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 develop a formal account of digit serial number representations by describing them as strings from a language. A prefix of a string represents an interval approximating a number by enclosure. Standard on-line representations are shown to be a special case of the general digit serial representations. Matrices are introduced as representations of intervals and a finite-state transducer is used for mapping strings into intervals. Homographic and bi-homographic functions are used for representing basic arithmetic operations on digit serial numbers, and finally a digit serial representation of floating point numbers is introduced. Key Words: Computer Arithmetic, On-line Computation, Number Representations, Redundant Digit sets, Continued Fractions, Intervals. Category: B.2 1 Introduction A number is usually represented as a string of digits belonging to some digit set \Sigma . The number representation specifies a function that maps the string to its value. In the context of this pa...

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In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.

Computers manipulate approximations of real numbers, called floating-point numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floating-point operations lose so much precision that they quickly become irrelevant. In this article, we review some of the problems one can encounter, focussing on the IEEE754-1985 norm. We give a (sketch of a) semantics of its basic operations then abstract them (in the sense of abstract interpretation) to extract information about the possible loss of precision. The expected application is abstract debugging of software ranging from simple on-board systems (which use more and more on-the-shelf micro-processors with floating-point units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1

Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency", \mediant " and \convergent " from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representation-induced precision loss for any real number by a xed length representable number of the system is shown to be at most 19 % of bit word length, with no precision loss whatsoever induced in the representation of any reasonably sized rational number. The representation is supported by a computer arithmetic system implementing exact rational and approximate real computations in an on-line fashion.

this paper. By introducing an extra termination symbol, which signals that an operand was merely terminated due to its length exceeding some bound, operands can be kept as intervals, representing an imprecise operand. Operands terminated in the ordinary way can be taken to represent exact numbers. The cube modeling a function of two variables, can be generalized to a hypercube modeling a poly-homographic function of n variables. For n = 3 the function is defined as:

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