I am solving a problem involving the line with the set of points $(x_3,y_3)$ that are equidistant to two given points $(x_1,y_1)$ and $(x_2,y_2)$. The equation for this line is$$(x_3 - x_1)^2 + (y_3 - y_1)^2 = (x_3 - x_2)^2 + (y_3 - y_2)^2$$which boils down to$$2 (x_1 - x_2) x_3 + 2 (y_1 - y_2) y_3 = x_1^2 + y_1^2 - x_2^2 - y_2^2$$The question I have is quite simple: how should I write the right hand side of this equation? I know that there is more than one way to do it and that some alternatives might be more numerically accurate than others, due to floating point errors.\begin{align}    &x_1^2 + y_1^2 - x_2^2 - y_2^2 = \\    &(x_1^2 + y_1^2) - (x_2^2 + y_2^2) = \\    &(x_1^2 - x2^2) + (y_1^2 - y_2^2) = \\    &(x_1 + x_2) (x_1-x_2) + (y_1 + y_2) (y_1 - y_2) = \\    & \vdots\end{align}How do I compute the right hand sidein the most accurate way?
How do I add some floating point numbers, keeping numerical accuracy in mind?
numerical analysis;numerics;stability;floating point