Fix sincos for reals that are not AbstractFloats

[dpsanders]

Fixes #22095.

This PR adds a generic fallback for sincos.
This allows exp(a + im*b) to work correctly when the type of a and b is a non-standard subtype of Real (e.g. ArbFloat or Interval).

There is a test provided with the simplest such non-standard type.

[Note that this -- EDIT: that is, exp(a + im*b) -- worked in Julia 0.6, but does not work in current master, so this is in fact a bug fix for a regression and not a new feature.]

[giordano]

What about irrational?

[pkofod]

That's covered by the old definition that's kept below.

[giordano]

Yes, but irrationals don't use the faster version, right? (I'm on my phone, I can't check myself right now)

[pkofod]

Oh, my bad, of course you're right, it should be {Integer, Rational, Irrational}. The current changes makes sincos(pi) slower than before.

[dpsanders]

Ah, good point, thanks, I'll change it.

I don't understand how to interpret the test errors -- can anybody help me?

[fredrikekre]

LoadError("sysimg.jl", 320, LoadError("math.jl", 988, LoadError("special/trig.jl", 201, UndefVarError(:Irrational))))

[pkofod]

Is irrational not defined at this point?

[dpsanders]

Apparently not. Also, in the future we may want to define sin(pi) etc explicitly. So I think it's better to leave the generic definition to cover irrationals.

[pkofod]

Practically speaking this should be of no real importance, but do you then propose that sincos(someotherrational) should be slow? Or do you want to define sin for all irrationals?

[dpsanders]

I have now implemented @yuyichao's approach from #22095 (comment)

and left a generic fallback.
I think this is the best solution for now.

[dpsanders]

The current failures seem unrelated (being in the show tests, not the math tests).

[fredrikekre]

It is related

Error in testset show:
Test Failed at /tmp/julia/share/julia/test/show.jl:580
  Expression: contains(Base.url(first(methods(sin))), "https://github.com/JuliaLang/julia/tree/$((Base.GIT_VERSION_INFO).commit)/base/missing.jl#L")

[JeffBezanson]

That is a very unfortunately specific test. It should be revised or removed so this can be merged.

[vtjnash]

Change first(methods(sin)) ==> which(sin, (Float64,)), so it's a bit less arbitrary than first (new file is "special/trig.jl")?

[dpsanders]

I have now reimplemented the logic using multiple dispatch instead.
I fixed the overly-specific test following Jameson's suggestion.

Not sure what the test failures are about now. They seem unrelated?

[dpsanders]

Made a big rebase/merge mess again, sorry. How do I back myself out of this one?

[dpsanders]

Have now fixed the mess by cherry-picking onto latest master and force pushing.

[dpsanders]

Are the failures in Circle x86_64 related? If not, then this is ready to go.