Function Evaluation in Unnormalized Arithmetic

ABSTRACT

The evaluation of a function of one argument is a standard computational task. When an unnormalized number representation is used, it is appropriate that function evaluation to subject to certain “adjustment” criteria, defined independently of the computing method. In this paper some such criteria are developed, and their application described. In particular, consideration is given to questions of the extent to which general principles apply (i.e. for large classes of functions of a single argument) and the manner in which particular properties of the functions involved (e.g. the functions evaluated by standard library subroutines) may be invoked. The analysis, although framed in terms of unnormalized arithmetic, gives insight into the general nature of significance propagation through function evaluation, and provides a means for analyzing calculations carried out in the usual normalized arithmetic, particularly in the case where an “index of significance” is employed.

REFERENCES

• 1: R. L. Ashenhurst , N. Metropolis, Unnormalized Floating Point Arithmetic, Journal of the ACM (JACM), v.6 n.3, p.415-428, July 1959 [doi>10.1145/320986.320996]
• 2: ASHENHURST, R.L. The MANIAC III arithmetic system. Proc. Joint Comput. Conf., Zl (1962), 195-202.
• 3: Max Goldstein, Significance arithmetic on a digital computer, Communications of the ACM, v.6 n.3, p.111-117, March 1963 [doi>10.1145/366274.366339]
• 4: GRAY, HERBERT L., AND HARRISON, CHARLES, JR. Normalized floating-point arithtactic with an index of significance. Proc. Joint Comput. Conf., 16 (1959), 244--8.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Computer arithmetic

Additional Classification:
  F. Theory of Computation
  F.3 LOGICS AND MEANINGS OF PROGRAMS
      F.3.3 Studies of Program Constructs
          Subjects: Functional constructs

General Terms:
Measurement, Performance, Reliability, Theory