#ifndef HELMHOLTZ_H
#define HELMHOLTZ_H

#include <algorithm>
#include <array>
#include <vector>
#include "rapidjson_include.h"
//#include "Eigen/Core"
#include "time.h"
#include "CachedElement.h"
#include "Backends/Cubics/GeneralizedCubic.h"
#include <memory>
using std::shared_ptr;
#include "CPnumerics.h"

#if ENABLE_CATCH
#    include "MultiComplex/MultiComplex.hpp"
#endif

namespace CoolProp {

// #############################################################################
// #############################################################################
// #############################################################################
//                                RESIDUAL TERMS
// #############################################################################
// #############################################################################
// #############################################################################

#define LIST_OF_DERIVATIVE_VARIABLES   \
    X(alphar)                          \
    X(dalphar_ddelta)                  \
    X(dalphar_dtau)                    \
    X(d2alphar_ddelta2)                \
    X(d2alphar_dtau2)                  \
    X(d2alphar_ddelta_dtau)            \
    X(d3alphar_ddelta3)                \
    X(d3alphar_ddelta_dtau2)           \
    X(d3alphar_ddelta2_dtau)           \
    X(d3alphar_dtau3)                  \
    X(d4alphar_ddelta4)                \
    X(d4alphar_ddelta3_dtau)           \
    X(d4alphar_ddelta2_dtau2)          \
    X(d4alphar_ddelta_dtau3)           \
    X(d4alphar_dtau4)                  \
    X(delta_x_dalphar_ddelta)          \
    X(tau_x_dalphar_dtau)              \
    X(delta2_x_d2alphar_ddelta2)       \
    X(deltatau_x_d2alphar_ddelta_dtau) \
    X(tau2_x_d2alphar_dtau2)

struct HelmholtzDerivatives
{
#define X(name) CoolPropDbl name;
    LIST_OF_DERIVATIVE_VARIABLES
#undef X
    CoolPropDbl tau, delta, T_red, rhomolar_red;

    void reset(CoolPropDbl v){
#define X(name) name = v;
      LIST_OF_DERIVATIVE_VARIABLES
#undef X
    } HelmholtzDerivatives operator+(const HelmholtzDerivatives& other) const {
        HelmholtzDerivatives _new;
#define X(name) _new.name = name + other.name;
        LIST_OF_DERIVATIVE_VARIABLES
#undef X
        return _new;
    }
    HelmholtzDerivatives operator*(const CoolPropDbl& other) const {
        HelmholtzDerivatives _new;
#define X(name) _new.name = name * other;
        LIST_OF_DERIVATIVE_VARIABLES
#undef X
        return _new;
    }
    HelmholtzDerivatives() : tau(0.0), delta(0.0), T_red(_HUGE), rhomolar_red(_HUGE) {
        reset(0.0);
    };
    /// Retrieve a single value based on the number of derivatives with respect to tau and delta
    double get(std::size_t itau, std::size_t idelta) {
        if (itau == 0) {
            if (idelta == 0) {
                return alphar;
            } else if (idelta == 1) {
                return dalphar_ddelta;
            } else if (idelta == 2) {
                return d2alphar_ddelta2;
            } else if (idelta == 3) {
                return d3alphar_ddelta3;
            } else if (idelta == 4) {
                return d4alphar_ddelta4;
            } else {
                throw ValueError();
            }
        } else if (itau == 1) {
            if (idelta == 0) {
                return dalphar_dtau;
            } else if (idelta == 1) {
                return d2alphar_ddelta_dtau;
            } else if (idelta == 2) {
                return d3alphar_ddelta2_dtau;
            } else if (idelta == 3) {
                return d4alphar_ddelta3_dtau;
            } else {
                throw ValueError();
            }
        } else if (itau == 2) {
            if (idelta == 0) {
                return d2alphar_dtau2;
            } else if (idelta == 1) {
                return d3alphar_ddelta_dtau2;
            } else if (idelta == 2) {
                return d4alphar_ddelta2_dtau2;
            } else {
                throw ValueError();
            }
        } else if (itau == 3) {
            if (idelta == 0) {
                return d3alphar_dtau3;
            } else if (idelta == 1) {
                return d4alphar_ddelta_dtau3;
            } else {
                throw ValueError();
            }
        } else if (itau == 4) {
            if (idelta == 0) {
                return d4alphar_dtau4;
            } else {
                throw ValueError();
            }
        } else {
            throw ValueError();
        }
    }
};
#undef LIST_OF_DERIVATIVE_VARIABLES

/// The base class class for the Helmholtz energy terms
/**

 Residual Helmholtz Energy Terms:

 Term                               | Helmholtz Energy Contribution
 ----------                         | ------------------------------
 ResidualHelmholtzPower             | \f$ \alpha^r=\left\lbrace\begin{array}{cc}\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} & l_i=0\\ \displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}) & l_i\neq 0\end{array}\right.\f$
 ResidualHelmholtzExponential       | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\gamma_i\delta^{l_i}) \f$
 ResidualHelmholtzLemmon2005        | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}-\tau^{m_i})\f$
 ResidualHelmholtzGaussian          | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\tau-\gamma_i)^2)\f$
 ResidualHelmholtzGERG2008Gaussian  | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\delta-\gamma_i))\f$
 ResidualHelmholtzNonAnalytic       | \f$ \begin{array}{c}\alpha^r&=&\displaystyle\sum_i n_i \Delta^{b_i}\delta\psi \\ \Delta & = & \theta^2+B_i[(\delta-1)^2]^{a_i}\\ \theta & = & (1-\tau)+A_i[(\delta-1)^2]^{1/(2\beta_i)}\\ \psi & = & \exp(-C_i(\delta-1)^2-D_i(\tau-1)^2) \end{array}\f$
 ResidualHelmholtzSAFTAssociating   | \f$ \alpha^r = am\left(\ln X-\frac{X}{2}+\frac{1}{2}\right); \f$


 Ideal-Gas Helmholtz Energy Terms:

 Term                                        | Helmholtz Energy Contribution
 ----------                                  | ------------------------------
 IdealHelmholtzLead                          | \f$ \alpha^0 = n_1 + n_2\tau + \ln\delta \f$
 IdealHelmholtzEnthalpyEntropyOffset         | \f$ \alpha^0 = \displaystyle\frac{\Delta s}{R_u/M}+\frac{\Delta h}{(R_u/M)T}\tau \f$
 IdealHelmholtzLogTau                        | \f$ \alpha^0 = n_1\log\tau \f$
 IdealHelmholtzPower                         | \f$ \alpha^0 = \displaystyle\sum_i n_i\tau^{t_i} \f$
 IdealHelmholtzPlanckEinsteinGeneralized     | \f$ \alpha^0 = \displaystyle\sum_i n_i\log[c_i+d_i\exp(\theta_i\tau)] \f$
 */
class BaseHelmholtzTerm
{
   public:
    BaseHelmholtzTerm() {};
    virtual ~BaseHelmholtzTerm() {};

    /// Returns the base, non-dimensional, Helmholtz energy term (no derivatives) [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl base(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.alphar;
    };
    /// Returns the first partial derivative of Helmholtz energy term with respect to tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dTau(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.dalphar_dtau;
    };
    /// Returns the second partial derivative of Helmholtz energy term with respect to tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dTau2(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d2alphar_dtau2;
    };
    /// Returns the second mixed partial derivative (delta1,dtau1) of Helmholtz energy term with respect to delta and tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta_dTau(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d2alphar_ddelta_dtau;
    };
    /// Returns the first partial derivative of Helmholtz energy term with respect to delta [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.dalphar_ddelta;
    };
    /// Returns the second partial derivative of Helmholtz energy term with respect to delta [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta2(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d2alphar_ddelta2;
    };
    /// Returns the third mixed partial derivative (delta2,dtau1) of Helmholtz energy term with respect to delta and tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta2_dTau(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d3alphar_ddelta2_dtau;
    };
    /// Returns the third mixed partial derivative (delta1,dtau2) of Helmholtz energy term with respect to delta and tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta_dTau2(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d3alphar_ddelta_dtau2;
    };
    /// Returns the third partial derivative of Helmholtz energy term with respect to tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dTau3(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d3alphar_dtau3;
    };
    /// Returns the third partial derivative of Helmholtz energy term with respect to delta [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dDelta3(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d3alphar_ddelta3;
    };
    /// Returns the fourth partial derivative of Helmholtz energy term with respect to tau [-]
    /** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
     *  @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
     */
    virtual CoolPropDbl dTau4(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d4alphar_dtau4;
    };
    virtual CoolPropDbl dDelta_dTau3(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d4alphar_ddelta_dtau3;
    };
    virtual CoolPropDbl dDelta2_dTau2(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d4alphar_ddelta2_dtau2;
    };
    virtual CoolPropDbl dDelta3_dTau(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d4alphar_ddelta3_dtau;
    };
    virtual CoolPropDbl dDelta4(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        HelmholtzDerivatives deriv;
        all(tau, delta, deriv);
        return deriv.d4alphar_ddelta4;
    };

    virtual void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) = 0;
#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const {
        throw CoolProp::NotImplementedError("The mcx derivative function was not implemented");
    }
#endif
};

struct ResidualHelmholtzGeneralizedExponentialElement
{
    /// These variables are for the n*delta^d_i*tau^t_i part
    CoolPropDbl n, d, t;
    /// These variables are for the exp(u) part
    /// u is given by -c*delta^l_i-omega*tau^m_i-eta1*(delta-epsilon1)-eta2*(delta-epsilon2)^2-beta1*(tau-gamma1)-beta2*(tau-gamma2)^2
    CoolPropDbl c, l_double, omega, m_double, eta1, epsilon1, eta2, epsilon2, beta1, gamma1, beta2, gamma2;
    /// If l_i or m_i are integers, we will store them as integers in order to call pow(double, int) rather than pow(double, double)
    int l_int, m_int;
    /// If l is an integer, store a boolean flag so we can evaluate the correct pow() function
    bool l_is_int, m_is_int;

    ResidualHelmholtzGeneralizedExponentialElement() {
        n = 0;
        d = 0;
        t = 0;
        c = 0;
        l_double = 0;
        omega = 0;
        m_double = 0;
        eta1 = 0;
        epsilon1 = 0;
        eta2 = 0;
        epsilon2 = 0;
        beta1 = 0;
        gamma1 = 0;
        beta2 = 0;
        gamma2 = 0;
        l_int = 0;
        m_int = 0;
        l_is_int = false;
        m_is_int = true;
    };
};
/** \brief A generalized residual helmholtz energy container that can deal with a wide range of terms which can be converted to this general form
 *
 * \f$ \alpha^r=\sum_i n_i \delta^{d_i} \tau^{t_i}\exp(u_i) \f$
 *
 * where \f$ u_i \f$ is given by
 *
 * \f$ u_i = -c_i\delta^{l_i}-\omega_i\tau^{m_i}-\eta_{1,i}(\delta-\epsilon_{1,i})-\eta_{2,i}(\delta-\epsilon_{2,i})^2-\beta_{1,i}(\tau-\gamma_{1,i})-\beta_{2,i}(\tau-\gamma_{2,i})^2 \f$
 */
class ResidualHelmholtzGeneralizedExponential : public BaseHelmholtzTerm
{

   public:
    bool delta_li_in_u, tau_mi_in_u, eta1_in_u, eta2_in_u, beta1_in_u, beta2_in_u, finished;
    std::vector<CoolPropDbl> s;
    std::size_t N;

    // These variables are for the exp(u) part
    // u is given by -c*delta^l_i-omega*tau^m_i-eta1*(delta-epsilon1)-eta2*(delta-epsilon2)^2-beta1*(tau-gamma1)-beta2*(tau-gamma2)^2
    std::vector<double> n, d, t, c, l_double, omega, m_double, eta1, epsilon1, eta2, epsilon2, beta1, gamma1, beta2, gamma2;
    // If l_i or m_i are integers, we will store them as integers in order to call pow(double, int) rather than pow(double, double)
    std::vector<int> l_int, m_int;

    //Eigen::ArrayXd uE, du_ddeltaE, du_dtauE, d2u_ddelta2E, d2u_dtau2E, d3u_ddelta3E, d3u_dtau3E;

    std::vector<ResidualHelmholtzGeneralizedExponentialElement> elements;
    // Default Constructor
    ResidualHelmholtzGeneralizedExponential()
      : delta_li_in_u(false), tau_mi_in_u(false), eta1_in_u(false), eta2_in_u(false), beta1_in_u(false), beta2_in_u(false), finished(false), N(0) {};
    /** \brief Add and convert an old-style power (polynomial) term to generalized form
	 *
	 * Term of the format
	 * \f$ \alpha^r=\left\lbrace\begin{array}{cc}\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} & l_i=0\\ \displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}) & l_i\neq 0\end{array}\right.\f$
	 */
    void add_Power(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                   const std::vector<CoolPropDbl>& l) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.l_double = l[i];
            el.l_int = (int)el.l_double;
            if (el.l_double > 0)
                el.c = 1.0;
            else
                el.c = 0.0;
            elements.push_back(el);
        }
        delta_li_in_u = true;
    };
    /** \brief Add and convert an old-style exponential term to generalized form
	 *
	 * Term of the format
	 * \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-g_i\delta^{l_i}) \f$
	 */
    void add_Exponential(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                         const std::vector<CoolPropDbl>& g, const std::vector<CoolPropDbl>& l) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.c = g[i];
            el.l_double = l[i];
            el.l_int = (int)el.l_double;
            elements.push_back(el);
        }
        delta_li_in_u = true;
    }
    /** \brief Add and convert an old-style Gaussian term to generalized form
	 *
	 * Term of the format
	 * \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\tau-\gamma_i)^2)\f$
	 */
    void add_Gaussian(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                      const std::vector<CoolPropDbl>& eta, const std::vector<CoolPropDbl>& epsilon, const std::vector<CoolPropDbl>& beta,
                      const std::vector<CoolPropDbl>& gamma) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.eta2 = eta[i];
            el.epsilon2 = epsilon[i];
            el.beta2 = beta[i];
            el.gamma2 = gamma[i];
            elements.push_back(el);
        }
        eta2_in_u = true;
        beta2_in_u = true;
    };
    /** \brief Add and convert an old-style Gaussian term from GERG 2008 natural gas model to generalized form
	 *
	 * Term of the format
	 * \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\delta-\gamma_i))\f$
	 */
    void add_GERG2008Gaussian(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                              const std::vector<CoolPropDbl>& eta, const std::vector<CoolPropDbl>& epsilon, const std::vector<CoolPropDbl>& beta,
                              const std::vector<CoolPropDbl>& gamma) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.eta2 = eta[i];
            el.epsilon2 = epsilon[i];
            el.eta1 = beta[i];
            el.epsilon1 = gamma[i];
            elements.push_back(el);
        }
        eta2_in_u = true;
        eta1_in_u = true;
    };
    /** \brief Add and convert a term from Lemmon and Jacobsen (2005) used for R125
	 *
	 * Term of the format
	 * \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}-\tau^{m_i})\f$
	 */
    void add_Lemmon2005(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                        const std::vector<CoolPropDbl>& l, const std::vector<CoolPropDbl>& m) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.c = 1.0;
            el.omega = 1.0;
            el.l_double = l[i];
            el.m_double = m[i];
            el.l_int = (int)el.l_double;
            el.m_int = (int)el.m_double;
            elements.push_back(el);
        }
        delta_li_in_u = true;
        tau_mi_in_u = true;
    };
    /** \brief Add and convert a double-exponential term
     *
     * Term of the format
     * \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-gd_j\delta^{ld_j}-gt_j\tau^{lt_i})\f$
     */
    void add_DoubleExponential(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& d, const std::vector<CoolPropDbl>& t,
                               const std::vector<CoolPropDbl>& gd, const std::vector<CoolPropDbl>& ld, const std::vector<CoolPropDbl>& gt,
                               const std::vector<CoolPropDbl>& lt) {
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzGeneralizedExponentialElement el;
            el.n = n[i];
            el.d = d[i];
            el.t = t[i];
            el.c = gd[i];
            el.l_double = ld[i];
            el.l_int = (int)el.l_double;
            el.omega = gt[i];
            el.m_double = lt[i];
            elements.push_back(el);
        }
        delta_li_in_u = true;
        tau_mi_in_u = true;
    };

    void finish() {
        n.resize(elements.size());
        d.resize(elements.size());
        t.resize(elements.size());
        c.resize(elements.size());
        omega.resize(elements.size());
        l_double.resize(elements.size());
        l_int.resize(elements.size());
        m_double.resize(elements.size());
        m_int.resize(elements.size());
        epsilon2.resize(elements.size());
        eta2.resize(elements.size());
        gamma2.resize(elements.size());
        beta2.resize(elements.size());

        for (std::size_t i = 0; i < elements.size(); ++i) {
            n[i] = elements[i].n;
            d[i] = elements[i].d;
            t[i] = elements[i].t;
            c[i] = elements[i].c;
            omega[i] = elements[i].omega;
            l_double[i] = elements[i].l_double;
            l_int[i] = elements[i].l_int;
            m_double[i] = elements[i].m_double;
            m_int[i] = elements[i].m_int;
            epsilon2[i] = elements[i].epsilon2;
            eta2[i] = elements[i].eta2;
            gamma2[i] = elements[i].gamma2;
            beta2[i] = elements[i].beta2;

            // See if l is an integer, and store a flag if it is
            elements[i].l_is_int = (std::abs(static_cast<long>(elements[i].l_double) - elements[i].l_double) < 1e-14);
        }
        //        uE.resize(elements.size());
        //        du_ddeltaE.resize(elements.size());
        //        du_dtauE.resize(elements.size());
        //        d2u_ddelta2E.resize(elements.size());
        //        d2u_dtau2E.resize(elements.size());
        //        d3u_ddelta3E.resize(elements.size());
        //        d3u_dtau3E.resize(elements.size());

        finished = true;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc);

    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;
    //void allEigen(const CoolPropDbl &tau, const CoolPropDbl &delta, HelmholtzDerivatives &derivs) throw();

#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

struct ResidualHelmholtzNonAnalyticElement
{
    CoolPropDbl n, a, b, beta, A, B, C, D;
};
class ResidualHelmholtzNonAnalytic : public BaseHelmholtzTerm
{

   public:
    std::size_t N;
    std::vector<CoolPropDbl> s;
    std::vector<ResidualHelmholtzNonAnalyticElement> elements;
    /// Default Constructor
    ResidualHelmholtzNonAnalytic() {
        N = 0;
    };
    /// Destructor. No implementation
    ~ResidualHelmholtzNonAnalytic() {};
    /// Constructor
    ResidualHelmholtzNonAnalytic(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& a, const std::vector<CoolPropDbl>& b,
                                 const std::vector<CoolPropDbl>& beta, const std::vector<CoolPropDbl>& A, const std::vector<CoolPropDbl>& B,
                                 const std::vector<CoolPropDbl>& C, const std::vector<CoolPropDbl>& D) {
        N = n.size();
        s.resize(N);
        for (std::size_t i = 0; i < n.size(); ++i) {
            ResidualHelmholtzNonAnalyticElement el;
            el.n = n[i];
            el.a = a[i];
            el.b = b[i];
            el.beta = beta[i];
            el.A = A[i];
            el.B = B[i];
            el.C = C[i];
            el.D = D[i];
            elements.push_back(el);
        }
    };
    void to_json(rapidjson::Value& el, rapidjson::Document& doc);
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;
#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

class ResidualHelmholtzGeneralizedCubic : public BaseHelmholtzTerm
{
   protected:
    shared_ptr<AbstractCubic> m_abstractcubic;
    std::vector<double> z;  /// Vector of mole fractions, will be initialized to [1.0] since this is a pure fluid
   public:
    bool enabled;

    /// Default Constructor
    ResidualHelmholtzGeneralizedCubic() {
        enabled = false;
    };
    /// Constructor given an abstract cubic instance
    ResidualHelmholtzGeneralizedCubic(shared_ptr<AbstractCubic>& ac) : m_abstractcubic(ac) {
        enabled = true;
        z = std::vector<double>(1, 1);  // Init the vector to [1.0]
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc);
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

class ResidualHelmholtzGaoB : public BaseHelmholtzTerm
{
   protected:
    std::vector<double> n, t, d, eta, beta, gamma, epsilon, b;

   public:
    bool enabled;

    /// Default Constructor
    ResidualHelmholtzGaoB() {
        enabled = false;
    };

    /// Constructor given coefficients
    ResidualHelmholtzGaoB(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& t, const std::vector<CoolPropDbl>& d,
                          const std::vector<CoolPropDbl>& eta, const std::vector<CoolPropDbl>& beta, const std::vector<CoolPropDbl>& gamma,
                          const std::vector<CoolPropDbl>& epsilon, const std::vector<CoolPropDbl>& b)
      : n(n), t(t), d(d), eta(eta), beta(beta), gamma(gamma), epsilon(epsilon), b(b) {
        enabled = true;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc);
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;

#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

/// The generalized Lee-Kesler formulation of Xiang & Deiters: doi:10.1016/j.ces.2007.11.029
class ResidualHelmholtzXiangDeiters : public BaseHelmholtzTerm
{

   public:
    bool enabled;
    ResidualHelmholtzGeneralizedExponential phi0, phi1, phi2;
    CoolPropDbl Tc, pc, rhomolarc, acentric, R, theta;
    /// Default Constructor
    ResidualHelmholtzXiangDeiters() : Tc(_HUGE), pc(_HUGE), rhomolarc(_HUGE), acentric(_HUGE), R(_HUGE), theta(_HUGE) {
        enabled = false;
    };
    /// Constructor
    ResidualHelmholtzXiangDeiters(const CoolPropDbl Tc, const CoolPropDbl pc, const CoolPropDbl rhomolarc, const CoolPropDbl acentric,
                                  const CoolPropDbl R);
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;
#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

class ResidualHelmholtzSAFTAssociating : public BaseHelmholtzTerm
{

   protected:
    double a, m, epsilonbar, vbarn, kappabar;

    CoolPropDbl Deltabar(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl dDeltabar_ddelta__consttau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2Deltabar_ddelta2__consttau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl dDeltabar_dtau__constdelta(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2Deltabar_dtau2__constdelta(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2Deltabar_ddelta_dtau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3Deltabar_dtau3__constdelta(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3Deltabar_ddelta_dtau2(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3Deltabar_ddelta3__consttau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3Deltabar_ddelta2_dtau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;

    CoolPropDbl X(const CoolPropDbl& delta, const CoolPropDbl& Deltabar) const;
    CoolPropDbl dX_dDeltabar__constdelta(const CoolPropDbl& delta, const CoolPropDbl& Deltabar) const;
    CoolPropDbl dX_ddelta__constDeltabar(const CoolPropDbl& delta, const CoolPropDbl& Deltabar) const;
    CoolPropDbl dX_dtau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl dX_ddelta(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2X_dtau2(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2X_ddeltadtau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d2X_ddelta2(const CoolPropDbl& tau, const CoolPropDbl& delta) const;

    CoolPropDbl d3X_dtau3(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3X_ddelta3(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3X_ddeltadtau2(const CoolPropDbl& tau, const CoolPropDbl& delta) const;
    CoolPropDbl d3X_ddelta2dtau(const CoolPropDbl& tau, const CoolPropDbl& delta) const;

    CoolPropDbl g(const CoolPropDbl& eta) const;
    CoolPropDbl dg_deta(const CoolPropDbl& eta) const;
    CoolPropDbl d2g_deta2(const CoolPropDbl& eta) const;
    CoolPropDbl d3g_deta3(const CoolPropDbl& eta) const;
    CoolPropDbl eta(const CoolPropDbl& delta) const;

   public:
    /// Default constructor
    ResidualHelmholtzSAFTAssociating() : a(_HUGE), m(_HUGE), epsilonbar(_HUGE), vbarn(_HUGE), kappabar(_HUGE) {
        disabled = true;
    };

    // Constructor
    ResidualHelmholtzSAFTAssociating(double a, double m, double epsilonbar, double vbarn, double kappabar)
      : a(a), m(m), epsilonbar(epsilonbar), vbarn(vbarn), kappabar(kappabar) {
        disabled = false;
    };

    bool disabled;

    //Destructor. No Implementation
    ~ResidualHelmholtzSAFTAssociating() {};

    void to_json(rapidjson::Value& el, rapidjson::Document& doc);

    CoolPropDbl dTau4(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        return 1e99;
    };
    CoolPropDbl dDelta_dTau3(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        return 1e99;
    };
    CoolPropDbl dDelta2_dTau2(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        return 1e99;
    };
    CoolPropDbl dDelta3_dTau(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        return 1e99;
    };
    CoolPropDbl dDelta4(const CoolPropDbl& tau, const CoolPropDbl& delta) throw() {
        return 1e99;
    };

    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& deriv) throw();
};

class BaseHelmholtzContainer
{
   protected:
    std::array<double, 16> cache = create_filled_array<double, 16>(_HUGE);
    std::array<bool, 16> is_cached = create_filled_array<bool, 16>(false);
    constexpr static std::size_t i00 = 0, i01 = 1, i02 = 2, i03 = 3, i04 = 4, i10 = 5, i11 = 6, i12 = 7, i13 = 8, i20 = 9, i21 = 10, i22 = 11,
                                 i30 = 12, i31 = 13, i40 = 14;

    bool cache_valid(std::size_t i) const {
        return is_cached[i];
    }

   public:
    void clear() {
        memset(cache.data(), 0, sizeof(cache));
        memset(is_cached.data(), false, sizeof(is_cached));
    };

    virtual void empty_the_EOS() = 0;
    virtual HelmholtzDerivatives all(const CoolPropDbl tau, const CoolPropDbl delta, bool cache_values) = 0;

    CoolPropDbl base(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i00) || dont_use_cache)
            return all(tau, delta, false).alphar;
        else
            return cache[i00];
    };
    CoolPropDbl dDelta(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i10) || dont_use_cache)
            return all(tau, delta, false).dalphar_ddelta;
        else
            return cache[i10];
    };
    CoolPropDbl dTau(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i01) || dont_use_cache)
            return all(tau, delta, false).dalphar_dtau;
        else
            return cache[i01];
    };
    CoolPropDbl dDelta2(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i20) || dont_use_cache)
            return all(tau, delta, false).d2alphar_ddelta2;
        else
            return cache[i20];
    };
    CoolPropDbl dDelta_dTau(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i11) || dont_use_cache)
            return all(tau, delta, false).d2alphar_ddelta_dtau;
        else
            return cache[i11];
    };
    CoolPropDbl dTau2(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i02) || dont_use_cache)
            return all(tau, delta, false).d2alphar_dtau2;
        else
            return cache[i02];
    };
    CoolPropDbl dDelta3(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i30) || dont_use_cache)
            return all(tau, delta, false).d3alphar_ddelta3;
        else
            return cache[i30];
    };
    CoolPropDbl dDelta2_dTau(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i21) || dont_use_cache)
            return all(tau, delta, false).d3alphar_ddelta2_dtau;
        else
            return cache[i21];
    };
    CoolPropDbl dDelta_dTau2(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i12) || dont_use_cache)
            return all(tau, delta, false).d3alphar_ddelta_dtau2;
        else
            return cache[i12];
    };
    CoolPropDbl dTau3(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        if (!cache_valid(i03) || dont_use_cache)
            return all(tau, delta, false).d3alphar_dtau3;
        else
            return cache[i03];
    };
    CoolPropDbl dDelta4(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        return all(tau, delta, false).d4alphar_ddelta4;
    };
    CoolPropDbl dDelta3_dTau(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        return all(tau, delta, false).d4alphar_ddelta3_dtau;
    };
    CoolPropDbl dDelta2_dTau2(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        return all(tau, delta, false).d4alphar_ddelta2_dtau2;
    };
    CoolPropDbl dDelta_dTau3(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        return all(tau, delta, false).d4alphar_ddelta_dtau3;
    };
    CoolPropDbl dTau4(CoolPropDbl tau, CoolPropDbl delta, const bool dont_use_cache = false) {
        return all(tau, delta, false).d4alphar_dtau4;
    };
};

class ResidualHelmholtzContainer : public BaseHelmholtzContainer
{
   public:
    ResidualHelmholtzNonAnalytic NonAnalytic;
    ResidualHelmholtzSAFTAssociating SAFT;
    ResidualHelmholtzGeneralizedExponential GenExp;
    ResidualHelmholtzGeneralizedCubic cubic;
    ResidualHelmholtzXiangDeiters XiangDeiters;
    ResidualHelmholtzGaoB GaoB;

    void empty_the_EOS() {
        NonAnalytic = ResidualHelmholtzNonAnalytic();
        SAFT = ResidualHelmholtzSAFTAssociating();
        GenExp = ResidualHelmholtzGeneralizedExponential();
        cubic = ResidualHelmholtzGeneralizedCubic();
        XiangDeiters = ResidualHelmholtzXiangDeiters();
        GaoB = ResidualHelmholtzGaoB();
    };

    HelmholtzDerivatives all(const CoolPropDbl tau, const CoolPropDbl delta, bool cache_values = false) {
        HelmholtzDerivatives derivs;  // zeros out the elements
        GenExp.all(tau, delta, derivs);
        NonAnalytic.all(tau, delta, derivs);
        SAFT.all(tau, delta, derivs);
        cubic.all(tau, delta, derivs);
        XiangDeiters.all(tau, delta, derivs);
        GaoB.all(tau, delta, derivs);
        if (cache_values) {
            cache[i00] = derivs.alphar;
            cache[i10] = derivs.dalphar_ddelta;
            cache[i01] = derivs.dalphar_dtau;
            cache[i20] = derivs.d2alphar_ddelta2;
            cache[i02] = derivs.d2alphar_dtau2;
            cache[i11] = derivs.d2alphar_ddelta_dtau;
            cache[i30] = derivs.d3alphar_ddelta3;
            cache[i03] = derivs.d3alphar_dtau3;
            cache[i21] = derivs.d3alphar_ddelta2_dtau;
            cache[i12] = derivs.d3alphar_ddelta_dtau2;
            memset(is_cached.data(), true, sizeof(is_cached));
        }
        return derivs;
    };
};

// #############################################################################
// #############################################################################
// #############################################################################
//                                 IDEAL GAS TERMS
// #############################################################################
// #############################################################################
// #############################################################################

/// The leading term in the EOS used to set the desired reference state
/**
\f[
\alpha^0 = \log(\delta)+a_1+a_2\tau
\f]
*/
class IdealHelmholtzLead : public BaseHelmholtzTerm
{

   private:
    CoolPropDbl a1, a2;
    bool enabled;

   public:
    // Default constructor
    IdealHelmholtzLead() : a1(_HUGE), a2(_HUGE), enabled(false) {}

    // Constructor
    IdealHelmholtzLead(CoolPropDbl a1, CoolPropDbl a2) : a1(a1), a2(a2), enabled(true) {}

    bool is_enabled() const {
        return enabled;
    }

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealHelmholtzLead", doc.GetAllocator());
        el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
        el.AddMember("a2", static_cast<double>(a2), doc.GetAllocator());
    };

    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

/// The term in the EOS used to shift the reference state of the fluid
/**
\f[
\alpha^0 = a_1+a_2\tau
\f]
*/
class IdealHelmholtzEnthalpyEntropyOffset : public BaseHelmholtzTerm
{
   private:
    CoolPropDbl a1, a2;  // Use these variables internally
    std::string reference;
    bool enabled;

   public:
    IdealHelmholtzEnthalpyEntropyOffset() : a1(_HUGE), a2(_HUGE), enabled(false) {}

    // Constructor
    IdealHelmholtzEnthalpyEntropyOffset(CoolPropDbl a1, CoolPropDbl a2, const std::string& ref) : a1(a1), a2(a2), reference(ref), enabled(true) {}

    // Set the values in the class
    void set(CoolPropDbl a1, CoolPropDbl a2, const std::string& ref) {
        // If it doesn't already exist, just set the values
        if (enabled == false) {
            this->a1 = a1;
            this->a2 = a2;
            enabled = true;
        } else if (ref == "DEF") {
            this->a1 = 0.0;
            this->a2 = 0.0;
            enabled = false;
        } else {
            // Otherwise, increment the values
            this->a1 += a1;
            this->a2 += a2;
            enabled = true;
        }
        this->reference = ref;
    }

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealHelmholtzEnthalpyEntropyOffset", doc.GetAllocator());
        el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
        el.AddMember("a2", static_cast<double>(a2), doc.GetAllocator());
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

/**
\f[
\alpha^0 = a_1\ln\tau
\f]
*/
class IdealHelmholtzLogTau : public BaseHelmholtzTerm
{
   private:
    CoolPropDbl a1;
    bool enabled;

   public:
    /// Default constructor
    IdealHelmholtzLogTau() : a1(_HUGE), enabled(false) {}

    // Constructor
    IdealHelmholtzLogTau(CoolPropDbl a1) : a1(a1), enabled(true) {}

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealHelmholtzLogTau", doc.GetAllocator());
        el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

/**
\f[
\alpha^0 = \displaystyle\sum_i n_i\tau^{t_i}
\f]
*/
class IdealHelmholtzPower : public BaseHelmholtzTerm
{

   private:
    std::vector<CoolPropDbl> n, t;  // Use these variables internally
    std::size_t N;
    bool enabled;

   public:
    IdealHelmholtzPower() : N(0), enabled(false) {};
    // Constructor
    IdealHelmholtzPower(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& t) : n(n), t(t), N(n.size()), enabled(true) {};

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealHelmholtzPower", doc.GetAllocator());
        cpjson::set_long_double_array("n", n, el, doc);
        cpjson::set_long_double_array("t", t, el, doc);
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

/**
\f[
\alpha^0 = \displaystyle\sum_i n_i\log[c_i+d_i\exp(\theta_i\tau)]
\f]

To convert conventional Plank-Einstein forms, given by
\f$
\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( b_k/T \right)^2\exp \left( b_k/T \right)}{\left(\exp \left(b_k/T\right) - 1 \right)^2}
\f$
and
\f$
\alpha^0 = a_k\ln \left[1 - \exp \left( \frac{-b_k\tau}{T_c} \right) \right]
\f$
use \f$c = 1\f$, \f$d = -1\f$, \f$n = a\f$, \f$\theta = -\displaystyle\frac{b_k}{T_c}\f$

To convert the second form of Plank-Einstein terms, given by
\f$
\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( -b_k/T \right)^2\exp \left( b_k/T \right)}{c\left(\exp \left(-b_k/T\right) + 1 \right)^2}
\f$
and
\f$
\alpha^0 = a_k\ln \left[c + \exp \left( \frac{b_k\tau}{T_c} \right) \right]
\f$
use \f$c = 1\f$, \f$d = 1\f$, \f$n = -a\f$, \f$\theta = \displaystyle\frac{b_k}{T_c}\f$

Converting Aly-Lee tems is a bit more complex

Aly-Lee starts as
\f[\frac{c_p^0}{R_u} = A + B\left(\frac{C/T}{\sinh(C/T)}\right)^2 + D\left(\frac{E/T}{\cosh(E/T)}\right)^2\f]

Constant is separated out, and handled separately.  sinh part can be expanded as
\f[B\left(\frac{C/T}{\sinh(C/T)}\right)^2 = \frac{B(-2C/T)^2\exp(-2C/T)}{(1-\exp(-2C/T))^2}\f]
where
\f[n_k = B\f]
\f[\theta_k = -\frac{2C}{T_c}\f]
\f[c_k = 1\f]
\f[d_k = -1\f]

cosh part can be expanded as
\f[D\left(\frac{E/T}{\cosh(E/T)}\right)^2 = \frac{D(-2E/T)^2\exp(-2E/T)}{(1+\exp(-2E/T))^2}\f]
where
\f[n_k = -D\f]
\f[\theta_k = -\frac{2E}{T_c}\f]
\f[c_k = 1\f]
\f[d_k = 1\f]
*/
class IdealHelmholtzPlanckEinsteinGeneralized : public BaseHelmholtzTerm
{

   private:
    std::vector<CoolPropDbl> n, theta, c, d;  // Use these variables internally
    std::size_t N;
    bool enabled;

   public:
    IdealHelmholtzPlanckEinsteinGeneralized() : N(0), enabled(false) {}
    // Constructor with std::vector instances
    IdealHelmholtzPlanckEinsteinGeneralized(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& theta,
                                            const std::vector<CoolPropDbl>& c, const std::vector<CoolPropDbl>& d)
      : n(n), theta(theta), c(c), d(d), N(n.size()), enabled(true) {}

    // Extend the vectors to allow for multiple instances feeding values to this function
    void extend(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& theta, const std::vector<CoolPropDbl>& c,
                const std::vector<CoolPropDbl>& d) {
        this->n.insert(this->n.end(), n.begin(), n.end());
        this->theta.insert(this->theta.end(), theta.begin(), theta.end());
        this->c.insert(this->c.end(), c.begin(), c.end());
        this->d.insert(this->d.end(), d.begin(), d.end());
        N += n.size();
    }

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealHelmholtzPlanckEinsteinGeneralized", doc.GetAllocator());
        cpjson::set_long_double_array("n", n, el, doc);
        cpjson::set_long_double_array("theta", theta, el, doc);
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

class IdealHelmholtzCP0Constant : public BaseHelmholtzTerm
{

   private:
    double cp_over_R, Tc, T0, tau0;  // Use these variables internally
    bool enabled;

   public:
    /// Default constructor
    IdealHelmholtzCP0Constant() : cp_over_R(_HUGE), Tc(_HUGE), T0(_HUGE), tau0(_HUGE) {
        enabled = false;
    };

    /// Constructor with just a single double value
    IdealHelmholtzCP0Constant(CoolPropDbl cp_over_R, CoolPropDbl Tc, CoolPropDbl T0) : cp_over_R(cp_over_R), Tc(Tc), T0(T0) {
        enabled = true;
        tau0 = Tc / T0;
    };

    /// Destructor
    ~IdealHelmholtzCP0Constant() {};

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc) {
        el.AddMember("type", "IdealGasHelmholtzCP0Constant", doc.GetAllocator());
        el.AddMember("cp_over_R", cp_over_R, doc.GetAllocator());
        el.AddMember("Tc", Tc, doc.GetAllocator());
        el.AddMember("T0", T0, doc.GetAllocator());
    };

    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) throw();
};

class IdealHelmholtzCP0PolyT : public BaseHelmholtzTerm
{
   private:
    std::vector<CoolPropDbl> c, t;
    CoolPropDbl Tc, T0, tau0;  // Use these variables internally
    std::size_t N;
    bool enabled;

   public:
    IdealHelmholtzCP0PolyT() : Tc(_HUGE), T0(_HUGE), tau0(_HUGE), N(0), enabled(false) {}

    /// Constructor with std::vectors
    IdealHelmholtzCP0PolyT(const std::vector<CoolPropDbl>& c, const std::vector<CoolPropDbl>& t, double Tc, double T0)
      : c(c), t(t), Tc(Tc), T0(T0), tau0(Tc / T0), N(c.size()), enabled(true) {
        assert(c.size() == t.size());
    }

    void extend(const std::vector<CoolPropDbl>& c, const std::vector<CoolPropDbl>& t) {
        this->c.insert(this->c.end(), c.begin(), c.end());
        this->t.insert(this->t.end(), t.begin(), t.end());
        N += c.size();
    }

    bool is_enabled() const {
        return enabled;
    };

    void to_json(rapidjson::Value& el, rapidjson::Document& doc);
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;
#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};
/**

*/
class IdealHelmholtzGERG2004Sinh : public BaseHelmholtzTerm
{
   private:
    std::vector<CoolPropDbl> n, theta;
    CoolPropDbl Tc, _Tr;
    std::size_t N;
    bool enabled;

   public:
    IdealHelmholtzGERG2004Sinh() : Tc(_HUGE), _Tr(_HUGE), N(0), enabled(false) {}

    /// Constructor with std::vectors
    IdealHelmholtzGERG2004Sinh(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& theta, double Tc)
      : n(n), theta(theta), Tc(Tc), _Tr(_HUGE), N(n.size()), enabled(true) {
        assert(n.size() == theta.size());
    }

    void extend(const std::vector<CoolPropDbl>& c, const std::vector<CoolPropDbl>& t) {
        this->n.insert(this->n.end(), n.begin(), n.end());
        this->theta.insert(this->theta.end(), theta.begin(), theta.end());
        N += c.size();
    }
    void set_Tred(CoolPropDbl Tr) {
        this->_Tr = Tr;
    }

    bool is_enabled() const {
        return enabled;
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;

#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

class IdealHelmholtzGERG2004Cosh : public BaseHelmholtzTerm
{
   private:
    std::vector<CoolPropDbl> n, theta;
    CoolPropDbl Tc, _Tr;
    std::size_t N;
    bool enabled;

   public:
    IdealHelmholtzGERG2004Cosh() : Tc(_HUGE), _Tr(_HUGE), N(0), enabled(false) {}

    /// Constructor with std::vectors
    IdealHelmholtzGERG2004Cosh(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& theta, double Tc)
      : n(n), theta(theta), Tc(Tc), _Tr(_HUGE), N(n.size()), enabled(true) {
        assert(n.size() == theta.size());
    }

    void extend(const std::vector<CoolPropDbl>& n, const std::vector<CoolPropDbl>& theta) {
        this->n.insert(this->n.end(), n.begin(), n.end());
        this->theta.insert(this->theta.end(), theta.begin(), theta.end());
        N += n.size();
    }
    void set_Tred(CoolPropDbl Tr) {
        this->_Tr = Tr;
    }

    bool is_enabled() const {
        return enabled;
    };
    void all(const CoolPropDbl& tau, const CoolPropDbl& delta, HelmholtzDerivatives& derivs) override;

#if ENABLE_CATCH
    virtual mcx::MultiComplex<double> one_mcx(const mcx::MultiComplex<double>& tau, const mcx::MultiComplex<double>& delta) const override;
#endif
};

///// Term in the ideal-gas specific heat equation that is based on Aly-Lee formulation
///** Specific heat is of the form:
//\f[
//\frac{c_p^0}{R_u} = A + B\left(\frac{C/T}{\sinh(C/T)}\right)^2 + D\left(\frac{E/T}{\cosh(E/T)}\right)^2
//\f]
//Second partial of ideal-gas Helmholtz energy given directly by specific heat (\f$\displaystyle\alpha_{\tau\tau}^0=-\frac{1}{\tau^2}\frac{c_p^0}{R_u} \f$) - this is obtained by real gas \f$c_p\f$ relationship, and killing off residual Helmholtz terms
//\f[
//\alpha^0_{\tau\tau} = -\frac{A}{\tau^2} - \frac{B}{\tau^2}\left(\frac{C/T}{\sinh(C/T)}\right)^2 - \frac{D}{\tau^2}\left(\frac{E/T}{\cosh(E/T)}\right)^2
//\f]
//or in terms of \f$ \tau \f$:
//\f[
//\alpha^0_{\tau\tau} = -\frac{A}{\tau^2} - \frac{BC^2}{T_c^2}\left(\frac{1}{\sinh(C\tau/T_c)}\right)^2 - \frac{DE^2}{T_c^2}\left(\frac{1}{\cosh(E\tau/T_c)}\right)^2
//\f]
//Third partial:
//\f[
//\alpha^0_{\tau\tau\tau} = 2\frac{A}{\tau^3} + 2\frac{BC^3}{T_c^3}\frac{\cosh(C\tau/T_c)}{\sinh^3(C\tau/T_c)} +2 \frac{DE^3}{T_c^3}\frac{\sinh(E\tau/T_c)}{\cosh^3(E\tau/T_c)}
//\f]
//Now coming back to the ideal gas Helmholtz energy definition:
//\f[
//\alpha^0 = -\tau\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau+\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau
//\f]
//Applying derivative
//\f[
//\alpha^0_{\tau} = -\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau-\tau\frac{\partial}{\partial \tau}\left[\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau \right]+\frac{\partial}{\partial \tau}\left[\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau \right]
//\f]
//Fundamental theorem of calculus
//\f[
//\alpha^0_{\tau} = -\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau-\tau \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau+\frac{1}{\tau}\frac{c_p^0}{R_u}
//\f]
//Last two terms cancel, leaving
//\f[
//\alpha^0_{\tau} = -\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau
//\f]
//Another derivative yields (from fundamental theorem of calculus)
//\f[
//\alpha^0_{\tau\tau} = - \frac{1}{\tau^2}\frac{c_p^0}{R_u}
//\f]
//
//see also Jaeschke and Schley, 1995, (http://link.springer.com/article/10.1007%2FBF02083547#page-1)
//*/
///*
//class IdealHelmholtzCP0AlyLee : public BaseHelmholtzTerm{
//private:
//    std::vector<CoolPropDbl> c;
//    CoolPropDbl Tc, tau0, T0; // Use these variables internally
//    bool enabled;
//public:
//    IdealHelmholtzCP0AlyLee(){enabled = false;};
//
//    /// Constructor with std::vectors
//    IdealHelmholtzCP0AlyLee(const std::vector<CoolPropDbl> &c, double Tc, double T0)
//    :c(c), Tc(Tc), T0(T0)
//    {
//        tau0=Tc/T0;
//        enabled = true;
//    };
//
//    /// Destructor
//    ~IdealHelmholtzCP0AlyLee(){};
//
//    bool is_enabled() const {return enabled;};
//
//    void to_json(rapidjson::Value &el, rapidjson::Document &doc);
//
//
//    /// The antiderivative given by \f$ \displaystyle\int \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau \f$
//    /**
//    sympy code for this derivative:
//
//        from sympy import *
//        a1,a2,a3,a4,a5,Tc,tau = symbols('a1,a2,a3,a4,a5,Tc,tau', real = True)
//        integrand = a1 + a2*(a3/Tc/sinh(a3*tau/Tc))**2 + a4*(a5/Tc/cosh(a5*tau/Tc))**2
//        integrand = integrand.rewrite(exp)
//        antideriv = trigsimp(integrate(integrand,tau))
//        display(antideriv)
//        print latex(antideriv)
//        print ccode(antideriv)
//
//    \f[
//    \displaystyle\int \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau = -\frac{a_0}{\tau}+\frac{2a_1a_2}{T_c\left[\exp\left(-\frac{2a_2\tau}{T_c}\right)-1\right]}+\frac{2a_3a_4}{T_c\left[\exp\left(-\frac{2a_4\tau}{T_c}\right)+1\right]}
//    \f]
//    */
//    CoolPropDbl anti_deriv_cp0_tau2(const CoolPropDbl &tau);
//
//    /// The antiderivative given by \f$ \displaystyle\int \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau \f$
//    /**
//    sympy code for this derivative:
//
//        a_0,a_1,a_2,a_3,a_4,Tc,tau = symbols('a_0,a_1,a_2,a_3,a_4,Tc,tau', real = True)
//        integrand = a_0/tau + a_1/tau*(a_2*tau/Tc/sinh(a_2*tau/Tc))**2 + a_3/tau*(a_4*tau/Tc/cosh(a_4*tau/Tc))**2
//
//        term2 = a_1/tau*(a_2*tau/Tc/sinh(a_2*tau/Tc))**2
//        term2 = term2.rewrite(exp)  # Unpack the sinh to exp functions
//        antideriv2 = trigsimp(integrate(term2,tau))
//        display(antideriv2)
//        print latex(antideriv2)
//        print ccode(antideriv2)
//
//        term3 = a_3/tau*(a_4*tau/Tc/cosh(a_4*tau/Tc))**2
//        term3 = term3.rewrite(exp)  # Unpack the cosh to exp functions
//        antideriv3 = factor(trigsimp(integrate(term3,tau).rewrite(exp)))
//        display(antideriv3)
//        print latex(antideriv3)
//        print ccode(antideriv3)
//
//    Can be broken into three parts (trick is to express \f$sinh\f$ and \f$cosh\f$ in terms of \f$exp\f$ function)
//
//    Term 2:
//    \f[
//    \displaystyle\int \frac{a_1a_2^2}{T_c^2}\frac{\tau}{\sinh\left(\displaystyle\frac{a_2\tau}{T_c}\right)^2} d\tau = \frac{2 a_{1} a_{2} \tau}{- Tc + Tc e^{- \frac{2 a_{2}}{Tc} \tau}} + a_{1} \log{\left (-1 + e^{- \frac{2 a_{2}}{Tc} \tau} \right )} + \frac{2 a_{1}}{Tc} a_{2} \tau
//    \f]
//
//    Term 3:
//    \f[
//    \displaystyle\int \frac{a_1a_2^2}{T_c^2}\frac{\tau}{\cosh\left(\displaystyle\frac{a_2\tau}{T_c}\right)^2} d\tau = - \frac{a_{3}}{Tc \left(e^{\frac{2 a_{4}}{Tc} \tau} + 1\right)} \left(Tc e^{\frac{2 a_{4}}{Tc} \tau} \log{\left (e^{\frac{2 a_{4}}{Tc} \tau} + 1 \right )} + Tc \log{\left (e^{\frac{2 a_{4}}{Tc} \tau} + 1 \right )} - 2 a_{4} \tau e^{\frac{2 a_{4}}{Tc} \tau}\right)
//    \f]
//    */
//    CoolPropDbl anti_deriv_cp0_tau(const CoolPropDbl &tau);
//
//    CoolPropDbl base(const CoolPropDbl &tau, const CoolPropDbl &delta) throw();
//    CoolPropDbl dDelta(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dTau(const CoolPropDbl &tau, const CoolPropDbl &delta) throw();
//    CoolPropDbl dDelta2(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dDelta_dTau(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dTau2(const CoolPropDbl &tau, const CoolPropDbl &delta) throw();
//    CoolPropDbl dDelta3(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dDelta2_dTau(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dDelta_dTau2(const CoolPropDbl &tau, const CoolPropDbl &delta) throw(){return 0.0;};
//    CoolPropDbl dTau3(const CoolPropDbl &tau, const CoolPropDbl &delta) throw();
//    CoolPropDbl dTau4(const CoolPropDbl &tau, const CoolPropDbl &delta) throw();
//
//};

class IdealHelmholtzContainer : public BaseHelmholtzContainer
{
   private:
    double _prefactor;

   public:
    IdealHelmholtzLead Lead;
    IdealHelmholtzEnthalpyEntropyOffset EnthalpyEntropyOffsetCore, EnthalpyEntropyOffset;
    IdealHelmholtzLogTau LogTau;
    IdealHelmholtzPower Power;
    IdealHelmholtzPlanckEinsteinGeneralized PlanckEinstein;

    IdealHelmholtzCP0Constant CP0Constant;
    IdealHelmholtzCP0PolyT CP0PolyT;
    IdealHelmholtzGERG2004Cosh GERG2004Cosh;
    IdealHelmholtzGERG2004Sinh GERG2004Sinh;

    IdealHelmholtzContainer() : _prefactor(1.0) {};

    void set_prefactor(double prefactor) {
        _prefactor = prefactor;
    }

    void set_Tred(double T_red) {
        GERG2004Cosh.set_Tred(T_red);
        GERG2004Sinh.set_Tred(T_red);
    }

    void empty_the_EOS() {
        Lead = IdealHelmholtzLead();
        EnthalpyEntropyOffsetCore = IdealHelmholtzEnthalpyEntropyOffset();
        EnthalpyEntropyOffset = IdealHelmholtzEnthalpyEntropyOffset();
        LogTau = IdealHelmholtzLogTau();
        Power = IdealHelmholtzPower();
        PlanckEinstein = IdealHelmholtzPlanckEinsteinGeneralized();
        CP0Constant = IdealHelmholtzCP0Constant();
        CP0PolyT = IdealHelmholtzCP0PolyT();
        GERG2004Cosh = IdealHelmholtzGERG2004Cosh();
        GERG2004Sinh = IdealHelmholtzGERG2004Sinh();
    };

    HelmholtzDerivatives all(const CoolPropDbl tau, const CoolPropDbl delta, bool cache_values = false) {
        HelmholtzDerivatives derivs;  // zeros out the elements
        Lead.all(tau, delta, derivs);
        EnthalpyEntropyOffsetCore.all(tau, delta, derivs);
        EnthalpyEntropyOffset.all(tau, delta, derivs);
        LogTau.all(tau, delta, derivs);
        Power.all(tau, delta, derivs);
        PlanckEinstein.all(tau, delta, derivs);
        CP0Constant.all(tau, delta, derivs);
        CP0PolyT.all(tau, delta, derivs);
        GERG2004Cosh.all(tau, delta, derivs);
        GERG2004Sinh.all(tau, delta, derivs);

        if (cache_values) {
            cache[i00] = derivs.alphar * _prefactor;
            cache[i10] = derivs.dalphar_ddelta * _prefactor;
            cache[i01] = derivs.dalphar_dtau * _prefactor;
            cache[i20] = derivs.d2alphar_ddelta2 * _prefactor;
            cache[i02] = derivs.d2alphar_dtau2 * _prefactor;
            cache[i11] = derivs.d2alphar_ddelta_dtau * _prefactor;
            cache[i30] = derivs.d3alphar_ddelta3 * _prefactor;
            cache[i03] = derivs.d3alphar_dtau3 * _prefactor;
            cache[i21] = derivs.d3alphar_ddelta2_dtau * _prefactor;
            cache[i12] = derivs.d3alphar_ddelta_dtau2 * _prefactor;
            memset(is_cached.data(), true, sizeof(is_cached));
        }
        return derivs * _prefactor;
    };
};
}; /* namespace CoolProp */

#endif