IEEE 754 standard

Definition

The IEEE 754 standard is the technical standard for floating-point arithmetic of the Institute of Electrical and Electronics Engineers. Its main requirements are the following.

1. Formats: a format has a base $\beta \in \set{2, 10}$, a precision $p$ and an exponent range $[e_{\min}, e_{\max}]$ with $e_{\min} = 1 - e_{\max}$. The formats binary32 and binary64 have $(p, e_{\max}) = (24, 127)$ and $(53, 1023)$.
2. Special values: each format contains $+0$ and $-0$, $+\infty$ and $-\infty$, NaN ("not a number") values, and subnormal numbers.
3. Encoding: a binary64 number has $1$ sign bit, $11$ bits of biased exponent (bias $1023$) and $52$ bits of fraction. The leading significand bit of a normalized number is implicit.
4. Correct rounding: the operations $+$, $-$, $\times$, $/$, square root and fused multiply-add must return the exact result rounded by the current rounding mode.
5. Rounding modes: the default is round to nearest with ties to even. The other modes are ties away from zero and directed rounding toward $0$, $+\infty$ and $-\infty$.