, Volume 35, Issue 2, pp 127–139 | Cite as

Relative error in floating-point multiplication

  • R. H. Goodman
  • A. Feldstein
  • J. Bustoz


A model of the relative error in floating point multiplication is developed and is analyzed stochastically for various choices of computer design parameters. These parameters include the base, the type of rounding rule, the number of guard digits, and whether the post-arithmetic normalization shift (if needed) is done before or after rounding. Under the assumption of logarithmic distribution for the fraction (mantissa), the major stochastic conclusions are:
  1. 1.

    The average relative error in multiplication increases as the base increases.

  2. 2.

    This error is minimized by selecting the machine base to be binary (better yet, binary with a hidden bit) and is rather large for machines with base 16.

  3. 3.

    The classical relative error bounds are pessimistic. The average overestimation by those bounds increases as the base increases.


AMS Subject Classifications

65G05 65G10 


Relative error computer arithmetic floating point multiplication normalization options guard digits floating point numbers floating point precision and significance round-off error fraction error mean and standard deviation of errors logarithmically distributed numbers 

Relative Fehler bei der Gleitkomma-Multiplikation


Es wird ein Modell für den relativen Fehler bei der Gleitkomma-Multiplikation entwickelt unf für verschiedene Kombinationen der Arithmetikparameter stochastisch analysiert. Die Parameter sind die Basis, die Rundungsart, die Anzahl der Schutzstellen, und ob die Normalisierung vor oder nach dem Runden erfolgt. Bei einer angenommenen logarithmischen Verteilung für die Mantisse kommt man zu folgenden Schlüssen:
  1. 1.

    Der durchschnittliche relative Fehler bei der Multiplikation wächst mit der Basis.

  2. 2.

    Dieser Fehler wird minimal für die Basis 2 (am besten mit verborgenem ersten Bit) und recht groß für die Basis 16.

  3. 3.

    Die klassischen Schranken für den relativen Fehler sind pessimistisch. Ihre durchschnittliche Überschätzung wächst mit der Basis.



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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • R. H. Goodman
    • 1
  • A. Feldstein
    • 2
  • J. Bustoz
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsArizona State UniversityTempeUSA

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