Features of a hardware implementation of an optimal arithmetic

We give a brief review of the definition of the arithmetic operations on a computer by semimorphisms. Then we display the 15 fundamental operations that are most useful and convenient for an implementation of semimorphic operations in computer representable subsets of the most commonly used linear spaces and their interval sets. Techniques for the implementation of twelve of these operations: addition, subtraction, multiplication and division with three roundings are well known and are common knowledge nowadays. The paper focuses, therefore, on the implementation of scalar products with maximum accuracy and diverse roundings. We sketch several possibilities for a hardware realization of optimal scalar products. We give an algorithmic and flow chart description of one such hardware unit and discuss the natural parallelisms in scalar products, Finally, we comment on the pipelining of the 15 fundamental arithmetic operations.