Peter Kornerup

Mapping integers and hensel codes onto Farey fractions

The order-N Farey fractions, whereN is the largest integer satisfyingN≦√((p−1)/2), can be mapped onto a proper subset of the integers {0, 1,...,p−1} in a one-to-one and onto fashion. However, no completely satisfactory algorithm for affecting the inverse mapping (the mapping of the integers back onto the order-N Farey fractions) appears in the literature.A new algorithm for the inverse mapping problem is described which is based on the Euclidean Algorithm. This algorithm solves the inverse mapping problem for both integers and the Hensel codes of Krishnamurthy et al.

Finite precision lexicographic continued fraction number systems

Lexicographic continued fraction binary (LCF) representation provides an order preserving bitstring representation of the non negative real numbers where every rational number has a finite length bitstring representation. We investigate the precision of k-bit LCF approximation. The maximum gap size over [0,1] for (k+1)-bit LCF representation is shown to be less than 2^−.81k, comparable to binary coded decimal in worst case representation efficiency. The distribution of gap sizes for (k+1)-bit LCF representation over [0,1] is shown on a logarithmic scale to be bell shaped between 2^−.81k and 2^−1.39k, becoming more peaked near the value corresponding to uniform spacing, 2^−k, with increasing k.

Foundations of Finite Precision Rational Arithmetic

Finite precision fraction number systems are characterized and their number theoretic foundations are developed. Closed approximate rational arithmetic in these systems is obtained by the natural canonical rounding obtained using the continued fraction theory concept of best rational approximation. These systems are shown to be natural finite precision number systems in that they are essentially independent of the apparatus of the representation. The specific fixed-slash and floating-slash fraction number systems are described and their feasibility and convenience for Computer implementation are discussed. The foundations of adaptive variable precision are explored. The overall goal is to better understand the inherent mathematical properties of finite precision arithmetic and to provide a most natural and convenient computation system for approximating real arithmetic on a Computer.

An order preserving finite binary encoding of the rationals

We describe a new binary encoding for numbers termed lexicographic continued fraction (LCF) representation that provides a one-to-one order preserving finite bit string representation for every rational. Conversion either way between binary integer numerator-denominator pair representation and LCF representation is shown feasible in time linear with bit string length, given registers of length sufficient to hold the numerator and denominator. LCF bit string length is about 2 max{log2p, log2q} for the irreducible fraction p/q. Realization of arithmetic (+, −, ×, ÷ on LCF bit string encoded operands is shown feasible. Some relations between the theory of best rational approximation and the values represented by truncated LCF bit strings are noted to assess the feasibility of a finite precision arithmetic based on LCF representation.

Interpretation and code generation based on intermediate languages

The possibility of supporting high level languages through intermediate languages to be used for direct interpretation and as intermediate forms in compilers is investigated. An accomplished project in the construction of an interpreter and a code generator using one common intermediate form is evaluated. The subject is analysed in general, and a proposal for an improved design scheme is given.

Approximate rational arithmetic systems: Analysis of recovery of simple fractions during expression evaluation

Closed approximate rational arithmetic systems are described and their number theoretic foundations are surveyed. The arithmetic is shown to implicitly contain an adaptive single-to-double precision natural rounding behavior that acts to recover true simple fractional results. The probability of such recovery is investigated and shown to be quite favorable.

A feasibility analysis of fixed-slash rational arithmetic

An investigation of the feasibility of a finite precision approximate rational arithmetic based on fixed-slash representation of rational numbers is presented. Worst-case and average-case complexity analyses of the involved rounding algorithm (an extended shift-subtract gcd algorithm) are presented. The results are applied to a proposed hardware realization of a fixed-slash arithmetic unit.

A bit-serial arithmetic unit for rational arithmetic

We describe a binary implementation of an algorithm of Gosper to compute the sum, difference, product, quotient and certain rational functions of two rational operands applicable to integrated approximate and exact rational computation. The arithmetic unit we propose is an eight register computation cell with bit serial input and output employing the binary lexicographic continued fraction (LCF) representation of the rational operands. The operands and results are processed in a most-significant-bit first on-line fashion with bit level logic leading to less delay in the computation cell when compared to operation on the full partial quotients of the standard continued fraction representation. Minimization of delay is investigated with the aim of supporting greater throughput in cascaded parallel computation with such computation cells.

An integrated rational arithmetic unit

Based on the classical Euclidian Algorithm, we develop the foundations of an arithmetic unit performing Add, Subtract, Multiply and Divide on rational operands. The unit uses one unified algorithm for all operations, including rounding. A binary implementation, based on techniques known from the SRT division, is described. Finally, a hardware implementation using ripple-free, carry-save addition is analyzed, and adapted to a floating-slash representation of the rational operands.

A note on rational arithmetic

A recent paper in SIGMICRO [1] contained a comparison of the accuracy of floating point vs. rational representations, which is very unfair to the latter. The format chosen for rational numbers utilizes 16 bits for numerators and 16 bits for denominators. This implies that the spacing between consecutive numbers in the system is in most cases of the order 2-32. Only around simple rational numbers (e.g. 1/1, 2/3) is the spacing of the order 2-16. However the rounding algorithm presented in [1] will almost certainly introduce a rounding error of the order 2-16, i.e. introduce an error which in most cases is of the order 216 larger than necessary.