Hyb Error: A Hybrid Metric Combining Absolute and Relative Errors

TL;DR

This paper introduces Hyb Error, a new metric combining absolute and relative errors to provide a balanced and robust measure of approximation accuracy, especially useful for sequences.

Contribution

The paper proposes Hyb Error, a hybrid metric that effectively combines absolute and relative errors, addressing their individual limitations in numerical approximation assessments.

Findings

Hyb Error approaches absolute error near zero values.

Hyb Error approaches relative error for large values.

Maximum Element-wise Hyb Error (MEHE) reflects the most significant error.

Abstract

Suppose $x$ is an approximation of $y$. This paper proposes using $\frac{|x-y|}{1+|y|}$, named Hyb Error, to measure the error. This metric equals half the harmonic mean of absolute error and relative error, effectively combining their advantages while mitigating their limitations. For example, Hyb Error approaches absolute error as $|y|$ approaches 0, thereby avoiding the exaggeration of relative error, and approaches relative error as $|y|$ approaches infinity, thereby avoiding the exaggeration of absolute error. The Hyb Error of $\epsilon$ is equivalent to $|x-y|=\epsilon+\epsilon |y|$, which implies $\mathrm{isclose}(x,y,\epsilon,\epsilon)=\mathrm{True}$, where ``isclose'' is a common floating-point equality check function in numerical libraries. For sequences, this property makes the Maximum Element-wise Hyb Error (MEHE) a pragmatic error metric that reflects the most significant error and equals the decision boundary of the ``isclose'' function.