Multiplication and Division

Multiplication

• More complicated than addition

• A straightforward implementation will involve shifts and adds.

• More complex operation can lead to

• More space (on silicon) and/or

• More time (multiple cycles or longer clock cycle time)

Multiplying Integers (Unsigned/Signed)

Solution 1: (Unsigned Integers)

• Multiply using Straightforward Algorithm.

• The Straightforward algorithm only works for unsigned integers.

Solution 2: (Signed Integers)

• Multiply only magnitudes using Straightforward Algorithm.

• If signs are different, negate the answer.

Solution 3: (Signed Integers)

• Booth’s algorithm

• Developed by A.D. Booth in 1950 at Birkbeck College in Bloomsbury,

London.

• Works for signed integers.

Booth’s Algorithm

n – Required bits in 2’s complement representation

A - n-bits register

M - n-bits register

Q - n-bits register

Q-1

- 1-bit Register

Booth’s Algorithm

• If the multiplicand/multiplier is negative

• Represent in Two’s Complement Representation

• If any carry is generated in addition

• Discard the carry

n - Minimum required bits to represent both M and Q in Two’s Complement Representation

A - n-bits register

M - n-bits register

Q - n-bits register

Q-1

- 1-bit Register

Final Product in A,Q

Booth’s Algorithm

❑3X7

❑How many bits are required ?

❑Check Range: -2n-1

to (2n-1

-1)

❑4-bits

❑ Initialize values-

Booth’s Algorithm

Result (AQ) = 0001 0101 (Positive Integer) = 21

Booth’s Algorithm

n – Required bits in 2’s complement representation

A - n-bits register

M - n-bits register

Q - n-bits register

Q-1

- 1-bit Register

Multiplying Signed Integers

• Example: (3) X (-7) using 4-bit Two’s Complement.

• 0011 X 1001

A Q Q -1 M Operations Count

0000 0011 0 1001 -- 4

Q0Q-1 == 10,

A = A – M = A + 2’s complement of M

A = 0000 + 2’s complement of 1001 = 0000 + 0111 =

A = A – M = 0111

0111 0011 0 1001 Subtract (A = A – M)

0011 1001 1 1001 ARS 3

Multiplying Signed Integers

• Example: (3) X (-7) using 4-bit Two’s Complement.

• 0011 X 1001

A Q Q -1 M Operations Count

0000 0011 0 1001 -- 4

0111 0011 0 1001 Subtract (A = A – M)

0011 1001 1 1001 ARS 3

0001 1100 1 1001 ARS 2

A = A +M = 0001 + 1001 = 1010

1010 1100 1 1001 Add (A = A + M)

1101 0110 0 1001 ARS 1

1110 1011 0 1001 ARS 0

Multiplying Signed Integers

• Example: (3) X (-7) using 4-bit Two’s Complement.

• 0011 X 1001

A Q Q -1 M Operations Count

1110 1011 0 1001 ARS 0

Result = 1110 1011 (Negative Integer)

= Two’s complement of (1110 1011)

= - (0001 0101) = -21

Multiplying Signed Integers

Exercise-1

Multiply 20 and 15

Exercise-2

Multiply 20 and -15

Exercise-3

Multiply -20 and 15

Exercise-4

Multiply -20 and -15

Why Booth Algorithm works?

• The multiplicand/multiplier (M/Q) may be positive or negative.

• Positive (M/Q) - n-bit binary equivalent

• Negative (M/Q) - n-bit Two’s complement

Why Booth Algorithm works?

• Booth's algorithm performs an Arithmetic Right Shift for all combinations of

0 and 1

• 00, 11, 01, 10

• When it encounters Q0Q-1 = 01

• It performs an addition

• When it encounters the end of the block of ones (01)

• When it encounters Q0Q-1 = 10

• It performs a subtraction

• When it encounters the first digit of a block of ones (10)

Why Booth Algorithm works?

Why Booth Algorithm works?

Property: The sum of series of numbers (Which can be represented in continuous

power of 2) can be reduced to two operations i.e. one addition and one subtraction

2n +2n-1 + ———— +2n-K = 2n+1 - 2n-K

Example

• (24 + 23 + 22 + 21) = (25 - 21)

• (27 + 26 + 23 + 22) = (28 – 26 + 24 - 22)

• 22 = (23 - 22)

Why Booth Algorithm works?

• First check for unsigned numbers,

• 19X 30

• M X Q

• 0001 0011 X 0001 1110

• M X 0001 1110

• 0001 0011 X (24 + 23 + 22 + 21)

• 0001 0011 X (30)

• 0001 0011 X (32-2)

• 0001 0011 X (25 - 21)

• 19X (25 - 21)

Why Booth Algorithm works?

Positive Integers

Due to previous property, the product can be generated by one addition and one

subtraction of the multiplicand.

• M X Q

• 0001 0011 X 0001 1110

• 0001 0011 X (24 + 23 + 22 + 21)

• 0001 0011X (25 - 21)

• Check it using Booth’s Algorithm

This scheme extends to any number of blocks of 1s in a multiplier, including

the case in which a single 1 is treated as a block.

• 0001 1110 X 0001 0011

• 0001 1110 X (24 + 21 + 20)

• 0001 1110 X (25 - 24 + 22 - 20)

• Check it using Booth’s Algorithm

Why Booth Algorithm works?

Negative Integers

The MSB is 1 for negative integers.

• X = 1xn-2xn-3…….x1x0

• X = -2n-1 + xn-2*2n-2 + xn-3*2n-3 ……… + x1*21 + x0*20

Example:

(-97)10 = -(0110 0001) = 1001 1111

1001 1111 = -27 + 1*24 + 1*23 + 1*22 + 1*21 + 1*20

1001 1111 = -128 + 16 + 8 + 4 + 2 + 1

1001 1111 = -128 + 31 = -97

Why Booth Algorithm works?

Assume X = 1 1 1….110xk-1

xk-2

….x1x

0

• X = -2n-1 + 2n-2 + 2n-3 +…… 2k+1 + xk-1*2k-1 + xk-2*2k-2 +…… + x1*21 +

x0*20

• X = -2n-1 + 2n-2+1 - 2k+1 + xk-1*2k-1 + xk-2*2k-2 +…… + x1*21 + x0*20

• X = - 2k+1 + xk-1*2k-1 + xk-2*2k-2 +…… + x1*21 + x0*20

As the algorithm scans over the leftmost 0 and encounters the next 1 (2k + 1

), a 1–0 transition occurs and a

subtraction takes place ( - 2k + 1

).

Why Booth Algorithm works?

Negative Integers

In Two's complement representation, using an 8-bit word, (-6) is represented as 1111 1010

M*(-6) = M* (Two’s complement of 6) = M*(One’s complement of 6 + 1)

= M*1111 1010

1111 1010 = -27 +2

6 +2

5 +2

4 +2

3 +2

1

M*(11111010) = M*(-27 +2

6 +2

5 +2

4 +2

3+ 2

1) = M*(-2

7 +2

7 - 2

3+ 2

2 -2

1)

M*(11111010)=M*(-23 + 2

2 -2

1)

Replace M = 7 = 0000 0111, Compute using Booth’s Algorithm

Why Booth Algorithm works?

Negative Integers

In twos complement representation, using an 8-bit word, (-15) is represented as 1111 0001

M*(-63) = M* (Two complement of 63) = M*(One’s complement of 63 + 1)

= M* 1111 0001

1111 0001 = -27 +2

6 +2

5+2

4 +2

0

M*(1100 0001) = M*(-27 +2

6 +2

5 +2

4 +2

0) = M*(-2

7 +2

7 - 2

4+ 2

1 -2

0)

M*(1100 0001) = M*(-24 + 2

1 -2

0)

Replace M = 7 = 0000 0111, Compute using Booth’s Algorithm