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}

You might want to adjust the second argument of Rationalize[] if seen fit. — J. M.'s missing motivation, Aug 05 '15 at 20:30
@belisarius, try this: Interval[{0., 1.}] // Map[Rationalize[#, 0.0000000000000001] &, #, {2}] & — alancalvitti, Aug 05 '15 at 20:33
@Guesswhoitis., most positive values of 2nd parameters seem to work, though note above counterexample. But I want to know why this issue is Interval specific. — alancalvitti, Aug 05 '15 at 20:34
Interesting also is Interval[{0., 1.}]/2 // Rationalize for demonstrating how bad my floating-point intuition is. — Patrick Stevens, Aug 05 '15 at 20:36
@PatrickStevens Perhaps Interval[{0., 1.}]/2 // Rationalize // Trace and $MinMachineNumber shows that "though this be madness, yet there is method in't." — Michael E2, Aug 05 '15 at 23:00
Again follows from standard evaluation semantics. If you want an exact zero, either use an exact zero or Rationalize before evaluating Interval. — Daniel Lichtblau, Aug 06 '15 at 00:29

Andy Ross · Answer 1 · 2015-08-05T23:12:50.907

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.

What's the transitive closure? This is a disaster waiting to happen in temporal queries or real time systems. — alancalvitti, Aug 05 '15 at 21:20

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

1 Answers1

Linked