1. CDA 3101 Summer 2007 Introduction to Computer Organization
Division
14 June 2007

2. Review
Multiplication can be implemented using simple shift and add hardware
Algorithm is similar to the grade-school method
Booth’s algorithm for two’s complement
Handles all cases of positive/negative operands
More efficient than conventional algorithm
2n + 2n-1 + 2n-2 + … + 2k = 2n+1 - 2k
Further optimization is possible
MIPS multiply instructions ignore overflow
Multu: Hi = 0
Mult: Hi = replicated sign of Lo

3. Dividend = Quotient x Divisor + Remainder
Division
Long division of unsigned binary integers
1 0 0
Quotient
Dividend
Divisor
Partial remainders
Remainder
Dividend = Quotient x Divisor + Remainder

4. Unsigned Division M Divisor Q Dividend Count n, A 0 START
Shift left: A, Q
A A - M No
Yes
A < 0? Q
A A + M Q
Count Count - 1 No
Yes
Quotient in Q
Remainder in A
Count = 0 ?
END

5. Division Hardware Divisor M31 . . . M0 32-bit ALU 4’ Add 2 Subtract
4’ write 0
Control
4 write 1
1 SLL 3
A A0
Q Q0
Dividend

6. Examples
A Q
M = 0011
7 / 3 :
Initial values
Shift
A = A - M
A = A + M 1
Shift
A = A - M
A = A + M 2
Shift
A = A - M
Q0 = 1 3
Shift
A = A - M
A = A + M 4

7. Signed Division Simplest solution
Negate the quotient if the signs of divisor and dividend disagree
Remainder and dividend must have the same signs
Remainder = (Dividend – Quotient * Divisor)
(+7) / (+3): Q = 2; R = 1
(-7) / (+3): Q = -2; R = -1
(+7) / (-3): Q = -2; R = 1
(-7) / (-3): Q = 2; R = -1

8. Signed Division Algorithm
A, Q Dividend
M Divisor, Count n
START
Shift left: A, Q
S A31, Count Count - 1 No
Yes
A A + M
M31 = A31?
A A - M No
Yes
S = A31?
Yes No
A=0 and
Q=0 ? Q
Restore A
Yes
Quotient in Q
Remainder in A No
Count = 0 ?
END

9. Examples (1/2)
A Q
1101
1110
0000
M = 0011
A Q
1101
1110
0000
M = 1101
Initial values
Initial values
Shift
Subtract
Restore
Shift
Add
Restore 1 1
Shift
Subtract
Restore
Shift
Add
Restore 2 2
Shift
Subtract
Q0 = 1
Shift
Add
Q0 = 1 3 3
Shift
Subtract
Restore
Shift
Add
Restore 4 4
(7) / (3)
(7) / (-3)

10. Examples (2/2)
A Q
0010
0001
1111
M = 0011
A Q
0010
0001
1111
M = 1101
Initial values
Initial values
Shift
Add
Restore
Shift
Subtract
Restore 1 1
Shift
Add
Restore
Shift
Subtract
Restore 2 2
Shift
Add
Q0 = 1
Shift
Subtract
Q0 = 1 3 3
Shift
Add
Restore
Shift
Subtract
Restore 4 4
(-7) / (3)
(-7) / (-3)

11. MIPS Multiply and divide use existing hardware
ALU and shifter
Extra hardware: 64-bit register able to SLL/SRA
Hi contains the remainder (mfhi)
Lo contains the quotient (mflo)
Instructions
Div: signed divide
Divu: unsigned divide
MIPS ignores overflow ?
Division by 0 must be checked in software

12. MIPS Processor ALU Sign extend Zero Overflow Registers Memory address
IR:
Control Unit
Sign extend 16 32
ALU
Zero
Overflow
Registers
Memory address
Sub
Operation M Lo Hi
SLL/SRA

13. Have a Great Weekend!!