We detail an algorithm that -for all but a 1 Ω(log(dH)) fraction of f ∈ Z[x] with exactly 3 monomial terms, degree d, and all coefficients in {−H, . . . , H} -produces an approximate root (in the sense of Smale) for each real root of f in deterministic time log 4+o(1) (dH) in the classical Turing model. (Each approximate root is a rational with logarithmic height O(log(dH)).) The best previous deterministic bit complexity bounds were exponential in log d. We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree d trinomial f ∈ Z[x] with coefficients in {−H, . . . , H}, at a point r ∈ Q of logarithmic height log H, in (deterministic) time log O(1) (dH). We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction 1 − 1 Ω(log(dH)) of the inputs (f, r).