Concept

hypot — where it appears

The routine that computes the length of a two-component vector without squaring either component first. It exists because the square of a representable number need not be representable, and its absence is the usual way a norm overflows before its answer does.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A norm that overflows before it is a norm

The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.

arithmetic · overflow

The units that overflow before the answer does

A change of variable that is exact in the algebra requires γ² times a matrix to be a number the format can hold. In binary64 that is a bound nobody meets by accident. In binary32 it arrives at 10¹⁹ and in fp16 at 256, and past it there is no answer rather than a poor one.

arithmetic · overflow

Named alongside it

The objects these essays reach for when they reach for this one.

Overflow bfloat16 Binary32 Catastrophic cancellation Dynamic range Exponent range Fp16 Half precision Matrix polynomial Quadratic eigenvalue problem Scaled norm Scaling

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