An analysis of carry transmission in computer addition

Since all arithmetic operations reduce ultimately to addition, a basic consideration in designing a high speed computing automation is fast addition; and a basic difficulty in achieving fast addition is the carry problem, inasmuch as a single carry may propagate through an entire addition. Further, the expected number of carries from the addition of words of equivalent information is greatest in the binary number system, which is commonly used in the high speed scientific computer because of high storage efficiency and easy mechanization. Nor is the problem avoided altogether by going to a decimal radix: for a binary coded decimal or biquinary representation is usually used, and the binary problem still exists in altered form. It is the purpose of this paper to determine the effect of various methods of increasing the speed of binary addition by "carry sensing", designed to increase the speed with which carries are propagated. Any procedure which alters the expected length of a carry will be called a "carry transformation" T; by T:c i --~ cj, we mean a carry transformation T which transforms a carry of i places into a carry of j places. Assuming that we are dealing with words of n bits (binary digits), the longest possible carry is n places; then T:c n-~ c k means that the longest possible carry in the equivalent transformed system is k places. By determining the various possible carry transformations T, we can accomplish several objectives., Perhaps most important, any information we possess about the binary system (which is the easiest number system to deal with) is automatically transformed into equivalent information about the transformed system. Further, it may be desirable to use more than one kind of transformation on the same circuit, or to determine what succession of transformations is most efficient under a given set of rules. Finally, any number system used may be regarded simply as a transformed binary system so that information about the binary system only is necessary to solve carry problems in any radix. (The basic information required includes the longest expected carry as a function of the number of digits used, and the probabilities Pm that the longest carry does not exceed m places; these probabilities may be calculated recursively as a function of m and n, but the computation is difficult even in the