Avoid convert in ^ for literal powers

[lbenet]

Fix #618

[lbenet]

This is related to the test that fails:

julia> f(x)  = interval(2)^x
f (generic function with 1 method)

julia> f′(x) = log(interval(2)) * f(x)
f′ (generic function with 1 method)

julia> df(t) = ForwardDiff.derivative(f, t)
df (generic function with 1 method)

julia> f′(0) === df(0)
false

# But
julia> isequal_interval(f′(0), df(0))
true

The subtlety in === is that df(0) is NG (because of ForwardDiff).

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (21e84ff) 83.56% compared to head (255d9b2) 83.41%.
Report is 1 commits behind head on master.

❗ Your organization needs to install the Codecov GitHub app to enable full functionality.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #619      +/-   ##
==========================================
- Coverage   83.56%   83.41%   -0.16%     
==========================================
  Files          25       25              
  Lines        2142     2146       +4     
==========================================
  Hits         1790     1790              
- Misses        352      356       +4     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[lbenet]

I will need to add tests. Perhaps it is also good time to increase the tests related to all the power functions and power modes.

[lbenet]

Followed @OlivierHnt suggestion here; it seems to work nicely, distinguishing literal powers from whatever. (All previous changes have been reverted.)

[OlivierHnt]

Looks good to me!

Note (just as a memo): while Val{n} can have n to be a float or a bool, the call to Base.literal_pow only occurs when n is a literal integer which makes this implementation correct.
Also, the issue is not entirely fixed for large literal integers:

julia> interval(1)^2305843009213693951
[1.0, 1.0]_com

julia> interval(1)^2305843009213693952 # any integer larger than this one will also have the NG flag
[1.0, 1.0]_com_NG

Whether or not this is a bug, it is on Julia's side.

[OlivierHnt]

I am on board with merging this PR.

@Kolaru any thoughts?

[lbenet]

Note (just as a memo): while Val{n} can have n to be a float or a bool, the call to Base.literal_pow only occurs when n is a literal integer which makes this implementation correct. Also, the issue is not entirely fixed for large literal integers:

julia> interval(1)^2305843009213693951
[1.0, 1.0]_com

julia> interval(1)^2305843009213693952 # any integer larger than this one will also have the NG flag
[1.0, 1.0]_com_NG

Whether or not this is a bug, it is on Julia's side.

Good catch! I will check if I can find what's going on, but I guess this is some sort of rounding issue (due perhaps to the conversion to Float64), as commented here.

[Kolaru]

For me it can be merged once the tests are there :)

[OlivierHnt]

At the end of test/interval_tests/power.jl, we can just add something like

@testset "Literal powers" begin
    x = interval(1)
    n = 2
    @test isguaranteed(x ^ 2)
    @test !isguaranteed(x ^ n)
    @test_broken isguaranteed(x ^ 2.0)
    @test_broken isguaranteed(x ^ 2305843009213693952)
end