Unbounded Spigot Algorithms for the Digits of Pi

Rabinowitz and Wagon call their algorithm a spigot algorithm, because it yields digits incrementally and does not reuse digits after they have been computed. The digits drip out one by one, as if from a leaky tap. In contrast, most algorithms for computing the digits of π execute inscrutably, delivering no output until the whole computation is completed. However, the Rabinowitz–Wagon algorithm has its weaknesses. In particular, the computation is inherently bounded: one has to commit in advance to computing a certain number of digits. Based on this commitment, the computation proceeds on an appropriate finite prefix of the infinite series (1). In fact, it is essentially impossible to determine in advance how big that finite prefix should be for a given number of digits—specifically, a computation that terminates with nines for the last few digits of the output is inconclusive, because there may be a “carry” from the first few truncated terms. Rabinowitz and Wagon suggest that “in practice, one might ask for, say, six extra digits, reducing the odds of this problem to one in a million” [8, p. 197], a not entirely satisfactory recommendation. Indeed, the implementation printed at the end of their paper is not quite right [1, p. 82], sometimes printing an incorrect last digit because the finite approximation of the infinite series is one term too short. We propose a different algorithm, based on the same series (1) for π but avoiding these problems. We also show the same technique applied to other characterizations of π. No commitment need be made in advance to the number of digits to be computed; given enough memory, the programs will generate digits ad infinitum. Once more (necessarily, in fact, given the previous property), the programs are spigot algorithms in Rabinowitz and Wagon’s sense: they yield digits incrementally and do not reuse them after producing them. Of course, no algorithm using a bounded amount of memory can generate a nonrepeating sequence such as the digits of π indefinitely, so we have to allow arbitrary-precision arithmetic, or some other manifestation of dynamic memory allocation. Like Rabinowitz and Wagon’s algorithm, our proposals are not competitive with state-of-the-art arithmetic-geometric mean algorithms for computing π