Overflow, Underflow, and Severe Loss of Significance in Floating-Point Addition and Subtraction

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Abstract
In this paper it is shown that, under the assumption of the logarithmic distribution of numbers, floating-point addition and subtraction can result in overflow or underflow with alarming frequency—a frequency that increases rapidly with machine speed unless the range of the exponent is also increased. If numbers are assumed to be distributed in accordance with Sweeney's (1965) experiments, then severe loss of significance occurs with large probability in floating point subtraction. These results have implications for computer design and lead to the suggestion of a long word format which will reduce the risks to acceptable levels.

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