Reworking ComplexInfinity

[PatrickHaecker]

Ok, this is my take on fixing the problems around ComplexInfinity by storing the angle in an UInt64. With this, they should be accurate whenever the users wants them to be accurate. They should be bijective, so no more multiple internal representations which mean the same. No more different quantizations depending on where you are on the complex circle.
So overall they should now behave as similar to Base as possible. However, it's not all perfect. The angle Float64 interface which is important in Base is not a natural fit to this representation and the conversion is not really what I would call elegant.

Update:
Running through the issues and actually using the package showed up to be very useful. I fixed some inconsistencies and simplified the printing.

Details from the 🤖:

Makes a direction and a value the same thing. Two ComplexInfinitys pointing the same way were not equal and did not hash alike:

julia> ComplexInfinity(0.5) == ComplexInfinity(2.5)   # same direction
false

The representation

The field is a UInt64 counting turns in units of 2^-64, wrapping where the circle does. Every count names exactly one direction, the group operation is a machine add, and the resolution is uniform around the circle.

Most code needs no constructor, since multiplying by ∞ takes the direction from the other operand:

julia> im*∞
0 + ∞*im

julia> ComplexInfinity(0x4000000000000000) ≡ ComplexInfinity(halfturns = 1//2) ≡ im*∞
true

halfturns and the x*∞ forms go through angle, so they round. They are exact on the axes and diagonals, but exp(im*π/8)*∞ lands 256 counts past a sixteenth turn. Pass the UInt64 where another direction has to be exact, and read it back with reinterpret(UInt64, x).

Breaking

	master	this PR
ComplexInfinity(0.5)	exp(0.5*im*π)∞	MethodError, use halfturns = 0.5
the type	ComplexInfinity{Float64}	ComplexInfinity, no parameter
5 < ComplexInfinity()	true	MethodError
div(ComplexInfinity(), 5)	exp(false*im*π)∞	MethodError
angle(ComplexInfinity(halfturns = 1.5))	4.71238898038469	-1.5707963267948966
angle(∞), angle(ℵ₀)	0	0.0
im*∞	exp(0.5*im*π)∞	0 + ∞*im
∞ + im*∞	NotANumber()	∞ + ∞*im
NaN * ComplexInfinity()	NotANumber()	NotANumber() + NotANumber()*im

angle now reports on Base's branch (-π, π] and always returns a float. Ordering and the integer operations are gone, because the complex plane has neither, exactly as for Complex.

The type parameter is gone

It described how the direction had been spelled, never the value. Whether an infinity lies on the real axis is now asked of the value:

julia> isreal(ComplexInfinity()), isreal((1+im)*∞)
(true, false)

julia> RealInfinity((1+im)*∞)
ERROR: InexactError

Consistent with Base's Complex

On the axes and diagonals each part is infinite or exactly zero, so addition works part by part as it does for Complex. That makes complex(x, y) == x + im*y hold with infinite parts too:

julia> ∞ + im*∞ ≡ complex(∞, ∞) ≡ complex(0.0, Inf) + ∞
true

Any other two different directions give an undefined result. These eight directions print as their parts, the way Base prints 0.0 + Inf*im. Other directions print as cispi(h)∞ where that reads back exactly, and as the count otherwise, so every printed form evaluates to the value it came from.

Worth a look

Equality and hash read the count, never the angle. The count has 64 bits and a Float64 angle has 53, so directions a float cannot tell apart must still compare unequal. Ordinary arithmetic reaches them: ComplexInfinity(halfturns = 0.5) * ComplexInfinity(halfturns = 2.0^-63).

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 100.00%. Comparing base (08ccb94) to head (1d36d39).

Additional details and impacted files

@@            Coverage Diff            @@
##            master       #87   +/-   ##
=========================================
  Coverage   100.00%   100.00%           
=========================================
  Files            6         6           
  Lines          336       352   +16     
=========================================
+ Hits           336       352   +16     

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[PatrickHaecker]

Thanks, @dlfivefifty. So I rebased #87 on master, so it is ready for review.