Make abs, abs_imag, inv, and / resistant to under/overflow

[sethaxen]

This PR is the companion of JuliaGeometry/Quaternions.jl#122 and fixes #4. It primarily ensures abs, abs_imag, inv, and / do not under/overflow unnecessarily, following the implementations of the corresponding functions for Complex. It also implements RealDot.realdot as in JuliaGeometry/Quaternions.jl#119.

A quick before/after benchmark demonstrating the peformance cost of these changes.

julia> using BenchmarkTools, Octonions

julia> q, w = rand(OctonionF64, 2);

Before:

julia> @btime abs2($q);
  1.923 ns (0 allocations: 0 bytes)

julia> @btime abs($q);
  1.924 ns (0 allocations: 0 bytes)

julia> @btime $(Octonions.abs_imag)($q);
  1.926 ns (0 allocations: 0 bytes)

julia> @btime inv($q);
  12.884 ns (0 allocations: 0 bytes)

julia> @btime $q / $w;
  30.243 ns (0 allocations: 0 bytes)

After:

julia> @btime abs2($q);
  1.687 ns (0 allocations: 0 bytes)

julia> @btime abs($q);
  18.967 ns (0 allocations: 0 bytes)

julia> @btime $(Octonions.abs_imag)($q);
  16.890 ns (0 allocations: 0 bytes)

julia> @btime inv($q);
  26.269 ns (0 allocations: 0 bytes)

julia> @btime $q / $w;
  49.940 ns (0 allocations: 0 bytes)

[codecov]

Codecov Report

Merging #14 (8760cf0) into main (ae650c6) will not change coverage.
The diff coverage is 100.00%.

@@            Coverage Diff            @@
##              main       #14   +/-   ##
=========================================
  Coverage   100.00%   100.00%           
=========================================
  Files            2         2           
  Lines           68        92   +24     
=========================================
+ Hits            68        92   +24     

Impacted Files	Coverage Δ
src/Octonions.jl	100.00% <ø> (ø)
src/octonion.jl	100.00% <100.00%> (ø)

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[sethaxen]

Note that the above benchmarks are for v1.9-beta3. For v1.8 and earlier, abs and abs_imag are much slower, because hypot was much slower. We should probably do I did what we did in JuliaGeometry/Quaternions.jl#122 and basically ported modern _hypot.

[hyrodium]

LGTM!