tanpi produces NaN for large complex argument

[Youjack]

Current definition of tanpi produces NaN for large complex argument. For example,

julia> tanpi(500 * exp(im * 3pi/4))
NaN + NaN*im

The reason is that sinpi and cospi produce Inf with this argument,

julia> sinpi(500 * exp(im * 3pi/4))
Inf + Inf*im

julia> cospi(500 * exp(im * 3pi/4))
Inf - Inf*im

while tanpi is defined as

julia/base/special/trig.jl

Line 943 in 0163991

tanpi(x::Complex) = sinpi(x) / cospi(x) # Is there a better way to do this?

This NaN causes NaN in some special functions. For example,

julia> using SpecialFunctions

julia> digamma(500 * exp(im * 3pi/4))
NaN + NaN*im

See the related issue in SpecialFunctions.jl repo for details.

Inspired by how cotpi is defined in mpmath:
https://github.com/mpmath/mpmath/blob/5e57cb8039a3a4c9f222901a5eb2e08c53b2b2f8/mpmath/libfp.py#L153-L161
I suggest defining tanpi as

tanpi(x::Complex) = imag(x) > 10 ? 1.0im : imag(x) < 10 ? -1.0im : sinpi(x) / cospi(x)

This issue is copied from #28943 (comment).

[nsajko]

Potentially related: #46315?