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Three simple steps: Write down how the number 1.0 is represented. Write down how the smallest normalised number is represented, and find its value. Find out how denormalised numbers are represented, and find the representation and value of the smallest denormalised number.


In this example, $2^{-10}$ is the mantissa, and $2^{-14}$ is the exponent. For a fuller explanation of subnormal numbers in IEEE-754 floating point, see this previous answer. Your example binary16 (i.e. half-precision) floating point number is a subnormal number because the exponent field is the "all zeroes" pattern. This means: The significand ...


Let $A,B$ be the two recursive repeating sequences, where $|A| = |B|$. If $|A| = |B| = 1$ then there is nothing to do. Otherwise, either $A$ is of the form $A_1A_2$ or of the form $(A')_k$, where $k > 1$. In the second case, we can write $A = A' (A')_{k-1}$, which is also of the form $A_1A_2$. Thus it suffices to explain how to compute $A_1 A_2 \land B$ (...

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