JuliaMath / JuliaMath/FixedPointNumbers.jl

Negative number of fractional bits

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add-on enhancement
Dominant language
Julia
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Description

With `Fixed` we can reduce the number of fractional bits to trade range for precision:
```julia
julia> Fixed{Int8, 2} .|> (typemin, typemax, eps)
(-32.0Q5f2, 31.75Q5f2, 0.25Q5f2)

julia> Fixed{Int8, 1} .|> (typemin, typemax, eps)
(-64.0Q6f1, 63.5Q6f1, 0.5Q6f1)

julia> Fixed{Int8, 0} .|> (typemin, typemax, eps)
(-128.0Q7f0, 127.0Q7f0, 1.0Q7f0)
```

However, currently support stops here. It would be useful to have a
```julia
julia> Fixed{Int8, -1} .|> (typemin, typemax, eps)
(-256.0Q8f-1, 254.0Q8f-1, 2.0Q8f-1)
```
and all the other types with a negative number of fractional bits available, when range demands are high, precision demands are low and you want to distribute the available values uniformly in the range or want to use integer arithmetic (so floating points are not an option).

This would not really match the packet's name as they are not really fixed-point numbers (with the "point" being outside of the represented bits). On the other hand, an implementation of these numbers (scaled integers) seems to be completely identical in representing the type (struct) and should cover both positive and negative numbers of fractional bits if some methods are generalized or their domain restrictions are relaxed.

So what would these scaled integers be? Would they use a slightly generalized version of `Fixed`, would they be a `ScaledInteger` within the `FixedPointNumbers.jl` package or would they be the topic of a `ScaledIntegers.jl` package?

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