  There is a rich theory of the approximability of NP optimization problems (See [Arora &amp; Barak, 2009]).  For example, the Subset Sum problem mentioned above is an NP-complete problem.  Most likely it requires exponential time to tell whether a given Subset Sum problem has an exact solution.  However, if we only want to see if we can reach the target up to a fixed number of digits of accuracy, then the problem is quite easy, i.e., Subset Sum is hard, but very easy to approximate.