document use of interval arithmetic

[was]

This email snippet pretty much describes the issue:

On Tue, May 5, 2009 at 11:16 PM, Henryk Trappmann <bo198214@googlemail.com> wrote:
>
> Ah now I see, you mean though it displays 1/384 it is internally still
> the above sum, which is computed when evaluated with n.

True.  In Sage right now the internal form of the expression (not the simplified form) is used by the "n" command.  This will change when Sage switches from using maxima-based symbolics to Pynac. 

By the way, you can use the real interval field to get a floating point approximation to any symbolic expression to a given number of digits of precision.  Every digit is definitely right except the one right before the question mark:

sage: a = 1/(48*sqrt(1)) - 7/(96*1**(3/2)) + 3/(32*1**(5/2)) - 5/ (128*1**(7/2))
sage: RealIntervalField(100)(a)             # increase 100 for more digits
0.002604166666666666666666666667?

I think the n function (numerical_approx) for symbolics should be changed to using interval arithmetic.  This will be a huge improvement.  Notice, e.g., that interval arithmetic very nicely gives the right answer for the infamous sin(10^50):

sage: a = sin(10^50)
sage: a.n(53)
-0.480500143493759
sage: a.n(100)
0.60974154556722786199645650055
sage: RealIntervalField(100)(a)
0.?
sage: RealIntervalField(1000)(a)
-0.7896724934293100827102895399174077539600834046214027191457808736221899969800609898633436757589688470442999273506152178357769064871103469499564331175635613221319397479785737324994506546860108913238488404198306006819757685879489185272089985858148036954222175628785469474395231359019098600625732453528693?
sage: RealIntervalField(2000)(a)
-0.78967249342931008271028953991740775396008340462140271914578087362218999698006098986334367575896884704429992735061521783577690648711034694995643311756356132213193974797857373249945065468601089132384884041983060068197576858794891852720899858581480369542221756287854694743952313590190986006257324535286926640214204183176856658976160340849634781130568053474154330242776565926107540133198976420887112928640131582614537425282391078909233424580311555104358881651194953182665408243214532152322603956371555619997139323527489307648072219268176687894373677502675114853503742816202001868587837402822439060931321957?

[kcrisman]

Disagree, primarily because it's been this way for quite a while now, and because a lot of users would be really confused by the question mark. We certainly could update the documentation to make it clear how to get much more accurate answers using RIF.