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CPE 335 Computer Organization MIPS Arithmetic – Part I Content from Chapter 3 and Appendix B

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CPE 335 Computer OrganizationMIPS Arithmetic – Part IContent from Chapter 3 and Appendix B

Dr. Iyad Jafar

Adatped from Dr. Gheith Abandah Slides

http://www.abandah.com/gheith/Courses/CPE335_S08/index.html

- Computer programs calculate both positive a negative numbers.
- On approach is to use the sign and magnitude representation.
- Use separate bit for the sign
- Shortcomings:
- Where to put the sign ?
- Positive and negative zeros !
- Need complex hardware to perform arithmetic

- Alternative: use the complement notation; specifically the two’s complement !

maxint

MSB

LSB

minint

- 32-bit signed numbers (2’s complement):0000 0000 0000 0000 0000 0000 0000 0000two = 0ten0000 0000 0000 0000 0000 0000 0000 0001two = + 1ten...
0111 1111 1111 1111 1111 1111 1111 1110two = + 2,147,483,646ten0111 1111 1111 1111 1111 1111 1111 1111two = + 2,147,483,647ten1000 0000 0000 0000 0000 0000 0000 0000two = – 2,147,483,648ten1000 0000 0000 0000 0000 0000 0000 0001two = – 2,147,483,647ten...

1111 1111 1111 1111 1111 1111 1111 1110two = – 2ten1111 1111 1111 1111 1111 1111 1111 1111two = – 1ten

- If we use N bits to represent a signed number using two’s complement, then
- The maximum number is 2N-1 – 1
- The minimum number is -2N-1

1011

and add a 1

1010

complement all the bits

-23 =

-(23 - 1) =

- Negate

23 - 1 =

- Converting signed numbers to decimal
- (1111 1110 )2=
- -1*2^7 + 1*2^6 + 1*2^5 + 1*2^4 + 1*2^3 + 2^2 + 1*2^1 + 0*2^0 = -2

- Converting <32-bit values into 32-bit values
- Sign Extension: copy the most significant bit (the sign bit) into the “empty” bits0010 -> 0000 00101010 -> 1111 1010
- Zero Extension: place zeros in the extended bits.
- 0010 -> 0000 00101010 -> 0000 1010

- How to negate a number ?
- There is no special instruction
- Suppose we have x = - x
- sub $s0 , $zero , $s0

- This is in contrast to complementing a number !
- x = ~x
- Bitwise complement of x
- There is no single instruction
- Recall the XOR operation x 1 = x’
- addi$t0, $zero, -1
- xor$s0, $s0, $t0

- addvsaddu, subvssubu, and addivsaddiu
- Addition/subtraction is performed in the same manner
- Is overflow exception generated ?

- lbvslbu
- lb sign extend the additional bits
- lbuzero extend the additional bits

- sltvssltu and sltivssltiu
- sltandslti perform signed comparison with a constant
- sltuandsltiuperform unsigned comparison with a constant

- Suppose that
- $s0 = 1111 1111 1111 1111 1111 1111 1111 1111
- $s1 = 0000 0000 0000 0000 0000 0000 0000 0001
- then what is the value stored in $t0 in the following cases :
- slt $t0, $s1, $s0
- Signed comparison
- $t0 = 0 since $s1 = 1 and $s0 = -1

- sltu $t0, $s1, $s0
- Unsigned comparison
- $t0 = 1 since $s1 = 1 and $s0 = 2^32 -1

- slt $t0, $s1, $s0

- Binary addition is simple !
- 0 + 0 = 0 and 0 carry
- 0 + 1 = 1 and 0 carry
- 1 + 0 = 1 and 0 carry
- 1 + 1 = 0 and 1 carry

- Add corresponding bits and propagate the carry, if any, to the next bit.

S = A B carry_in

carry_out = A&B | A&carry_in | B&carry_in

carry_in

A

1-bit Full Adder

S

B

carry_out

- How can we use it to build a 32-bit adder?
- How can we modify it easily to build an adder/subtractor?

add/sub

c0=carry_in

A0

1-bit FA

S0

B0

c1

control

(0=add,1=sub)

A1

1-bit FA

B0 if control = 0, !B0 if control = 1

S1

B0

B1

c2

A2

1-bit FA

S2

B2

c3

. . .

c31

A31

1-bit FA

S31

B31

c32=carry_out

- Remember 2’s complement is just
- complement all the bits
- add a 1 in the least significant bit

- Subtraction is equivalent to adding the negative of the number

A 0111 0111 B - 0110 +

1001

1

0001

1 0001

0

1

1

1

1

0

0

1

1

1

7

1

1

0

0

–4

+

0

0

1

1

3

+

1

0

1

1

– 5

1

0

1

0

0

1

- Overflow: the result is too large to represent in 32 bits
- Overflow occurs when
- adding two positives yields a negative
- or, adding two negatives gives a positive
- or, subtract a negative from a positive gives a negative
- or, subtract a positive from a negative gives a positive

- On your own: Prove that you can detect overflow by:
- Carry into MSB XOR Carry out of MSB, ex for 4 bit signed numbers

– 6

1

1

7

- Need to support the logic operations
- Need to support arithmetic operations
- Need to support the set-on-less-than instruction
- Need to support test for equality
- Immediates are sign or zero extended outside the ALU with wiring (i.e., no logic needed)

zero

ovf

1

1

A

32

ALU

result

32

B

32

4

m (operation)

- Must support the Arithmetic/Logic operations of the ISA
add, addi, addiu, addu

sub, subu, neg

mult, multu, div, divu

sqrt

and, andi, nor, or, ori, xor, xori

beq, bne, slt, slti, sltiu, sltu

- With special handling for
- sign extend – addi, addiu andi, ori, xori, slti, sltiu
- zero extend – lbu, addiu, sltiu
- no overflow detected – addu, addiu, subu, multu, divu, sltiu, sltu

- Start with 1-bit ALU
- Can easily implement the logic instruction ANDandOR since they map directly to hardware.
- Perform all possible operations in parallel then use a multiplexer to select the result based on the instruction type.
- The control signal Operation is issued by the control unit

- For the ADD instruction, use a full adder. The CarryIn input will be used later on to expand the 1-bit ALU to n-Bit.
- Expand the multiplexer inputs and select lines to accommodate for the add instruction.

- For the subtract instruction, we use 2’s complement subtraction.
- We need to complement B and add 1.
- Define Binvert to select between B and B’ and set CarryIn to 1.
- Combine Binvertand
CarryInin one signal

Bnegatesince they

have the same value

all the time.

- Supporting the NOR operation requires no separate gate.
- Use Demorgan’s theorem and the AND gate and define the signal Ainvert
- (A+B)’ = A’.B’

- Constructing 32-bit ALU
- Replicate the 1-bit ALU and
connect the CarryIn signals

- All cells receive the
same control signals

- Supporting the SLT instruction
- Expand the multiplexer for one more input.
- Subtract the two registers and feed the sign bit (the result of bit 31) back to the least significant bit.
- The slt input of the multiplexer is connected to 0 for remaining bits .

LSB

MSB

- 32-bit ALU with SLT support.

- Supporting conditional branch instructions
- Need to generate a signal that indicates whether the result is zero or not.
- Simply OR the result bits and take the complement.
- This signal will be used to make the selection between the branch address and the PC.

- Example on using the Zero signal on selecting the address for BEQ instruction

- Final ALU with overflow detection

- Control signals values and corresponding operations

- The ripple-carry adder is slow
- We have to wait until the carry is propagated to the final position in order to read out the addition/subtraction result.
- Carry generation is associated with two levels of gates at each bit position (Coi = AiBi + AiCini + BiCini).
- Total delay = gate delay x 2 x number of bits
- Example: 16 bit adder delay is 32 delay units

- Need fast way to find the carry
- Design a separate unit that computes carries for different bits in parallel !

- In a 4 bit adder, the equations of the carries are
c1 = (b0 . c0) + (a0 . c0) + (a0 . b0)

c2 = (b1 . c1) + (a1 . c1) + (a1 . b1)

c3 = (b2 . c2) + (a2 . c2) + (a2 . b2)

c4 = (b3 . c3) + (a3 . c3) + (a3 . b3)

- By substitution
c2 = (a1 . a0 . b0) + (a1 . a0 . c0) + (a1 . b0 . c0) + (b1 . a0 . b0) + (b1 . a0 . c0 ) + (b1 . b0 . c0) + (a1.b1)

c3 = (b2 . a1 . a0 . b0) + (b2 . a1 . a0 . c0) + (b2 . a1 . b0 . c0) + (b2 . b1 . a0 . b0) + (b2 . b1 . a0 . c0 ) + (b2 . b1 . b0 . c0) + (b2 . a1 . b1) + (a2 . a1 . a0 . b0) + (a2 . a1 . a0 . c0) + (a2 . a1 . b0 . c0) + (a2 . b1 . a0 . b0) + (a2 . b1 . a0 . c0 ) + (a2 . b1 . b0 . c0) + (a2 . a1 . b1) + (a2.b2)

c4 = ……

- Imagine the equation if the adder is 32 bits ?? .

- We can reduce the logic cost by simple simplification
- ci+1 = (bi . ci) + (ai . ci) + (ai . bi)
= (ai . bi) + (ai + bi) . ci

= gi + pi . ci

- gi : carry generate
- pi : carry propagate

- ci+1 = (bi . ci) + (ai . ci) + (ai . bi)
- Carry equations for 4 bit adder
- c1 = g0 + p0 . c0
- c2 = g1 + (p1 . g0) + (p1 . p0 . c0)
- c3 = g2 + (p2 . g1) + (p2 . p1 . g0) + (p2 . p1 . p0 . c0)
- c4 = g3 + (p3 . g2) + (p3 . p2 . g1) + (p3 . p2 . p1 . g0) +
(p3 . p2 . p1 . p0 . c0)

- Still cost is high for larger adders ! ! !

- Assume 16 bit adder that consists of 4 single 4-bit adders with carry-lookahead implementation
- We can generate the carries using three levels of gates in parallel
- Delay to generate C4 is 3 gates

- Need to generate the carry propagate and generate signals at higher level
- Think of each 4-bit adder block as a single
unit that can either generate of propagate

a carry.

- Super propagate signals
- P0 = p3⋅p2⋅p1⋅p0
- P1 = p7⋅p6⋅p5⋅p4
- P2 = p11⋅p10⋅p9⋅p8
- P3 = p15⋅p14⋅p13⋅p12

- Super generate signals
- G0 = g3+(p3 ⋅ g2)+(p3⋅p2⋅g1)+(p3⋅p2⋅p1⋅g0)
- G1 = g7+(p7 ⋅ g6)+(p7⋅p6⋅g5)+(p7⋅p6⋅p5⋅g4)
- G2 = g11+(p11 ⋅ g10)+(p11⋅p10⋅g9)+(p11⋅p10⋅p9⋅g8)
- G3 = g15+(p15 ⋅ g14)+(p15⋅p14⋅g13)+(p15⋅p14⋅p13⋅g12)

- Carry signal at higher levels are
- C1 = G0 + (P0 ⋅ c0)
- C2 = G1 + (P1 ⋅ G0) + (P1⋅P0⋅c0)
- C3 = G2 + (P2 ⋅ G1) + (P2⋅P1⋅G0) + (P2⋅P1⋅P0⋅c0)
- C4 = G3 + (P3 ⋅ G2) + (P3⋅P2⋅G1) + (P3⋅P2⋅P1⋅G0)
+ (P3⋅P2⋅P1⋅P0⋅c0)

- Each supper carry signal is two level implementation in terms of Pi and Gi
- Pi is one level of gates while Gi is two and expressed in terms of pi and gi
- pi and gi are one level of gates
- Total delay is 2 + 2 + 1 = 5
- 16-bit CLA is 5 times faster than the 16-bit ripple carry adder