In this paper, we incorporate a representation of the non-negative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.

We present the design of the Boost interval arithmetic library, a C++ library designed to efficiently handle mathematical intervals in a generic way. Interval computations are an essential tool for reliable computing. Increasingly a number of mathematical proofs have relied on global optimization problems solved using branch-andbound algorithms with interval computations; it is therefore extremely important to have a mathematically correct implementation of interval arithmetic. Various implementations exist with diverse semantics. Our design is unique in that it uses policies to specify three independent variable behaviors: rounding, checking, comparisons. As a result, with the proper policies, our interval library is able to emulate almost any of the specialized libraries available for interval arithmetic, without any loss of performance nor sacrificing the ease of use. This library is openly available at www.boost.org.

Abstract not found

There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make "howlers" or not to simplify enough. In this paper we outline the "unwinding number" approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have not yet reduced the simplifiation process to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.

Abstract not found

Abstract not found

Abstract not found

Abstract: Interval arithmetic is a basic tool for certified mathematical computations, it is presented in many references. We describe here the formal proposal to include interval arithmetic in the C++ standard library. Key-words: C++, standardization, library, interval arithmetic, certification