This paper introduces the concept of precision-sensitive algorithms, in analogy to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach based on ideas in [CSY], and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of 0(2-r) in a time polynomial in r and 1/6, where 6 denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precision-sensitive algorithm then we can show that USAT (unambiguous SAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.