template<class T, class Policies> inline
T width(const interval<T, Policies>& x)
{
if (interval_lib::detail::test_input(x)) return static_cast<T>(0);
typename Policies::rounding rnd;
return rnd.sub_up(x.upper(), x.lower());
}
template<class T, class Policies> inline
interval<T, Policies> div_zero_part1(const interval<T, Policies>& x,
const interval<T, Policies>& y, bool& b)
{
// assert(y.lower() < 0 && y.upper() > 0);
if (is_zero(x.lower()) && is_zero(x.upper()))
{
b = false;
return x;
}
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
const T& xl = x.lower();
const T& xu = x.upper();
const T& yl = y.lower();
const T& yu = y.upper();
typedef typename I::checking checking;
const T& inf = checking::inf();
if (is_neg(xu))
{
b = true;
return I(-inf, rnd.div_up(xu, yu), true);
}
else if (is_neg(xl))
{
b = false;
return I(-inf, inf, true);
}
else
{
b = true;
return I(-inf, rnd.div_up(xl, yl), true);
}
}
template<class T, class Policies> inline
interval<T, Policies> cos(const interval<T, Policies>& x)
{
if (interval_lib::detail::test_input(x))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
typedef typename interval_lib::unprotect<I>::type R;
// get lower bound within [0, pi]
const R pi2 = interval_lib::pi_twice<R>();
R tmp = fmod((const R&)x, pi2);
if (width(tmp) >= pi2.lower())
return I(static_cast<T>(-1), static_cast<T>(1), true); // we are covering a full period
if (tmp.lower() >= interval_lib::constants::pi_upper<T>())
return -cos(tmp - interval_lib::pi<R>());
T l = tmp.lower();
T u = tmp.upper();
BOOST_USING_STD_MIN();
// separate into monotone subintervals
if (u <= interval_lib::constants::pi_lower<T>())
return I(rnd.cos_down(u), rnd.cos_up(l), true);
else if (u <= pi2.lower())
return I(static_cast<T>(-1), rnd.cos_up(min BOOST_PREVENT_MACRO_SUBSTITUTION(rnd.sub_down(pi2.lower(), u), l)), true);
else
return I(static_cast<T>(-1), static_cast<T>(1), true);
}
template<class T, class Policies> inline
interval<T, Policies> widen(const interval<T, Policies>& x, const T& v)
{
if (interval_lib::detail::test_input(x))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
return interval<T, Policies>(rnd.sub_down(x.lower(), v),
rnd.add_up (x.upper(), v), true);
}
template<class T, class Policies> inline
interval<T, Policies> operator-(const T& x, const interval<T, Policies>& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
return interval<T,Policies>(rnd.sub_down(x, y.upper()),
rnd.sub_up (x, y.lower()), true);
}
template<class T, class Policies> inline
interval<T, Policies> exp(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x))
return I::empty();
typename Policies::rounding rnd;
return I(rnd.exp_down(x.lower()), rnd.exp_up(x.upper()), true);
}
template<class I> inline
I div(const typename I::base_type& x, const typename I::base_type& y)
{
typedef typename I::traits_type Policies;
if (detail::test_input<typename I::base_type, Policies>(x, y) || user::is_zero(y))
return I::empty();
typename Policies::rounding rnd;
return I(rnd.div_down(x, y), rnd.div_up(x, y), true);
}
template<class T, class Policies> inline
interval<T, Policies> acosh(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x) || x.upper() < static_cast<T>(1))
return I::empty();
typename Policies::rounding rnd;
T l = x.lower() <= static_cast<T>(1) ? static_cast<T>(0) : rnd.acosh_down(x.lower());
return I(l, rnd.acosh_up(x.upper()), true);
}
template<class T, class Policies> inline
interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
T n = rnd.int_down(rnd.div_down(x.lower(), y));
return (const I&)x - n * I(y);
}
template<class T, class Policies> inline
T median(const interval<T, Policies>& x)
{
if (interval_lib::detail::test_input(x)) {
typedef typename Policies::checking checking;
return checking::nan();
}
typename Policies::rounding rnd;
return rnd.median(x.lower(), x.upper());
}
template<class T, class Policies> inline
interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
const T& yb = interval_lib::detail::is_neg(x) ? y.lower() : y.upper();
T n = rnd.int_down(rnd.div_down(x, yb));
return x - n * (const I&)y;
}
template<class T, class Policies> inline
interval<T, Policies>& interval<T, Policies>::operator-=(const T& r)
{
if (interval_lib::detail::test_input(*this, r))
set_empty();
else {
typename Policies::rounding rnd;
set(rnd.sub_down(low, r), rnd.sub_up(up, r));
}
return *this;
}
template<class T, class Policies> inline
interval<T, Policies> div_non_zero(const T& x, const interval<T, Policies>& y)
{
// assert(!in_zero(y));
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
const T& yl = y.lower();
const T& yu = y.upper();
if (::boost::numeric::interval_lib::user::is_neg(x))
return I(rnd.div_down(x, yl), rnd.div_up(x, yu), true);
else
return I(rnd.div_down(x, yu), rnd.div_up(x, yl), true);
}
template<class T, class Policies> inline
interval<T, Policies> operator/(const interval<T, Policies>& x, const T& y)
{
if (interval_lib::detail::test_input(x, y) || interval_lib::user::is_zero(y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
const T& xl = x.lower();
const T& xu = x.upper();
if (interval_lib::user::is_neg(y))
return interval<T, Policies>(rnd.div_down(xu, y), rnd.div_up(xl, y), true);
else
return interval<T, Policies>(rnd.div_down(xl, y), rnd.div_up(xu, y), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_zero_part2(const interval<T, Policies>& x,
const interval<T, Policies>& y)
{
// assert(::boost::numeric::interval_lib::user::is_neg(y.lower()) && ::boost::numeric::interval_lib::user::is_pos(y.upper()) && (div_zero_part1(x, y, b), b));
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
typedef typename Policies::checking checking;
if (::boost::numeric::interval_lib::user::is_neg(x.upper()))
return I(rnd.div_down(x.upper(), y.lower()), checking::pos_inf(), true);
else
return I(rnd.div_down(x.lower(), y.upper()), checking::pos_inf(), true);
}
template<class T, class Policies> inline
interval<T, Policies> log(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x) ||
!interval_lib::user::is_pos(x.upper()))
return I::empty();
typename Policies::rounding rnd;
typedef typename Policies::checking checking;
T l = !interval_lib::user::is_pos(x.lower())
? checking::neg_inf() : rnd.log_down(x.lower());
return I(l, rnd.log_up(x.upper()), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_positive(const T& x, const T& yu)
{
// assert(yu > T(0));
typedef interval<T, Policies> I;
if (is_zero(x)) return I(0, 0, true);
typename Policies::rounding rnd;
typedef typename Policies::checking checking;
const T& inf = checking::inf();
if (is_neg(x))
return I(-inf, rnd.div_up(x, yu), true);
else
return I(rnd.div_down(x, yu), inf, true);
}
template<class T, class Policies> inline
interval<T, Policies> div_positive(const T& x, const T& yu)
{
// assert(::boost::numeric::interval_lib::user::is_pos(yu));
typedef interval<T, Policies> I;
if (::boost::numeric::interval_lib::user::is_zero(x))
return I(static_cast<T>(0), static_cast<T>(0), true);
typename Policies::rounding rnd;
typedef typename Policies::checking checking;
if (::boost::numeric::interval_lib::user::is_neg(x))
return I(checking::neg_inf(), rnd.div_up(x, yu), true);
else
return I(rnd.div_down(x, yu), checking::pos_inf(), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_zero_part2(const interval<T, Policies>& x,
const interval<T, Policies>& y)
{
// assert(y.lower() < 0 && y.upper() > 0 && (div_zero_part1(x, y, b), b));
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
typedef typename I::checking checking;
const T& inf = checking::inf();
if (is_neg(x.upper()))
return I(rnd.div_down(x.upper(), y.lower()), inf, true);
else
return I(rnd.div_down(x.lower(), y.upper()), inf, true);
}
template<class T, class Policies> inline
interval<T, Policies> atanh(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x)
|| x.upper() < static_cast<T>(-1) || x.lower() > static_cast<T>(1))
return I::empty();
typename Policies::rounding rnd;
typedef typename Policies::checking checking;
T l = (x.lower() <= static_cast<T>(-1))
? checking::neg_inf() : rnd.atanh_down(x.lower());
T u = (x.upper() >= static_cast<T>(1) )
? checking::pos_inf() : rnd.atanh_up (x.upper());
return I(l, u, true);
}
template<class T, class Policies> inline
interval<T, Policies> asin(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x)
|| x.upper() < static_cast<T>(-1) || x.lower() > static_cast<T>(1))
return I::empty();
typename Policies::rounding rnd;
T l = (x.lower() <= static_cast<T>(-1))
? -interval_lib::constants::pi_half_upper<T>()
: rnd.asin_down(x.lower());
T u = (x.upper() >= static_cast<T>(1) )
? interval_lib::constants::pi_half_upper<T>()
: rnd.asin_up (x.upper());
return I(l, u, true);
}
template<class T, class Policies> inline
interval<T, Policies> operator*(const T& x, const interval<T, Policies>& y)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x, y))
return I::empty();
typename Policies::rounding rnd;
const T& yl = y.lower();
const T& yu = y.upper();
// x is supposed not to be infinite
if (interval_lib::user::is_neg(x))
return I(rnd.mul_down(x, yu), rnd.mul_up(x, yl), true);
else if (interval_lib::user::is_zero(x))
return I(static_cast<T>(0), static_cast<T>(0), true);
else
return I(rnd.mul_down(x, yl), rnd.mul_up(x, yu), true);
}
template<class T, class Policies> inline
interval<T, Policies> tan(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x))
return I::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<I>::type R;
// get lower bound within [-pi/2, pi/2]
const R pi = interval_lib::pi<R>();
R tmp = fmod((const R&)x, pi);
const T pi_half_d = interval_lib::constants::pi_half_lower<T>();
if (tmp.lower() >= pi_half_d)
tmp -= pi;
if (tmp.lower() <= -pi_half_d || tmp.upper() >= pi_half_d)
return I::whole();
return I(rnd.tan_down(tmp.lower()), rnd.tan_up(tmp.upper()), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_negative(const interval<T, Policies>& x, const T& yl)
{
// assert(yl < T(0));
if (is_zero(x.lower()) && is_zero(x.upper()))
return x;
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
const T& xl = x.lower();
const T& xu = x.upper();
typedef typename Policies::checking checking;
const T& inf = checking::inf();
if (is_neg(xu))
return I(rnd.div_down(xu, yl), inf, true);
else if (is_neg(xl))
return I(-inf, inf, true);
else
return I(-inf, rnd.div_up(xl, yl), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_positive(const interval<T, Policies>& x, const T& yu)
{
// assert(::boost::numeric::interval_lib::user::is_pos(yu));
if (::boost::numeric::interval_lib::user::is_zero(x.lower()) &&
::boost::numeric::interval_lib::user::is_zero(x.upper()))
return x;
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
const T& xl = x.lower();
const T& xu = x.upper();
typedef typename Policies::checking checking;
if (::boost::numeric::interval_lib::user::is_neg(xu))
return I(checking::neg_inf(), rnd.div_up(xu, yu), true);
else if (::boost::numeric::interval_lib::user::is_neg(xl))
return I(checking::neg_inf(), checking::pos_inf(), true);
else
return I(rnd.div_down(xl, yu), checking::pos_inf(), true);
}
template<class T, class Policies> inline
interval<T, Policies> div_zero_part1(const interval<T, Policies>& x,
const interval<T, Policies>& y, bool& b)
{
// assert(::boost::numeric::interval_lib::user::is_neg(y.lower()) && ::boost::numeric::interval_lib::user::is_pos(y.upper()));
if (::boost::numeric::interval_lib::user::is_zero(x.lower()) && ::boost::numeric::interval_lib::user::is_zero(x.upper()))
{ b = false; return x; }
typename Policies::rounding rnd;
typedef interval<T, Policies> I;
const T& xl = x.lower();
const T& xu = x.upper();
const T& yl = y.lower();
const T& yu = y.upper();
typedef typename Policies::checking checking;
if (::boost::numeric::interval_lib::user::is_neg(xu))
{ b = true; return I(checking::neg_inf(), rnd.div_up(xu, yu), true); }
else if (::boost::numeric::interval_lib::user::is_neg(xl))
{ b = false; return I(checking::neg_inf(), checking::pos_inf(), true); }
else
{ b = true; return I(checking::neg_inf(), rnd.div_up(xl, yl), true); }
}
template<class T, class Policies> inline
interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (detail::test_input(x))
return I::empty();
T one = static_cast<T>(1);
typename Policies::rounding rnd;
if (in_zero(x)) {
typedef typename Policies::checking checking;
if (!detail::is_zero(x.lower()))
if (!detail::is_zero(x.upper()))
return I::whole();
else
return I(-checking::inf(), rnd.div_up(one, x.lower()), true);
else
if (!detail::is_zero(x.upper()))
return I(rnd.div_down(one, x.upper()), checking::inf(), true);
else
return I::empty();
} else
return I(rnd.div_down(one, x.upper()), rnd.div_up(one, x.lower()), true);
}
template<class T, class Policies> inline
interval<T, Policies> cosh(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x))
return I::empty();
typename Policies::rounding rnd;
if (interval_lib::user::is_neg(x.upper()))
return I(rnd.cosh_down(x.upper()), rnd.cosh_up(x.lower()), true);
else if (!interval_lib::user::is_neg(x.lower()))
return I(rnd.cosh_down(x.lower()), rnd.cosh_up(x.upper()), true);
else
return I(static_cast<T>(0), rnd.cosh_up(-x.lower() > x.upper() ? x.lower() : x.upper()), true);
}
template<class T, class Policies> inline
interval<T, Policies> operator*(const interval<T, Policies>& x,
const interval<T, Policies>& y)
{
BOOST_NUMERIC_INTERVAL_using_max(min);
BOOST_NUMERIC_INTERVAL_using_max(max);
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x, y))
return I::empty();
typename Policies::rounding rnd;
const T& xl = x.lower();
const T& xu = x.upper();
const T& yl = y.lower();
const T& yu = y.upper();
if (interval_lib::detail::is_neg(xl))
if (interval_lib::detail::is_pos(xu))
if (interval_lib::detail::is_neg(yl))
if (interval_lib::detail::is_pos(yu)) // M * M
return I(min(rnd.mul_down(xl, yu), rnd.mul_down(xu, yl)),
max(rnd.mul_up (xl, yl), rnd.mul_up (xu, yu)), true);
else // M * N
return I(rnd.mul_down(xu, yl), rnd.mul_up(xl, yl), true);
else
if (interval_lib::detail::is_pos(yu)) // M * P
return I(rnd.mul_down(xl, yu), rnd.mul_up(xu, yu), true);
else // M * Z
return I(0, 0, true);
else
if (interval_lib::detail::is_neg(yl))
if (interval_lib::detail::is_pos(yu)) // N * M
return I(rnd.mul_down(xl, yu), rnd.mul_up(xl, yl), true);
else // N * N
return I(rnd.mul_down(xu, yu), rnd.mul_up(xl, yl), true);
else
if (interval_lib::detail::is_pos(yu)) // N * P
return I(rnd.mul_down(xl, yu), rnd.mul_up(xu, yl), true);
else // N * Z
return I(0, 0, true);
else
if (interval_lib::detail::is_pos(xu))
if (interval_lib::detail::is_neg(yl))
if (interval_lib::detail::is_pos(yu)) // P * M
return I(rnd.mul_down(xu, yl), rnd.mul_up(xu, yu), true);
else // P * N
return I(rnd.mul_down(xu, yl), rnd.mul_up(xl, yu), true);
else
if (interval_lib::detail::is_pos(yu)) // P * P
return I(rnd.mul_down(xl, yl), rnd.mul_up(xu, yu), true);
else // P * Z
return I(0, 0, true);
else // Z * ?
return I(0, 0, true);
}
template<class T, class Policies> inline
interval<T, Policies> operator*(const interval<T, Policies>& x,
const interval<T, Policies>& y)
{
BOOST_USING_STD_MIN();
BOOST_USING_STD_MAX();
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x, y))
return I::empty();
typename Policies::rounding rnd;
const T& xl = x.lower();
const T& xu = x.upper();
const T& yl = y.lower();
const T& yu = y.upper();
if (interval_lib::user::is_neg(xl))
if (interval_lib::user::is_pos(xu))
if (interval_lib::user::is_neg(yl))
if (interval_lib::user::is_pos(yu)) // M * M
return I(min BOOST_PREVENT_MACRO_SUBSTITUTION(rnd.mul_down(xl, yu), rnd.mul_down(xu, yl)),
max BOOST_PREVENT_MACRO_SUBSTITUTION(rnd.mul_up (xl, yl), rnd.mul_up (xu, yu)), true);
else // M * N
return I(rnd.mul_down(xu, yl), rnd.mul_up(xl, yl), true);
else
if (interval_lib::user::is_pos(yu)) // M * P
return I(rnd.mul_down(xl, yu), rnd.mul_up(xu, yu), true);
else // M * Z
return I(static_cast<T>(0), static_cast<T>(0), true);
else
if (interval_lib::user::is_neg(yl))
if (interval_lib::user::is_pos(yu)) // N * M
return I(rnd.mul_down(xl, yu), rnd.mul_up(xl, yl), true);
else // N * N
return I(rnd.mul_down(xu, yu), rnd.mul_up(xl, yl), true);
else
if (interval_lib::user::is_pos(yu)) // N * P
return I(rnd.mul_down(xl, yu), rnd.mul_up(xu, yl), true);
else // N * Z
return I(static_cast<T>(0), static_cast<T>(0), true);
else
if (interval_lib::user::is_pos(xu))
if (interval_lib::user::is_neg(yl))
if (interval_lib::user::is_pos(yu)) // P * M
return I(rnd.mul_down(xu, yl), rnd.mul_up(xu, yu), true);
else // P * N
return I(rnd.mul_down(xu, yl), rnd.mul_up(xl, yu), true);
else
if (interval_lib::user::is_pos(yu)) // P * P
return I(rnd.mul_down(xl, yl), rnd.mul_up(xu, yu), true);
else // P * Z
return I(static_cast<T>(0), static_cast<T>(0), true);
else // Z * ?
return I(static_cast<T>(0), static_cast<T>(0), true);
}