Re: complex exponentials: also see Why is this Julia code considerably slower than Matlab.
As I understand, Julia uses a custom implementation of trig functions instead of optimized libraries, can it be the reason?
As of 0.7, Julia does use native Julia versions of trigonometric functions, ported from openlibm. This generally caused speedups, not slowdowns over 0.6, but openlibm’s implementations are definitely not the fastest.