Floating Point Representation - Details, Types & Needs

What is Floating Point Representation?

Binary numbers can also be expressed in exponential form. The representation of binary integers in exponential form is known as floating point representation. The floating point representation divides the number into two parts: the left side is a signed, fixed-point integer known as a mantissa, and the right side is the exponent.

Floating point representation can also have a sign, with 0 denoting a positive value and 1 denoting a negative value. The IEEE (Institute of Electrical and Electronics Engineers) has developed a floating point representation standard.

Floating Point Representation Types

The IEEE specifies two types of formats in floating point representation that are:

• Single precision(32-bit)
• Double precision(64-bit) 

Single Precision Floating Point Representation

The single-precision floating-point representation (also known as FP32 or float32) is a computer number format that uses a floating radix point to express a wide dynamic range of numeric values. The IEEE 754 standard defines a binary32 as having the following characteristics:

• 1 bit for sign
• 8-bit for exponent
• The precision of significand: 24 bits (23 explicitly stored)

The structure of single precision floating point representation is as follows:

Exponent calculation

In the IEEE 754 standard, the single-precision floating-point representation, the exponent is encoded using an offset-binary encoding, with the zero offset being 127; this is known as exponent bias.

E_min = 01H – 7FH = −126

E_max = FEH – 7FH = 127

Exponent bias = 7FH = 127

Thus, the offset of 127 must be removed from the recorded exponent to obtain the real exponent as described by the offset-binary representation.

Double Precision Floating Point Representation

The double precision floating point representation (also known as FP64 or float64) is a computer number format that uses a floating radix point to express a wide dynamic range of numeric values. The IEEE 754 standard defines a binary64 as having the following characteristics:

• 1 bit for sign
• 11-bit for exponent
• The precision of significand: 53 bits (52 explicitly stored)

The structure of double precision floating point representation is as follows:

Need for Floating Point Representation

A fixed point representation will not be sufficient when representing extremely small or extremely big numbers. The precision will be lost. As a result, you must examine floating point representations, in which the binary point is believed to be floating.

Consider the decimal value 12.34 * 10⁷, which may alternatively be written as 0.1234 * 10⁹, where 0.1234 is the fixed-point mantissa. The other portion is the exponent value, and it shows that the actual position of the binary point in the fraction is 9 places to the right (left) of the specified binary point.

A floating point representation is so named because the binary point can be shifted to any place and the exponent value can be modified accordingly. By convention, you should use a normalized form, with the floating point to the right of the first nonzero (significant) digit.