underflow

Underflow is a condition in computing in which a calculation yields a magnitude closer to zero than the smallest value representable by the chosen numeric format. The term is most often used in floating-point arithmetic but can also describe behavior of integer types and other numeric representations when the result lies outside the allowable range toward zero.

In IEEE 754 floating-point systems, underflow occurs when the computed magnitude is smaller than the smallest normal positive number. The result may be a subnormal (denormal) number, bridging the gap between normal numbers and zero, or it may be rounded to zero. Subnormals preserve some precision but can incur performance penalties on some hardware. In many systems, underflow also triggers an underflow exception or sets a status flag.

In fixed-size integer arithmetic, underflow occurs when a value would lie below the minimum representable value (or below zero for unsigned types). Depending on the language, this can lead to wraparound, saturating behavior, exceptions, or undefined results (notably for signed integers in C/C++).

Mitigating underflow involves range checks before operations, using larger or arbitrary-precision types, or employing libraries that provide safe arithmetic. Floating-point underflow is sometimes acceptable, but for critical numeric code, designers consider gradual underflow, subnormal handling, and appropriate error reporting.

See also overflow; subnormal numbers; numerical analysis.