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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 representat ..."
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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. Key Words: Computer Arithmetic, Online 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...
Lazy Arithmetic
 IEEE TRANSACTIONS ON COMPUTERS
, 1994
"... Finiteprecision leads to many problems in geometric methods from CAD or Computational Geometry. Until now, using exact rational arithmetic was a simple, yet much too slow solution to be of any practical use in realscale applications. A recent optimization  the lazy rational arithmetic ([4])  s ..."
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Cited by 9 (5 self)
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Finiteprecision leads to many problems in geometric methods from CAD or Computational Geometry. Until now, using exact rational arithmetic was a simple, yet much too slow solution to be of any practical use in realscale applications. A recent optimization  the lazy rational arithmetic ([4])  seems promising: It defers exact computations until they become either unnecessary (in most cases) or unavoidable; in such a context, only indispensable computations are performed exactly, that is: Those without which any given decision cannot be reached safely using only floatingpoint arithmetic. This paper takes stock of the lazy arithmetic paradigm: Principles, functionalities and limits, speed, possible variants and extensions, difficulties, problems solved or left unresolved.
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 provide a found ..."
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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", \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 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.
R.: Arithmetic Unit Based on Continued Fraction, to appear in proceedings of ECI’2006 conference
, 2006
"... We introduce architecture of an arithmetic unit that is based on continued fractions and allows computing any linear rational function of two variables, including basic arithmetic operations like addition, subtraction, multiplication and division. Such a unit can easily exploit the parallel nature o ..."
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Cited by 2 (1 self)
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We introduce architecture of an arithmetic unit that is based on continued fractions and allows computing any linear rational function of two variables, including basic arithmetic operations like addition, subtraction, multiplication and division. Such a unit can easily exploit the parallel nature of continued fraction arithmetic and accelerate the otherwise low performance of its software implementation. The proposed architecture uses continued logarithms, which are an adapted variant of continued fractions and which suit hardware implementation more naturally through their bitlevel granularity. We have used FPGA to implement such a unit and we present here some experimental results, which demonstrate promising properties of the proposed architecture.
Number systems and Digit Serial Arithmetic
, 1997
"... 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. T ..."
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Cited by 1 (1 self)
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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 polyhomographic function of n variables. For n = 3 the function is defined as:
Precision of SemiExact Redundant Continued Fraction Arithmetic for VLSI
 VLSI, SPIE ADVANCED SIGNAL PROCESSING ALGORITHMS, ARCHITECTURES, AND IMPLEMENTATIONS IX
, 1999
"... Continued fractions (CFs) enable straightforward representation of elementary functions and rational approximations. We improve the positional algebraic algorithm, which computes homographic functions such as y = ax+b cx+d , given redundant continued fractions x; y, and integers a; b; c; d. The im ..."
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Cited by 1 (1 self)
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Continued fractions (CFs) enable straightforward representation of elementary functions and rational approximations. We improve the positional algebraic algorithm, which computes homographic functions such as y = ax+b cx+d , given redundant continued fractions x; y, and integers a; b; c; d. The improved algorithm for the linear fractional transformation produces exact results, given regular continued fraction input. In case the input is in redundant continued fraction form, our improved linear algorithm increases the percentage of exact results with 12bit state registers from 78% to 98%. The maximal error of nonexact results is improved from 1 to 2 8 . Indeed, by detecting a small number of cases, we can add a nal correction step to improve the guaranteed accuracy of nonexact results. We refer