Refinement of rational end-points real numbers by means of floating-point numbers

This paper addresses the topic of the refinement of exact real numbers. It presents a three-steps formal development towards the implementation of exact real numbers. It considers real numbers as intervals whose end-points are rational numbers. We investigate the possibility to represent these intervals by floating-point numbers as end-points in order to increase the efficiency of the implementation and to use the hardware resources. We show on an extension of the PCF language that this result can be carried out but by losing the adequacy property as defined in (Escardo, 1996). However, we show that it is possible to introduce a weak version of the adequacy property described by a Galois connection defining an abstract interpretation. Soundness and completeness properties are proved in this context. Accuracy analysis by a program analysis of the representation allows to choose between different representations of real numbers.