standard book ok computer architecture

Binary Multiplication
CMPE 325 Computer
Architecture II

• At each step we multiply the multiplicand by a single digit from the multiplier.
– In binary, we multiply by either 1 or 0 (much simpler than decimal).

Cem Ergün
Eastern Mediterranean University

• Keep a running sum instead of storing and adding all the partial products at the end.

Multiplication & Division Algorithms
Cem Ergun

Multiplication

Implementing Multiplication

• Long-hand multiplication (12 × 9 = 108)
1 1 0 0 = 12ten
× 1 0 0 1 = 9ten
----------------------1 1 0 0
0 0 0 0 0
0 0 0 0 0 0
+ 1 1 0 0 0 0 0
----------------------1 1 0 1 1 0 0 = 108ten

2
Multiplication & Division Algorithms

• Several different ways to implement
• Things to note about previous method

(n bit Multiplicand)
(m bit Multiplier)

– At each step we either copied or set the value to 0
– LSBs of the product don’t change once computed
⇒ Can form product by shifting and adding

• Solution #1
– Shift multiplicand left at each step (same as adding 0’s)
– Shift multiplier right so we can always consider the 0’th bit

(n + m bit Product)

• The result is a number that is n + m - 1 bits long

Cem Ergun

3
Multiplication & Division Algorithms

Solution #1:
Operations

• Multiplication without sign (12 × 9 = 108)
(n bit Multiplicand)
(m bit Multiplier)

Multiplicand

0 0 0 0
1 1 1

Multiplier

0 0 0 0
1 1

Multiplier0 = 1

1. Test
Multiplier0

Multiplier0 = 0

1a. Add multiplicand to product and place the result in Product register

2. Shift the Multiplicand register left 1 bit

3. Shift the Multiplier register right 1 bit

1 0 0 1

Product

4
Multiplication & Division Algorithms

Start

Solution #1
1 1 0 0 = 12ten
× 1 0 0 1 = 9ten
----------------------1 1 0 0
0 0 0 0 0
0 0 0 0 0 0
+ 1 1 0 0 0 0 0
----------------------1 1 0 1 1 0 0 = 108ten

Cem Ergun

1 1 0 0
0 0

32nd repetition?

No: < 32 repetitions

Yes: 32