Towards Optimal Use of Multi-Precision Arithmetic: A Remark

If standard-precision computations do not lead to the desired accuracy, then it is reasonable to increase precision until we reach this accuracy. What is the optimal way of increasing precision? One possibility is to choose a constant q > 1, so that if the precision which requires the time t did not lead to a success, we select the next precision that requires time q · t. It was shown that among such strategies, the optimal (worst-case) overhead is attained when q = 2. In this paper, we show that this “time-doubling” strategy is optimal among all possible strategies, not only among the ones in which we always increase time by a constant q > 1. Formulation of the problem. In multi-precision arithmetic, it is possible to pick a precision and make all computations with this precision; see, e.g., [1, 2]. If we use validated computations, then after the corresponding computations, we learn the accuracy of the results. Usually, we want to compute the result of an algorithm with a given accuracy. We can start with a certain precision. If this precision leads to the desired results accuracy, we are done; if not, we repeat the computations with the increased precision, etc. The question is: what is the best approach to increasing precision? A natural approach to solving this problem. We usually have some idea of how the computation time t depends on the precision: e.g., for addition, the computation time grows as the number d of digits; for the standard multiplication, the time grows as d, etc. In view of this known dependence, we can easily transform the precision (number of digits) into time and vice versa. Therefore, the problem of selecting the precision d can be reformulated as the problem of selecting the corresponding computation time t. In other words, what we need to describe is as follows: if computations with time t are not sufficient, then we must select computations with larger precision which require a longer computation time t′(t) > t. The question is: given t, what t′(t) > t should we choose. Continuous approximation. In reality, we can only choose between integer number of digits 1, 2, 3, . . .. However, the need for high precision arises only when normal accuracy, with 32, 64, or 128 bits, is not sufficient. For the resulting huge number of bits, the difference between, say, 128 and 129 bit precision is not so large, so we can safely ignore the discreteness and assume that d (and hence t) can take all real values. ∗published in Reliable Computing, 12:365–369, 2006