Floating-point arithmetic: Difference between revisions

Revision as of 12:55, 27 March 2025

Revision as of 17:19, 27 March 2025

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The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the [[discretization error]] and is limited by the [[machine epsilon]].

The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a [[unit in the last place]] (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c2245A70C22_hex16 and 1.45a70c2445A70C24_hex16, the ULP is 2×16⁻⁸, or 2⁻³¹. For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2⁻²³ or about 10⁻⁷ in single precision, and exactly 2⁻⁵³ or about 10⁻¹⁶ in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.

=== Rounding modes ===