1. IT11004: Data Representation and Organization
IT11004: Data Representation and Organization
Floating Point Representation

2. normalized form
A real number is called normalized, if it is in the form:
d0.d1d2d3…x10n
where n is an integer, d1d2d3… are the digits of the number in base 10, and d0 is not zero.
As examples,
the number in normalized form is
x102
the number in normalized form is
x10-3
Clearly, any non-zero real number can be normalized.

3. Floating Point Precisions s
exp
frac
Encoding
MSB is sign bit
exp field encodes E
frac field encodes M
Sizes
Single precision: 8 exp bits, 23 frac bits
32 bits total
Double precision: 11 exp bits, 52 frac bits
64 bits total
Extended precision: 15 exp bits, 63 frac bits
Only found in Intel-compatible machines
Stored in 80 bits
1 bit wasted

4. Single-precision floating-point format (binary32)
Single-precision floating-point format (binary32)
A computer number format that occupies 4 bytes (32 bits) in computer memory and represents a wide dynamic range of values by using a floating point.
One of the first programming languages to provide single- and double-precision floating-point data types was Fortran.
Single-precision binary floating-point is used due to its wider range over fixed point
Single precision is known as
float in C, C++, C#, Java[1], and Haskell, and
single in Pascal, Visual Basic, and MATLAB.

5. IEEE single-precision binary floating-point format: binary32
IEEE single-precision binary floating-point format: binary32

6. convert a base 10 real number into binary32 format
consider a real number with an integer and a fraction part such as
Convert the integer part into binary
convert the fraction part using the following technique
add the two results and adjust them to produce a proper final conversion
Conversion of the fractional part
0.375 x 2 = 0.750
0.750 x 2 = 1.500
0.500 x 2 = fraction = 0.000, terminate
(0.375)10 can be exactly represented in binary as (0.011)2
Therefore (12.375)10 = (12)10 + (0.375)10 = (1100)2 + (0.011)2 = ( )2
In normalized form (12.375)10 = x23

7. convert a base 10 real number into binary32 format…
convert a base 10 real number into binary32 format…
In normalized form (12.375)10 = x23
From which we deduce:
The exponent is 3
(and in the biased form it is therefore =130 = )
The fraction is (looking to the right of the binary point)
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of as:
= H

8. Ex 1 Consider a value 0.25 . We can see that : (0.25)10 =(1.0)2x2-2
0.25 x 2 = 0.5
0.5 x 2 = 1.0
 = 0.012
Consider a value
We can see that : (0.25)10 =(1.0)2x2-2
From which we deduce :
The exponent is −2
(and in the biased form it is 127+(−2)= 125 = )
The fraction is 0
(looking to the right of binary point in 1.0 is all zeros)
From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.25 as:
= 3e800000H

9. Ex 2
Convert into binary 32 floating point format

10. Double-precision floating-point format (binary64)
a computer number format that occupies two adjacent storage locations in computer memory.
A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point
binary64 is having:
Sign bit: 1 bit
Exponent width: 11 bits
Significand precision: 52 bits