roundoff

Roundoff error is the discrepancy that occurs when representing real numbers with finite precision. In computing and numerical analysis, roundoff refers to the errors that arise from rounding operands and results to the available number of digits or bits. These errors are inherent to digital representations and can affect the accuracy of numerical computations.

Rounding modes specify how to map a real number to a representable value. Common modes include round to nearest, round toward zero, round toward positive infinity, and round toward negative infinity. In many floating-point systems, notably IEEE 754, round to nearest with ties to even (banker's rounding) is the default. Decimal arithmetic often uses round half up or round half away from zero. The choice of mode can influence bias and stability of computations.

Key concepts related to roundoff include the unit in the last place (ULP), which is the spacing between adjacent representable values, and machine epsilon, the upper bound on relative rounding error for a given format. Roundoff error is the difference between the exact result of a calculation and the value produced by finite-precision arithmetic. Such errors can accumulate over sequences of operations and affect the reliability of numerical algorithms.

Roundoff is distinct from truncation error, which arises when a mathematical expression is approximated by omitting terms or digits. In floating-point computation, both rounding and truncation contribute to total error, but rounding is the deliberate mapping to a limited set of representable values.

To mitigate roundoff, techniques include using higher precision, selecting appropriate rounding modes, applying compensated algorithms (such as Kahan summation), or employing arbitrary-precision arithmetic where necessary.