Fractal programming. Any way to optimize this code for real-time rendering?

Question

I have code that I would like to optimize, if possible, while reducing the maximum number of iterations. I heard that there is a way to detect cyclic, but I tried to implement it in different ways, and either it became slower or it made garbage. Display functions are not displayed, as this does not cause a slowdown.

#pragma once
#include <SFML/Graphics/Rect.hpp>
#include <SFML/System/Vector2.hpp>
#include <cstdint>
#include <complex>
#include <functional>
#include <vector>

using namespace std;

template<class T>
class Fractal
{
public:
    Fractal(void);
    ~Fractal(void);

    //the most important function
    vector<uint32_t> evaluate(const sf::Rect<T>& area, const sf::Vector2u& subdivisions);

    //set the iterative function
    typedef function<void(complex<T>&)> iterative_function;
    void setIterativeFunction(iterative_function func);

    //set the domain function
    typedef function<bool(complex<T>&)> domain_function;
    void setDomainFunction(domain_function func);

    //set the maximum number of escape iterations
    void setMaxIterations(const uint32_t iterations);

    //get maximum iterations
    uint32_t getMaxIterations() const;

    //a coordinates generator
    //generates the coordinates to evaluate the fractal
    class CoordinatesGenerator
    {
    public:
        CoordinatesGenerator(const sf::Rect<T>& area, const sf::Vector2u& subdivisions);
        ~CoordinatesGenerator();

        complex<T> operator()();
    private:
        const sf::Rect<T>& area_;
        const sf::Vector2u& subdivisions_;
        complex<T> coord_;
        sf::Vector2u pixel_;
    };
private:
    //the number of escape iterations
    uint32_t max_iterations_;

    //the tolerance where z must change
    T tolerance_;

    //the formula used for the iterative system
    iterative_function iter_function_;

    //the formula that decides either the given complex is inside or not the domain
    domain_function domain_function_;

    //returns the number of iterations that z has to do to escape
    uint32_t getIterations(complex<T> z) const;
};

template<class T>
Fractal<T>::Fractal()
{
    //setting max iterations to 1000 by default
    max_iterations_ = 1000;

    //setting standard Manderbot iterative function
    iter_function_ = iterative_function([](complex<T>& z)
    {
        z = z*z + complex<T>(1,0);
    });

    //setting standard Manderbot domain function
    domain_function_ = domain_function([](complex<T>& z)
    {
        return abs(z) < 2;
    });
}

// Fractal<T>::setIterativeFunction
// iterative_function func : the function on which the system iterates
// must match this signature : void(Complex<T>&)
template<class T>
void Fractal<T>::setIterativeFunction(iterative_function func)
{
    iter_function_ = func;
}

// Fractal<T>::setDomainFunction
// domain_function func : the function that determines if complex is inside domain
// must match this signature : bool(Complex<T>&)
template<class T>
void Fractal<T>::setDomainFunction(domain_function func)
{
    domain_function_ = func;
}

// Fractal<T>::setMaxIterations
// iterations : set the maximum iterations for escape
template<class T>
void Fractal<T>::setMaxIterations(const uint32_t iterations)
{
    max_iterations_ = iterations;
}

// vector<uint32_t> Fractal<T>::evaluate(const sf::Rect<T>& area, const sf::Vector2u& subdivisions)
// area: the fractal area to evaluate
// subdivisions : the number of subdivisions to evaluate
// return a vector of the number of iterations
// the vector is construction from x = 0 ... n, y = 0 ... n
template<class T>
vector<uint32_t> Fractal<T>::evaluate(const sf::Rect<T>& area, const sf::Vector2u& subdivisions)
{
    uint32_t temp;
    complex<T> z(area.left,area.top);
    uint32_t num_coordinates = (subdivisions.x)*(subdivisions.y);

    vector<uint32_t> result;
    vector<complex<T>> coordinates(num_coordinates);
    CoordinatesGenerator generator(area,subdivisions);
    generate(coordinates.begin(),coordinates.end(),generator);

    for(auto& z: coordinates)
    {
        temp = getIterations(z);
        result.push_back(temp);
    }
    return result;
}

// uint32_t Fractal<T>::getIterations(complex<T> z) const
// z : the complex number to evaluate
// return the number of iterations that z escapes domain
// using iterative and domain functions
template<class T>
uint32_t Fractal<T>::getIterations(complex<T> z) const
{
    static uint32_t result;
    result = 0;

    while(domain_function_(z) && result < max_iterations_)
    {
        iter_function_(z);
        result++;
    }
    return result;
}

// Fractal<T>::CoordinatesGenerator::CoordinatesGenerator(const sf::Rect<T>& area, const sf::Vector2u& subdivisions)
// area : the fractal area to evaluate
// subdivisions : the number of subdivisions
// used by STL algorithm
template<class T>
Fractal<T>::CoordinatesGenerator::CoordinatesGenerator(const sf::Rect<T>& area, const sf::Vector2u& subdivisions):
    area_(area),subdivisions_(subdivisions)
{
    coord_ = complex<T>(area_.left,area_.top);
    pixel_.x = 0;
    pixel_.y = 0;
}

template<class T>
Fractal<T>::CoordinatesGenerator::~CoordinatesGenerator()
{
}

// complex<T> Fractal<T>::CoordinatesGenerator::operator()()
// Generate coordinates to evaluate the fractal
// used by STL algorithm
template<class T>
complex<T> Fractal<T>::CoordinatesGenerator::operator()()
{
    //getting the variation of X and Y
    T deltaX = area_.width/static_cast<T>(subdivisions_.x);
    T deltaY = area_.height/static_cast<T>(subdivisions_.y);

    //creating the coordinate
    coord_ = complex<T>(static_cast<T>(pixel_.x)*deltaX + area_.left,static_cast<T>(pixel_.y)*deltaY + area_.top);

    //applying some changes to generate the next coordinate
    pixel_.x++;

    if(pixel_.x >= subdivisions_.x)
    {
        pixel_.y++;
        pixel_.x = 0;
    }

    return coord_;
}

template<class T>
Fractal<T>::~Fractal()
{
}

template<class T>
uint32_t Fractal<T>::getMaxIterations() const
{
    return max_iterations_;
}

Answer 1

I noticed that your function returns

vector<uint32_t> 

Please make sure that you are using the C ++ 11 compiler, as you are likely to benefit from move semantics.