Before revision:  We choose F(y)=y^{-2}\sin^2 y and show that the expectation can be approximated within relative error 0< \epsilon < 1 in quasi-polynomial time (m+n)^{O(\ln (m+n)-\ln \epsilon)}, provided each form q_i depends on not more than r real variables, has common variables with at most r-1 other forms and satisfies |q_i(x)| \leq \gamma \|x\|^2/r, where \gamma >0 is an absolute constant  .
After revision:  We choose F(y)=y^{-2}\sin^2 y and show that the expectation can be approximated within relative error 0< \epsilon < 1 in quasi-polynomial time (m+n)^{O(\ln (m+n)-\ln \epsilon)}, provided each form q_i depends on not more than r real variables, has common variables with at most r-1 other forms and satisfies |q_i(x)| \leq \gamma \|x\|^2/r, where \gamma >0 is an absolute constant . This allows us to distinguish between "easily solvable" and "badly unsolvable" systems in some non-trivial situations .