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+// 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.hpp>
+#include <boost/math/tools/config.hpp>
+#include <boost/math/tools/promotion.hpp>
+#include <boost/multiprecision/rational_adaptor.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<decay_t<T>>;
+
+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<decay_t<T>>;
+
+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<decay_t<T>>;
+
+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 {
+ 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 boost::enable_if<boost::is_arithmetic<decay_t<T>>>::type>
+ 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
+};
+
+// C++11 compatibility
+#ifdef BOOST_NO_CXX17_IF_CONSTEXPR
+#define BOOST_AUTODIFF_IF_CONSTEXPR
+#else
+#define BOOST_AUTODIFF_IF_CONSTEXPR constexpr
+#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]);
+ if BOOST_AUTODIFF_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);
+ if BOOST_AUTODIFF_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();
+ if BOOST_AUTODIFF_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 if BOOST_AUTODIFF_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];
+ if BOOST_AUTODIFF_IF_CONSTEXPR (Order < Order2)
+ for (size_t i = Order + 1; i <= Order2; ++i)
+ retval.v[i] = cr.v[i];
+ else if BOOST_AUTODIFF_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];
+ if BOOST_AUTODIFF_IF_CONSTEXPR (Order < Order2)
+ for (auto i = Order + 1; i <= Order2; ++i)
+ retval.v[i] = -cr.v[i];
+ else if BOOST_AUTODIFF_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;
+ if BOOST_AUTODIFF_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();
+ if BOOST_AUTODIFF_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 if BOOST_AUTODIFF_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();
+ if BOOST_AUTODIFF_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) {
+ using std::pow;
+ 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)};
+ 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)};
+ if BOOST_AUTODIFF_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);
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto const d1 = make_fvar<root_type, order - 1>(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 multiprecision::exp2;
+ 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 multiprecision::exp2;
+ 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));
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto c = cos(make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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);
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto y = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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);
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_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);
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_IF_CONSTEXPR (order == 0)
+ return fvar<RealType, Order>(d0);
+ else {
+ auto x = make_fvar<root_type, order - 1>(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));
+ if BOOST_AUTODIFF_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);
+ if BOOST_AUTODIFF_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);
+ if BOOST_AUTODIFF_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
+ if BOOST_AUTODIFF_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));
+ if BOOST_AUTODIFF_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;
+ if BOOST_AUTODIFF_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