eating back

it depends on what method you are using...you have to understand the method.

if you include exercise in your activity level (TDEE method) then you would obviously be double dipping if you ate back exercise calories...an estimate of them have been included in your targets given they are in your activity level.

if you don't include exercise activity in your activity level (NEAT method...MFP method) then that activity would obviously be unaccounted for and since activity increases your calorie requisites, you would obviously have to account for them somewhere...which is why MFP gives you calories to "eat back" after exercise...it's simply a way of accounting for activity that is otherwise not accounted for. one caveat here is that it is difficult to measure calories out...machines, databases, HRMs, etc are all estimates so you should have some kind of allowance for estimation error.