Generalise Broadcasts using ChainRules.jl

[max-vassili3v]

This pull request adds generalised broadcasting for DualVectors based on ChainRules.jl for:

• single-argument functions that operate on reals
• binary operations broadcasted between a Real and a DualVector
• binary operations broadcasted between a Dual and a DualVector

As well as this, some other changes have been made:

• Split into many files for maintainability
• Add a more descriptive readme
• Fix a bug with vcat
• Organise existing unit tests as well as add more for broadcasting

[codecov]

Codecov Report

❌ Patch coverage is 82.14286% with 20 lines in your changes missing coverage. Please review.
⚠️ Please upload report for BASE (main@e30580e). Learn more about missing BASE report.

Files with missing lines	Patch %	Lines
src/utilities.jl	50.00%	14 Missing ⚠️
src/arithmetic.jl	95.16%	3 Missing ⚠️
src/indexing.jl	72.72%	3 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##             main      #17   +/-   ##
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  Coverage        ?   82.14%           
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  Files           ?        5           
  Lines           ?      112           
  Branches        ?        0           
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  Hits            ?       92           
  Misses          ?       20           
  Partials        ?        0           

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[dlfivefifty]

Is this a standard pattern? I haven't seen this before

[max-vassili3v]

I think so, I wasn't able to find another way in Julia to get the function from the typeof(function) provided in the frule signature. I've seen/used a similar pattern in various LeetCode problems for example

[dlfivefifty]

Can you add a citation as a comment to where you saw this pattern?

I'm wary of using obscure patterns. I would look at the packages that depend on ChainRules.jl to see if there's an established pattern. But if you can't find anything, it's fine: I tend to use a "test-driven development" point of view the unit tests are more important than the details of the implementation (which can be modified freely as long as tests pass).

[max-vassili3v]

I had another look at the implementation and realised the lookup table was unnecessary