Path: csiph.com!eternal-september.org!feeder.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Paul Rubin <no.email@nospam.invalid>
Newsgroups: comp.lang.forth
Subject: Re: real to rational number
Date: Wed, 19 Apr 2017 00:41:09 -0700
Organization: A noiseless patient Spider
Lines: 53
Message-ID: <87zifcetey.fsf@nightsong.com>
References: <d34926ea-ca84-4918-aa2f-82ebb21c6470@googlegroups.com> <87efwpf7cw.fsf@nightsong.com> <718428df-9808-4765-a9a1-48d43a971d24@googlegroups.com>
Mime-Version: 1.0
Content-Type: text/plain
Injection-Info: mx02.eternal-september.org; posting-host="47e0bd6af43efb397f12aac8c938d83c"; logging-data="12518"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18hpfNq3orKPTZHdL9sO4/x"
User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/24.3 (gnu/linux)
Cancel-Lock: sha1:Dv5ghFRz2Sr7xh3q6si7tU4GFfk= sha1:bmeDCEyjV1uoBPeGsPsVrPwXluY=
Xref: csiph.com comp.lang.forth:57584

krishna.myneni@ccreweb.org writes:
> Just so the problem is clear, I have a double precision value for r,
> which means I know it to about 16 significant digits. I know r, with
> infinite precision, is a rational number, but my machine
> representation is a rounded version to the limited precision. Now, I
> want to find "a" and "b" where r = a/b. Dividing the floating point
> representations of a and b, ra and rb, should give me a number equal
> to r in the machine representation.

For the r you gave, it's 8/105 in 64-bit gforth:

 7.61904761904762e-02 fvariable r  
 8e 105e f/ fvariable q
 q r f- f. 

=>   0.  ok

If you want to find the ratio by continued fractions, here's Haskell
code that I just wrote:

    import Data.Ratio
    import Control.Monad (forM_)

    r = 7.61904761904762e-02

    -- turn real number into continued fraction sequence
    contFrac :: RealFrac a => a -> [Integer]
    contFrac x = a : contFrac (1/b) where
      (a,b) = properFraction x

    -- turn truncated continued fraction into a rational number
    unFrac :: Integral a => [a] -> Ratio Integer
    unFrac [c] = fromIntegral c
    unFrac (c:cs) = fromIntegral c + 1/unFrac cs

    -- make rational from first n terms of cont frac of a real
    approx :: RealFrac a => Int -> a -> Ratio Integer
    approx n x = unFrac . take n . contFrac $ x

    -- print first 4 rational approximations of r
    main = forM_ [1..4] $ \i -> do
              print (approx i r)

The output for the first 4 terms is:

    *Main> main
    0 % 1
    1 % 13
    8 % 105
    70368744177665 % 923589767331853

You can see the 4th approximant is basically noise.  The 5th is much
worse noise, several dozen digits.