An Exact Arithmetic Package for Ml

We present the design and implementation of an exact rational arithmetic package for the programming language Caml, an ML dialect close to the original LCF-ML. Our arithmetic, written almost entirely in Caml, combines eeciency, power and exibility. For eeciency reasons, the package provides several diier-ent basic types to represent numbers, and to reinforce exibility, it implements many functions operating on these types. To master this complexity and to keep the package as simple as possible for the casual user, we design a super type of numbers which provides easy access to exact arithmetic. Nevertheless, the integration of the package into an existing ML compiler raises interesting problems: rational numbers introduce a new data-type which is not a free algebra. This implies modifying the pattern matching algorithm of ML. The proliferation of basic types forces us to nd a new way to input numbers; existing libraries are not designed for the new numerical types. We must adapt them to the numerical package to avoid a tremendous number of coercions or type clashes; nally, since basic operators operate on many diierent numerical types (the mathematical operation \+" has nine incompatible types in the package) we need a mechanism to reconcile them. We want to use the common mathematical notation for arithmetic expressions in order to maintain program readability. In the following, we brieey describe the Caml implementation of the package and some of the algorithms we use. In particular, suitable normalization of rational numbers is discussed. We then present the super type of numbers and state some of its properties. We present problems and propose solutions to integrate the package into an existing ML compiler. Then we give some interesting examples of numeric computation using the package, and nally we give some benchmarks comparing our arithmetic to other, well-known packages. 1 In defense of a sound arithmetic First of all, we would like to convince the reader that exact arithmetic is a desirable feature for a modern programming language. Hardware limitations to the range of representable integers are well known and, although irritating, overrows and underrows of operations are not too diicult to detect, so that programs may explicitly fail in these cases and never return mathematically unsound results. Evidently, this is not the case for oating point arithmetic: partial results are always rounded oo, and these successive roundoo errors may lead to completely erroneous answers. Yet, when computing with real numbers, …