You could rewrite 1 - cos(x)
as 2 * sin(x/2)^2 and that might give you more precision? On Wed, Aug 16, 2023, 01:50 Leonard Mada via R-help <r-help@r-project.org> wrote: > 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 > > ______________________________________________ > R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
