The formula I am using says that it is a non-relativistic equation and to calculate velocities near the speed of light I have to transform it somehow to make it relativistically accurate (something like that)

A non-relativistic equation is invariant under Galilean boosts. (See here.) A relativistic equation is invariant under Lorentz boosts. (See here.) When the velocities in the problem you're considering are much smaller than c, Lorentz transformations and Galilean transformations become approximately the same thing. (More formally, Lorentz transformations and Galilean transformations are equal in the limit where (1/c)->0.)