Floating-Point Number Representation

numeric

Definition

Floating-Point Number Representation

Floating-point representation expresses a real number in the form

$$x=(-1{)}^s\cdot m\cdot b^e,s\in \{0,1\},b\in {\mathbb{Z}}_{\ge 2},e\in \mathbb{Z},$$

where $b$ is the base, $e$ the exponent, and $m$ the mantissa (significand) written with $p$ digits in base $b$. The mantissa is normalised: its leading digit is nonzero, so for $b=2$ that leading digit is necessarily $1$ and need not be stored — the binary implicit leading bit. The stored layout is a bit word partitioned into a sign bit, an exponent field, and the mantissa field.

Floating-point is the standard number representation of real numbers on a computer, trading exactness for the ability to span a wide range of magnitudes with fixed storage.

Exponent Notation

A real number admits many decompositions $x=m\cdot b^e$ — the pair $(m,e)$ is not determined by $x$ alone.

Non-unique decomposition

$$0.00123=123\cdot 10^{-5}=12.3\cdot 10^{-4}=1.23\cdot 10^{-3}$$

Every form is arithmetically equal; they differ only in where the radix point sits.

To make the representation unique in memory, the mantissa is normalised: exactly one nonzero digit sits before the radix point. The example then collapses to a single canonical form:

$$0.00123=1.23\cdot 10^{-3}.$$

For $b=2$ the normalisation becomes free storage: the leading digit of $m$ must be $1$, so it carries no information and need not be stored — the binary implicit leading bit already named in the definition.

Layout

The first bit stores the sign of the represented number:

• $0$ means positive, and
• $1$ means negative.

The exponent field stores the scale factor $b^e$. The mantissa field stores the significant digits as a fixed-point number with exactly one digit before the radix point; for normalised binary numbers, that leading digit is implicit.

IEEE 754

Definition

IEEE 754

IEEE 754 is the dominant standard for floating-point number representation on computers, fixing $b=2$ and a family of precisions via the parameter set

$$\mathbb{F}(2,p,e_{min},e_{max},\text{denorm}).$$

Each format allocates a bit word into a one-bit sign, an exponent field of $w$ bits, and a mantissa field of $p-1$ bits (the leading $1$ of a normalised mantissa is implicit, so only the fractional $p-1$ bits are stored). The exponent is encoded with an excess (bias) $\beta =-e_{min}+1$, so the stored bits represent $e+\beta$.

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Rounding

Rounding is important in numeric since computers can only store a certain number of bits, meaning a limited precision. There are multiple methods of rounding for floating point numbers.

Truncate

Truncating means removing a part of or the whole fractional part from a number.

Example: Truncating $(-1.626{)}_{10}$ to 2 decimal digits is $(-1.62{)}_{10}$.

Directed Rounding

Directed rounding refers to rounding an number in a specific direction:

1. Round Towards Zero: Always rounds the number closer to zero.
2. Round Away from Zero: Always rounds the number farther from zero.

Examples:

• Rounding Towards Zero: $\text{down}((1.524{)}_{10})=(1.52{)}_{10}$
• Round Away from Zero: $\text{up}((1.524{)}_{10})=(1.53{)}_{10}$

Round to Nearest

Rounding to the nearest refers to rounding a number to the closest value at a specified level of precision, such as the nearest integer, tenth, hundredth, or any other place value.

Precision of Rounding