The mass-energy formula E=mc^2 is thought to be derived by Einstein from special relativity. The present study shows that since the formula has also been derived from classical physics by Einstein, it has a deep-rooted connection with classical physics. The formula is implied by Maxwell’s electromagnetic momentum P=E/c and the Newtonian definition of momentum P=mv. It can be derived from classical physics with c as the constant velocity of light in its medium ether. The present study also shows that within the framework of classical physics, this classical physics based formula is correct in other inertial frames that move relative to the ether frame as well. In contrast, Einstein’s derivation in 1905 seems logically flawed as a relativistic proof, because it relies on the classical kinetic energy definition, approximates at low velocity and fails to show mass-energy equivalence in the same reference frame. Therefore, treating E=mc^2 as a quantity like the momentum P=mv , which applies to both classical physics and special relativity if relativistic mass is used in the equation, appears to be more consistent with the logic of special relativity. Then the truly relativistic formula should be E=E_0/√(1-v^2/c^2) derived by Laue and Klein, which corresponds to the formula of relativistic mass.