Tweaking Ramanujan's Approximation of n!

About 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation $n!$. He gave a good formula which asymptotic to Since then hundreds papers have given alternative proofs his result and improved upon it, including notably by Burside, Gosper, Mortici. However, Srinivasa Ramanujan remarkably better formula. Hirschhorn Villarino nice proof Ramanujan's an error estimate for approximation. In recent years there been several improvements Nemes, Windschitl, Chen. Here it shown (i) how all these results can be easily verified; (ii) Villarino's argument allows tweaking give approximation; (iii) that new formulae obtained further result. Tables are calculated displaying each approximations $n$ up one million.