[R] Why does sin(pi) not return 0?

[Martin Maechler]

>>>>> Suzen, Mehmet <msuzen at gmail.com>
>>>>>     on Thu, 26 Sep 2013 14:48:52 +0200 writes:

    > On 26 September 2013 11:30, Rainer M Krug <Rainer at krugs.de> wrote:
    >>>>> Why doesn't return me 0?

    > It isn't R question at all. You might want to read about representing
    > real numbers in a computer using floating point
    > http://en.wikipedia.org/wiki/Floating_point

    > If you want more precision for some reason, you may want to use Rmpfr
    > package from CRAN, for example
    >> require(Rmpfr)
    >> pii <- mpfr(pi, 1200)
    >> sin(pii)
    > 1 'mpfr' number of precision  1200   bits
    > [1] 1.22464679914735317722606593227499799708305390129979194948825771626086960997325810377509325527569013655456428540074414189136673810003656057935764118217436637676835016019778833613838580470703060741630570066750947925902443295873487819032259435513861185501796412843027607796970259523768923503206248925733373776859085615900203929142965774524665617260404787862664073939e-16

Thank you, Suzen, for mentioning Rmpfr.
In that case, I think you should also show how to get a more
accurate approximation of 0:

The most convenient way with Rmpfr is to use the builtin pi, in
the following :

> pii <- Const("pi") #  default precision of 120 bits
> sin(pii)
1 'mpfr' number of precision  120   bits 
[1] -5.5059930380881434362856026294599943387e-37
> ## or
> pii <- Const("pi", prec = 240)
> pii
1 'mpfr' number of precision  240   bits 
[1] 3.1415926535897932384626433832795028841971693993751058209749445923078164072
## and indeed the above is pi accurate to about 74 decimal digits
> sin(pii)
1 'mpfr' number of precision  240   bits 
[1] -8.9200160854418520294613808989866371392121893805952954481681649396960358864e-73
> asNumeric(sin(pii))
[1] -8.920016e-73
> 

So you see, the more accurate the approximation  pii  for the true
\pi, the more accurate is the result  of  sin(pii) 


Martin Maechler, ETH Zurich
(and author of Rmpfr)