### Representing and manipulating float point number

IEEE floating-point representation $ V= (-1)^s M 2^E $

`s`

determines whether the number is negative or positive- The significand
`M`

is a fractional binary number that ranges either between 1 and $2-\epsilon $ or between 0 and $1- \epsilon$ - The exponent
`E`

weights the value by power of 2 - In a single-precision floating-point format, fields
`s`

,`exp`

and`frac`

are`1`

,`k=8`

and`n=23`

; for a double-precision floating-point format, fields`s`

,`exp`

and`frac`

are`1`

,`k=11`

, and`n=52`

bit - Normalized value: when the bit pattern of
`exp`

is neither all zeros nor all ones. In this case, the exponent field is interpreted as representing a signed integer in biased form $E=e-Bias$ , where $Bias=2^{k-1}-1$. The fraction field is interpreted as representing the fractional value f, where`0<f<1`

. The significand is defined to be`M=1+f`

. - Denormalized value: when the exponent field is all zeros. In this case, the exponent value $E=1-Bias$, and $M=f$.
- Special value: when the exponent field is all ones.