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
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