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Abstract. This paper presents the complete design of a reversible arithmetic logic unit (ALU) that can be part of a programmable reversible computing device ...
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SOF'I'WARE-PRACTICE
AND EXPERIENCE, VOL. 8, 171-176 (1978)
Rational Arithmetic for Minicomputers BERTHOLD K. P. HORN
Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 545 Technology Square, Cambridge, Mass. 02139 U.S.A. SUMMARY A representation for numbers using two computer words is discussed, where the value represented is the ratio of the corresponding integers. This allows for better dynamic range and relative accuracy than single-precision fixed point, yet is less costly than floating point arithmetic. The scheme is easy to implement and particularly well suited for minicomputer applications that call for a great deal of numerical computation. The techniques described have been used to implement a mathematical function subroutine package for a minicomputer as well as a number of applications programs in the machine vision and machine manipulation area. KEY WORDS
Rational arithmetic Minicomputers Number representation
INTRODUCTION Typically, minicomputers come equipped with arithmetic instructions that operate on single words. Frequently double-word addition and subtraction is also provided for. Unfortunately, integers are inconvenient for many calculations and require the programmer (or at least the compiler) to remember scale factors relating the number represented to the integer actually stored. Even with scale factors one frequently runs out of dynamic range or has difficulty because of the need to estimate the approximate magnitude of variables ahead of time. Floating-point representation is the obvious, but costly answer. Software to perform the common floating-point operations requires substantial amounts of memory and is usually very slow, while hardware that carries out the same operations-if availableis expensive. RATIONAL A R I T H M E T I C An oft-overlooked alternative that lies between fixed-point and floating-point arithmetic in complexity, cost and capability is rational arithmetic (see Figure 1). Variables are stored in two words and the number represented is the ratio of the integers in these words. This allows a fair dynamic range and possibly higher precision than single-word integer arithmetic, without the complexity of extracting exponents and mantissas as required for floating-point arithmetic. I t is also easy to provide software that performs the usual arithmetic operations at relatively high speed. Since the number of instructions required is relatively small, one may elect to compile them in line instead of providing them as subroutine calls. Similarly, it should be easy to microprogram a machine to provide these operations directly as part of its instruction repertoire. Notice that we are not discussing exact rational arithmetic which requires the 0038-6644/78/02084171 SO1.OO @ 1978 by John Wiley & Sons, Ltd.
