Size-depth tradeoffs for algebraic formulae

Some tradeoffs between the size and depth of algebraic formulas are proved. It is shown that, for any fixed ∈>0, any algebraic formula of size S can be converted into an equivalent formula of depth O(log S) and size O(S^1+∈). This result is an improvement over previously known results where, to obtain the same depth bound, the formula size is Ω(S^α), with α⩾2