Girish Mahajan (Editor)

Unum (number format)

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The unum (universal number) format is a format similar to floating point, proposed by John Gustafson as an alternative to the now ubiquitous IEEE 754 format. The proposal and justification are explained in his book The End of Error.

The two defining features of the unum 1.0 format (while unum 2.0 is different) are:

  • a variable-width storage format for both the significand and exponent, and
  • a u-bit, which determines whether the unum corresponds to an exact number (u=0), or an interval between consecutive exact unums (u=1). In this way, the unums cover the entire extended real number line [−∞,+∞].

    • For performing computation with the format, Gustafson proposes using interval arithmetic with a pair of unums, what he calls a ubound, providing the guarantee that the resulting interval contains the exact solution.

    Unum implementations have been explored in Julia. including unum 2.0 (or at least a modified version of his new proposal). Recently, unum has been explored in MATLAB.

    William Kahan and John Gustafson discussed unums at the Arith23 conference on July 12, 2016.

    Critique

    William Kahan, the principal architect of IEEE 754-1985 criticizes unums on the following grounds:

  • Unum computation does not always deliver correct results.
  • The description of unums sidesteps using unum for solving calculus problems.
  • Unums can be expensive in terms of time and power consumption.
  • Each computation in unum space is likely to change the bit length of the structure. This requires either unpacking them into a fixed-size space, or data allocation, deallocation, and garbage collection during unum operations, similar to the issues for dealing with variable-length records in mass storage.
  • Unums provide only two kinds of numerical exception, quiet and signaling NaN (Not-a-Number).
  • Unum computation may deliver overly loose bounds from the selection of an algebraically correct but numerically unstable algorithm.
  • The costs and benefits of unum over short precision floating point for problems requiring low precision are not obvious.
  • Solving differential equations and evaluating integrals with unums guarantee correct answers but may not be as fast as methods that usually work.

    • References

    Unum (number format) Wikipedia
    (Text) CC BY-SA