The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. Motivated by the fact that the variability from randomization can lead to inaccurate outputs, we propose a deterministic approach. First, we establish several computational complexity results for the exact bootstrap method, in the case of the sample mean. Second, we present the first efficient, deterministic approximation algorithm (FPTAS) for producing exact bootstrap confidence intervals which, unlike traditional methods, has guaranteed bounds on the approximation error. Third, we develop a simple exact algorithm for exact bootstrap confidence intervals based on polynomial multiplication. We provide empirical evidence involving several hundreds (and in some cases over one thousand) data points that the proposed deterministic algorithms can quickly produce confidence intervals that are substantially more accurate compared to those from randomized methods, and are thus practical alternatives in applications such as clinical trials.