The Look-Ahead AdderLecture 22Section C.6

Robb T. Koether

Hampden-Sydney College

Wed, Oct 23, 2019

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1 Look-Ahead Adders

2 Propagate and Generate

3 A 16-bit Adder

4 Subtraction

5 Assignment

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Outline

1 Look-Ahead Adders

2 Propagate and Generate

3 A 16-bit Adder

4 Subtraction

5 Assignment

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The Carry Bits

In the case of the ripple adder, we saw some of the operationscould be done simultaneously, such as adding ai and bi .Other operations had to wait for the carry-in bit from the previouscolumn.The look-ahead adder pushes this idea much further.It puts a premium on computing the carry-in bits for all columns asquickly as possible.

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Outline

1 Look-Ahead Adders

2 Propagate and Generate

3 A 16-bit Adder

4 Subtraction

5 Assignment

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The Carry Bits

In the case of the ripple adder, we saw that the carry-out bit maybe computed as

ab + ac + bc,

where a and b are the two bits to be added and c is the carry-inbit.This formula will give the carry-out bit for each column of amulti-bit adder.

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The Carry Bits

The Carry Bits

c0 = 0,c1 = b0c0 + a0c0 + a0b0 = (a0 + b0)c0 + a0b0,

c2 = b1c1 + a1c1 + a1b1 = (a1 + b1)c1 + a1b1,

c3 = b2c2 + a2c2 + a2b2 = (a2 + b2)c2 + a2b2,

c4 = b3c3 + a3c3 + a3b3 = (a3 + b3)c3 + a3b3,

...

ci represents the carry-in bit and ci+1 is the carry-out bit for thei-th column.

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 7 / 28

The Carry Bits

The Carry Bits

c1 = b0c0 + a0c0 + a0b0,

c2 = b1(b0c0 + a0c0 + a0b0) + a1(b0c0 + a0c0 + a0b0) + a1b1,

= b1b0c0 + b1a0c0 + b1a0b0 + a1b0c0 + a1a0c0 + a1a0b0 + a1b1,

etc.

We may substitute the formula for c1 into the expression for c2,and so on.Thus, we may express each ci in terms of the ai ’s, bi ’s, and c0.This will quickly get out of hand.

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The Generate and Propagate Bits

The Generate and Propagate Bits

ci+1 = aibi + (ai + bi)ci ,

gi = aibi ,

pi = ai + bi .

Instead, we will calculate the generate and propagate bits.In column i , a carry of 1 is generated if aibi = 1.In column i , a carry of 1 is propagated from the previous column ifai + bi = 1.

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The Generate and Propagate Bits

The Carry Bits

ci+1 = gi + pici .

The carry bit ci+1 is 1 if, in the i-th column, eitherGenerates a carry (gi = 1), orPropagates a carry (pi = 1) from the previous column (ci = 1), orBoth.

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The Generate and Propagate Bits

The Carry Bits

c0 = c0,

c1 = g0 + p0c0,

c2 = g1 + p1c1

= g1 + p1(g0 + p0c0)

= g1 + p1g0 + p1p0c0,

c3 = g2 + p2c2

= g2 + p2 (g1 + p1g0 + p1p0c0)

= g2 + p2g1 + p2p1g0 + p2p1p0c0,

We now have more compact expressions for ci in terms of c0.

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 11 / 28

The Generate and Propagate Bits

The Carry Bits

c4 = g3 + p3c3,

= g3 + p3(g2 + p2g1 + p2p1g0 + p2p1p0c0)

= g3 + p3g2 + p3p2g1 + p3p2p1g0 + p3p2p1p0c0,

etc.

We now have more compact expressions for ci in terms of c0.

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The Generate and Propagate Bits

Using the logic of the generate and propagate bits, we firstcompute gi = aibi and pi = ai + bi for all columns simultaneously.We can see that c1 through c4 can be calculated using only 5more levels of logic.

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A Four-bit Adder

1

2 3 4 etc.

p0

p0c0 p1(p0c0) p3(p2p1p0c0) . . .

g0

p1g0 p2(p1g0) (g1 + p1g0) + (p1p0c0) . . .

p1

p2g1 p3(p2g1) (g2 + p2g1) + (p2p1g0) . . .

g1

p3g2 g1 + (p1g0) (g3 + p3g2) + (p3p2g1) . . .

p2

p2p1 g2 + (p2g1)...

g2

p3p2 g3 + (p3g2)

p3

... (p2p1)(p0c0)

g3

...

Using the logic of the generate and propagate bits, we firstcompute aibi and ai + bi for all positions simultaneously.How many levels of logic are required to compute s0, s1, s2, s3,and c4?

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 14 / 28

A Four-bit Adder

1 2

3 4 etc.

p0 p0c0

p1(p0c0) p3(p2p1p0c0) . . .

g0 p1g0

p2(p1g0) (g1 + p1g0) + (p1p0c0) . . .

p1 p2g1

p3(p2g1) (g2 + p2g1) + (p2p1g0) . . .

g1 p3g2

g1 + (p1g0) (g3 + p3g2) + (p3p2g1) . . .

p2 p2p1

g2 + (p2g1)...

g2 p3p2

g3 + (p3g2)

p3...

(p2p1)(p0c0)

g3

...

Using the logic of the generate and propagate bits, we firstcompute aibi and ai + bi for all positions simultaneously.How many levels of logic are required to compute s0, s1, s2, s3,and c4?

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 14 / 28

A Four-bit Adder

1 2 3

4 etc.

p0 p0c0 p1(p0c0)

p3(p2p1p0c0) . . .

g0 p1g0 p2(p1g0)

(g1 + p1g0) + (p1p0c0) . . .

p1 p2g1 p3(p2g1)

(g2 + p2g1) + (p2p1g0) . . .

g1 p3g2 g1 + (p1g0)

(g3 + p3g2) + (p3p2g1) . . .

p2 p2p1 g2 + (p2g1)

...

g2 p3p2 g3 + (p3g2)

p3... (p2p1)(p0c0)

g3...

Using the logic of the generate and propagate bits, we firstcompute aibi and ai + bi for all positions simultaneously.How many levels of logic are required to compute s0, s1, s2, s3,and c4?

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 14 / 28

A Four-bit Adder

1 2 3 4

etc.

p0 p0c0 p1(p0c0) p3(p2p1p0c0)

. . .

g0 p1g0 p2(p1g0) (g1 + p1g0) + (p1p0c0)

. . .

p1 p2g1 p3(p2g1) (g2 + p2g1) + (p2p1g0)

. . .

g1 p3g2 g1 + (p1g0) (g3 + p3g2) + (p3p2g1)

. . .

p2 p2p1 g2 + (p2g1)...

g2 p3p2 g3 + (p3g2)

p3... (p2p1)(p0c0)

g3...

Using the logic of the generate and propagate bits, we firstcompute aibi and ai + bi for all positions simultaneously.How many levels of logic are required to compute s0, s1, s2, s3,and c4?

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 14 / 28

A Four-bit Adder

1 2 3 4 etc.p0 p0c0 p1(p0c0) p3(p2p1p0c0) . . .g0 p1g0 p2(p1g0) (g1 + p1g0) + (p1p0c0) . . .p1 p2g1 p3(p2g1) (g2 + p2g1) + (p2p1g0) . . .g1 p3g2 g1 + (p1g0) (g3 + p3g2) + (p3p2g1) . . .

p2 p2p1 g2 + (p2g1)...

g2 p3p2 g3 + (p3g2)

p3... (p2p1)(p0c0)

g3...

Using the logic of the generate and propagate bits, we firstcompute aibi and ai + bi for all positions simultaneously.How many levels of logic are required to compute s0, s1, s2, s3,and c4?

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 14 / 28

Outline

1 Look-Ahead Adders

2 Propagate and Generate

3 A 16-bit Adder

4 Subtraction

5 Assignment

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 15 / 28

A 16-bit Adder

It is not practical to continue in this manner through 16 bits, to saynothing of 32 bits or 64 bits.Instead, we view the four-bit adder as a unit.A 16-bit adder consists of 4 such units.Each unit has its own carry-in and carry-out bits.We apply the same logic to these 4 units.

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 16 / 28

A 16-bit Adder

A Four-bit Adder

Four-bitAdder

a0 b0a1 b1a2 b2a3 b3

s0s1s2s3

c4 c0

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A 16-bit Adder

Higher-level Generate and Propagate Bits

C0 = c0,

P0 = p3p2p1p0,

G0 = g3 + p3g2 + p3p2g1 + p3p2p1g0.

The 4-bit adder may generate the carry bit c4 or it may propagateit from c0.We define P0 and G0 as shown above.Then we compute c4 as

C1 = G0 + P0C0.

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 18 / 28

A 16-bit Adder

Higher-level Generate and Propagate Bits

C0 = c0,

P0 = p3p2p1p0,

G0 = g3 + p3g2 + p3p2g1 + p3p2p1g0,

C1 = c4

= g3 + p3g2 + p3p2g1 + p3p2p1g0 + p3p2p1p0c0,

= G0 + P0C0.

Robb T. Koether (Hampden-Sydney College) The Look-Ahead Adder Wed, Oct 23, 2019 19 / 28

A 16-bit Adder

A 4-bit Adder

Four-bitAdder

a0 b0a1 b1a2 b2a3 b3

s0s1s2s3

C0

G0 P0

Therefore, the 4-bit adder will output P0 and G0 instead of c4.

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A 16-bit Adder

Four 4-bit Adders

Four-bitAdder

a4 b4a5 b5a6 b6a7 b7

s4s5s6s7

C1

G1 P1

Four-bitAdder

a0 b0a1 b1a2 b2a3 b3

s0s1s2s3

C0

G0 P0

Four-bitAdder

a12b12a13b13a14b14a15b15

s12s13s14s15

C3

G3 P3

Four-bitAdder

a8 b8a9 b9a10b10a11b11

s8s9s10s11

C2

G2 P2

Join together four such units.

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A 16-bit Adder

A 16-bit Adder

Four-bitAdder

a0..3 b0..3

s0..3

C0

G0 P0C1

Four-bitAdder

a4..7 b4..7

s4..7

G1 P1C2

Four-bitAdder

a8..11 b8..11

s8..11

G2 P2

Four-bitAdder

a12..15b12..15

s12..15

G3 P3 C3C4

carry out

Carry Look-ahead Unit

Add the carry look-ahead unit to compute the carry-in to each4-bit adder.The final carry C4 is the carry-out of the 16-bit adder.

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A 64-bit Adder

How would we design a 32-bit look-ahead adder?How would we design a 64-bit look-ahead adder?

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Outline

1 Look-Ahead Adders

2 Propagate and Generate

3 A 16-bit Adder

4 Subtraction

5 Assignment

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Assignment

AssignmentRead Section C.6.

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