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1. Csci 136 Computer Architecture IICsci 136 Computer Architecture II
–– Constructing An Arithmetic Logic UnitConstructing An Arithmetic Logic Unit
Xiuzhen Cheng
[email protected]

2. Announcement
Homework assignment #4: Due on Feb 22, before
class
Readings: Sections 3.2, 3.3, B.5, B.6
Problems: 3.7, 3.9, 3.10, 3.12, 3.18-3.22
Project #1 is due on 11:59PM, Feb 13.
Project #2 is due on 11:59PM, March 10.

3. What Are We Going To Do?
Implementing the MIPS ISA architecture!
Must support the arithmetic/logic operations
Tradeoffs of cost and speed based on frequency of occurrence,
hardware budget.
32
32
32
operation
result
a
b
ALU

4. Negating a two's complement number: invert all
bits and add 1
remember: “negate” and “invert” are quite different!
Converting n bit numbers into numbers with more
than n bits:
MIPS 16 bit immediate gets converted to 32 bits for arithmetic
copy the most significant bit (the sign bit) into the other bits
0010 -> 0000 0010
1010 -> 1111 1010
"sign extension" (lbu vs. lb)
Review on Two's Complement Operations

5. Just like in grade school (carry/borrow 1s)
0111 0111 0110
+
 0110
-
 0110
-
 0101
Two's complement operations easy
subtraction using addition of negative numbers
0111
+
 1010
Overflow (result too large or too small for finite
computer word):
Example: -8 <= 4-bit binary number <=7
Review on Addition & Subtraction

6. No overflow when adding a positive and a negative
number
No overflow when signs are the same for
subtraction
Overflow occurs when the value affects the sign:
overflow when adding two positives yields a negative
or, adding two negatives gives a positive
or, subtract a negative from a positive and get a negative
or, subtract a positive from a negative and get a positive
Detecting Overflow

7. An exception (interrupt) occurs
Control jumps to predefined address for exception
Interrupted address is saved for possible resumption
Details based on software system / language
Don't always want to detect overflow
— new MIPS instructions: addu, addiu,
subu
note: addiu still sign-extends!
note: sltu, sltiu for unsigned comparisons
Effects of Overflow

8. Not easy to decide the “best” way to build something
Don't want too many inputs to a single gate
Don’t want to have to go through too many gates
For our purposes, ease of comprehension is important
Let's look at a 1-bit ALU for addition:
Building blocks: AND, OR, Inverter, MUX, etc.
How could we build a 1-bit ALU for add, and, and or?
How could we build a 32-bit ALU?
Different Implementations for ALU
c
out
= a b + a c
in
+ b c
in
sum = a xor b xor c
in
Sum
CarryIn
CarryOut
a
b

9. A 1-bit ALU
AND and OR
A logic unit performing logic
AND and OR.
Full Adder
A 1-bit Full Adder ((3,2)
adder).
Implementation of a 1-bit
adder
A 1-bit ALU that performs
AND, OR, and addition
Figure B.5.6

10. A 32-bit ALU, Ripple Carry Adder
A 32-bit ALU for AND,
OR and ADD operation:
connecting 32 1-bit ALUs

11. What About Subtraction
Invert each bit (by inverter) of b and add 1
How do we implement?
A very clever solution: a + (-b) = a + (b’ +1)

12. What About NOR Operation?
Explore existing
hardware in the ALU
NOR (a,b) = not (a or b) =
not(a) and not(b)
Only need to add an
inverter for input a

13. 32-Bit ALU for AND, OR, ADD, SUB, NOR
Binvert
Ainvert

14. In-Class Question
Prove that you can detect overflow by
CarryIn31 xor CarryOut31
that is, an overflow occurs if the CarryIN to the most
significant bit is not the same as the CarryOut of the
most significant bit

15. Set on Less Than Operation
Idea: For
slt $t0, $s1, $s2
Check $s1-$s2. If negative,
then set the LSB of $t0 to 1,
set all other bits of $t0 to 0;
otherwise, set all bits of $t0 to
0.
How to set these bits? – Less
input line
Implementation: Connect
Result31 to Less
Overflow detection

16. Set on Less Than Operation
Question:
In figure B.5.11, Set connects
directly to Less0.
Can you find out any problem
in this implementation?
How to fix it?

17. Conditional Branch
MIPS Instruction:
beq $S1, $s2,
label
Idea:
Test $s1-$s2. Use
an OR gate to test
whether the result is
0 or not. It $s1=$s2,
set a zero detector.

18. A Final 32-bit ALU
Operations supported: and, or, nor, add, sub, slt, beq/bnq
ALU Control lines: 2 operation control signal for and, or, add,
and slt, 2 control line for sub, nor, and slt
ALU Control Lines Function
0000 And
0001 Or
0010 Add
0110 Sub
0111
1100
Slt
NOR

19. SPEED of the 32-bit Ripple Carry Adder
CarryOut, Result
The critical path
Path traversed by CARRY: contains 32 and gates, 32 or gates.
We must sequentially evaluate all 1-bit adders.
 Ripple-carry is too low to be used in time-critical hardware
Speedup: anticipate the carry!
 needs more hardware for parallel operation
An extreme case – sum of products
“Infinite” hardware through two-level logic
An example implementation – too expensive!
 the number of gates grows exponentially!
 How many gates are needed to compute
c1: c2: c3: c4:

20. Carry-Lookahead Adder
The concept of propagate and generate
c(i+1) = (ai . bi) + (ai . ci) + (bi . ci) = (ai . bi) + ((ai+bi).ci)
pi = ai + bi;gi = ai . bi
Write down c1, c2, c3, c4.
Why fast?
First Level of abstraction: pi, gi
Still too expensive! Why?
 still large number of gates
 How many gates are needed to compute
c1: c2: c3: c4:

21. Carry-Lookahead Adder
CarryOut is 1 if
some earlier
adder generates
a carry and all
intermediary
adders propagate
the carry.

22. Build Bigger Adders
Can’t build a 16-bit adder with carry lookahead!
Could use ripple carry of 4-bit adder
Use carry lookahead at higher levels
“Super” Propagate Pi vs. “Super” Generate Gi
Group concept, 4-bit adder as a building block
“Super” Propagate/Generate definition
c4 = g3 + (p3.g2) + (p3.p2.g1) + (p3.p2.p1.g0) +
(p3.p2.p1.p0.c0) = C1

23. Super Propagate and Generate
A “super” propagate is
true only if all propagates
in the same group is true
P0= P1=P2=P3=
A “super” generate is
true only if at least one
generate in its group is
true and all the
propagates downstream
from that generate are
true.
G0= G1= G2= G3=

24. A 16-Bit Adder
Give the equations for
C1, C2, C3, C4?
Better: use the CLA
principle again!
Second-level of
abstraction!

25. An Example
Determine gi, pi, Gi, Pi, and C1, C2, C3, C4 for the
following two 16-bit numbers:
a:0010 1001 0011 0010
b:1101 0101 1110 1011
Do it yourself

26. Speed of Ripple Carry vs. Carry Lookahead
Example: Assume each AND and OR gate take the
same time. Gate Delay is defined to be the number of
gates along the critical path. Compare the gate delays
of three 16-bit adder, one using ripple carry, one
using first-level carry lookahead, and one using two-
level carry lookahead.

27. Summary
Traditional ALU can be built from a multiplexor plus a
few gates that are replicated 32 times
To tailor to MIPS ISA, we expand the traditional ALU
with hardware for slt, beq, and overflow detection
Carry lookahead is much faster than ripple carry!
CLA principle can be applied multiple times!

28. Questions?