FRPOLY: A Benchmark Revisited

The FRPOLY Lisp performance benchmark [3] was based on a circa1968 piece of code for computing powers of polynomials. We address two questions: (a) What algorithm would you use if you really wanted to compute powers of polynomials fast? and (b) Given that Common Lisp supports many types of data structures other than the simple lists used for the benchmark, what more efficient representations might be appropriate to use for polynomials? 1 The origin of the original benchmark The FRPOLY Lisp benchmark was devised by the author to test various Lisp systems on computations typical for a symbolic algebra system – at least one that is written in Lisp. The name for the benchmark combined the dialect in which it was originally submitted (Franz Lisp), and the nature of the computation – polynomial arithmetic. The code originated in a polynomial arithmetic package written in about 1968 by MIT Prof. William A. Martin for use in the Macsyma computer algebra system. Although major and minor variations on representations and algorithms have appeared in the programming literature, Martin’s design still seems to be fairly sturdy. A good deal of the algorithmic portion of Macsyma depends directly or indirectly on this code. Martin tuned his code in various ways to minimize the number of conscells required for temporary storage. When the code was written for the 1.2 megabyte (256k word) address-space PDP-6, the MIT Macsyma group initially viewed the program space as huge – which it was, compared to the previous limits encountered on a 32k word IBM 7090. Therefore code size was considered a secondary consideration compared to speed. 1More information on this general area is readily available [1]. 2A trademark of Symbolics Inc. 3Some discussion of the design criteria may be found in the literature on symbolic mathematical computation (e.g. [1]). 4In fact, the Macsyma code exceeded that address space within 3 years, leading to schemes utilizing load-on-command files.