For the laymen who has not much of a picture of how E=mc² fails, I’ll give a simplest of example. Energy may most accurately (even when it involves energy of flying protons within those most advanced Large Hadron Collider, the LHC) be estimated by how man kettles of water that may be brought to boil with an amount of energy. A chemist can do a very good estimate of how many kettle of water may be brought to boil by burning 1 litre of gasoline (after some estimates of heat lost). Engineers too have good estimates of how many miles a gallon of petrol may give to a model of a motorbike.
What about the physicists at the CERN running the Large Hadron Collider? Say there is a process which produces an amount of energy. The CERN physicists make an estimate of how much that amount of energy is, in the number of kettles of water that may be brought to boil; say their estimate is 15,000 kettles. Then a gadget is invented to convert the amount of energy of the process to boil kettles of water (assuming no significant energy wastage). It is found that the amount of energy can only boil 1 kettle of water – not 15,000 kettles! There is no excuse for the CERN physicists to make an estimate that is out by a factor of 15,000! What really is the problem is that they are using invalid formulas; had they stick to using the old formula for kinetic energy as what high school students are taught, their estimate would come out right – 1 kettle of water.
The method of computing the energy through heat produced is called calorimetry.