RoundTowardZero

RoundTowardZero, often abbreviated RTZ and also described as round toward zero, is a rounding mode used in floating-point arithmetic. Under this mode, the exact mathematical result of a computation is approximated by discarding the fractional part and keeping the sign unchanged. In other words, positive values are rounded downward toward zero and negative values are rounded upward toward zero. Examples: 3.7 becomes 3, while -3.7 becomes -3. If the exact result is already an integer, it remains unchanged.

In IEEE 754, RTZ is one of the standard rounding modes used to determine the final representable floating-point value after an operation such as addition, subtraction, multiplication, division, or square root. The operation is performed with infinite precision, and the final result is selected by rounding toward zero. This means the returned value has a magnitude not greater than the exact result and the same sign as the exact result. There are no ties in RTZ, since any non-integer exact result maps to a single truncated value.

Relation to integer conversion and language semantics: RTZ is closely aligned with how many programming languages convert floating-point numbers to integers. For example, converting a floating-point value to an integer in languages like C and Java truncates toward zero. In other languages, or in different contexts, rounding behavior for conversion may vary, so it is important to distinguish explicit RTZ rounding from other forms of truncation or rounding.

Uses and considerations: RTZ is useful when a predictable, sign-preserving truncation is desired, such as certain fixed-point or iterative numerical methods. It introduces a consistent, asymmetric rounding error relative to the exact result and can accumulate bias in some computations, especially compared with round-to-nearest modes.