Why doesn't Rationalize work for 0. in Interval?

Question

Same motivation as this q. Why doesn't Rationalize work for 0.?

Interval[{0., 1.}] // Map[Rationalize, #, {2}] &

Interval[{-4.45015*10^-308,1}]

There's nothing wrong with 0. by itself:

{0., 1.} // Map[Rationalize]

{0, 1}

Answer 1

I believe this has to do with the fact that intervals "grow" just a bit on evaluation with machine numbers to ensure that values at the endpoint will be included in the interval.

NestList[Interval @@ # &, Interval[{0., 1.}], 5] // InputForm

(*
{Interval[{-2.2250738585072014*^-308, 1.0000000000000002}], 
 Interval[{-4.450147717014403*^-308, 1.0000000000000004}], 
 Interval[{-6.675221575521604*^-308, 1.0000000000000007}], 
 Interval[{-8.900295434028806*^-308, 1.0000000000000009}], 
 Interval[{-1.1125369292536007*^-307, 1.000000000000001}], 
 Interval[{-1.3350443151043208*^-307, 1.0000000000000013}]}
*)

I'm only guessing at why it is designed this way. The docs give a pretty vague description.

"For approximate machine- or arbitrary-precision numbers x, Interval[x] yields an interval reflecting the uncertainty in x."

Edit:

Just to make it a little more interesting, this is what it appears to be doing to the endpoints.

NestList[# + {$MinMachineNumber, $MachineEpsilon} &, {0., 1.}, 5] // InputForm

(*
{{0., 1.}, 
 {2.2250738585072014*^-308, 1.0000000000000002}, 
 {4.450147717014403*^-308, 1.0000000000000004}, 
 {6.675221575521604*^-308, 1.0000000000000007}, 
 {8.900295434028806*^-308, 1.0000000000000009}, 
 {1.1125369292536007*^-307, 1.000000000000001}}
*)

I suspect that in general the step is $MachineEpsilon but zero is a special case as it often is in Mathematica.