[R] : Ramanujan and the accuracy of floating point computations - using Rmpfr in R

[Aditya Singh]

Ravi

I am a chemical engineer by training. Is there not something like law of corresponding states in numerical analysis?

Aditya



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On Thu 2 Jul, 2015 7:28 AM PDT Ravi Varadhan wrote:

>Hi,
>
>Ramanujan supposedly discovered that the number, 163, has this interesting property that exp(sqrt(163)*pi), which is obviously a transcendental number, is real close to an integer (close to 10^(-12)).
>
>If I compute this using the Wolfram alpha engine, I get:
>262537412640768743.99999999999925007259719818568887935385...
>
>When I do this in R 3.1.1 (64-bit windows), I get:
>262537412640768256.0000
>
>The absolute error between the exact and R's value is 488, with a relative error of about 1.9x10^(-15).
>
>In order to replicate Wolfram Alpha, I tried doing this in "Rmfpr" but I am unable to get accurate results:
>
>library(Rmpfr)
>
>
>> exp(sqrt(163) * mpfr(pi, 120))
>
>1 'mpfr' number of precision  120   bits
>
>[1] 262537412640767837.08771354274620169031
>
>The above answer is not only inaccurate, but it is actually worse than the answer using the usual double precision.  Any thoughts as to what I am doing wrong?
>
>Thank you,
>Ravi
>
>
>
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