Floating Point

Overview

The IEEE binary floating-point standard defines an encoding used to represent numbers of form

where $s$ denotes the sign bit, $M$ the significand, and $E$ the exponent. The binary representation of floating-point numbers are segmented into three fields: the sign bit, the exponent field, and the fraction field. Furthermore, there are three classes these fields are interpreted with respect to:

• Normalized Form
  • Here the exponent field is neither all 0s nor all 1s.
  • The significand is $1+f$, where $f$ denotes the fractional part.
  • $E=e-Bias$ where $e$ is the unsigned interpretation of the exponent field.
• Denormalized Form
  • Here the exponent field is all 0s.
  • The significand is $f$, where $f$ denotes the fractional part.
  • $E=1-Bias$, defined for smooth transition between normalized and denormalized values.
• Special Values
  • Here the exponent field is all 1s.
  • If the fraction field is all 0s, we have an $\mathrm{\infty }$ value.
  • If the fraction field is not all 0s, we have $NaN$.

The $Bias$ in the first two forms is set to $2^{k-1}-1$ where $k$ denotes the number of bits that make up the exponent field.

Rounding

Because floating-point arithmetic can't represent every real number, it must round results to the "nearest" representable number, however "nearest" is defined. The IEEE floating-point standard defines four rounding modes to influence this behavior:

• Round-to-even (also known as round-to-nearest) rounds numbers to the closest representable value. In the case of values equally between two representations, it rounds to the number with an even least significant digit.
• Round-toward-zero rounds downward for positive values and upward for negative values.
• Round-down always rounds downward.
• Round-up always rounds upward.

Arithmetic

Floating-point arithmetic operations (e.g. ${+}^f$, ${\ast }^f$, etc.) are neither commutative nor associative. This is a consequence of representations being an approximation of the fractional values they are meant to represent and the corresponding rounding errors.