rtol

rtol, short for relative tolerance, is a parameter used in numerical computing to specify when two floating-point values are considered close. It is often paired with an absolute tolerance (atol) to form a closeness criterion that accounts for both relative and absolute differences.

A common formulation compares two numbers a and b using the condition |a - b| <= atol + rtol * |b| (or, in some variations, rtol * max(|a|, |b|)). This means that allowable error scales with the magnitude of the values, while the absolute tolerance handles near-zero comparisons. Different libraries may adopt slightly different variants of the formula, but the same general idea applies: rtol handles relative error, while atol handles fixed, magnitude-independent error.

rtol is widely used in testing and numerical verification. Functions such as isclose and allclose in scientific computing libraries expose rtol (and atol) as parameters. For example, numpy.isclose(a, b, rtol=1e-5, atol=1e-8) uses a small relative tolerance and a tiny absolute tolerance to determine closeness. PyTorch provides similar options in its allclose function.

Choosing values for rtol depends on the numerical method and precision. Typical rtol values range from 1e-5 to 1e-8, with smaller values for higher precision computations or more sensitive results. When the reference value b is zero, the criterion relies entirely on atol, making a nonzero rtol ineffective in that case.

See also: isclose, allclose, tolerance, numerical precision.