Mandelbrot's algorithm not giving the desired result without using complex numbers

Question

when i use the following algorithm for using the mandelbrot algorithm i get the a image similar to the image i want but it looks more like an outline of the final image and when i use the algorithm with complex numbers it gives a proper sharp image

the code is written in c++

//pseudocode 
//z=x+yi    //x=0,y=0
//c=x0+y0i  //x=scaled(px),y=scaled(py)
//do
//z2=x2+(2*xy)i-y2
//z=z2+c   //z=x2-y2+x0+(2*xy+y0)i
//x=x2-y2+x0   //y=2*xy+y0
//x2=x*x       //y2=y*y
//if(sqrt(x2+y2)<2 )
// break
//iteration++
//while( iteration<max_iteration)

//code without complex numbers
int mandelbrot::_test_num_iteration( double px, double py ) {
double x0{ scale_x( px ) }, x{ 0 };
double y0{ scale_y( py ) }, y{ 0 };
double x2{ 0 }, y2{ 0 };
//z=x+iy        c=x0+iy0
int iteration{ 0 };
while (iteration < max_iteration) {         //for instance max_iteration=1000
    x2 = x * x;             y2 = y * y;
    x = x2  + x0 - y2;      y = 2 * x * y + y0;
    x2 = (x * x);           y2 = (y * y);
    if (sqrt(( x2 + y2 )) > 2)   
        break;
    iteration++;
}
return iteration;
}

 //code with complex numbers
    int mandelbrot::get_iteration( double px, double py) {
    //x+iy=z
    //z2=x2+(2*xyi)-y2
    //c=x0+iy0
    //x0&y0=scaled x and y coordinates
    double x{ scale_x( px ) };
    double y{ scale_y( py ) };
    complex::complex_numbers z;
    complex::complex_numbers c{ x,y };
    int iteration{ 0 };
    while (iteration < max_iteration) {
        z = z * z + c;
        if (mod( z ) > 2)   //mod is a function which returns sqrt(x2+y2)
            break;
        iteration++;
    }

    return iteration;
    }

Answer 1

The problem is that you're not calculating both new values from the old values.
Here,

x = x2  + x0 - y2;      y = 2 * x * y + y0;

you're using the new x for the new y.
You need a couple of variables for temporary storage of the updated values.

double new_x = x2 - y2 + x0;      
double new_y = 2 * x * y + y0;
x = new_x;
y = new_y;