  Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973)
and Brent (JACM 1974) show that any algebraic formula of size s can be
converted to one of depth O(log s) with only a polynomial blow-up in size. In
this paper, we consider a fine-grained version of this result depending on the
degree of the polynomial computed by the algebraic formula. Given a homogeneous
algebraic formula of size s computing a polynomial P of degree d, we show that
P can also be computed by an (unbounded fan-in) algebraic formula of depth
O(log d) and size poly(s). Our proof shows that this result also holds in the
highly restricted setting of monotone, non-commutative algebraic formulas. This
improves on previous results in the regime when d is small (i.e., d<<s). In
particular, for the setting of d=O(log s), along with a result of Raz (STOC
2010, JACM 2013), our result implies the same depth reduction even for
inhomogeneous formulas. This is particularly interesting in light of recent
algebraic formula lower bounds, which work precisely in this ``low-degree" and
``low-depth" setting. We also show that these results cannot be improved in the
monotone setting, even for commutative formulas.
