IEEE floating-point representation

The bit representation of a floating-point number is divided into three fields to encode these values:

• The single sign bit $s$ directly encodes the sign of the number.
• The $k$-bit exponent field $\exp =e_{k-1}\cdots e_1{e}_0$ encodes the exponent $E$.
• The $n$-bit fraction field $\mathrm{frac}=f_{n-1}\cdots f_1{f}_0$ encodes the significand/mantissa $M$, but the value encoded also depends on whether or not the exponent field equals 0.

The actual floating-point number represented is: $n=\mathrm{sign}\cdot 1.\mathrm{frac}\cdot 2^{\exp -127}$

Single-precision floating-point

In single-precision floating-point values (32-bit), there are $k=8$ bits for the exponent field and $n=23$ bits for the mantissa / fraction field.

In cases where a computation fails, we might end up with special values: