missing Complex expm1 and log1p

[stevengj]

The expm1 and log1p functions are not yet implemented for Complex types.

In principle, I think these are straightforward: use a Taylor expansion for small arguments z (e.g. it is cheapest to check whether |real(z)|+|imag(z)| is small), and explicit function exp/log calls otherwise. The only annoyance is that you have to change the "small" cutoff and the order of the Taylor series depending upon the precision. For Float64 and Float32 this is simple, but for BigFloat matters are hairier.

[bsxfan]

Bump. The absence of these functions breaks complex step differentiation of code that uses these functions.

[stevengj]

Now that #6539 is merged (thanks to @simonbyrne), we only need a log1p function. There are a couple of tricks to implement this accurately that should work for complex values, I think (though this should be checked). Either:

log1p(x) = 2*atanh(x/(x+2))

or (due to Kahan, apparently)

function log1p(x)
    u = 1+x
    u == 1 ? x : log(u)*x/(u-1)
end

though some care is required in the latter version to ensure type stability (e.g. for x::Complex{Int}).

[stevengj]

On my machine, the atanh version above is slightly faster for complex x. However, the atanh version is wrong for x == -2 + 0im, and is inaccurate near -2.

Some care is required with the branch cuts, in both versions, e.g. to have the right sign of zero along the negative real axis.

[stevengj]

Actually, both versions are inaccurate near x == -2. Grrr.