### 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.