# [R] Odd underflow(?) error

Thomas Lumley tlumley at u.washington.edu
Sat Dec 4 00:22:07 CET 2004

```On Fri, 3 Dec 2004, William Faulk wrote:

> I'm still trying to install R on my Irix machine.  Now I have a new problem
> that crops up during the checks.  I've found the root cause, and it's that R
> is returning zero for certain things for reasons I don't understand.
>
> 2.225073859e-308, entered directly into R, responds "2.225074e-308".
> 2.225073858e-308 responds "0".
>
> Their negative values respond similarly, so it would appear that somewhere in
> there is the smallest absolute value that that installation of R will hold.

Yes.  .Machine\$double.xmin tells you the smallest number representable to
full precision, which is 2.225074e-308 (I think on all machines where R
works)

> On another machine where the checks passed, both responses are correct, not
> just the first one.  The underflow there is significantly lower, with much
> less accuracy, as opposed to what seems to be good accuracy on what looks
> like the broken one.  The values there are:
>
> 2.4703282293e-324 gives 4.940656e-324
> 2.4703282292e-324 gives 0

Machines can differ in what they do with numbers smaller than
and so lose precision.  Suppose you had a 4-digit base 10 machine with 2
digits of exponent.  The smallest number representable to full accuracy
would be
1.000e-99
but by allowing the leading digits to be zero you could represent
0.001e-99
ie, 1e-102, to one digit accuracy (these are called "denormalized"
numbers).

My Mac laptop denormalizes, and agrees with your other computer, giving
the smallest representable number as 4.940656e-324. It is
.Machine\$double.xmin/2^52.   This number has very few bits left to
represent values, so for example
> (a/2^52)*1.3==(a/2^52)
[1] TRUE
where a is .Machine\$double.xmin

Both your machines should be correct. I don't think we deliberately
require denormalized numbers to work anywhere.

-thomas

```