Audio Version available in YouTube Link : www.youtube.com/Aksharam subscribe the channel Computer Architecture and Organization V semester Anna University By Babu M, Assistant Professor Department of ECE RMK College of Engineering and Technology Chennai

1. INTRODUCTION TO
FLOATING POINT ARITHMETIC

2. FLOATING POINT NUMBERS
As the name implies, floating point numbers are numbers that contain floating decimal points.
For example,
The numbers 5.5, 0.001, and -2,345.6789 are floating point numbers.
-2,345.6789 = -2345.6789 x 100
-234.56789 x 101
-23.456789 x 102
-2.3456789 x 103
-23456.789 x 10-1
-234567.89 x 10-2
-2345678.9 x 10-3
(-1)S x F x 10E

3. BINARY FLOATING POINT NUMBERS
10011.10= 10011.10 x 20
1001.110 x 21
100.1110 x 22
10.01110 x 23
100111.0 x 2-1
1001110.0 x 2-2
10011100.0 x 2-3
(-1)S x F x 2E 1.001110 x 24

4. NORMALIZATION
(-1)S x 1.F x 2E
10011.10= 10011.10 x 20
1001.110 x 21
100.1110 x 22
10.01110 x 23
100111.0 x 2-1
1001110.0 x 2-2
1.001110 x 24
Scientific Notation Which has no leading zero is known as a Normalized Number
The number which
has a single
digit to the left of
the decimal point
0.1001110 x 25
110.11= 1.1011 x 22
0.001010= 1.010 x 2-3
101.01 x 2-1= 1.0101 x 21
0.00010 x 24= 1.0 x 20

5. TRUNCATION
1001.110 x 24 = 1.001110 x 27
1.001 x 27
Decimal Binary
5456.450 x 102 = 5.456450 x 105
5.456 x 105
Truncate for 4 (Digits)bits / 3 decimal places

6. ROUNDING
1001.10011 x 24 = 1.01010011 x 27
1.01010011 x 27
Decimal Binary
5456.87032 x 102
= 5.45687032 x 105
5.45687032 x 105
GUARD BIT ROUND BIT STICKY
BITS
GUARD BIT(G) ROUND
BIT(R)
STICKY BITS (S)
5.45687 x 105 5.457 x 105 1.011 x 27
Round for 4 (Digits)bits / 3 decimal places
G R S Rounding Action
0 0 0 Truncate
0 0 1 Truncate
0 1 0 Truncate
0 1 1 Truncate
1 0 0 Round to Even
1 0 1 Round Up
1 1 0 Round Up
1 1 1 Round Up

7. ROUNDING EXAMPLEG R S Rounding Action
0 0 0 Truncate
0 0 1 Truncate
0 1 0 Truncate
0 1 1 Truncate
1 0 0 Round to Even
1 0 1 Round Up
1 1 0 Round Up
1 1 1 Round Up
1.01010011 x 27
Round for 4 bits / 3 decimal places
= 1.011 x 27
1.01111000 x 27 = 1.100 x 27
1.01010000 x 27 = 1.010 x 27
1.01110000 x 27 = 1.100 x 27
1.00010001 x 27 = 1.001 x 27
1.11110001 x 27 = 10.001 x 27