Bug in `log()` for negative reals?

[trahflow]

I think there is a bug in Base.log(q::Quaternion) if q is a negative real.

For example:

julia> q = quat(-1.0)
QuaternionF64(-1.0, 0.0, 0.0, 0.0, true)

julia> exp(log(q)) == q
false

note that this is not due to floating point (im-)precision:

julia> exp(log(q))
QuaternionF64(1.0, 0.0, 0.0, 0.0, true)

julia> log(q)
QuaternionF64(0.0, 0.0, 0.0, 0.0, false)

Related answer on math.stackexchange: https://math.stackexchange.com/a/3750503

Note in particular:

For negative reals, the imaginary component of the branches has the form (2n+1)πu^. Thus (log(k),0,0,0) is not a valid value of log(k,0,0,0).

I think a fix would be to choose the branch of the logarithm that corresponds to the logarithm on complex numbers, again quoting above math.stackexchange answer:

But, for example, if you choose u^=(1,0,0) then you get log(−2,0,0,0)=(log(|−2|),(2n+1)π,0,0). This is exactly the logarithm on C.

[sethaxen]

Indeed this is a bug that it appears was introduced in #56, where we incorrectly removed this check:

Quaternions.jl/src/Quaternion.jl

Lines 134 to 139 in 2ea186f

	if M > 0
	M = th / M
	return Quaternion(log(a), q.v1 * M, q.v2 * M, q.v3 * M)
	else
	return Quaternion(log(a), th, 0.0, 0.0)
	end

We kept that check in extend_analytic but use a separate implementation for log. I'll push a fix.