2-bit fast adder; full subtractor from half subtractors

2-bit binary fast adder

A fast adder (carry look-ahead adder) is a parallel adder in which all carries are produced directly from the input bits and the input carry, instead of waiting for the carry to ripple through each stage. This makes the addition time almost independent of the number of bits.

For each bit:

$$G_i = A_iB_i \ (\text{carry generate}), \qquad P_i = A_i \oplus B_i \ (\text{carry propagate})$$

Carries for 2 bits ($A_1A_0 + B_1B_0 + C_0$):

$$\begin{aligned} C_1 &= G_0 + P_0C_0 \\ C_2 &= G_1 + P_1C_1 = G_1 + P_1G_0 + P_1P_0C_0 \end{aligned}$$

Sums:

$$S_0 = P_0 \oplus C_0, \qquad S_1 = P_1 \oplus C_1$$

A0,B0 -[XOR]-P0   -[AND]-G0
A1,B1 -[XOR]-P1   -[AND]-G1

C1 = G0 + P0.C0           (1 AND, 1 OR)
C2 = G1 + P1.G0 + P1.P0.C0 (2 AND, 1 OR)

S0 = P0 xor C0 ; S1 = P1 xor C1  (2 XOR)
C2 is the output carry.

Since $C_2$ comes from a two-level AND–OR circuit, it is ready after the same delay as $C_1$.

Full subtractor using half subtractors

$$D = A \oplus B \oplus B_{in}, \qquad B_o = A'B + (A \oplus B)'B_{in}$$

• HS1 ($A$, $B$): $D_1 = A \oplus B$, $B_1 = A'B$
• HS2 ($D_1$, $B_{in}$): $D = D_1 \oplus B_{in}$, $B_2 = D_1'B_{in}$
• $B_o = B_1 + B_2$ (one OR gate)

A --->[ HS1 ]--D1-->[ HS2 ]----------> D
B --->[     ]  Bin->[     ]
         |B1           |B2
         +----->[OR]<--+
                  +------------------> Bo