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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/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<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 std::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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_IF_CONSTEXPR (Order < Order2)
     for (size_t i = Order + 1; i <= Order2; ++i)
       retval.v[i] = cr.v[i];
   else BOOST_MATH_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_MATH_IF_CONSTEXPR (Order < Order2)
     for (auto i = Order + 1; i <= Order2; ++i)
       retval.v[i] = -cr.v[i];
   else BOOST_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_IF_CONSTEXPR (order == 0)
     return fvar<RealType, Order>(*derivatives);
   else {
     root_type numerator = root_type(0.5);
     root_type powers = 1;
 #ifndef BOOST_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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_MATH_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<detail::autodiff_fvar_type<RealType0, Order0>,
                       detail::autodiff_fvar_type<RealType1, Order1>> {
   using type = detail::autodiff_fvar_type<typename promote_args<RealType0, RealType1>::type,
 #ifndef BOOST_MATH_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<detail::autodiff_fvar_type<RealType0, Order0>, RealType1> {
   using type = detail::autodiff_fvar_type<typename promote_args<RealType0, RealType1>::type, Order0>;
 };
 
 template <typename RealType0, typename RealType1, size_t Order1>
 struct promote_args<RealType0, detail::autodiff_fvar_type<RealType1, Order1>> {
   using type = detail::autodiff_fvar_type<typename promote_args<RealType0, RealType1>::type, Order1>;
 };
 
 template <typename destination_t, typename RealType, std::size_t Order>
 inline constexpr destination_t real_cast(detail::autodiff_fvar_type<RealType, Order> const& from_v)
     noexcept(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 std::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 std::conditional<Policy::promote_double_type::value, long double, double>::type, Order>;
 };
 
 }  // namespace policies
 }  // namespace math
 }  // namespace boost
 
 #ifdef BOOST_MATH_NO_CXX17_IF_CONSTEXPR
 #include "autodiff_cpp11.hpp"
 #endif
 
 #endif  // BOOST_MATH_DIFFERENTIATION_AUTODIFF_HPP