merged lin_reg python files, added comment on Ramanujan's version of gamma

function to sterling_approx;

1 files changed, 12 insertions, 6 deletions

diff --git a/sterling_approx.awk b/sterling_approx.awk
index 5abe41d..ca119ac 100644
--- a/sterling_approx.awk
+++ b/sterling_approx.awk
@@ -1,17 +1,23 @@
 #!/usr/bin/awk -f
 
+
+# https://en.wikipedia.org/wiki/Sterling_Approximation
+# An alternative approximation for the Gamma function stated by Srinivasa
+# Ramanujan (Ramanujan 1988) is
+#    Gamma(1+x) ~= sqrt(pi)((x/e)^x)(8x^3 + 4x^2 + x + 1/30)^(1/6)
+# for x >= 0. The equivalent approximation for ln(n!) has an asymptotic error
+# of 1/(1400*n^3) ...
+
+
 ### sterling_approx.awk
 # https://en.wikipedia.org/wiki/Stirling%27s_approximation
 
 BEGIN {
     ARGV[1] ? n = ARGV[1] : n = 0
     pi = 4*atan2(1,1)
-    p = 0
+    f = 0
     if (n > 0) {
-        p = 1
-        for (m=n; m>0; m--)
-            p *= n*exp(-1)
-        p = sqrt(2*pi*n)*p*(1 + 1/(12*n) + 1/(288*n*n) - 139/(51840*n*n*n) - 571/(2488320*n*n*n*n))
+        f = sqrt(2*pi*n)*exp(n*log(n*exp(-1)))*(1 + 1/(12*n) + 1/(288*n*n) - 139/(51840*n*n*n) - 571/(2488320*n*n*n*n))
     }
-    printf(OFMT ORS, p)
+    printf(OFMT ORS, f)
 }