[Scilab-users] Strange floating point issue below %eps

[Pinçon Bruno]

Le 18/09/2017 à 20:54, Samuel Gougeon a écrit :
>
> Now comes the issue:
> In (A), the relative difference 1/2^53 is too small (< %eps) to be 
> recorded and to change the number. OK.
> Since 1 / (2^53 +2) is even smaller than 1 / (2^53), it should nor 
> make a difference. Yet, it does:
>
> --> (2^53 + 2^1) + 1 == (2^53 + 2^1)
>  ans  =
>   F
>
> How is this possible ??!
>

Hi all,

    no issue here but simply the round to even rule. Every real number
   should be approximated by the nearest floating point number but in 
case of
   a number exactly between 2 successive floats the one with an even end
   digit win. You can see this rule as a mean to approximate those 
ambiguous cases,
   one in two down and one in two up, but has more subtle features in it 
such
   (it is explained in Knuth I).


   OK that2^53 + 2^1 is exactly represented
   (it is a double float number). When computing :

           x = (2^53 + 2^1) + 1

   it is not a float but is exactly at the same distance between the two 
floats :

2^53 + 2^1   and 2^53 + 2^2  but this last one ends with a even digit (0)
   so :
          fl( (2^53 + 2^1) + 1) = 2^53 + 2^2

   hth
  Bruno

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