October 18th, 2010, 7:10 pm
I do wait for a concrete suggestion by renorm to learn about coding ...Concerning the (relative) errors: hoping for err < 0.5 ULP and all tweaks is likehoping for the quality of already implemented library solution, which may berelated to hardware ... unlikely to get it faster.The error of 2 EPS mainly has two sources:- evaluating a polynomial (for the central region) is not exact using the givenHorner form, even if the given polynomial is a minimax solution with err belowESP/8 (or so, and as exp(0) = 1 relative and abs error are the 'same'). The reason is, that the error increases by degree*EPS as a rule of thumb.Therefore one wants low degrees (or smaller ranges ---> table method).- the additive range reduction x - integer*ln(2) [or in more educated words:the modulus of x in Reals / Integers*ln(2)] is not exact enough (BTW thatalso occurs for trigonometric function as remainders of Pi, where the wayis Cody-Waite as the old and classical approach).I think, that both can be healed (as the libraries show), but not at cost=0(essentially one artificially goes beyond working precision in a 'local manner',i.e. for the problem only but not passing to extending precision in general).But costs for interval methods are tremendous and usually give estimates,not correctly rounded values.