Floating Point Representation

Apr 3, 2023

Blog: Introduction to Floating Point Number

• 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
  • $2^E * M$

• 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.

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.

• 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