[R] Numerical stability of: 1/(1 - cos(x)) - 2/x^2

[Leonard Mada]

Dear R-Users,

I tried to compute the following limit:
x = 1E-3;
(-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
# 0.4299226
log(2)/2 + 1/12
# 0.4299069

However, the result diverges as x decreases:
x = 1E-4
(-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
# 0.9543207
# correct: 0.4299069

I expected the precision to remain good with x = 1E-4 or x = 1E-5.

This part blows up - probably some significant loss of precision of 
cos(x) when x -> 0:
1/(cos(x) - 1) - 2/x^2

Maybe there is some room for improvement.

Sincerely,

Leonard
==========
The limit was part of the integral:
up = pi/5;
integrate(function(x) 1 / sin(x)^3 - 1/x^3 - 1/(2*x), 0, up)
(log( (1 - cos(up)) / (1 + cos(up)) ) +
     + 1/(cos(up) - 1) + 1/(cos(up) + 1) + 2*log(2) - 1/3) / 4 +
     + (1/(2*up^2) - log(up)/2);

# see:
https://github.com/discoleo/R/blob/master/Math/Integrals.Trig.Fractions.Poly.R