRACHMBITS */
uint32_t frach;
p->ext_frach++;
frach = p->ext_frach;
#endif /* EXT_FRACHMBITS */
if (frach == 0) {
#ifdef EXT_FRACLMBITS
uint64_t fracl;
fracl = ((uint64_t)p->ext_fraclm << EXT_FRACLBITS) |
p->ext_fracl;
fracl++;
p->ext_fraclm = fracl >> EXT_FRACLBITS;
p->ext_fracl = fracl & 0x00000000ffffffff;
#else /* EXT_FRACLMBITS */
uint32_t fracl;
p->ext_fracl++;
fracl = p->ext_fracl;
#endif /* EXT_FRACLMBITS */
if (fracl == 0) {
p->ext_exp++;
p->ext_frach |= LDBL_NBIT;
}
}
return x;
}
/* Return (x - ulp) for normal positive x. Assumes no underflow. */
static inline long double
dec(long double x)
{
struct ieee_ext *p = (struct ieee_ext *)&x;
#ifdef EXT_FRACLMBITS
uint64_t fracl;
fracl = ((uint64_t)p->ext_fraclm << EXT_FRACLBITS) | p->ext_fracl;
fracl--;
p->ext_fraclm = fracl >> EXT_FRACLBITS;
p->ext_fracl = fracl & 0x00000000ffffffff;
#else /* EXT_FRACLMBITS */
uint32_t fracl;
p->ext_fracl--;
fracl = p->ext_fracl;
#endif /* EXT_FRACLMBITS */
if (fracl == 0) {
#ifdef EXT_FRACHMBITS
uint64_t frach;
frach = ((uint64_t)p->ext_frach << EXT_FRACHMBITS) |
p->ext_frachm;
frach--;
p->ext_frach = frach >> EXT_FRACHMBITS;
p->ext_frachm = frach & 0x00000000ffffffff;
#else /* EXT_FRACHMBITS */
uint32_t frach;
p->ext_frach--;
frach = p->ext_frach;
#endif /* EXT_FRACHMBITS */
if (frach == LDBL_NBIT) {
p->ext_exp--;
p->ext_frach |= LDBL_NBIT;
}
}
return x;
}
/*
* This is slow, but simple and portable. You should use hardware sqrt
* if possible.
*/
long double
sqrtl(long double x)
{
union {
long double e;
struct ieee_ext bits;
} u;
int k, r;
long double lo, xn;
u.e = x;
/* If x = NaN, then sqrt(x) = NaN. */
/* If x = Inf, then sqrt(x) = Inf. */
/* If x = -Inf, then sqrt(x) = NaN. */
if (u.bits.ext_exp == LDBL_MAX_EXP * 2 - 1)
return (x * x + x);
/* If x = +-0, then sqrt(x) = +-0. */
if ((u.bits.ext_frach
#ifdef EXT_FRACHMBITS
| u.bits.ext_frachm
#endif /* EXT_FRACHMBITS */
#ifdef EXT_FRACLMBITS
| u.bits.ext_fraclm
#endif /* EXT_FRACLMBITS */
| u.bits.ext_fracl | u.bits.ext_exp) == 0)
return (x);
/* If x < 0, then raise invalid and return NaN */
if (u.bits.ext_sign)
return ((x - x) / (x - x));
if (u.bits.ext_exp == 0) {
/* Adjust subnormal numbers. */
u.e *= 0x1.0p514;
k = -514;
} else {
k = 0;
}
/*
* u.e is a normal number, so break it into u.e = e*2^n where
* u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
*/
if ((u.bits.ext_exp - 0x3ffe) & 1) { /* n is odd. */
k += u.bits.ext_exp - 0x3fff; /* 2k = n - 1. */
u.bits.ext_exp = 0x3fff; /* u.e in [1,2). */
} else {
k += u.bits.ext_exp - 0x4000; /* 2k = n - 2. */
u.bits.ext_exp = 0x4000; /* u.e in [2,4). */
}
/*
* Newton's iteration.
* Split u.e into a high and low part to achieve additional precision.
*/
xn = sqrt(u.e); /* 53-bit estimate of sqrtl(x). */
#if LDBL_MANT_DIG > 100
xn = (xn + (u.e / xn)) * 0.5; /* 106-bit estimate. */
#endif
lo = u.e;
u.bits.ext_fracl = 0; /* Zero out lower bits. */
#ifdef EXT_FRACLMBITS
u.bits.ext_fraclm = 0;
#endif /* EXT_FRACLMBITS */
lo = (lo - u.e) / xn; /* Low bits divided by xn. */
xn = xn + (u.e / xn); /* High portion of estimate. */
u.e = xn + lo; /* Combine everything. */
u.bits.ext_exp += (k >> 1) - 1;
feclearexcept(FE_INEXACT);
r = fegetround();
fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
xn = x / u.e; /* Chopped quotient (inexact?). */
if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
if (xn == u.e) {
fesetround(r);
return (u.e);
}
/* Round correctly for inputs like x = y**2 - ulp. */
xn = dec(xn); /* xn = xn - ulp. */
}
if (r == FE_TONEAREST) {
xn = inc(xn); /* xn = xn + ulp. */
} else if (r == FE_UPWARD) {
u.e = inc(u.e); /* u.e = u.e + ulp. */
xn = inc(xn); /* xn = xn + ulp. */
}
u.e = u.e + xn; /* Chopped sum. */
fesetround(r); /* Restore env and raise inexact */
u.bits.ext_exp--;
return (u.e);
}
DEF_STD(sqrtl);
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