# Floating Point Representation

Apr 3, 2023

• sign s
• negative: s=1
• positive: s=0
• exponent E
• weights the value by a (possibly negative) power of 2
• significand M
• fractional binary number ranges between 1 and $2 - \epsilon$ (Normalized) or between 0 and $1 - \epsilon$ (Denormalized)
• result V
• $2^E * M$

## # Case 1: Normalized Values

• When bit pattern of exp is neither all zeros nor all ones. Then the exponent field is interpreted as in biased form.
• The exponent value is $E = e - Bias$
• where e is the unsigned number having bit representation $e_{k-1}\cdots e_{1}e_{0}$,
• and Bias is a bias value $2^{k-1} - 1$ (where k is the number of bits in the exponent, 127 for single precision and 1023 for double)
• This yields exponent ranges from −126 to +127 for single precision and −1022 to +1023 for double precision.
• The fraction field frac is interpreted as representing the fractional value f, where $0 \leq f < 1$, having binary representation $0.f_{n-1} \cdots f_1 f_0$ .
• The significand is defined to be $M = 1 + f$
• This is sometimes called an implied leading 1 representation, because we can view M to be the number with binary representation $1.f_{n-1} \cdots f_1 f_0$
• This representation is a trick for getting an additional bit of precision for free, since we can always adjust the exponent E so that significand M is in the range $1 \leq M < 2$ . We therefore do not need to explicitly represent the leading bit, since it always equals 1.

## # Case 2: Denormalized Values

When the exponent field is all zeros, the represented number is in denormalized form.
the exponent value is $E = 1 − Bias$, and the significand value is $M = f$ , that is, the value of the fraction field without an implied leading 1.

• Why?
• they provide a way to represent numeric value 0, since with a normalized number we must always have $M \geq 1$ , and hence we cannot represent 0. We even have +0.0 and -0.0
• represent numbers that are very close to 0.0. They provide a property known as gradual underflow in which possible numeric values are spaced evenly near 0.0.

## # Case 3: Special Values

• When the exponent field is all ones
• When the fraction field is all zeros, the resulting values represent infinity
• When the fraction field is nonzero, the resulting value is called a NaN, short for “not a number.”

Floating Point Rounding