Relative error ($|x_{\mathrm{approx}}-x|/|x|$)

For a nonzero reference value $x$, the relative error is the absolute error divided by the reference's absolute value:It is dimensionless and expresses the size of the error compared with the quantity being approximated. Approximating $1/3$ by $0.33$ has absolute error $1/300$ and relative error $1/100=1\%$. The definition is unavailable when the reference is zero; use an absolute error or a separately stated scale in that case.

The signed relative error is $\epsilon =(x_{\mathrm{approx}}-x)/x$ for a nonzero reference $x$. Its absolute value is the relative error. For a positive reference, a negative signed relative error means underestimation, and a positive one means overestimation. For a negative reference that interpretation reverses, so retain the displayed definition when interpreting its sign.