[Rd] floor and ceiling can't handle more than 15 decimal pla (PR#8591)

[ted.harding@nessie.mcc.ac.uk]

On 12-Feb-06 benphalan at gmail.com wrote:
> Full_Name: Ben Phalan
> Version: 2.2.1
> OS: Win XP
> Submission from: (NULL) (131.111.111.231)
> 
> 
> I have noticed that floor returns the wrong number when there are more
> than 15
> decimal places:
> 
>> floor(6.999999999999999)
> [1] 6
>> floor(6.9999999999999999)
> [1] 7
> 
> There is a similar problem with ceiling, so this may apply to all/most
> rounding functions?
> 
>> ceiling (2.000000000000001)
> [1] 3
>> ceiling (2.0000000000000001)
> [1] 2

This is not a problem (nor a bug) with 'floor' or 'ceiling'.
The "problem" (in quotes because the real problem is the user's)
is in R, intrinsic to the finite-length floating-point arithmetic
which is used. See:

  > 6.999999999999999 - 7
  [1] -8.881784e-16
  > 6.9999999999999999 - 7
  [1] 0
  > 2.000000000000001 - 2
  [1] 8.881784e-16
  > 2.0000000000000001 - 2
  [1] 0

so, in fact, R cannot "see" the 16th decimal place when you enter
a number to that precision -- it is simply lost. Exactly the same
"problem" would arise at some point whatever the finite precision
to which a floating-point number is stored. The effect is not
confined to functions 'floor' and 'ceiling' or any similar
"rounding" functions. It applies to all functions; it is simply
more obvious with the rounding functions.

Enter

  .Machine

and the first two items in the output are:

  $double.eps
  [1] 2.220446e-16

  $double.neg.eps
  [1] 1.110223e-16

showing that the smallest difference which can be "seen" by R
is greater than 1-^(-16).

So, when you type it in, you *think* you have entered

    2.0000000000000001

into R, but you have not. So the user has to face the problem of
how to cope with the finite-length representation in any situation
where the distinction between 2 and 2.0000000000000001 really
matters.

Hoping this helps,
Ted.

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Date: 12-Feb-06                                       Time: 15:37:53
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