feat: add ability to define and query function inverses

[AayushSabharwal]

Close #1337

I'd like input on the NaNMath inverses. They violate the contract that inverse(inverse(f)) == f. We could leave it be, define wrapper functions, or add to the macro the ability to wrap the provided inverse function in an anonymous function. For example,

@register_inverse f g wrap_inverse = true

would define

var"some_gensymed_name"(x) = g(x)
inverse(::typeof(f)) = var"some_gensymed_name"
inverse(::typeof(var"some_gensymed_name")) = f

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Codecov Report

Attention: Patch coverage is 45.71429% with 19 lines in your changes missing coverage. Please review.

Project coverage is 77.77%. Comparing base (3e31a75) to head (affad5d).
Report is 14 commits behind head on master.

Files with missing lines	Patch %	Lines
src/inverse.jl	45.71%	19 Missing ⚠️

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Additional details and impacted files

@@             Coverage Diff             @@
##           master    #1338       +/-   ##
===========================================
+ Coverage    6.69%   77.77%   +71.08%     
===========================================
  Files          47       49        +2     
  Lines        4618     4801      +183     
===========================================
+ Hits          309     3734     +3425     
+ Misses       4309     1067     -3242     

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[ChrisRackauckas]

I still think we should lowering sqrt to Pow

[AayushSabharwal]

Yes, but why would that preclude defining the inverse?

[ChrisRackauckas]

In that case we wouldn't need to, since the function would not be in the representation.

[ChrisRackauckas]

why?

[AayushSabharwal]

If f: X -> Y is an injective function and g: Y -> X is its left inverse, then I think it can be proven that g is surjective and has f as (one of) its right inverse(s) (and vice versa)?

[ChrisRackauckas]

I see, never thought about that.