Response Within the context of our study; that i s , the d i r e c t s u b s t i t u t i o n of a pure rational scheme for a f l o a t i n g point scheme of arithmetic and storage within an extant FORTRAN system with no changes to the compiler (and thus to the storage space available for a--single numerical q u a n t i t y ) , we continue to claim that our results are valid. We again wish to note that our investigation was l i m i t e d to the pure rational scheme using the same storage space available to a f l o a t i n g point scheme. The point made about spacing between consecutive numbers being on the order of 2-32 in most cases is correct only under the assumption that the range of representable numbers is of the order [ 0 , I ] . However, that is not the case with our study. Granted an a l t e r n a t i v e worthy of investigation would be a system where the integer and f r a c t i o n a l parts are separated. We would also l i k e to point out that an e r r o r of 2-16 is the worst case error (assuming an interval of [ 0 , I ] ) with our implementation. Using the process of mediant rounding as presented in your reference [2] shows that "accuracy e s s e n t i a l l y varies from N to 2N b i t s . . . . " Therefore, i f applied to our implementation, the worst case error would be 2- . We did not go as far as [2] however in analyzing what the average error would be. In our implementation, the errors appear to be much larger for the actual test cases. This is due to the representations not f a l l i n g in the interval [ 0 , I ] . From t h i s one can see that a pure rational system would not be feasible. In f u r t h e r defence of our decision to truncate one-bit of both the numerator and the denominator with a subsequent r e - t r y of the greatest common d i v i s o r algorithm in those cases where the pure rational representation could not be stored within the space available (as defined by the f l o a t ing point data type of the extant FORTRAN d i a l e c t ) results in the preservation and use of more information in deriving the f i n a l approximate pure rational quantity than would immediate rounding except in those cases where t h i s truncation/GCD search is unsuccessfull over the f u l l number of i t e r a t i v e attempts necessary to achieve storage in the space available. I t should be noted that one b i t binary truncation is equivalent to rounding in those cases where rounding is "down"; that i s , in one-half of the cases. Please note that we are f u l l y convinced of the u t i l i t y of t h i s , and of other, rational schemes when the basic storage unit can e i t h e r : ( I ) be dynamically adjusted in size to accomodate the information necessary for exact value representation; or (2) consist of an integral portion and a r a t i o n a l l y represented f r a c t i o n a l portion. The scheme being investigated by Kornerup and Matula has merit. We hope t h e i r work is continued. W. I. Thacker G. W. Gorsline

/lp/association-for-computing-machinery/response-ub7bLR071v