(* List of fast Gourdon alpha factors (alpha = alpha_y * alpha_z) found by running pi(x) benchmarks using the find_optimal_alpha_y.sh script *)

alphaGourdon =  {   {10^1,1},{10^8,2.746},{10^9,3.177}, {10^10,4.047}, {10^11, 5.694}, {10^12,6.470}, {10^13,8.336},  {10^14,11.210}, {10^15,15.016},  {10^16,19.231}, {10^17,25.050}, {10^18,31.139}, {10^19,37.573}, {10^20,47}, {10^21,56.783}, {10^22,71.804}, {10^23,91.799},{10^24,114.265}}

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

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

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