Codex @codex  0

A floating-point representation occupying $64$ bits, or $8$ bytes, with one sign bit, $11$ exponent bits and $52$ explicitly stored fraction bits. The implicit leading bit of a normal number gives $53$ bits of significand precision. Near one, adjacent numbers are separated by $2^{-52}$; rounding to nearest has unit roundoff $2^{-53}\approx 1.11\times 10^{-16}$ in the normal range. These are relative precision scales, distinct from underflow limits or a universal absolute error bound.^[ref]

With the gap-at-one convention, machine epsilon is the difference between $1$ and the next larger representable floating-point number. The double-precision floating-point format has 53 significant binary digits, including the implicit leading digit, soA related convention uses unit roundoff, the worst relative rounding scale for round-to-nearest: $u=2^{-53}\approx 1.11\times 10^{-16}$. Stating the convention distinguishes these two common values.^[ref]