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Semantics of Exact Real Arithmetic
, 1997
"... In this paper, we incorporate a representation of the nonnegative 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 exten ..."
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Cited by 29 (8 self)
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In this paper, we incorporate a representation of the nonnegative 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.
MSBFirst Digit Serial Arithmetic
, 1995
"... 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 online representations are shown to be a special case of the general digit serial representati ..."
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Cited by 18 (1 self)
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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 online representations are shown to be a special case of the general digit serial representations. Matrices are introduced as representations of intervals and a finitestate transducer is used for mapping strings into intervals. Homographic and bihomographic functions are used for representing basic arithmetic operations on digit serial numbers, and finally a digit serial representation of floating point numbers is introduced.
Static Analyses of FloatingPoint Operations
 In SAS’01, volume 2126 of LNCS
, 2001
"... Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint operations lose so much precision that they quickly become irrelevant. In thi ..."
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Cited by 8 (0 self)
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Computers manipulate approximations of real numbers, called floatingpoint numbers. The calculations they make are accurate enough for most applications. Unfortunately, in some (catastrophic) situations, the floatingpoint 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 IEEE7541985 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 onboard systems (which use more and more ontheshelf microprocessors with floatingpoint units) to scientific codes. The abstract analysis is demonstrated on simple examples and compared with related work. 1
Exact Arithmetic on the SternBrocot Tree
 NIJMEEGS INSTITUUT VOOR INFORMATICA EN INFORMATEIKUNDE, 2003. HTTP://WWW.CS.RU.NL/RESEARCH/REPORTS/FULL/NIIIR0325.PDF
, 2003
"... 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 ..."
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Cited by 8 (2 self)
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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.
LCF: A lexicographic binary representation of the rationals
 J. Universal Comput. Sci
, 1995
"... 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 pro ..."
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Cited by 7 (0 self)
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Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency&quot;, \mediant &quot; and \convergent &quot; from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representationinduced 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 online fashion.
On Progress of Investigations in Continued Logarithm Arithmetic ∗
"... Abstract. This is a workinprogress on research of exact real arithmetic using continued fraction paradigm. We introduce a new redundant extension of continued logarithm representation, which offers the lacking real number computability. In a uniform way, we contrast this new representation with ex ..."
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Abstract. This is a workinprogress on research of exact real arithmetic using continued fraction paradigm. We introduce a new redundant extension of continued logarithm representation, which offers the lacking real number computability. In a uniform way, we contrast this new representation with existing alternatives and using few examples we discuss their practical aspects. It follows from this comparison that continued logarithms have a severe competitor in a redundant admissible, continued fraction representation, but they are still worth to continue in their research.
and
, 1996
"... This paper describes and discusses the use of a massively parallel SIMD (single instruction, multiple data) computer system as a computer arithmetic laboratory. ..."
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This paper describes and discusses the use of a massively parallel SIMD (single instruction, multiple data) computer system as a computer arithmetic laboratory.