Equalvalued

Equalvalued is an adjective used to describe two or more expressions that evaluate to the same numerical value under a given interpretation or domain. The term is not standardized in mathematics, but it appears in informal explanations and teaching materials to emphasize that the focus is on outputs (values) rather than symbolic form.

In mathematics, two expressions or functions f and g are considered equal-valued on a domain D if f(x) = g(x) for every x in D. When this holds, the expressions produce identical outputs for all inputs within the domain. This concept is closely related to, but not identical with, formal identity: a true algebraic identity states a relationship that holds for all inputs without restriction, whereas equal-valued can describe equality of values over a specified domain or under a particular interpretation.

Equal-valued can be used to describe situations where two different-looking expressions yield the same results in practice, even if they are not algebraically identical. For example, simplified forms or numerical approximations may be equal-valued within a certain tolerance or over a chosen range of x.

In computer science and numerical analysis, equal-valued considerations arise in code optimization, symbolic computation, and error analysis. Expressions that are equal-valued under a given precision may be treated interchangeably in optimizations or proofs of correctness.

See also: equality, functional identity, pointwise equality, numerical equivalence, value comparison.