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); }