This note discusses the main issues in performing correctly rounded decimal-to-binary and binary-to-decimal conversions. It reviews recent work by Clinger and by Steele and White on these conversions and describes some efficiency enhancements. Computational experience with several kinds of arithmetic suggests that the average computational cost for correct rounding can be small for typical conversions. Source for conversion routines that support this claim is available from netlib.

Converting decimal scientific notation into binary floating point is nontrivial, but this conversion can be performed with the best possible accuracy without sacrificing efficiency. 1.

For scientific computations on a digital computer the set of real numbers is usually approximated by a finite set F of “floating-point ” numbers. We compare the numerical accuracy possible with different choices of F having approximately the same range and requiring the same word length. In particular, we compare different choices of base (or radix) in the usual floating-point systems. The emphasis is on the choice of F, not on the details of the number representation or the arithmetic, but both rounded and truncated arithmetic are considered. Theoretical results are given, and some simulations of typical floating point-computations (forming sums, solving systems of linear equations, finding eigenvalues) are described. If the leading fraction bit of a normalized base 2 number is not stored explicitly (saving a bit), and the criterion is to minimise the mean square roundoff error, then base 2 is best. If unnormalized numbers are allowed, so the first bit must be stored explicitly, then base 4 (or sometimes base 8) is the best of the usual systems. Index Terms: Base, floating-point arithmetic, radix, representation error, rms error, rounding error, simulation.

1 2. INTRODUCTION 1 2.1. Portability and Purity 2 2.2. Goals of Borneo 3 2.3. Brief Description of an IEEE 754 Machine 3 2.4. Language Features for Floating Point Computation 6 3. FUTURE WORK 9 3.1. Incorporating Java 1.1 Features 9 3.2. Unicode Support 10 3.3. Flush to Zero 10 3.4. Variable Trapping Status 10 3.5. Parametric Polymorphism 10 4. CONCLUSION 10 5. ACKNOWLEDGMENTS 11 6. BORNEO LANGUAGE SPECIFICATION 13 6.1. indigenous 13 6.2. Floating Point Literals 16 6.3. Float, Double, and Indigenous classes 17 6.4. New Numeric Types 18 6.5. Floating Point System Properties 20 + This material is based upon work supported under a National Science Foundation Graduate Fellowship. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. ii 6.6. Fused mac 21 6.7. Rounding Modes 21 6.8. Floating Point Exception Handling 31 6.9. Operator Overloading 51 6.10...

1 Introduction A real number x is usually approximated in a digital computer by an element fl(x) of a finite set F of "floating-point " numbers. We regard the elements of F as exactly representable real numbers, and take fl(x) as the floating-point number closest to x. The definition of "closest", rules for breaking ties, and the possibility of truncating instead of rounding are discussed later. We restrict our attention to binary computers in which floating-point numbers are represented in a word (or multiple word) of fixed length w bits, using some convenient (possibly redundant) code. Usually F is a set of numbers of the form

† Formerly known as Teak. In the future will be known as Kalimantan.