// Copyright Matthew Pulver 2018 - 2019. // Distributed under the Boost Software License, Version 1.0. // (See accompanying file LICENSE_1_0.txt or copy at // https://www.boost.org/LICENSE_1_0.txt) #ifndef BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP #define BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP #include <boost/cstdfloat.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/special_functions/trunc.hpp> #include <boost/math/special_functions/round.hpp> #include <boost/math/special_functions/acosh.hpp> #include <boost/math/special_functions/asinh.hpp> #include <boost/math/special_functions/atanh.hpp> #include <boost/math/special_functions/digamma.hpp> #include <boost/math/special_functions/polygamma.hpp> #include <boost/math/special_functions/erf.hpp> #include <boost/math/special_functions/lambert_w.hpp> #include <boost/math/tools/config.hpp> #include <boost/math/tools/promotion.hpp> #include <algorithm> #include <array> #include <cmath> #include <functional> #include <limits> #include <numeric> #include <ostream> #include <tuple> #include <type_traits> namespace boost { namespace math { namespace differentiation { // Automatic Differentiation v1 inline namespace autodiff_v1 { namespace detail { template <typename RealType, typename... RealTypes> struct promote_args_n { using type = typename tools::promote_args_2<RealType, typename promote_args_n<RealTypes...>::type>::type; }; template <typename RealType> struct promote_args_n<RealType> { using type = typename tools::promote_arg<RealType>::type; }; } // namespace detail template <typename RealType, typename... RealTypes> using promote = typename detail::promote_args_n<RealType, RealTypes...>::type; namespace detail { template <typename RealType, size_t Order> class fvar; template <typename T> struct is_fvar_impl : std::false_type {}; template <typename RealType, size_t Order> struct is_fvar_impl<fvar<RealType, Order>> : std::true_type {}; template <typename T> using is_fvar = is_fvar_impl<typename std::decay<T>::type>; template <typename RealType, size_t Order, size_t... Orders> struct nest_fvar { using type = fvar<typename nest_fvar<RealType, Orders...>::type, Order>; }; template <typename RealType, size_t Order> struct nest_fvar<RealType, Order> { using type = fvar<RealType, Order>; }; template <typename> struct get_depth_impl : std::integral_constant<size_t, 0> {}; template <typename RealType, size_t Order> struct get_depth_impl<fvar<RealType, Order>> : std::integral_constant<size_t, get_depth_impl<RealType>::value + 1> {}; template <typename T> using get_depth = get_depth_impl<typename std::decay<T>::type>; template <typename> struct get_order_sum_t : std::integral_constant<size_t, 0> {}; template <typename RealType, size_t Order> struct get_order_sum_t<fvar<RealType, Order>> : std::integral_constant<size_t, get_order_sum_t<RealType>::value + Order> {}; template <typename T> using get_order_sum = get_order_sum_t<typename std::decay<T>::type>; template <typename RealType> struct get_root_type { using type = RealType; }; template <typename RealType, size_t Order> struct get_root_type<fvar<RealType, Order>> { using type = typename get_root_type<RealType>::type; }; template <typename RealType, size_t Depth> struct type_at { using type = RealType; }; template <typename RealType, size_t Order, size_t Depth> struct type_at<fvar<RealType, Order>, Depth> { using type = typename conditional<Depth == 0, fvar<RealType, Order>, typename type_at<RealType, Depth - 1>::type>::type; }; template <typename RealType, size_t Depth> using get_type_at = typename type_at<RealType, Depth>::type; // Satisfies Boost's Conceptual Requirements for Real Number Types. // https://www.boost.org/libs/math/doc/html/math_toolkit/real_concepts.html template <typename RealType, size_t Order> class fvar { protected: std::array<RealType, Order + 1> v; public: using root_type = typename get_root_type<RealType>::type; // RealType in the root fvar<RealType,Order>. fvar() = default; // Initialize a variable or constant. fvar(root_type const&, bool const is_variable); // RealType(cr) | RealType | RealType is copy constructible. fvar(fvar const&) = default; // Be aware of implicit casting from one fvar<> type to another by this copy constructor. template <typename RealType2, size_t Order2> fvar(fvar<RealType2, Order2> const&); // RealType(ca) | RealType | RealType is copy constructible from the arithmetic types. explicit fvar(root_type const&); // Initialize a constant. (No epsilon terms.) template <typename RealType2> fvar(RealType2 const& ca); // Supports any RealType2 for which static_cast<root_type>(ca) compiles. // r = cr | RealType& | Assignment operator. fvar& operator=(fvar const&) = default; // r = ca | RealType& | Assignment operator from the arithmetic types. // Handled by constructor that takes a single parameter of generic type. // fvar& operator=(root_type const&); // Set a constant. // r += cr | RealType& | Adds cr to r. template <typename RealType2, size_t Order2> fvar& operator+=(fvar<RealType2, Order2> const&); // r += ca | RealType& | Adds ar to r. fvar& operator+=(root_type const&); // r -= cr | RealType& | Subtracts cr from r. template <typename RealType2, size_t Order2> fvar& operator-=(fvar<RealType2, Order2> const&); // r -= ca | RealType& | Subtracts ca from r. fvar& operator-=(root_type const&); // r *= cr | RealType& | Multiplies r by cr. template <typename RealType2, size_t Order2> fvar& operator*=(fvar<RealType2, Order2> const&); // r *= ca | RealType& | Multiplies r by ca. fvar& operator*=(root_type const&); // r /= cr | RealType& | Divides r by cr. template <typename RealType2, size_t Order2> fvar& operator/=(fvar<RealType2, Order2> const&); // r /= ca | RealType& | Divides r by ca. fvar& operator/=(root_type const&); // -r | RealType | Unary Negation. fvar operator-() const; // +r | RealType& | Identity Operation. fvar const& operator+() const; // cr + cr2 | RealType | Binary Addition template <typename RealType2, size_t Order2> promote<fvar, fvar<RealType2, Order2>> operator+(fvar<RealType2, Order2> const&) const; // cr + ca | RealType | Binary Addition fvar operator+(root_type const&) const; // ca + cr | RealType | Binary Addition template <typename RealType2, size_t Order2> friend fvar<RealType2, Order2> operator+(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr - cr2 | RealType | Binary Subtraction template <typename RealType2, size_t Order2> promote<fvar, fvar<RealType2, Order2>> operator-(fvar<RealType2, Order2> const&) const; // cr - ca | RealType | Binary Subtraction fvar operator-(root_type const&) const; // ca - cr | RealType | Binary Subtraction template <typename RealType2, size_t Order2> friend fvar<RealType2, Order2> operator-(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr * cr2 | RealType | Binary Multiplication template <typename RealType2, size_t Order2> promote<fvar, fvar<RealType2, Order2>> operator*(fvar<RealType2, Order2> const&)const; // cr * ca | RealType | Binary Multiplication fvar operator*(root_type const&)const; // ca * cr | RealType | Binary Multiplication template <typename RealType2, size_t Order2> friend fvar<RealType2, Order2> operator*(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr / cr2 | RealType | Binary Subtraction template <typename RealType2, size_t Order2> promote<fvar, fvar<RealType2, Order2>> operator/(fvar<RealType2, Order2> const&) const; // cr / ca | RealType | Binary Subtraction fvar operator/(root_type const&) const; // ca / cr | RealType | Binary Subtraction template <typename RealType2, size_t Order2> friend fvar<RealType2, Order2> operator/(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // For all comparison overloads, only the root term is compared. // cr == cr2 | bool | Equality Comparison template <typename RealType2, size_t Order2> bool operator==(fvar<RealType2, Order2> const&) const; // cr == ca | bool | Equality Comparison bool operator==(root_type const&) const; // ca == cr | bool | Equality Comparison template <typename RealType2, size_t Order2> friend bool operator==(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr != cr2 | bool | Inequality Comparison template <typename RealType2, size_t Order2> bool operator!=(fvar<RealType2, Order2> const&) const; // cr != ca | bool | Inequality Comparison bool operator!=(root_type const&) const; // ca != cr | bool | Inequality Comparison template <typename RealType2, size_t Order2> friend bool operator!=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr <= cr2 | bool | Less than equal to. template <typename RealType2, size_t Order2> bool operator<=(fvar<RealType2, Order2> const&) const; // cr <= ca | bool | Less than equal to. bool operator<=(root_type const&) const; // ca <= cr | bool | Less than equal to. template <typename RealType2, size_t Order2> friend bool operator<=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr >= cr2 | bool | Greater than equal to. template <typename RealType2, size_t Order2> bool operator>=(fvar<RealType2, Order2> const&) const; // cr >= ca | bool | Greater than equal to. bool operator>=(root_type const&) const; // ca >= cr | bool | Greater than equal to. template <typename RealType2, size_t Order2> friend bool operator>=(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr < cr2 | bool | Less than comparison. template <typename RealType2, size_t Order2> bool operator<(fvar<RealType2, Order2> const&) const; // cr < ca | bool | Less than comparison. bool operator<(root_type const&) const; // ca < cr | bool | Less than comparison. template <typename RealType2, size_t Order2> friend bool operator<(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // cr > cr2 | bool | Greater than comparison. template <typename RealType2, size_t Order2> bool operator>(fvar<RealType2, Order2> const&) const; // cr > ca | bool | Greater than comparison. bool operator>(root_type const&) const; // ca > cr | bool | Greater than comparison. template <typename RealType2, size_t Order2> friend bool operator>(typename fvar<RealType2, Order2>::root_type const&, fvar<RealType2, Order2> const&); // Will throw std::out_of_range if Order < order. template <typename... Orders> get_type_at<RealType, sizeof...(Orders)> at(size_t order, Orders... orders) const; template <typename... Orders> get_type_at<fvar, sizeof...(Orders)> derivative(Orders... orders) const; const RealType& operator[](size_t) const; fvar inverse() const; // Multiplicative inverse. fvar& negate(); // Negate and return reference to *this. static constexpr size_t depth = get_depth<fvar>::value; // Number of nested std::array<RealType,Order>. static constexpr size_t order_sum = get_order_sum<fvar>::value; explicit operator root_type() const; // Must be explicit, otherwise overloaded operators are ambiguous. template <typename T, typename = typename std::enable_if<std::is_arithmetic<typename std::decay<T>::type>::value>> explicit operator T() const; // Must be explicit; multiprecision has trouble without the std::enable_if fvar& set_root(root_type const&); // Apply coefficients using horner method. template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> apply_coefficients(size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const; template <typename Func> fvar apply_coefficients(size_t const order, Func const& f) const; // Use when function returns derivative(i)/factorial(i) and may have some infinite derivatives. template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> apply_coefficients_nonhorner(size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const; template <typename Func> fvar apply_coefficients_nonhorner(size_t const order, Func const& f) const; // Apply derivatives using horner method. template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> apply_derivatives(size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const; template <typename Func> fvar apply_derivatives(size_t const order, Func const& f) const; // Use when function returns derivative(i) and may have some infinite derivatives. template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> apply_derivatives_nonhorner(size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const; template <typename Func> fvar apply_derivatives_nonhorner(size_t const order, Func const& f) const; private: RealType epsilon_inner_product(size_t z0, size_t isum0, size_t m0, fvar const& cr, size_t z1, size_t isum1, size_t m1, size_t j) const; fvar epsilon_multiply(size_t z0, size_t isum0, fvar const& cr, size_t z1, size_t isum1) const; fvar epsilon_multiply(size_t z0, size_t isum0, root_type const& ca) const; fvar inverse_apply() const; fvar& multiply_assign_by_root_type(bool is_root, root_type const&); template <typename RealType2, size_t Orders2> friend class fvar; template <typename RealType2, size_t Order2> friend std::ostream& operator<<(std::ostream&, fvar<RealType2, Order2> const&); // C++11 Compatibility #ifdef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RootType> void fvar_cpp11(std::true_type, RootType const& ca, bool const is_variable); template <typename RootType> void fvar_cpp11(std::false_type, RootType const& ca, bool const is_variable); template <typename... Orders> get_type_at<RealType, sizeof...(Orders)> at_cpp11(std::true_type, size_t order, Orders... orders) const; template <typename... Orders> get_type_at<RealType, sizeof...(Orders)> at_cpp11(std::false_type, size_t order, Orders... orders) const; template <typename SizeType> fvar epsilon_multiply_cpp11(std::true_type, SizeType z0, size_t isum0, fvar const& cr, size_t z1, size_t isum1) const; template <typename SizeType> fvar epsilon_multiply_cpp11(std::false_type, SizeType z0, size_t isum0, fvar const& cr, size_t z1, size_t isum1) const; template <typename SizeType> fvar epsilon_multiply_cpp11(std::true_type, SizeType z0, size_t isum0, root_type const& ca) const; template <typename SizeType> fvar epsilon_multiply_cpp11(std::false_type, SizeType z0, size_t isum0, root_type const& ca) const; template <typename RootType> fvar& multiply_assign_by_root_type_cpp11(std::true_type, bool is_root, RootType const& ca); template <typename RootType> fvar& multiply_assign_by_root_type_cpp11(std::false_type, bool is_root, RootType const& ca); template <typename RootType> fvar& negate_cpp11(std::true_type, RootType const&); template <typename RootType> fvar& negate_cpp11(std::false_type, RootType const&); template <typename RootType> fvar& set_root_cpp11(std::true_type, RootType const& root); template <typename RootType> fvar& set_root_cpp11(std::false_type, RootType const& root); #endif }; // Standard Library Support Requirements // fabs(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> fabs(fvar<RealType, Order> const&); // abs(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> abs(fvar<RealType, Order> const&); // ceil(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> ceil(fvar<RealType, Order> const&); // floor(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> floor(fvar<RealType, Order> const&); // exp(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> exp(fvar<RealType, Order> const&); // pow(cr, ca) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> pow(fvar<RealType, Order> const&, typename fvar<RealType, Order>::root_type const&); // pow(ca, cr) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> pow(typename fvar<RealType, Order>::root_type const&, fvar<RealType, Order> const&); // pow(cr1, cr2) | RealType template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> pow(fvar<RealType1, Order1> const&, fvar<RealType2, Order2> const&); // sqrt(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> sqrt(fvar<RealType, Order> const&); // log(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> log(fvar<RealType, Order> const&); // frexp(cr1, &i) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> frexp(fvar<RealType, Order> const&, int*); // ldexp(cr1, i) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> ldexp(fvar<RealType, Order> const&, int); // cos(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> cos(fvar<RealType, Order> const&); // sin(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> sin(fvar<RealType, Order> const&); // asin(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> asin(fvar<RealType, Order> const&); // tan(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> tan(fvar<RealType, Order> const&); // atan(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> atan(fvar<RealType, Order> const&); // atan2(cr, ca) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> atan2(fvar<RealType, Order> const&, typename fvar<RealType, Order>::root_type const&); // atan2(ca, cr) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> atan2(typename fvar<RealType, Order>::root_type const&, fvar<RealType, Order> const&); // atan2(cr1, cr2) | RealType template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> atan2(fvar<RealType1, Order1> const&, fvar<RealType2, Order2> const&); // fmod(cr1,cr2) | RealType template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> fmod(fvar<RealType1, Order1> const&, fvar<RealType2, Order2> const&); // round(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> round(fvar<RealType, Order> const&); // iround(cr1) | int template <typename RealType, size_t Order> int iround(fvar<RealType, Order> const&); template <typename RealType, size_t Order> long lround(fvar<RealType, Order> const&); template <typename RealType, size_t Order> long long llround(fvar<RealType, Order> const&); // trunc(cr1) | RealType template <typename RealType, size_t Order> fvar<RealType, Order> trunc(fvar<RealType, Order> const&); template <typename RealType, size_t Order> long double truncl(fvar<RealType, Order> const&); // itrunc(cr1) | int template <typename RealType, size_t Order> int itrunc(fvar<RealType, Order> const&); template <typename RealType, size_t Order> long long lltrunc(fvar<RealType, Order> const&); // Additional functions template <typename RealType, size_t Order> fvar<RealType, Order> acos(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> acosh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> asinh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> atanh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> cosh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> digamma(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> erf(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> erfc(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> lambert_w0(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> lgamma(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> sinc(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> sinh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> tanh(fvar<RealType, Order> const&); template <typename RealType, size_t Order> fvar<RealType, Order> tgamma(fvar<RealType, Order> const&); template <size_t> struct zero : std::integral_constant<size_t, 0> {}; } // namespace detail template <typename RealType, size_t Order, size_t... Orders> using autodiff_fvar = typename detail::nest_fvar<RealType, Order, Orders...>::type; template <typename RealType, size_t Order, size_t... Orders> autodiff_fvar<RealType, Order, Orders...> make_fvar(RealType const& ca) { return autodiff_fvar<RealType, Order, Orders...>(ca, true); } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR namespace detail { template <typename RealType, size_t Order, size_t... Is> auto make_fvar_for_tuple(std::index_sequence<Is...>, RealType const& ca) { return make_fvar<RealType, zero<Is>::value..., Order>(ca); } template <typename RealType, size_t... Orders, size_t... Is, typename... RealTypes> auto make_ftuple_impl(std::index_sequence<Is...>, RealTypes const&... ca) { return std::make_tuple(make_fvar_for_tuple<RealType, Orders>(std::make_index_sequence<Is>{}, ca)...); } } // namespace detail template <typename RealType, size_t... Orders, typename... RealTypes> auto make_ftuple(RealTypes const&... ca) { static_assert(sizeof...(Orders) == sizeof...(RealTypes), "Number of Orders must match number of function parameters."); return detail::make_ftuple_impl<RealType, Orders...>(std::index_sequence_for<RealTypes...>{}, ca...); } #endif namespace detail { #ifndef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RealType, size_t Order> fvar<RealType, Order>::fvar(root_type const& ca, bool const is_variable) { if constexpr (is_fvar<RealType>::value) { v.front() = RealType(ca, is_variable); if constexpr (0 < Order) std::fill(v.begin() + 1, v.end(), static_cast<RealType>(0)); } else { v.front() = ca; if constexpr (0 < Order) v[1] = static_cast<root_type>(static_cast<int>(is_variable)); if constexpr (1 < Order) std::fill(v.begin() + 2, v.end(), static_cast<RealType>(0)); } } #endif template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> fvar<RealType, Order>::fvar(fvar<RealType2, Order2> const& cr) { for (size_t i = 0; i <= (std::min)(Order, Order2); ++i) v[i] = static_cast<RealType>(cr.v[i]); BOOST_IF_CONSTEXPR (Order2 < Order) std::fill(v.begin() + (Order2 + 1), v.end(), static_cast<RealType>(0)); } template <typename RealType, size_t Order> fvar<RealType, Order>::fvar(root_type const& ca) : v{{static_cast<RealType>(ca)}} {} // Can cause compiler error if RealType2 cannot be cast to root_type. template <typename RealType, size_t Order> template <typename RealType2> fvar<RealType, Order>::fvar(RealType2 const& ca) : v{{static_cast<RealType>(ca)}} {} /* template<typename RealType, size_t Order> fvar<RealType,Order>& fvar<RealType,Order>::operator=(root_type const& ca) { v.front() = static_cast<RealType>(ca); if constexpr (0 < Order) std::fill(v.begin()+1, v.end(), static_cast<RealType>(0)); return *this; } */ template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> fvar<RealType, Order>& fvar<RealType, Order>::operator+=(fvar<RealType2, Order2> const& cr) { for (size_t i = 0; i <= (std::min)(Order, Order2); ++i) v[i] += cr.v[i]; return *this; } template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::operator+=(root_type const& ca) { v.front() += ca; return *this; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> fvar<RealType, Order>& fvar<RealType, Order>::operator-=(fvar<RealType2, Order2> const& cr) { for (size_t i = 0; i <= Order; ++i) v[i] -= cr.v[i]; return *this; } template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::operator-=(root_type const& ca) { v.front() -= ca; return *this; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> fvar<RealType, Order>& fvar<RealType, Order>::operator*=(fvar<RealType2, Order2> const& cr) { using diff_t = typename std::array<RealType, Order + 1>::difference_type; promote<RealType, RealType2> const zero(0); BOOST_IF_CONSTEXPR (Order <= Order2) for (size_t i = 0, j = Order; i <= Order; ++i, --j) v[j] = std::inner_product(v.cbegin(), v.cend() - diff_t(i), cr.v.crbegin() + diff_t(i), zero); else { for (size_t i = 0, j = Order; i <= Order - Order2; ++i, --j) v[j] = std::inner_product(cr.v.cbegin(), cr.v.cend(), v.crbegin() + diff_t(i), zero); for (size_t i = Order - Order2 + 1, j = Order2 - 1; i <= Order; ++i, --j) v[j] = std::inner_product(cr.v.cbegin(), cr.v.cbegin() + diff_t(j + 1), v.crbegin() + diff_t(i), zero); } return *this; } template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::operator*=(root_type const& ca) { return multiply_assign_by_root_type(true, ca); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> fvar<RealType, Order>& fvar<RealType, Order>::operator/=(fvar<RealType2, Order2> const& cr) { using diff_t = typename std::array<RealType, Order + 1>::difference_type; RealType const zero(0); v.front() /= cr.v.front(); BOOST_IF_CONSTEXPR (Order < Order2) for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, --j, --k) (v[i] -= std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero)) /= cr.v.front(); else BOOST_IF_CONSTEXPR (0 < Order2) for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, j && --j, --k) (v[i] -= std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero)) /= cr.v.front(); else for (size_t i = 1; i <= Order; ++i) v[i] /= cr.v.front(); return *this; } template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::operator/=(root_type const& ca) { std::for_each(v.begin(), v.end(), [&ca](RealType& x) { x /= ca; }); return *this; } template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::operator-() const { fvar<RealType, Order> retval(*this); retval.negate(); return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> const& fvar<RealType, Order>::operator+() const { return *this; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator+( fvar<RealType2, Order2> const& cr) const { promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval; for (size_t i = 0; i <= (std::min)(Order, Order2); ++i) retval.v[i] = v[i] + cr.v[i]; BOOST_IF_CONSTEXPR (Order < Order2) for (size_t i = Order + 1; i <= Order2; ++i) retval.v[i] = cr.v[i]; else BOOST_IF_CONSTEXPR (Order2 < Order) for (size_t i = Order2 + 1; i <= Order; ++i) retval.v[i] = v[i]; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::operator+(root_type const& ca) const { fvar<RealType, Order> retval(*this); retval.v.front() += ca; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> operator+(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return cr + ca; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator-( fvar<RealType2, Order2> const& cr) const { promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval; for (size_t i = 0; i <= (std::min)(Order, Order2); ++i) retval.v[i] = v[i] - cr.v[i]; BOOST_IF_CONSTEXPR (Order < Order2) for (auto i = Order + 1; i <= Order2; ++i) retval.v[i] = -cr.v[i]; else BOOST_IF_CONSTEXPR (Order2 < Order) for (auto i = Order2 + 1; i <= Order; ++i) retval.v[i] = v[i]; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::operator-(root_type const& ca) const { fvar<RealType, Order> retval(*this); retval.v.front() -= ca; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> operator-(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { fvar<RealType, Order> mcr = -cr; // Has same address as retval in operator-() due to NRVO. mcr += ca; return mcr; // <-- This allows for NRVO. The following does not. --> return mcr += ca; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator*( fvar<RealType2, Order2> const& cr) const { using diff_t = typename std::array<RealType, Order + 1>::difference_type; promote<RealType, RealType2> const zero(0); promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval; BOOST_IF_CONSTEXPR (Order < Order2) for (size_t i = 0, j = Order, k = Order2; i <= Order2; ++i, j && --j, --k) retval.v[i] = std::inner_product(v.cbegin(), v.cend() - diff_t(j), cr.v.crbegin() + diff_t(k), zero); else for (size_t i = 0, j = Order2, k = Order; i <= Order; ++i, j && --j, --k) retval.v[i] = std::inner_product(cr.v.cbegin(), cr.v.cend() - diff_t(j), v.crbegin() + diff_t(k), zero); return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::operator*(root_type const& ca) const { fvar<RealType, Order> retval(*this); retval *= ca; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> operator*(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return cr * ca; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> promote<fvar<RealType, Order>, fvar<RealType2, Order2>> fvar<RealType, Order>::operator/( fvar<RealType2, Order2> const& cr) const { using diff_t = typename std::array<RealType, Order + 1>::difference_type; promote<RealType, RealType2> const zero(0); promote<fvar<RealType, Order>, fvar<RealType2, Order2>> retval; retval.v.front() = v.front() / cr.v.front(); BOOST_IF_CONSTEXPR (Order < Order2) { for (size_t i = 1, j = Order2 - 1; i <= Order; ++i, --j) retval.v[i] = (v[i] - std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero)) / cr.v.front(); for (size_t i = Order + 1, j = Order2 - Order - 1; i <= Order2; ++i, --j) retval.v[i] = -std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero) / cr.v.front(); } else BOOST_IF_CONSTEXPR (0 < Order2) for (size_t i = 1, j = Order2 - 1, k = Order; i <= Order; ++i, j && --j, --k) retval.v[i] = (v[i] - std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(k), zero)) / cr.v.front(); else for (size_t i = 1; i <= Order; ++i) retval.v[i] = v[i] / cr.v.front(); return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::operator/(root_type const& ca) const { fvar<RealType, Order> retval(*this); retval /= ca; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> operator/(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { using diff_t = typename std::array<RealType, Order + 1>::difference_type; fvar<RealType, Order> retval; retval.v.front() = ca / cr.v.front(); BOOST_IF_CONSTEXPR (0 < Order) { RealType const zero(0); for (size_t i = 1, j = Order - 1; i <= Order; ++i, --j) retval.v[i] = -std::inner_product( cr.v.cbegin() + 1, cr.v.cend() - diff_t(j), retval.v.crbegin() + diff_t(j + 1), zero) / cr.v.front(); } return retval; } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator==(fvar<RealType2, Order2> const& cr) const { return v.front() == cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator==(root_type const& ca) const { return v.front() == ca; } template <typename RealType, size_t Order> bool operator==(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca == cr.v.front(); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator!=(fvar<RealType2, Order2> const& cr) const { return v.front() != cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator!=(root_type const& ca) const { return v.front() != ca; } template <typename RealType, size_t Order> bool operator!=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca != cr.v.front(); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator<=(fvar<RealType2, Order2> const& cr) const { return v.front() <= cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator<=(root_type const& ca) const { return v.front() <= ca; } template <typename RealType, size_t Order> bool operator<=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca <= cr.v.front(); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator>=(fvar<RealType2, Order2> const& cr) const { return v.front() >= cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator>=(root_type const& ca) const { return v.front() >= ca; } template <typename RealType, size_t Order> bool operator>=(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca >= cr.v.front(); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator<(fvar<RealType2, Order2> const& cr) const { return v.front() < cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator<(root_type const& ca) const { return v.front() < ca; } template <typename RealType, size_t Order> bool operator<(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca < cr.v.front(); } template <typename RealType, size_t Order> template <typename RealType2, size_t Order2> bool fvar<RealType, Order>::operator>(fvar<RealType2, Order2> const& cr) const { return v.front() > cr.v.front(); } template <typename RealType, size_t Order> bool fvar<RealType, Order>::operator>(root_type const& ca) const { return v.front() > ca; } template <typename RealType, size_t Order> bool operator>(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { return ca > cr.v.front(); } /*** Other methods and functions ***/ #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // f : order -> derivative(order)/factorial(order) // Use this when you have the polynomial coefficients, rather than just the derivatives. E.g. See atan2(). template <typename RealType, size_t Order> template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_coefficients( size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); size_t i = (std::min)(order, order_sum); promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_coefficients( order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...); while (i--) (accumulator *= epsilon) += cr.apply_coefficients( order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...); return accumulator; } #endif // f : order -> derivative(order)/factorial(order) // Use this when you have the polynomial coefficients, rather than just the derivatives. E.g. See atan(). template <typename RealType, size_t Order> template <typename Func> fvar<RealType, Order> fvar<RealType, Order>::apply_coefficients(size_t const order, Func const& f) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); #ifndef BOOST_NO_CXX17_IF_CONSTEXPR size_t i = (std::min)(order, order_sum); #else // ODR-use of static constexpr size_t i = order < order_sum ? order : order_sum; #endif fvar<RealType, Order> accumulator = f(i); while (i--) (accumulator *= epsilon) += f(i); return accumulator; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // f : order -> derivative(order) template <typename RealType, size_t Order> template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_coefficients_nonhorner( size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_coefficients_nonhorner( order, [&f](auto... indices) { return f(0, static_cast<std::size_t>(indices)...); }, std::forward<Fvars>(fvars)...); size_t const i_max = (std::min)(order, order_sum); for (size_t i = 1; i <= i_max; ++i) { epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0); accumulator += epsilon_i.epsilon_multiply( i, 0, cr.apply_coefficients_nonhorner( order - i, [&f, i](auto... indices) { return f(i, static_cast<std::size_t>(indices)...); }, std::forward<Fvars>(fvars)...), 0, 0); } return accumulator; } #endif // f : order -> coefficient(order) template <typename RealType, size_t Order> template <typename Func> fvar<RealType, Order> fvar<RealType, Order>::apply_coefficients_nonhorner(size_t const order, Func const& f) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i fvar<RealType, Order> accumulator = fvar<RealType, Order>(f(0u)); #ifndef BOOST_NO_CXX17_IF_CONSTEXPR size_t const i_max = (std::min)(order, order_sum); #else // ODR-use of static constexpr size_t const i_max = order < order_sum ? order : order_sum; #endif for (size_t i = 1; i <= i_max; ++i) { epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0); accumulator += epsilon_i.epsilon_multiply(i, 0, f(i)); } return accumulator; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // f : order -> derivative(order) template <typename RealType, size_t Order> template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_derivatives( size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); size_t i = (std::min)(order, order_sum); promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_derivatives( order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...) / factorial<root_type>(static_cast<unsigned>(i)); while (i--) (accumulator *= epsilon) += cr.apply_derivatives( order - i, [&f, i](auto... indices) { return f(i, indices...); }, std::forward<Fvars>(fvars)...) / factorial<root_type>(static_cast<unsigned>(i)); return accumulator; } #endif // f : order -> derivative(order) template <typename RealType, size_t Order> template <typename Func> fvar<RealType, Order> fvar<RealType, Order>::apply_derivatives(size_t const order, Func const& f) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); #ifndef BOOST_NO_CXX17_IF_CONSTEXPR size_t i = (std::min)(order, order_sum); #else // ODR-use of static constexpr size_t i = order < order_sum ? order : order_sum; #endif fvar<RealType, Order> accumulator = f(i) / factorial<root_type>(static_cast<unsigned>(i)); while (i--) (accumulator *= epsilon) += f(i) / factorial<root_type>(static_cast<unsigned>(i)); return accumulator; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // f : order -> derivative(order) template <typename RealType, size_t Order> template <typename Func, typename Fvar, typename... Fvars> promote<fvar<RealType, Order>, Fvar, Fvars...> fvar<RealType, Order>::apply_derivatives_nonhorner( size_t const order, Func const& f, Fvar const& cr, Fvars&&... fvars) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i promote<fvar<RealType, Order>, Fvar, Fvars...> accumulator = cr.apply_derivatives_nonhorner( order, [&f](auto... indices) { return f(0, static_cast<std::size_t>(indices)...); }, std::forward<Fvars>(fvars)...); size_t const i_max = (std::min)(order, order_sum); for (size_t i = 1; i <= i_max; ++i) { epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0); accumulator += epsilon_i.epsilon_multiply( i, 0, cr.apply_derivatives_nonhorner( order - i, [&f, i](auto... indices) { return f(i, static_cast<std::size_t>(indices)...); }, std::forward<Fvars>(fvars)...) / factorial<root_type>(static_cast<unsigned>(i)), 0, 0); } return accumulator; } #endif // f : order -> derivative(order) template <typename RealType, size_t Order> template <typename Func> fvar<RealType, Order> fvar<RealType, Order>::apply_derivatives_nonhorner(size_t const order, Func const& f) const { fvar<RealType, Order> const epsilon = fvar<RealType, Order>(*this).set_root(0); fvar<RealType, Order> epsilon_i = fvar<RealType, Order>(1); // epsilon to the power of i fvar<RealType, Order> accumulator = fvar<RealType, Order>(f(0u)); #ifndef BOOST_NO_CXX17_IF_CONSTEXPR size_t const i_max = (std::min)(order, order_sum); #else // ODR-use of static constexpr size_t const i_max = order < order_sum ? order : order_sum; #endif for (size_t i = 1; i <= i_max; ++i) { epsilon_i = epsilon_i.epsilon_multiply(i - 1, 0, epsilon, 1, 0); accumulator += epsilon_i.epsilon_multiply(i, 0, f(i) / factorial<root_type>(static_cast<unsigned>(i))); } return accumulator; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // Can throw "std::out_of_range: array::at: __n (which is 7) >= _Nm (which is 7)" template <typename RealType, size_t Order> template <typename... Orders> get_type_at<RealType, sizeof...(Orders)> fvar<RealType, Order>::at(size_t order, Orders... orders) const { if constexpr (0 < sizeof...(Orders)) return v.at(order).at(static_cast<std::size_t>(orders)...); else return v.at(order); } #endif #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // Can throw "std::out_of_range: array::at: __n (which is 7) >= _Nm (which is 7)" template <typename RealType, size_t Order> template <typename... Orders> get_type_at<fvar<RealType, Order>, sizeof...(Orders)> fvar<RealType, Order>::derivative( Orders... orders) const { static_assert(sizeof...(Orders) <= depth, "Number of parameters to derivative(...) cannot exceed fvar::depth."); return at(static_cast<std::size_t>(orders)...) * (... * factorial<root_type>(static_cast<unsigned>(orders))); } #endif template <typename RealType, size_t Order> const RealType& fvar<RealType, Order>::operator[](size_t i) const { return v[i]; } template <typename RealType, size_t Order> RealType fvar<RealType, Order>::epsilon_inner_product(size_t z0, size_t const isum0, size_t const m0, fvar<RealType, Order> const& cr, size_t z1, size_t const isum1, size_t const m1, size_t const j) const { static_assert(is_fvar<RealType>::value, "epsilon_inner_product() must have 1 < depth."); RealType accumulator = RealType(); auto const i0_max = m1 < j ? j - m1 : 0; for (auto i0 = m0, i1 = j - m0; i0 <= i0_max; ++i0, --i1) accumulator += v[i0].epsilon_multiply(z0, isum0 + i0, cr.v[i1], z1, isum1 + i1); return accumulator; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::epsilon_multiply(size_t z0, size_t isum0, fvar<RealType, Order> const& cr, size_t z1, size_t isum1) const { using diff_t = typename std::array<RealType, Order + 1>::difference_type; RealType const zero(0); size_t const m0 = order_sum + isum0 < Order + z0 ? Order + z0 - (order_sum + isum0) : 0; size_t const m1 = order_sum + isum1 < Order + z1 ? Order + z1 - (order_sum + isum1) : 0; size_t const i_max = m0 + m1 < Order ? Order - (m0 + m1) : 0; fvar<RealType, Order> retval = fvar<RealType, Order>(); if constexpr (is_fvar<RealType>::value) for (size_t i = 0, j = Order; i <= i_max; ++i, --j) retval.v[j] = epsilon_inner_product(z0, isum0, m0, cr, z1, isum1, m1, j); else for (size_t i = 0, j = Order; i <= i_max; ++i, --j) retval.v[j] = std::inner_product( v.cbegin() + diff_t(m0), v.cend() - diff_t(i + m1), cr.v.crbegin() + diff_t(i + m0), zero); return retval; } #endif #ifndef BOOST_NO_CXX17_IF_CONSTEXPR // When called from outside this method, z0 should be non-zero. Otherwise if z0=0 then it will give an // incorrect result of 0 when the root value is 0 and ca=inf, when instead the correct product is nan. // If z0=0 then use the regular multiply operator*() instead. template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::epsilon_multiply(size_t z0, size_t isum0, root_type const& ca) const { fvar<RealType, Order> retval(*this); size_t const m0 = order_sum + isum0 < Order + z0 ? Order + z0 - (order_sum + isum0) : 0; if constexpr (is_fvar<RealType>::value) for (size_t i = m0; i <= Order; ++i) retval.v[i] = retval.v[i].epsilon_multiply(z0, isum0 + i, ca); else for (size_t i = m0; i <= Order; ++i) if (retval.v[i] != static_cast<RealType>(0)) retval.v[i] *= ca; return retval; } #endif template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::inverse() const { return static_cast<root_type>(*this) == 0 ? inverse_apply() : 1 / *this; } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::negate() { if constexpr (is_fvar<RealType>::value) std::for_each(v.begin(), v.end(), [](RealType& r) { r.negate(); }); else std::for_each(v.begin(), v.end(), [](RealType& a) { a = -a; }); return *this; } #endif // This gives log(0.0) = depth(1)(-inf,inf,-inf,inf,-inf,inf) // 1 / *this: log(0.0) = depth(1)(-inf,inf,-inf,-nan,-nan,-nan) template <typename RealType, size_t Order> fvar<RealType, Order> fvar<RealType, Order>::inverse_apply() const { root_type derivatives[order_sum + 1]; // LCOV_EXCL_LINE This causes a false negative on lcov coverage test. root_type const x0 = static_cast<root_type>(*this); *derivatives = 1 / x0; for (size_t i = 1; i <= order_sum; ++i) derivatives[i] = -derivatives[i - 1] * i / x0; return apply_derivatives_nonhorner(order_sum, [&derivatives](size_t j) { return derivatives[j]; }); } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::multiply_assign_by_root_type(bool is_root, root_type const& ca) { auto itr = v.begin(); if constexpr (is_fvar<RealType>::value) { itr->multiply_assign_by_root_type(is_root, ca); for (++itr; itr != v.end(); ++itr) itr->multiply_assign_by_root_type(false, ca); } else { if (is_root || *itr != 0) *itr *= ca; // Skip multiplication of 0 by ca=inf to avoid nan, except when is_root. for (++itr; itr != v.end(); ++itr) if (*itr != 0) *itr *= ca; } return *this; } #endif template <typename RealType, size_t Order> fvar<RealType, Order>::operator root_type() const { return static_cast<root_type>(v.front()); } template <typename RealType, size_t Order> template <typename T, typename> fvar<RealType, Order>::operator T() const { return static_cast<T>(static_cast<root_type>(v.front())); } #ifndef BOOST_NO_CXX17_IF_CONSTEXPR template <typename RealType, size_t Order> fvar<RealType, Order>& fvar<RealType, Order>::set_root(root_type const& root) { if constexpr (is_fvar<RealType>::value) v.front().set_root(root); else v.front() = root; return *this; } #endif // Standard Library Support Requirements template <typename RealType, size_t Order> fvar<RealType, Order> fabs(fvar<RealType, Order> const& cr) { typename fvar<RealType, Order>::root_type const zero(0); return cr < zero ? -cr : cr == zero ? fvar<RealType, Order>() // Canonical fabs'(0) = 0. : cr; // Propagate NaN. } template <typename RealType, size_t Order> fvar<RealType, Order> abs(fvar<RealType, Order> const& cr) { return fabs(cr); } template <typename RealType, size_t Order> fvar<RealType, Order> ceil(fvar<RealType, Order> const& cr) { using std::ceil; return fvar<RealType, Order>(ceil(static_cast<typename fvar<RealType, Order>::root_type>(cr))); } template <typename RealType, size_t Order> fvar<RealType, Order> floor(fvar<RealType, Order> const& cr) { using std::floor; return fvar<RealType, Order>(floor(static_cast<typename fvar<RealType, Order>::root_type>(cr))); } template <typename RealType, size_t Order> fvar<RealType, Order> exp(fvar<RealType, Order> const& cr) { using std::exp; constexpr size_t order = fvar<RealType, Order>::order_sum; using root_type = typename fvar<RealType, Order>::root_type; root_type const d0 = exp(static_cast<root_type>(cr)); return cr.apply_derivatives(order, [&d0](size_t) { return d0; }); } template <typename RealType, size_t Order> fvar<RealType, Order> pow(fvar<RealType, Order> const& x, typename fvar<RealType, Order>::root_type const& y) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const x0 = static_cast<root_type>(x); root_type derivatives[order + 1]{pow(x0, y)}; if (fabs(x0) < std::numeric_limits<root_type>::epsilon()) { root_type coef = 1; for (size_t i = 0; i < order && y - i != 0; ++i) { coef *= y - i; derivatives[i + 1] = coef * pow(x0, y - (i + 1)); } return x.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; }); } else { for (size_t i = 0; i < order && y - i != 0; ++i) derivatives[i + 1] = (y - i) * derivatives[i] / x0; return x.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> pow(typename fvar<RealType, Order>::root_type const& x, fvar<RealType, Order> const& y) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const y0 = static_cast<root_type>(y); root_type derivatives[order + 1]; *derivatives = pow(x, y0); root_type const logx = log(x); for (size_t i = 0; i < order; ++i) derivatives[i + 1] = derivatives[i] * logx; return y.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i]; }); } template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> pow(fvar<RealType1, Order1> const& x, fvar<RealType2, Order2> const& y) { BOOST_MATH_STD_USING using return_type = promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>>; using root_type = typename return_type::root_type; constexpr size_t order = return_type::order_sum; root_type const x0 = static_cast<root_type>(x); root_type const y0 = static_cast<root_type>(y); root_type dxydx[order + 1]{pow(x0, y0)}; BOOST_IF_CONSTEXPR (order == 0) return return_type(*dxydx); else { for (size_t i = 0; i < order && y0 - i != 0; ++i) dxydx[i + 1] = (y0 - i) * dxydx[i] / x0; std::array<fvar<root_type, order>, order + 1> lognx; lognx.front() = fvar<root_type, order>(1); #ifndef BOOST_NO_CXX17_IF_CONSTEXPR lognx[1] = log(make_fvar<root_type, order>(x0)); #else // for compilers that compile this branch when order == 0. lognx[(std::min)(size_t(1), order)] = log(make_fvar<root_type, order>(x0)); #endif for (size_t i = 1; i < order; ++i) lognx[i + 1] = lognx[i] * lognx[1]; auto const f = [&dxydx, &lognx](size_t i, size_t j) { size_t binomial = 1; root_type sum = dxydx[i] * static_cast<root_type>(lognx[j]); for (size_t k = 1; k <= i; ++k) { (binomial *= (i - k + 1)) /= k; // binomial_coefficient(i,k) sum += binomial * dxydx[i - k] * lognx[j].derivative(k); } return sum; }; if (fabs(x0) < std::numeric_limits<root_type>::epsilon()) return x.apply_derivatives_nonhorner(order, f, y); return x.apply_derivatives(order, f, y); } } template <typename RealType, size_t Order> fvar<RealType, Order> sqrt(fvar<RealType, Order> const& cr) { using std::sqrt; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type derivatives[order + 1]; root_type const x = static_cast<root_type>(cr); *derivatives = sqrt(x); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(*derivatives); else { root_type numerator = 0.5; root_type powers = 1; #ifndef BOOST_NO_CXX17_IF_CONSTEXPR derivatives[1] = numerator / *derivatives; #else // for compilers that compile this branch when order == 0. derivatives[(std::min)(size_t(1), order)] = numerator / *derivatives; #endif using diff_t = typename std::array<RealType, Order + 1>::difference_type; for (size_t i = 2; i <= order; ++i) { numerator *= static_cast<root_type>(-0.5) * ((static_cast<diff_t>(i) << 1) - 3); powers *= x; derivatives[i] = numerator / (powers * *derivatives); } auto const f = [&derivatives](size_t i) { return derivatives[i]; }; if (cr < std::numeric_limits<root_type>::epsilon()) return cr.apply_derivatives_nonhorner(order, f); return cr.apply_derivatives(order, f); } } // Natural logarithm. If cr==0 then derivative(i) may have nans due to nans from inverse(). template <typename RealType, size_t Order> fvar<RealType, Order> log(fvar<RealType, Order> const& cr) { using std::log; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = log(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto const d1 = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)).inverse(); // log'(x) = 1 / x return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> frexp(fvar<RealType, Order> const& cr, int* exp) { using std::exp2; using std::frexp; using root_type = typename fvar<RealType, Order>::root_type; frexp(static_cast<root_type>(cr), exp); return cr * static_cast<root_type>(exp2(-*exp)); } template <typename RealType, size_t Order> fvar<RealType, Order> ldexp(fvar<RealType, Order> const& cr, int exp) { // argument to std::exp2 must be casted to root_type, otherwise std::exp2 returns double (always) using std::exp2; return cr * exp2(static_cast<typename fvar<RealType, Order>::root_type>(exp)); } template <typename RealType, size_t Order> fvar<RealType, Order> cos(fvar<RealType, Order> const& cr) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = cos(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { root_type const d1 = -sin(static_cast<root_type>(cr)); root_type const derivatives[4]{d0, d1, -d0, -d1}; return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 3]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> sin(fvar<RealType, Order> const& cr) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = sin(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { root_type const d1 = cos(static_cast<root_type>(cr)); root_type const derivatives[4]{d0, d1, -d0, -d1}; return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 3]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> asin(fvar<RealType, Order> const& cr) { using std::asin; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = asin(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = sqrt((x *= x).negate() += 1).inverse(); // asin'(x) = 1 / sqrt(1-x*x). return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> tan(fvar<RealType, Order> const& cr) { using std::tan; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = tan(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto c = cos(make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr))); auto const d1 = (c *= c).inverse(); // tan'(x) = 1 / cos(x)^2 return cr.apply_coefficients_nonhorner(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> atan(fvar<RealType, Order> const& cr) { using std::atan; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = atan(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = ((x *= x) += 1).inverse(); // atan'(x) = 1 / (x*x+1). return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> atan2(fvar<RealType, Order> const& cr, typename fvar<RealType, Order>::root_type const& ca) { using std::atan2; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = atan2(static_cast<root_type>(cr), ca); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto y = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = ca / ((y *= y) += (ca * ca)); // (d/dy)atan2(y,x) = x / (y*y+x*x) return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> atan2(typename fvar<RealType, Order>::root_type const& ca, fvar<RealType, Order> const& cr) { using std::atan2; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = atan2(ca, static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = -ca / ((x *= x) += (ca * ca)); // (d/dx)atan2(y,x) = -y / (x*x+y*y) return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> atan2(fvar<RealType1, Order1> const& cr1, fvar<RealType2, Order2> const& cr2) { using std::atan2; using return_type = promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>>; using root_type = typename return_type::root_type; constexpr size_t order = return_type::order_sum; root_type const y = static_cast<root_type>(cr1); root_type const x = static_cast<root_type>(cr2); root_type const d00 = atan2(y, x); BOOST_IF_CONSTEXPR (order == 0) return return_type(d00); else { constexpr size_t order1 = fvar<RealType1, Order1>::order_sum; constexpr size_t order2 = fvar<RealType2, Order2>::order_sum; auto x01 = make_fvar<typename fvar<RealType2, Order2>::root_type, order2 - 1>(x); auto const d01 = -y / ((x01 *= x01) += (y * y)); auto y10 = make_fvar<typename fvar<RealType1, Order1>::root_type, order1 - 1>(y); auto x10 = make_fvar<typename fvar<RealType2, Order2>::root_type, 0, order2>(x); auto const d10 = x10 / ((x10 * x10) + (y10 *= y10)); auto const f = [&d00, &d01, &d10](size_t i, size_t j) { return i ? d10[i - 1][j] / i : j ? d01[j - 1] / j : d00; }; return cr1.apply_coefficients(order, f, cr2); } } template <typename RealType1, size_t Order1, typename RealType2, size_t Order2> promote<fvar<RealType1, Order1>, fvar<RealType2, Order2>> fmod(fvar<RealType1, Order1> const& cr1, fvar<RealType2, Order2> const& cr2) { using boost::math::trunc; auto const numer = static_cast<typename fvar<RealType1, Order1>::root_type>(cr1); auto const denom = static_cast<typename fvar<RealType2, Order2>::root_type>(cr2); return cr1 - cr2 * trunc(numer / denom); } template <typename RealType, size_t Order> fvar<RealType, Order> round(fvar<RealType, Order> const& cr) { using boost::math::round; return fvar<RealType, Order>(round(static_cast<typename fvar<RealType, Order>::root_type>(cr))); } template <typename RealType, size_t Order> int iround(fvar<RealType, Order> const& cr) { using boost::math::iround; return iround(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> long lround(fvar<RealType, Order> const& cr) { using boost::math::lround; return lround(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> long long llround(fvar<RealType, Order> const& cr) { using boost::math::llround; return llround(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> fvar<RealType, Order> trunc(fvar<RealType, Order> const& cr) { using boost::math::trunc; return fvar<RealType, Order>(trunc(static_cast<typename fvar<RealType, Order>::root_type>(cr))); } template <typename RealType, size_t Order> long double truncl(fvar<RealType, Order> const& cr) { using std::truncl; return truncl(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> int itrunc(fvar<RealType, Order> const& cr) { using boost::math::itrunc; return itrunc(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> long long lltrunc(fvar<RealType, Order> const& cr) { using boost::math::lltrunc; return lltrunc(static_cast<typename fvar<RealType, Order>::root_type>(cr)); } template <typename RealType, size_t Order> std::ostream& operator<<(std::ostream& out, fvar<RealType, Order> const& cr) { out << "depth(" << cr.depth << ")(" << cr.v.front(); for (size_t i = 1; i <= Order; ++i) out << ',' << cr.v[i]; return out << ')'; } // Additional functions template <typename RealType, size_t Order> fvar<RealType, Order> acos(fvar<RealType, Order> const& cr) { using std::acos; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = acos(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = sqrt((x *= x).negate() += 1).inverse().negate(); // acos'(x) = -1 / sqrt(1-x*x). return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> acosh(fvar<RealType, Order> const& cr) { using boost::math::acosh; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = acosh(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = sqrt((x *= x) -= 1).inverse(); // acosh'(x) = 1 / sqrt(x*x-1). return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> asinh(fvar<RealType, Order> const& cr) { using boost::math::asinh; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = asinh(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = sqrt((x *= x) += 1).inverse(); // asinh'(x) = 1 / sqrt(x*x+1). return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> atanh(fvar<RealType, Order> const& cr) { using boost::math::atanh; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = atanh(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); auto const d1 = ((x *= x).negate() += 1).inverse(); // atanh'(x) = 1 / (1-x*x) return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> cosh(fvar<RealType, Order> const& cr) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = cosh(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { root_type const derivatives[2]{d0, sinh(static_cast<root_type>(cr))}; return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 1]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> digamma(fvar<RealType, Order> const& cr) { using boost::math::digamma; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const x = static_cast<root_type>(cr); root_type const d0 = digamma(x); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { static_assert(order <= static_cast<size_t>((std::numeric_limits<int>::max)()), "order exceeds maximum derivative for boost::math::polygamma()."); return cr.apply_derivatives( order, [&x, &d0](size_t i) { return i ? boost::math::polygamma(static_cast<int>(i), x) : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> erf(fvar<RealType, Order> const& cr) { using boost::math::erf; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = erf(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); // d1 = 2/sqrt(pi)*exp(-x*x) auto const d1 = 2 * constants::one_div_root_pi<root_type>() * exp((x *= x).negate()); return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> erfc(fvar<RealType, Order> const& cr) { using boost::math::erfc; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = erfc(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { auto x = make_fvar<root_type, bool(order) ? order - 1 : 0>(static_cast<root_type>(cr)); // erfc'(x) = -erf'(x) auto const d1 = -2 * constants::one_div_root_pi<root_type>() * exp((x *= x).negate()); return cr.apply_coefficients(order, [&d0, &d1](size_t i) { return i ? d1[i - 1] / i : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> lambert_w0(fvar<RealType, Order> const& cr) { using std::exp; using boost::math::lambert_w0; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type derivatives[order + 1]; *derivatives = lambert_w0(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(*derivatives); else { root_type const expw = exp(*derivatives); derivatives[1] = 1 / (static_cast<root_type>(cr) + expw); BOOST_IF_CONSTEXPR (order == 1) return cr.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; }); else { using diff_t = typename std::array<RealType, Order + 1>::difference_type; root_type d1powers = derivatives[1] * derivatives[1]; root_type const x = derivatives[1] * expw; derivatives[2] = d1powers * (-1 - x); std::array<root_type, order> coef{{-1, -1}}; // as in derivatives[2]. for (size_t n = 3; n <= order; ++n) { coef[n - 1] = coef[n - 2] * -static_cast<root_type>(2 * n - 3); for (size_t j = n - 2; j != 0; --j) (coef[j] *= -static_cast<root_type>(n - 1)) -= (n + j - 2) * coef[j - 1]; coef[0] *= -static_cast<root_type>(n - 1); d1powers *= derivatives[1]; derivatives[n] = d1powers * std::accumulate(coef.crend() - diff_t(n - 1), coef.crend(), coef[n - 1], [&x](root_type const& a, root_type const& b) { return a * x + b; }); } return cr.apply_derivatives_nonhorner(order, [&derivatives](size_t i) { return derivatives[i]; }); } } } template <typename RealType, size_t Order> fvar<RealType, Order> lgamma(fvar<RealType, Order> const& cr) { using std::lgamma; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const x = static_cast<root_type>(cr); root_type const d0 = lgamma(x); BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(d0); else { static_assert(order <= static_cast<size_t>((std::numeric_limits<int>::max)()) + 1, "order exceeds maximum derivative for boost::math::polygamma()."); return cr.apply_derivatives( order, [&x, &d0](size_t i) { return i ? boost::math::polygamma(static_cast<int>(i - 1), x) : d0; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> sinc(fvar<RealType, Order> const& cr) { if (cr != 0) return sin(cr) / cr; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type taylor[order + 1]{1}; // sinc(0) = 1 BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(*taylor); else { for (size_t n = 2; n <= order; n += 2) taylor[n] = (1 - static_cast<int>(n & 2)) / factorial<root_type>(static_cast<unsigned>(n + 1)); return cr.apply_coefficients_nonhorner(order, [&taylor](size_t i) { return taylor[i]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> sinh(fvar<RealType, Order> const& cr) { BOOST_MATH_STD_USING using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; root_type const d0 = sinh(static_cast<root_type>(cr)); BOOST_IF_CONSTEXPR (fvar<RealType, Order>::order_sum == 0) return fvar<RealType, Order>(d0); else { root_type const derivatives[2]{d0, cosh(static_cast<root_type>(cr))}; return cr.apply_derivatives(order, [&derivatives](size_t i) { return derivatives[i & 1]; }); } } template <typename RealType, size_t Order> fvar<RealType, Order> tanh(fvar<RealType, Order> const& cr) { fvar<RealType, Order> retval = exp(cr * 2); fvar<RealType, Order> const denom = retval + 1; (retval -= 1) /= denom; return retval; } template <typename RealType, size_t Order> fvar<RealType, Order> tgamma(fvar<RealType, Order> const& cr) { using std::tgamma; using root_type = typename fvar<RealType, Order>::root_type; constexpr size_t order = fvar<RealType, Order>::order_sum; BOOST_IF_CONSTEXPR (order == 0) return fvar<RealType, Order>(tgamma(static_cast<root_type>(cr))); else { if (cr < 0) return constants::pi<root_type>() / (sin(constants::pi<root_type>() * cr) * tgamma(1 - cr)); return exp(lgamma(cr)).set_root(tgamma(static_cast<root_type>(cr))); } } } // namespace detail } // namespace autodiff_v1 } // namespace differentiation } // namespace math } // namespace boost namespace std { // boost::math::tools::digits<RealType>() is handled by this std::numeric_limits<> specialization, // and similarly for max_value, min_value, log_max_value, log_min_value, and epsilon. template <typename RealType, size_t Order> class numeric_limits<boost::math::differentiation::detail::fvar<RealType, Order>> : public numeric_limits<typename boost::math::differentiation::detail::fvar<RealType, Order>::root_type> { }; } // namespace std namespace boost { namespace math { namespace tools { namespace detail { template <typename RealType, std::size_t Order> using autodiff_fvar_type = differentiation::detail::fvar<RealType, Order>; template <typename RealType, std::size_t Order> using autodiff_root_type = typename autodiff_fvar_type<RealType, Order>::root_type; } // namespace detail // See boost/math/tools/promotion.hpp template <typename RealType0, size_t Order0, typename RealType1, size_t Order1> struct promote_args_2<detail::autodiff_fvar_type<RealType0, Order0>, detail::autodiff_fvar_type<RealType1, Order1>> { using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type, #ifndef BOOST_NO_CXX14_CONSTEXPR (std::max)(Order0, Order1)>; #else Order0<Order1 ? Order1 : Order0>; #endif }; template <typename RealType, size_t Order> struct promote_args<detail::autodiff_fvar_type<RealType, Order>> { using type = detail::autodiff_fvar_type<typename promote_args<RealType>::type, Order>; }; template <typename RealType0, size_t Order0, typename RealType1> struct promote_args_2<detail::autodiff_fvar_type<RealType0, Order0>, RealType1> { using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type, Order0>; }; template <typename RealType0, typename RealType1, size_t Order1> struct promote_args_2<RealType0, detail::autodiff_fvar_type<RealType1, Order1>> { using type = detail::autodiff_fvar_type<typename promote_args_2<RealType0, RealType1>::type, Order1>; }; template <typename destination_t, typename RealType, std::size_t Order> inline BOOST_MATH_CONSTEXPR destination_t real_cast(detail::autodiff_fvar_type<RealType, Order> const& from_v) BOOST_NOEXCEPT_IF(BOOST_MATH_IS_FLOAT(destination_t) && BOOST_MATH_IS_FLOAT(RealType)) { return real_cast<destination_t>(static_cast<detail::autodiff_root_type<RealType, Order>>(from_v)); } } // namespace tools namespace policies { template <class Policy, std::size_t Order> using fvar_t = differentiation::detail::fvar<Policy, Order>; template <class Policy, std::size_t Order> struct evaluation<fvar_t<float, Order>, Policy> { using type = fvar_t<typename conditional<Policy::promote_float_type::value, double, float>::type, Order>; }; template <class Policy, std::size_t Order> struct evaluation<fvar_t<double, Order>, Policy> { using type = fvar_t<typename conditional<Policy::promote_double_type::value, long double, double>::type, Order>; }; } // namespace policies } // namespace math } // namespace boost #ifdef BOOST_NO_CXX17_IF_CONSTEXPR #include "autodiff_cpp11.hpp" #endif #endif // BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP