floating point in computer architecture

Extreme cases of floating point Numbers:

1)When the exponent number becomes larger

Floating point numbers are represented in IEEE formats.

Let us consider IEEE 754 : 32 Bit Format

S	Biased Exponent	Mantissa
(1)	(8) Bias value=127	(23 bits)

• 32 bits are used to store the numbers.
• 23 bits are used for Mantissa.
• 8 bits are used for the Biased Exponent.
• 1 bit is used for the sign of the number.
• The Bias value is (127)₁₀
• The range is +1 x 10^-38 to +3 x 10³⁸ approximately.
• It is called a single precision format for floating point numbers.

For value 1.0,normalized form is

(-1)⁰ x 1.0 x 2⁰ 

True exponent(TE)=0

If:  TE = 0,    BE=127 ,   Representation = 0111 1111

If:  TE = 1,    BE=128 ,   Representation = 1000 0000

If:  TE = 2,    BE=129 ,   Representation =  1000 0000

…

…

…

If:  TE = 127,    BE=254 ,   Representation =  1111 1110

If:  TE = 128,    BE=255 ,   Representation =  1111 1111

If:  TE = 129,    BE=255 ,   Representation =  1111 1111

If:  TE = 130,    BE=255 ,   Representation =  1111 1111

• 8-bit based exponent cannot hold value more than 255.Hence, all cases where TE =128 or more,the BE will be represented as 1111 1111.
• This indicates an exception or error called overflow.
• The number is called NaN(Not a number).
• It is identified by exponents being all 1’s(1111 1111).
• Mantissa can be anything.

The extreme case of overflow is Infinity:

• It is not a regular NaN.
• The exponent will be 1111 1111.
• To differentiate NaN with infinity,the mantissa in infinity is 0000 0000
• Infinity is identified as exponent all 1’s and mantissa with all 0’s.

2)When the exponent number becomes small reaching towards zero

Let us assume a vale 0.1,normalised form is

(-1)⁰ x 1.0 x 2^-1

True exponent(TE)=-1

If:  TE = -1,    BE=126 ,   Representation = 0111 1110

If:  TE = -2,    BE=125 ,   Representation = 0111 1101

…

…

…

If:  TE = -126,    BE=1 ,   Representation =  0000 0001

If:  TE = -127,    BE=0 ,   Representation =  0000 0000

If:  TE = -128,    BE=0 ,   Representation =  0000 0000

If:  TE = -129,    BE=0 ,   Representation =  0000 0000

• 8-bit based exponent cannot hold value more than 255.Hence, all cases where TE =-127 or less,the BE will be represented as 0000 0000.
• This indicates an exception or error called underflow.
• The number is called De-Normal Number.
• It is identified by exponents being all 0’s(0000 0000).
• Mantissa can be anything.

The extreme case of underflow is Zero:

• It is not a regular De-Normal.
• The exponent will be 0000 0000.
• To differentiate de-normal with Zero,the mantissa in Zero is 0000 0000
• Zero is identified as exponent all 0’s and mantissa with all 0’s.

Summary

Number	Exception	Exponent	Mantissa
Normal Number	No error	0<E<255	Anything
NaN	overflow	1111 1111	Anything
Infinity	overflow	1111 1111	0000 0000
De-Normal number	underflow	0000 0000	Anything
Zero	underflow	0000 0000	0000 0000