dilogRemezApprox.m

(*
  Approximation of PolyLog[2,x]/x by a rational function using Remez
  algorithm on the interval [0, 1/2).
 *)

Needs["FunctionApproximations`"];

(* maximum number of terms in Li2 series,
  so that [0,0.5]^nMax << machine precision *)
nMax = 200;

(* claculation precision *)
prec = 100;

(* output precision *)
outPrec = 20;

(* series for PolyLog[2,x]/x *)
Li2x[x_, n_:nMax] := Sum[x^(i-1)/i^2, {i,1,n}];

{abscissa, {appox, maxErr}} =
MiniMaxApproximation[Li2x[x], {x, {0,1/2}, 7, 7},
                     WorkingPrecision -> prec,
                     MaxIterations -> 1000];

FormatCoeffs[expr_, x_, prec_] :=
    NumberForm[#, prec]& /@ CoefficientList[expr, x]

Print["Numerator coefficients: ",
      FormatCoeffs[#,x,outPrec]& @ Numerator[appox]];

Print["Denominator coefficients: ",
      FormatCoeffs[#,x,outPrec]& @ Denominator[appox]];

Print["max error: ", InputForm @ maxErr];