Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof / 5 / c

The smallest positive normal half-precision floating-point number uses $E=1$ and a zero fraction, while the largest uses $E=30$ and all ten fraction bits equal to one. HenceThe corresponding words are 0 00001 0000000000 and 0 11110 1111111111. The stored exponent $31$ is reserved and cannot extend the finite normal range.

For a subnormal number, $E=0$ and the implicit leading bit becomes zero, while the scale remains $2^{1-15}=2^{-14}$. The least positive nonzero fraction is $0.0000000001_2=2^{-10}$, givingUsing scale $2^{-15}$ here would be incorrect: the subnormal number convention uses the minimum normal scale, not the ordinary biased-exponent formula at $E=0$.