(* List of fast Deleglise-Rivat alpha factors found by running pi(x) benchmarks. *)

alpha = {(* {x, alpha} *) {1, 1},{10^7,2.210}, {10^9,3.590}, {10^10,4.900 }, {10^11, 6.211},  {10^12,8.154},{10^13, 9.857},{10^14, 11.715},{10^15,14.628},{10^16,18.138},{10^17, 23.375},{10^18, 28.992},{10^19,34.240},{10^20,42.741},{10^21,53.706},{10^22,67.605},{10^23,83.126}}

ListLogLinearPlot[alpha,Filling->Bottom, Joined-> True]

(* alpha is a tuning factor that balances the compuation of the easy special leaves and the hard special leaves. The formula below is used in the file src/common.cpp to calculate a fast alpha factor for the computation of pi(x). *)

NonlinearModelFit[alpha, a(Log[x])^3 +b(Log[x])^2 +c  Log[x] + d, {a,b, c, d}, x]