Representation of floating point arithmetic, Operation, Normalization, PIt Falls of floating point arithmetic

1. Floating point arithmetic
Representation of floating point
arithmetic
Operations
Normalization
Pit-Falls of floating point arithmetic

2.  There are two types of arithmetic
operations :-
Integer Arithmetic
Real – floating point arithmetic
Integer Arithmetic :- deals with
integer operands.
i.e. - num without fractional parts
Real Arithmetic :- use number with
fractional parts as operands and is
use
Real no. = mantissa * 10^exponent

3. COMPUTER
REPRESENTATION OF
FLOATING POINT NUMBERS
In the CPU, a 32-bit floating
point number is represented
using IEEE standard format as
follows:
S | EXPONENT | MANTISSA
where S is one bit, the EXPONENT
is 8 bits, and the MANTISSA is
23 bits.

4.  The mantissa
 represents the leading
significant bits in the number.
 It is made less than 1 and
greater than or equal to 0.1
 The exponent
 is used to adjust the position
of the binary point (as opposed
to a "decimal" point)
 In power of 10 and multiplies
with mantissa

5.  Normalization :-
Mantissa and Exponent have
their own independent signs
While storing no. of the
leading digit in the mantissa,
mantissa is always made non-
zero by appropriately shifting
it and adjusting the value of
the exponent
 The shifting of mantissa to the left
till its most significant digit is non-
zero is called normalization.

6. Arithmetic operations with
normalized floating point
numbers
 Addition
 Subtraction
 Multiplication
 Division

7. Addition :-
Consider a 4-digit decimal example
 9.999 × 101 + 1.610 × 10–1
1. Align decimal points
 Shift number with smaller exponent
 9.999 × 101 + 0.016 × 101
2. Add significant
 9.999 × 101 + 0.016 × 101 = 10.015 × 101
3. Normalize result & check for
over/underflow
4. Round and renormalize if necessary

8. Subtraction :-
Consider a 4-digit decimal example
 9.999 × 101 - 1.610 × 10–1
1. Align decimal points
 Shift number with smaller exponent
 9.999 × 101 - 0.016 × 101
2. Subtract significant
 9.999 × 101 - 0.016 × 101 = 9.983 × 101
3. Normalize result & check for
over/underflow
4. Round and renormalize if necessary

9. Multiplication :-
Consider a 4-digit decimal example
 1.110 × 1010 × 9.200 × 10–5
1. Add exponents
 New exponent = 10 + –5 = 5
2. Multiply significant
 1.110 × 9.200 = 10.212  10.212 × 105
3. Normalize result & check for over/underflow
 1.0212 × 106
4. Round and renormalize if necessary
 1.021 × 106
5. Determine sign of result from signs of
operands
 +1.021 × 106

10. Division :-
Consider a 4-digit decimal example
 1.110 × 1010 / 9.200 × 10–5
1. Subtract exponents
 New exponent = 10 – (–5) = 15
2. Divide significant
 1.110 / 9.200 = 0.12065  0.12065 ×
1015
3. Normalize result & check for
over/underflow
 1.2065 × 1014
4. Round and renormalize if necessary
5. Determine sign of result from signs of
operands