Yes, of course it’s possible to exhibit errors if you change the problem (though it should be noted that the source of error here is entirely in the fact that l is not representable). As soon as you allow representation error, of course, your tolerance-based approach goes out the window; what if the representation error in l and m is larger than your tolerance accounts for?

If I wanted a correct bound for your revised problem, I would require that l be an upper bound on the true value (by forcing it to be rounded up earlier computations, for example) and that m be a lower bound on the true value. Then the simple computation delivers a correct result regardless of the magnitude of the error in l or m.