Infinity arithmetic, comparisons and J

ABSTRACT

The principles of infinity arithmetic are explained. There is a need to bring J arithmetic and comparisons into line with these principles. Many of the results produced by J are mathematically anomalous and do not comply with the IEEE 754 Standard for floating-point arithmetic.Some proposals to generalise the concept of comparison tolerance are presented. There are also some tentative proposals regarding safety levels and precision.The paper is oriented towards J but many of the concepts are relevant to all dialects of APL.

REFERENCES

• Ber77: Robert Bernecky. Comparison Tolerance. SHARP APL Technical Notes 23, 1977.
• Eis80: Murray Eisenberg. Zero Divided by Zero: Zero or One or Neither. APL Quote Quad, 11(1):9-10, 1980.
• Fat92: Richard J. Fateman. A Review of Mathematica. j. Symbolic Compnt., 13(5):545-579, 1992.
• Gol91: David Goldberg, What every computer scientist should know about floating-point arithmetic, ACM Computing Surveys (CSUR), v.23 n.1, p.5-48, March 1991 [doi>10.1145/103162.103163]
• Hui92: Roger K.W. Hui. An Implementation of J. Iverson Software inc., Toronto, 1992.
• Hui94a: Roger K.W. Hui. Re: Indeterminate and Infinity. comp.lang.apl, 21 Nov, 1994.
• Hui94b: Roger K.W. Hui. Re: Testing for indeterminate in J. comp.lang.apl, 9 Dec, 1994.
• IEE85: 1EEE. IEEE Standard for Binary Floating- Point Arithmetic (ANSI/IEEE S~d 754-1985). IEEE, New York, 1985. Reprinted in SIGPLAN NOTICES vol. 22, No. 2, Feb 1987, pp9-25.
• Ive94: Kenneth E. Iverson. J Inlroduction and Diclionary. Iverson Software Inc., Toronto, 1994.
• McD76: E. E. McDonnell, Zero divided by zero, Proceedings of the eighth international conference on APL, p.295-296, September 22-24, 1976, Ottawa, Canada [doi>10.1145/800114.803689]
• MS80: E.E. McDonnell and J. O. Shallit. Extending APL to Infinity. APL8O Conference Proceedings, APL Quote Quad, 10, 1980.
• Ree87: Terence J. Reed, Functional programming, the IEEE floating point standard and APL, Proceedings of the international conference on APL: APL in transition, p.341-347, May 10-14, 1987, Dallas, Texas, United States
• Rei94: Clifford A. Reiter. Re: Testing for Indeterminate in J. comp.lang.apl, 8 Dec, 1994.

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: J

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

General Terms:
Algorithms, Languages, Standardization

Keywords:
IEEE 754 Standard, NaN, arithmetic, comparative, comparison, error-trapping, indeterminate, infinity, infinity arithmetic, limits, precision, tolerance