the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively. An even more amazing approximation
uses Catalan's constant and the Feigenbaum constant
is given by
which is accurate to within (D. Ross, cited in Pegg 2005).
A curious (although not particularly useful) approximation due to D. Barron is given by