Computing spinor equivalence for positive definite forms

Given an integral positive-definite rank $$n$$ quadratic form $$f$$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $$f$$ contains more than one spinor genus. My question is: given two (integral positive-definite) forms $$f,g$$ in the same genus, such that the genus contains more than one spinor genus, can one efficiently determine if the forms lie in distinct spinor genera?

I am interested in $$n\geq5$$.