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.
Comments
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.