1. Montek Singh Mon, Mar 28, 2011 Lecture 11
Computer Organization and Design Arithmetic Circuits: Multiplication and Division
Montek Singh
Mon, Mar 28, 2011
Lecture 11

2. Binary Multipliers Reading: Study Chapter 3.1-3.4
The key trick of multiplication is memorizing a digit-to-digit table… Everything else was just adding × 1 × 1 2 3 4 5 6 7 8 9 10 12 14 16 18 15 21 24 27 20 28 32 36 25 30 35 40 45 42 48 54 49 56 63 64 72 81
You’ve got to be kidding… It can’t be that easy!
Reading: Study Chapter
(including Booth coding [self study])

3. Binary Multiplication1 X
The “Binary” Multiplication Table
Binary multiplication is implemented using the same basic longhand algorithm that you learned in grade school.
Hey, that looks like an AND gate A3 A2 A1 A0 x B3 B2 B1 B0
A3B0
A2B0
A1B0
A0B0
AjBi is a “partial product”
A3B1
A2B1
A1B1
A0B1
A3B2
A2B2
A1B2
A0B2 +
A3B3
A2B3
A1B3
A0B3
Multiplying N-digit number by M-digit number gives (N+M)-digit result
Easy part: forming partial products (just an AND gate since BI is either 0 or 1)
Hard part: adding M, N-bit partial products

4. Multiplication: ImplementationS t a r t D o n e 1 . T s t M u l i p r a A d m c h P g 2 S f b 3 M u l t i p l i e r = 1 M u l t i p l i e r = t N o :
< 3 2 r e p e t i t i o n s 3 2 n d r e p e t i t i o n ? Y e s : 3 2 r e p e t i t i o n s

5. Second Version S t a r t M u l t i p l i c a n d D o n e 1 . T s t M ua A d m c h f P g 2 S b 3 3 2 b i t s M u l t i p l i e r = 1 M u l t i p l i e r = M u l t i p l i e r 3 2 - b i t A L U S h i f t r i g h t 3 2 b i t s S h i f t r i g h t P r o d u c t C o n t r o l t e s t W r i t e 6 4 b i t s i f t t h e M u l t i p l i e r r e g i s t e r r i g h t 1 b i t N o :
< 3 2 r e p e t i t i o n s 3 2 n d r e p e t i t i o n ? Y e s : 3 2 r e p e t i t i o n s

6. Example for second version
0010
Test true shift right 4
Test false shift right 3 2 1
1011
Initial
Product
Multiplicand
Multiplier
Step
Iteration

7. Final Version S t a r t D o n e 1 . T s t P r d u c a A m l i p h f g 2 S b 3 M u l t i p l i c a n d 3 2 b i t s P r o d u c t = 1 P r o d u c t = 3 2 - b i t A L U S h i f t r i g h t C o n t r o l P r o d u c t W r i t e t e s t 6 4 b i t s
The trick is to use the lower half of the product to hold the multiplier during the operation. N o :
< 3 2 r e p e t i t i o n s e p e t i t i o n ? Y e s : 3 2 r e p e t i t i o n s

8. What about the sign? Positive numbers are easy
How about negative numbers?
Please read Booth coding in textbook

9. Faster Multiply A1 & B A0 & B A2 & B A3 & B A31 & B P32-P63 P2 P1 P0

10. Simple Combinational Multiplier
tPD = 10 * tPD,FA
not 16 HA A
Co B S
tPD = (2*(N-1) + N) * tPD,FA
Components:
N * HA
N(N-1) * FA
The Logic of a Half- Adder CO
A B S
NB: this circuit only works for nonnegative operands

11. Carry-Save Combinational Multiplier
These Adders can be removed, and the AND gate outputs tied directly to the Carry inputs of the next stage.
Observation: Rather than propagating the sums across each row, the carries can instead be forwarded onto the next column of the following row
tPD = 8 * tPD,FA
This small improvement in performance hardly seems worth the effort, however, this design is easier to pipeline.
tPD = (N+N) * tPD,FA
Components:
N * HA
N2 * FA

12. Division
Start
1. Subtract Divisor from the Remainder leave the result in the Remainder
>=0
<0
Test Remainder
Restore Remainder by adding Divisor Shift Quotient to the left set its rightmost bit = 0
Shift Quotient to the left set its rightmost bit = 1
Shift Divisor Register right 1 bit
Repeat 33 times