Fix tan

[dpsanders]

No description provided.

[dpsanders]

Tests are failing for values right on quadrant boundaries, but I don't see what to do about that.
Tests are now passing.

[codecov-io]

Codecov Report

Merging #22 into master will increase coverage by <.01%.
The diff coverage is 100%.

@@            Coverage Diff             @@
##           master      #22      +/-   ##
==========================================
+ Coverage   92.14%   92.15%   +<.01%     
==========================================
  Files          22       22              
  Lines         917      918       +1     
==========================================
+ Hits          845      846       +1     
  Misses         72       72

Impacted Files	Coverage Δ
src/multidim/intervalbox.jl	81.25% <ø> (ø)	⬆️
src/intervals/trigonometric.jl	99.1% <100%> (ø)	⬆️

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[lbenet]

Tests are passing, so I think we should also merge this.

[lbenet]

I'm reopening this because the issue isn't really solved.

Current master yields the following:

julia>    # This examples appears in the tests
x = 2.0^1000
1.0715086071862673e301

julia> Interval(x)
Interval(1.0715086071862673e301, 1.0715086071862673e301)

julia> big(2.0)^1000 ∈ ans   # OK
true

julia> # This seems to be a corner case
x = Interval(2.0)^100
Interval(1.2676506002282294e30, 1.2676506002282294e30)

julia> tan(x)
Interval(-1.7829551493767193, -1.782955149376719)

julia> big(2.0)^100 ∈ ans    # WRONG
false

julia> # Another case
x = Interval(2.0)^60
Interval(1.152921504606847e18, 1.152921504606847e18)

julia> tan(x)  # USEFUL ?
Interval(-Inf, Inf)

julia> prevfloat(tan(x.lo)), nextfloat(tan(x.hi))
(1.4918373962705977, 1.4918373962705977)

julia> tan(big(2.0)^60) # this is contained by the bounds above 
1.491837396270597700887517480636152973049592658670418036396829287651786928528278

The problem seems to be in find_quadrant, more concretely in this line, which for intervals with large values returns results that are not accurate enough to be useful. (Note that currently, temp in that function corresponds to Interval(Inf,Inf), for Interval(2.0)^1000 in the tests, which makes no sense.)

[dpsanders]

I agree that there's an issue with large values of the argument, but I have no idea how to fix it!

I believe that Siegfried Rump has a method for this.

[lbenet]

For the case of tan I think I understand where our method is weak; that weekness should somehow also appear elsewhere.

[dpsanders]

All the trig functions have problems with large arguments. But I don't think it's worth spending that much effort on myself.