Introduction of Floating Point Representation

1. To convert the floating point into decimal, we have 3 elements in a 32-bit floating point representation:
i) Sign
ii) Exponent
iii) Mantissa

Sign bit is the first bit of the binary representation. ‘1’ implies negative number and ‘0’ implies positive number.
Example: 11000001110100000000000000000000 This is negative number.
Exponent is decided by the next 8 bits of binary representation. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where ‘k’ is the number of bits in exponent field.
There are 3 exponent bits in 8-bit representation and 8 exponent bits in 32-bit representation.

Thus

bias = 3 for 8 bit conversion (2^3-1 -1 = 4-1 = 3)
bias = 127 for 32 bit conversion. (2^8-1 -1 = 128-1 = 127)

Example: 01000001110100000000000000000000
10000011 = (131)₁₀
131-127 = 4

Hence the exponent of 2 will be 4 i.e. 2⁴ = 16.
Mantissa is calculated from the remaining 23 bits of the binary representation. It consists of ‘1’ and a fractional part which is determined by:
Example:

01000001110100000000000000000000

The fractional part of mantissa is given by:

1*(1/2) + 0*(1/4) + 1*(1/8) + 0*(1/16) +……… = 0.625

Thus the mantissa will be 1 + 0.625 = 1.625

The decimal number hence given as: Sign*Exponent*Mantissa = (-1)⁰*(16)*(1.625) = 26

2. To convert the decimal into floating point, we have 3 elements in a 32-bit floating point representation:
i) Sign (MSB)
ii) Exponent (8 bits after MSB)
iii) Mantissa (Remaining 23 bits)

Sign bit is the first bit of the binary representation. ‘1’ implies negative number and ‘0’ implies positive number.
Example: To convert -17 into 32-bit floating point representation Sign bit = 1
Exponent is decided by the nearest smaller or equal to 2ⁿ number. For 17, 16 is the nearest 2ⁿ. Hence the exponent of 2 will be 4 since 2⁴ = 16. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where ‘k’ is the number of bits in exponent field.
Thus bias = 127 for 32 bit. (2^8-1 -1 = 128-1 = 127)

Now, 127 + 4 = 131 i.e. 10000011 in binary representation.
Mantissa: 17 in binary = 10001.
Move the binary point so that there is only one bit from the left. Adjust the exponent of 2 so that the value does not change. This is normalizing the number. 1.0001 x 2⁴. Now, consider the fractional part and represented as 23 bits by adding zeros.

00010000000000000000000

Advantages:

Wide range of values: Floating factor illustration lets in for a extensive variety of values to be represented, along with very massive and really small numbers.

Precision: Floating factor illustration offers excessive precision, that is important for medical and engineering calculations.

Compatibility: Floating point illustration is extensively used in computer structures, making it well matched with a extensive variety of software and hardware.

Easy to use: Most programming languages offer integrated guide for floating factor illustration, making it smooth to use and control in laptop programs.

Disadvantages:

Complexity: Floating factor illustration is complex and can be tough to understand, mainly for folks that aren’t acquainted with the underlying mathematics.

Rounding errors: Floating factor illustration can result in rounding mistakes, where the real price of a number of is barely extraordinary from its illustration inside the computer.

Speed: Floating factor operations can be slower than integer operations, particularly on older or much less powerful hardware.

Limited precision: Despite its excessive precision, floating factor representation has a restrained number of sizeable digits, which could restrict its usefulness in some programs.

More questions on number representation:
https://www.geeksforgeeks.org/number-representation-gq/

This article is contributed by Kriti Kushwaha

Article Tags :

Digital Logic