IEEE 784 : Floating Point Representation

Floating Point Representation:

A floating point number is typically represented in binary as:

Sign bit

Biased-Exponent

Mantissa

Where,

·         Sign bit tells whether the number is positive or negative

For positive, sign bit=0

For negative, sign bit=1

·         Biased Exponent is a special integer which is added or subtracted from the exponent as described further.

·         Mantissa contains fraction part.

Typically in a computer, the size of each of these components are fixed.

Now, let us have a formal description of such numbers.

Let us consider a computer architecture which supports

·                     Sign bit of 1 bit

·                     Biased Exponent of 8 bits

·                     Mantissa of 23 bits

Hence,

            Total size=23+8+1=32-bits.

NOTE:

This configuration is a standard configuration and is typically known as IEEE-784 standard

Now, let us take an example and try to understand how it is represented.

For example, suppose we want to represent number 1234.625 into IEEE-784 standard.

To convert it following steps is followed.

STEP 1:

            Express the number purely in decimal i.e. without any exponent part.

            For e.g. if the number is 12.34*10³ then the number is represented as 12340.

            Then,

Convert the given number into its binary equivalent.

Hence, (1234.625)₁₀ = (0100 1101 0010.1010)₂

STEP 2:

            Now normalize the number.

            Number (0100 1101 0010.1010)₂ can be normalized as (1.0011 0100 1010 1000*2¹⁰)₂

STEP 3:

            Compute the biased exponent. The biased exponent is computed as

            For a representation, if exponent bit is represented in e bits.

            Then biased exponent will be

2^e-1-1

            For IEEE-784,

Biased exponent is

2^8-1-1, as e=8-bits

Biased exponent = 127

After computing the biased exponent, add this value to the exponent part of your number. Here,

Exponent is 10,

Hence biased exponent is 127+10=137

Convert the biased exponent to its binary equivalent

137 = 1000 1001

STEP 4:

            Depict it,

0

1000 1001

0011 0100 1010 1000 0000 000

Similarly, a given floating point number can be tracked down to its decimal equivalent.

For example,

                                          

0

1000 1001

0011 0100 1010 1000 0000 000

 Hence it is written as,

Firstly, the biased exponent is handled by subtracting the biased number from it.

Hence,

1000 1001 = 137

So, 127 is subtracted from 137.

Hence,

137-127 = 10

So, the given number is

1.0011 0100 1010 1 *2¹⁰  = (0100 1101 0010.1010)₂

Now, get its decimal equivalent,

Which is 1234.625

A general discussion on FLOATING POINT numbers

	Sign	Biased Exponent	Fraction	Value
Positive zero	0	0	0	0
Negative Zero	1	0	0	-0
Plus Infinity	0	255 ( all 1s)	0	∞
Minus Infinity	1	255 ( all 1s)	0	-∞
Not a Number (NaN)	0 or 1	255 ( all 1s)	Not equal to 0	NaN

With these restriction, there exists numbers which can’t be represented.

Let us try to compute these different boundaries,

Range of expressible positive numbers,

Let us consider a computer architecture which supports

·                     Sign bit of 1 bit (s = 1)

·                     Biased Exponent of 8 bits (e = 8)

·                     Mantissa of 23 bits (m= 23)

The maximum positive number expressible will be,

0

1111 1110

11111111111111111111111

Now, interpret this number,

Biased exponent is 2^e-1-1 = 127

Let say this number to be k,

Hence k=127

Now (1111 1110) = (254)

Hence our exponent is 254-k=254-127=127

Hence our number is,

1.11111111…1 * 2¹²⁷

Which can be written as,

1 followed by 23 1s and then followed by 104 0s, which is

1 11111111111111111111111 0000000000…0

( 104 zeros)

Now convert it into decimal, it would be

1*2¹⁰⁴ + 1*2¹⁰⁵ + 1*2¹⁰⁶ +…+ 1*2¹²⁷

=3.4028*10³⁸