Efficient method for generating the smoothing function

Question

The smoothing function of a boolean function with respect to one of its variables is the disjunction of its cofactors. For example given a Boolean function F(a,b,c) the cofactors with respect to a are F(0,b,c) and F(1,b,c), the smoothing function with respect to a therefore is G(b,c) = F(0,b,c) + F(1,b,c).

Now, given a circuit for F, one simple method for creating the circuit for G is to make two copies of the circuit for F, hardwire the inputs for a as 0 and 1 in the two circuits respectively and then OR the outputs. However this will double the number of gates needed. If one needs to do this for several variables, then the size of the resulting circuit grows exponentially in the number of variables removed. What bothers me the most about this is that we are creating simpler functions (as judged by the number of variables and consequently the size of the truth table) with increasing number of gates. This leads me to think that maybe there is a well known way for creating a circuit for the smoothing function using fewer gates than in the original circuit.

For the purpose of this question, let us assume that the truth tables are too big to enumerate and the functions under consideration are not linear.