Float Point Notation

$$ \pm S\times B^E \\ E=\text{Biased Exponent} - \text{bias} $$

$B$: Significand (base)

• Significand is normalized to start with 1.0 ($\pm1.bbbb...b\times 2^E$)
• So we only store digits to the right of the radix ($.bbbb...b)$

$E$: Exponent

• $\text{Biased exponent} = E + \text{bias}$
• $\text{bias} = 2^{k-1}-1$, where $k$ is the number of bits of the “biased exponent” part
• e.g. $k=8$, bias exponent has range 0~255, $E$ therefore has -127~128
• $\therefore$ Bias exponent turn $E$ unsigned → easier comparison

$\pm(S)$: Sign (1 bit)

IEEE Standards

• Single Precision bias: 127
• Double Precision bias: 1023

Encodings