#ifndef DEF_AUTOMATIC_DIFFERENTIATION
#define DEF_AUTOMATIC_DIFFERENTIATION

#include <cmath>
#include <ostream>

#include <Eigen/Dense>

/// Implementation of dual numbers for automatic differentiation.
/// This implementation uses vectors for b so that function gradients can be computed in one function call.
/// Set the index of every variable with the ::d(int i) function and call the function to be computed : f(x+DualVector::d(0), y+DualVector::d(1), z+DualVector::d(2), ...)
/// reference : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.89.7749&rep=rep1&type=pdf
template<typename Scalar, int N>
class DualVector
{
	public:
		using VectorT = Eigen::Matrix<Scalar, N, 1>;
		
		DualVector(const Scalar & _a, const VectorT & _b = VectorT::Zero())
			: a(_a),
			  b(_b)
			  {}
		
		DualVector(const Scalar & _a, const Scalar & _b)
			: a(_a)
		{
			b.fill(_b);
		}
		
		static DualVector d(int i = 0)
		{
			assert(i >= 0);
			assert(i < N);
			VectorT _d = VectorT::Zero();
			_d[i] = Scalar(1.);
			return DualVector(Scalar(0.), _d);
		}
		
		DualVector & operator+=(const DualVector & x)
		{
			a += x.a;
			b += x.b;
			return *this;
		}
		
		DualVector & operator-=(const DualVector & x)
		{
			a -= x.a;
			b -= x.b;
			return *this;
		}
		
		DualVector & operator*=(const DualVector & x)
		{
			b = a*x.b + b*x.a;
			a *= x.a;
			return *this;
		}
		
		DualVector & operator/=(const DualVector & x)
		{
			b = (x.a*b - a*x.b)/(x.a*x.a);
			a /= x.a;
			return *this;
		}
		
		DualVector operator+(const DualVector & x) const
		{
			DualVector res(*this);
			return (res += x);
		}
		
		DualVector operator+(void) const // +x
		{
			return (*this);
		}
		
		DualVector operator-(const DualVector & x) const {
			DualVector res(*this);
			return (res -= x);
		}
		
		DualVector operator-(void) const // -x
		{
			return DualVector(-a, -b);
		}
		
		DualVector operator*(const DualVector & x) const
		{
			DualVector res(*this);
			return (res *= x);
		}
		
		DualVector operator/(const DualVector & x) const
		{
			DualVector res(*this);
			return (res /= x);
		}
		
		Scalar a;	/// Real part
		VectorT b;	/// Infinitesimal parts
};

// Basic mathematical functions for Scalar numbers

// Trigonometric functions
template<typename Scalar> Scalar sec(const Scalar & x) {
	return Scalar(1.)/cos(x);
}

template<typename Scalar> Scalar cot(const Scalar & x) {
	return cos(x)/sin(x);
}

template<typename Scalar> Scalar csc(const Scalar & x) {
	return Scalar(1.)/sin(x);
}

// Inverse trigonometric functions
template<typename Scalar> Scalar asec(const Scalar & x) {
	return acos(Scalar(1.)/x);
}

template<typename Scalar> Scalar acot(const Scalar & x) {
	return atan(Scalar(1.)/x);
}

template<typename Scalar> Scalar acsc(const Scalar & x) {
	return asin(Scalar(1.)/x);
}

// Hyperbolic trigonometric functions
template<typename Scalar> Scalar sech(const Scalar & x) {
	return Scalar(1.)/cosh(x);
}

template<typename Scalar> Scalar coth(const Scalar & x) {
	return cosh(x)/sinh(x);
}

template<typename Scalar> Scalar csch(const Scalar & x) {
	return Scalar(1.)/sinh(x);
}

// Inverse hyperbolic trigonometric functions
template<typename Scalar> Scalar asech(const Scalar & x) {
	return log((Scalar(1.) + sqrt(Scalar(1.) - x*x))/x);
}

template<typename Scalar> Scalar acoth(const Scalar & x) {
	return Scalar(0.5)*log((x + Scalar(1.))/(x - Scalar(1.)));
}

template<typename Scalar> Scalar acsch(const Scalar & x) {
	return (x >= Scalar(0.)) ? log((Scalar(1.) + sqrt(Scalar(1.) + x*x))/x) : log((Scalar(1.) - sqrt(Scalar(1.) + x*x))/x);
}

template<typename Scalar> Scalar exp10(const Scalar & x) {
	return exp(x*log(Scalar(10.)));
}

template<typename Scalar> Scalar sign(const Scalar & x) {
	return (x >= Scalar(0.)) ? ((x > Scalar(0.)) ? Scalar(1.) : Scalar(0.)) : Scalar(-1.);
}

template<typename Scalar> Scalar heaviside(const Scalar & x) {
	return Scalar(x >= Scalar(0.));
}

// Basic mathematical functions for DualVector numbers
// f(a + b*d) = f(a) + b*f'(a)*d

// Trigonometric functions
template<typename Scalar, int N> DualVector<Scalar, N> cos(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(cos(x.a), -x.b*sin(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> sin(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sin(x.a), x.b*cos(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> tan(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(tan(x.a), x.b*sec(x.a)*sec(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> sec(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sec(x.a), x.b*sec(x.a)*tan(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> cot(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(cot(x.a), x.b*(-csc(x.a)*csc(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> csc(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(csc(x.a), x.b*(-cot(x.a)*csc(x.a)));
}

// Inverse trigonometric functions
template<typename Scalar, int N> DualVector<Scalar, N> acos(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acos(x.a), x.b*(-Scalar(1.)/sqrt(Scalar(1.)-x.a*x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> asin(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(asin(x.a), x.b*(Scalar(1.)/sqrt(Scalar(1.)-x.a*x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> atan(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(atan(x.a), x.b*(Scalar(1.)/(x.a*x.a + Scalar(1.))));
}

template<typename Scalar, int N> DualVector<Scalar, N> asec(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(asec(x.a), x.b*(Scalar(1.)/(sqrt(Scalar(1.)-Scalar(1.)/(x.a*x.a))*(x.a*x.a))));
}

template<typename Scalar, int N> DualVector<Scalar, N> acot(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acot(x.a), x.b*(-Scalar(1.)/((x.a*x.a)+Scalar(1.))));
}

template<typename Scalar, int N> DualVector<Scalar, N> acsc(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acsc(x.a), x.b*(-Scalar(1.)/(sqrt(Scalar(1.)-Scalar(1.)/(x.a*x.a))*(x.a*x.a))));
}

// Hyperbolic trigonometric functions
template<typename Scalar, int N> DualVector<Scalar, N> cosh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(cosh(x.a), x.b*sinh(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> sinh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sinh(x.a), x.b*cosh(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> tanh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(tanh(x.a), x.b*sech(x.a)*sech(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> sech(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sech(x.a), x.b*(-sech(x.a)*tanh(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> coth(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(coth(x.a), x.b*(-csch(x.a)*csch(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> csch(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(csch(x.a), x.b*(-coth(x.a)*csch(x.a)));
}

// Inverse hyperbolic trigonometric functions
template<typename Scalar, int N> DualVector<Scalar, N> acosh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acosh(x.a), x.b*(Scalar(1.)/sqrt((x.a*x.a)-Scalar(1.))));
}

template<typename Scalar, int N> DualVector<Scalar, N> asinh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(asinh(x.a), x.b*(Scalar(1.)/sqrt((x.a*x.a)+Scalar(1.))));
}

template<typename Scalar, int N> DualVector<Scalar, N> atanh(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(atanh(x.a), x.b*(Scalar(1.)/(Scalar(1.)-(x.a*x.a))));
}

template<typename Scalar, int N> DualVector<Scalar, N> asech(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(asech(x.a), x.b*(Scalar(-1.)/(sqrt(Scalar(1.)/(x.a*x.a)-Scalar(1.))*(x.a*x.a))));
}

template<typename Scalar, int N> DualVector<Scalar, N> acoth(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acoth(x.a), x.b*(-Scalar(1.)/((x.a*x.a)-Scalar(1.))));
}

template<typename Scalar, int N> DualVector<Scalar, N> acsch(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(acsch(x.a), x.b*(-Scalar(1.)/(sqrt(Scalar(1.)/(x.a*x.a)+Scalar(1.))*(x.a*x.a))));
}

// Exponential functions
template<typename Scalar, int N> DualVector<Scalar, N> exp(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(exp(x.a), x.b*exp(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> log(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(log(x.a), x.b/x.a);
}

template<typename Scalar, int N> DualVector<Scalar, N> exp10(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(exp10(x.a), x.b*(log(Scalar(10.))*exp10(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> log10(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(log10(x.a), x.b/(log(Scalar(10.))*x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> exp2(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(exp2(x.a), x.b*(log(Scalar(2.))*exp2(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> log2(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(log2(x.a), x.b/(log(Scalar(2.))*x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> pow(const DualVector<Scalar, N> & x, const DualVector<Scalar, N> & n) {
	return exp(n*log(x));
}

// Other functions
template<typename Scalar, int N> DualVector<Scalar, N> sqrt(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sqrt(x.a), x.b/(Scalar(2.)*sqrt(x.a)));
}

template<typename Scalar, int N> DualVector<Scalar, N> sign(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(sign(x.a), DualVector<Scalar, N>::VectorT::Zero());
}

template<typename Scalar, int N> DualVector<Scalar, N> abs(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(fabs(x.a), x.b*sign(x.a));
}

template<typename Scalar, int N> DualVector<Scalar, N> heaviside(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(heaviside(x.a), DualVector<Scalar, N>::VectorT::Zero());
}

template<typename Scalar, int N> DualVector<Scalar, N> floor(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(floor(x.a), DualVector<Scalar, N>::VectorT::Zero());
}

template<typename Scalar, int N> DualVector<Scalar, N> ceil(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(ceil(x.a), DualVector<Scalar, N>::VectorT::Zero());
}

template<typename Scalar, int N> DualVector<Scalar, N> round(const DualVector<Scalar, N> & x) {
	return DualVector<Scalar, N>(round(x.a), DualVector<Scalar, N>::VectorT::Zero());
}

template<typename Scalar, int N> std::ostream & operator<<(std::ostream & s, const DualVector<Scalar, N> & x)
{
	return (s << x.a);
}

#endif