Computing with Continued Logarithms

TL;DR

This paper introduces algorithms for arithmetic and transcendental functions on continued logarithms (CLs), offering a hardware-friendly, exact computation method that is simpler than continued fraction algorithms.

Contribution

The paper presents the first algorithms for computing trigonometric, exponential, and logarithmic functions directly on CLs, extending Gosper's arithmetic algorithms to this new representation.

Findings

Algorithms operate entirely in CL domain without floating-point arithmetic.

CL algorithms are simpler than CF counterparts.

Implemented in Haskell for practical use.

Abstract

Gosper developed an algorithm for performing arithmetic on continued fractions (CFs), and introduced continued logarithms (CLs) as a variant of continued fractions better suited to representing extremely large (or small) numbers. CLs are also well-suited to efficient hardware implementation. Here we present the algorithm for arithmetic on CLs, then extend it to the novel contribution of this paper, an algorithm for computing trigonometric, exponential, and log functions on CLs. These methods can be extended to other transcendental functions. As with the corresponding CF algorithms, computations are entirely in the domain of the CL representation, with no floating-point arithmetic; we read one CL input term at a time, producing the next CL term of the result as soon as it is determined. The CL algorithms are in fact simpler than their CF counterparts. We have implemented these algorithms in Haskell.