Computer Arithmetic | Set – 2

FLOATING POINT ADDITION AND SUBTRACTION

• FLOATING POINT ADDITION

To understand floating point addition, first we see addition of real numbers in decimal as same logic is applied in both cases.

For example, we have to add 1.1 * 10³ and 50.

We cannot add these numbers directly. First, we need to align the exponent and then, we can add significant.

After aligning exponent, we get 50 = 0.05 * 10³

Now adding significant, 0.05 + 1.1 = 1.15

So, finally we get (1.1 * 10³ + 50) = 1.15 * 10³

Here, notice that we shifted 50 and made it 0.05 to add these numbers.

Now let us take example of floating point number addition

We follow these steps to add two numbers:  

1. Align the significant

2. Add the significant  

3. Normalize the result

Let the two numbers be

x = 9.75  
y = 0.5625

Converting them into 32-bit floating point representation,

9.75’s representation in 32-bit format = 0 10000010 00111000000000000000000

0.5625’s representation in 32-bit format = 0 01111110 00100000000000000000000

Now we get the difference of exponents to know how much shifting is required.  

(10000010 – 01111110)2 = (4)10

Now, we shift the mantissa of lesser number right side by 4 units.

Mantissa of 0.5625 = 1.00100000000000000000000

(note that 1 before decimal point is understood in 32-bit representation)

Shifting right by 4 units, we get 0.00010010000000000000000

Mantissa of 9.75 = 1. 00111000000000000000000

Adding mantissa of both

0. 00010010000000000000000

+ 1. 00111000000000000000000

————————————————-

1. 01001010000000000000000

In final answer, we take exponent of bigger number

So, final answer consist of :

Sign bit = 0

Exponent of bigger number = 10000010

Mantissa = 01001010000000000000000

32 bit representation of answer = x + y = 0 10000010 01001010000000000000000

• FLOATING POINT SUBTRACTION

Subtraction is similar to addition with some differences like we subtract mantissa unlike addition and in sign bit we put the sign of greater number.

Let the two numbers be

x = 9.75  
y = – 0.5625

Converting them into 32-bit floating point representation

9.75’s representation in 32-bit format = 0 10000010 00111000000000000000000

– 0.5625’s representation in 32-bit format = 1 01111110 00100000000000000000000

Now, we find the difference of exponents to know how much shifting is required.

(10000010 – 01111110)2 = (4)10  
Now, we shift the mantissa of lesser number right side by 4 units.

Mantissa of – 0.5625 = 1.00100000000000000000000

(note that 1 before decimal point is understood in 32-bit representation)

Shifting right by 4 units, 0.00010010000000000000000

Mantissa of 9.75= 1. 00111000000000000000000

Subtracting mantissa of both

0. 00010010000000000000000

– 1. 00111000000000000000000

————————————————

1. 00100110000000000000000

Sign bit of bigger number = 0

So, finally the answer = x – y = 0 10000010 00100110000000000000000

This article has been contributed by Anuj Batham. 

In this continuation of the previous set, we will cover some more concepts and operations involved in computer arithmetic:

1. Signed vs. unsigned numbers: In computer arithmetic, numbers can be either signed or unsigned. Unsigned numbers only represent positive values, while signed numbers can represent both positive and negative values. Signed numbers are typically represented using two’s complement notation.
2. Shift operations: Shift operations are used to move the bits of a number left or right. A left shift multiplies the number by 2 raised to the power of the number of shifted bits, while a right shift divides the number by 2 raised to the power of the number of shifted bits.
3. Logical vs. arithmetic shifts: Shift operations can be either logical or arithmetic. Logical shifts insert zeros into the vacant bit positions, while arithmetic shifts preserve the sign bit when shifting a signed number.
4. Carry and borrow: In addition and subtraction operations, carry and borrow refer to the bit that is carried over or borrowed from the next digit position when the result of an operation exceeds the number of bits available.
5. Boolean algebra: Boolean algebra is a branch of mathematics that deals with logical operations on binary variables. It is widely used in computer arithmetic for operations such as bitwise AND, OR, and XOR.
6. Fixed-point arithmetic: Fixed-point arithmetic is used to perform arithmetic operations on numbers with a fixed number of decimal places. It is commonly used in applications such as digital signal processing.
7. Division algorithms: There are various algorithms used to perform division in computer arithmetic, including the restoring division algorithm, non-restoring division algorithm, and SRT division algorithm.
8. Multiplication algorithms: There are also various algorithms used to perform multiplication in computer arithmetic, including the Booth’s algorithm, the array multiplier, and the Wallace tree multiplier.

Overall, computer arithmetic is a complex and important field that underlies many aspects of modern computing. It involves a wide range of concepts and operations, from basic addition and subtraction to advanced algorithms for multiplication and division.