[Haskell-cafe] Canned routines for the first say thousand digits of pi, e, sqrt 2, etc?
Jeffrey Brown
jeffbrown.the at gmail.com
Fri Dec 18 20:20:54 UTC 2015
Thanks everybody! It turns out WolframAlpha will do the computation
<https://www.reddit.com/r/piano/comments/3uz8oj/the_melody_of_pi_226_digits_chromatic_%CF%80_base_12/cy39b5g>
for
me. (It won't let me copy the digits, so I have to transcribe them by hand,
but given how much time I'm spending reviewing the notes anyway, that is a
small part of the overall labor cost.)
On Fri, Dec 18, 2015 at 4:31 AM, Olaf Klinke <olf at aatal-apotheke.de> wrote:
> As it happens, I am just studying a presentation [1] Martin Escardo gave
> to students at the University of Birmingham. It contains Haskell code for
> exact real number computation. Among other things, there is a function that
> computes a signed digit representation of pi/32. It computes several
> thousand digits in a few seconds.
> I did not try it yet, but many irrational numbers are fixed points of
> simple arithmetical expressions. For example, the golden ratio is the fixed
> point of \x -> 1+1/x. Infinite streams of digits should be a type where
> such a fixed point is computable. Or you could use a sufficiently precise
> rational approximation and convert that do decimal in the usual way.
>
> import Data.Ratio
> import Data.List (iterate)
>
> -- one step of Heron's algorithm for sqrt(a)
> heron :: (Fractional a) => a -> a -> a
> heron a x = (x+a/x)/2
>
> -- infinite stream of approximations to sqrt(a)
> approx :: (Fractional a) => a -> [a]
> approx a = iterate (heron a) 1
>
> -- Find an interval with rational end-points
> -- for a signed-digit real number
> type SDReal = [Int] -- use digits [-1,0,1]
> interval :: Int -> SDReal -> (Rational,Rational)
> interval precision x = let
> f = foldr (\d g -> (a d).g) id (take precision x))
> a d = \x -> ((fromIntegral d)+x)/2
> in (f(-1),f(1))
>
> Cheers,
> Olaf
>
> [1] www.cs.bham.ac.uk/~mhe/.talks/phdopen2013/realreals.lhs
--
Jeffrey Benjamin Brown
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