Subnormal number

Remarks

Example: in IEEE binary64, $\beta = 2$, $p = 53$ and $e_{\min} = -1022$, so the subnormal numbers are $\pm k \, 2^{-1074}$ with $1 \le k \le 2^{52} - 1$; the smallest is $2^{-1074} \approx 4.9 \times 10^{-324}$. Non-example: $2^{-1022}$ is not subnormal in binary64; it is the smallest positive normalized number, with leading digit $1$. All subnormal numbers are multiples of the same spacing $\beta^{e_{\min} - p + 1}$, which is also the spacing of the normalized numbers in $[\beta^{e_{\min}}, \beta^{e_{\min}+1})$. A subnormal number has fewer than $p$ significant digits, so the relative rounding error in the subnormal range can exceed the unit roundoff. Other names are denormal and denormalized numbers.