[R] Odd underflow(?) error

[Thomas Lumley]

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 
.Machine$double.xmin. They can report zero, or they can add leading zeros 
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