Isclose

Isclose refers to the concept of determining whether two numerical values are approximately equal within a defined tolerance rather than exactly equal. In numerical computing, exact equality can be unreliable due to floating point representation, rounding errors, and measurement noise. An isclose test provides a robust way to compare values in algorithms, tests, and simulations.

In programming, isclose is implemented in several libraries with similar semantics. For example, Python’s math.isclose(a, b, rel_tol=1e-09, abs_tol=0.0) returns True if the absolute difference between a and b is at most the larger of rel_tol times the larger absolute value of a or b, and abs_tol. Formally, it checks whether abs(a - b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol). By default, NaN comparisons are False.

NumPy also provides isclose for array-like inputs: numpy.isclose(a, b, rtol=1e-05, atol=1e-08, equal_nan=False). It broadcasts over arrays and returns a boolean array indicating where elements are close. The same general idea applies, with rtol and atol controlling relative and absolute tolerance, and equal_nan allowing NaNs to be treated as equal when desired.

Practical guidance on tolerances varies by context. Relative tolerance (rtol) is suitable for large numbers, while absolute tolerance (atol) helps near zero. In high-stakes calculations, calibrate tolerances to the expected error bounds of the numerical method. Isclose is not a universal substitute for equality and should be used with awareness of its limitations and the non-transitive nature of approximate comparisons.

See also: relative error, machine epsilon, floating point arithmetic, numerical stability.