Fisher's theory of maximum likelihood estimation routinely provides approximate confidence intervals for a parameter of interest θ, the standard intervals θ ^ ± zασ ^ where θ ^ is the maximum likelihood estimator, ô is an estimate of standard error based on differentiation of the log likelihood function, and zα is a normal percentile point. Recent work has produced systems of better approximate confidence intervals, which look more like exact intervals when exact intervals exist, and in general have coverage probabilities an order of magnitude more accurate than the standard intervals. This paper develops an efficient and dependable algorithm for calculating highly accurate approximate intervals on a routine basis, for parameters θ defined in the framework of a multiparameter exponential family. The better intervals require only a few times as much computational effort as the standard intervals. A variety of numerical and theoretical arguments are used to show that the algorithm works well, and that the improvement over the standard intervals can be striking in realistic situations.