Abstract: In a floating point arithmetic execution unit, an additional adder unit and a selection network are added to the apparatus typically performing the arithmetic floating point function. The additional apparatus permits certain processes forming part of arithmetic operations to be executed in parallel. For selected arithmetic operations, the final result can be one of two values typically related by an intermediate shifting operation. By performing the processes in parallel and selecting the appropriate result, the execution time can be reduced when compared to the execution of the process in a serial implementation. The fundamental arithmetic operations of addition, subtraction, multiplication and division can each have the execution time decreased using the disclosed additional apparatus.

Abstract: A multifunction arithmetic logic circuit having comparison and numeric conversion circuitry, particularly adapted for use in graphics processing. The inventive architecture comprises a modular arithmetic logic unit in a pipelined architecture circuit. Functions performed are conversion of floating point numbers to fixed point numbers, and vise versa, arithmetic and logical operations, and numeric comparison operations. A visibility logic subcircuit is included for rapidly tracking numeric comparisons to indicate whether a graphics object is to be considered visible, partially visible, or invisible.

Abstract: : Differentiation arithmetic is an ordered-pair arithmetic which evaluates both the value and derivative of functions defined by formulas or subroutines, without symbolics or approximations. As in the case of complex arithmetic, multiplication and division are defined in terms of several real operations. Algorithms are given for evaluation of these operations with the same accuracy as real multiplication and division, that is, to the closest floating-point number. The same kind of optimal implementation is described for Taylor arithmetic, which permits calculation of Taylor coefficients of arbitrary order for functions defined by formulas or subroutines. Keywords: Automatic differentiation; Optimal computer arithmetic. (Author)

Abstract: This paper presents a design procedure to generate high-speed and easily testable implementations of functions realizable by iterative logic arrays. It is derived from the tree-oriented implementation technique for first-order recurrence relations developed by Abrabam and Gajski. A small number of control signals are introduced to facilitate test generation. The resulting circuit can be tested for arbitrary faults in individual cells of the tree structure, with a test set which grows as log2n, where n is the input operand size. Moreover, teats need to be generated for cells in only the first level of the tree. The remain ing tests are derived from these using simple vector operations defined by us in earlier work. This makes the test generation process virtually independent of the operand size. The design procedure is illustrated for functions realized by oneaad twdimensional iterative logic arrays, with an adder and a multiplier serving as representative examples. Comparisons are made with previous high-speed realizations of adders aad multipliers to highlight the advantages and disadvantages of our design method.

Abstract: Recently, a number of papers has been published on the subject of performing complex arithmetic in the residue number system. Methods have been proposed which reduce the arithmetic complexity of a complex multiply by more than 50 percent. In this correspondence it is shown that these methods achieve Winograd's lower bound, and how these efficient mappings can be derived in terms of polynomial rings.

Abstract: The addition and multiplication algorithms of two Hensel codes were presented in two earlier papers [1], [2] on p-adic arithmetic. It is shown here that these algorithms do not always generate a correct result having the same code word length as the two operands and the correct algorithms are given.

Abstract: The DMAC (Digital Multiplication by Analog Convolution) algorithm has been shown to be one technique for performing optical matrix-multiplication with improved precision. Past work in this area has addressed fixed-point arithmetic only. Presented in this paper is an extension of the DMAC algorithm for handling floating-point binary numbers as well. However, the technique employed for handling floating-point numbers is based on fixed-point concepts. For this reason we choose to call the arithmetic as being flixed-point, since it is a hybrid combination of both floating and fixed-point arithmetic. In this paper we also describe an acousto-optical time-integrating architecture using binary flixed-point arithmetic to perform matrix-vector multiplication. By employing an array of full-adders in conjunction with the photodetector array at the back-end of this architecture, it is possible to avoid generating mixed binary outputs that normally result through the use of the DMAC algorithm. Hence, we eliminate the need for analog-to-digital converters needed to convert mixed binary to pure binary. Preliminary experimental results are also presented.

Abstract: Examples of extended arithmetics include range, polynomial, dimensional, extended precision and fault tolerant arithmetic, and arithmetic on distributions of random variables. APL2's defined operators and general arrays enable the programmer to easily define such extensions. In this paper we will discuss some extended arithmetics and a prototype of arithmetic on statistical distributions that we have implemented in APL2 [1]. In this prototype, distributions take the place of numbers and the ordinary arithmetic scalar functions are replaced by derived functions that take distributions as arguments and return distributions as results. For ordinary addition, the natural derived function is convolution. Design issues and application examples are discussed.

Abstract: Many essential software functions in the mission critical computer resource application domain depend on floating point arithmetic. Numerically intensive functions associated with the Space Station project, such as emphemeris generation or the implementation of Kalman filters, are likely to employ the floating point facilities of Ada. Paranoia.Ada appears to be a valuabe program to insure that Ada environments and their underlying hardware exhibit the precision and correctness required to satisfy mission computational requirements. As a diagnostic tool, Paranoia.Ada reveals many essential characteristics of an Ada floating point implementation. Equipped with such knowledge, programmers need not tremble before the complex task of floating point computation.

Abstract: The effect of using fixed-point arithmetic in the digital implementation of two-level control algorithms is examined in this paper. Analytical expressions are developed to predict the change in the expected minimum cost and associated matrices. It is shown that there is a favourable match between the analytical predictions and averaged simulation experiments. The use of finite-precision machines increases the expected theoretical minimum cost and makes the two-level algorithm become slow and thus require excessive iterations to converge.

Abstract: Modular arithmetic provides a good introduction to genuinely mathematical ideas in arithmetic and to the basic concepts of abstract algebra. In this note we offer an algorithm for multiplying in modular arithmetic. To be precise, let n and I be positive integers with I < n and I prime to n. Then our algorithm will allow us to multiply by I modulo n without actually carrying out the multiplications in ordinary whole number arithmetic. We will also discuss a refinement of the algorithm which enables us to perform multiplications modulo n by all such numbers I in a simple natural sequence.

Abstract: Distributed arithmetic filters have been shown to be an effective method of implementing linear shift-invariant filters. In this paper, the overall roundoff error budgets of admissible distributed arithmetic filter structures are compared to conventional lumped parameter and to each other. In addition, dynamic range considerations are also introduced into the study.

Abstract: Improving the precision of optically performed computations is a critical aspect of photonic computing. One possible method for improving precision is through the use of modified signed-digit (MSD) arithmetic. Optical implementation of MSD arithmetic offers several important advantages over other optical techniques such as the digital multiplication by analog convolution (DMAC) algorithm or the use of residue arithmetic. These advantages include the parallel pipeline flow of digits due to carry-free addition and subtraction, fixed-point as well as floating-point capability, and the potential for performing divisions. We present a brief description of the modified signed-digit number system and suggest one optical architecture for implementing MSD fixed-point addition, subtraction, and multiplication.

Abstract: (1986). The Arithmetic of Differentiation. Mathematics Magazine: Vol. 59, No. 5, pp. 275-282.