Dan Piponi writes:
Jules Bean said:
If it is true of many Floating instances that (atan 1 * 4) is an accurate way to calculate pi (and it appears to be 'accurate enough' for Float and Double, on my computer) then adding it as a default doesn't appear to do any harm.
... For the case of power series as an instance of Num, using 4*atan 1 gives me the wrong thing as it triggers an infinite summation, whereas I'd want pi to simply equal the constant power series. Now you could counter that by saying that power series are an esoteric case. But apart from code in the libraries, most code I've seen that provides an instance of Num is doing something mildly esoteric.
For me series are not esoteric. But they can be implemented in a more cautious way, my own package behaves rationally: Prelude> :l Series [1 of 1] Compiling Series ( Series.hs, interpreted ) -- .... nasty messages omitted. Ok, modules loaded: Series. *Series> 4*atan 1 3.141592653589793 *Series> 4*atan 1 :: Series Double Loading package haskell98 ... linking ... done. [3.141592653589793,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0, ...] *Series> pi 3.141592653589793 *Series> pi :: Series Double [3.141592653589793,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0, ...] *Series> The implementation is, with (:-) as the cons-series operator, colon-dash for those whose browsers will convert it to some nice smiley...: zeros = 0:-zeros ... instance Floating a => Floating (Series a) where pi = pi:-zeros ... atan u@(u0:-_) = sInt (atan u0) (sDif u / (1+u*u)) etc. Of course, sDif differentiates and sInt integrates the series, see 2nd volume of Knuth for the algorithms, and papers, either a nice tutorial of Doug McIlroy, or my old paper for the implementation. The instance does it all. Default *could* help, and would give the same answer, but it would be much slower, so I would override it anyway. Jerzy Karczmarczuk