performance tuning - Exp is slower for some number ranges

Bug introduced in 9 and fixed in 11.1

I get a hit on performance with the Exp function depending on the range of my inputs. for example, with:

w1 = RandomComplex[-20000 I, {10000000}];

Exp[w1]; // Timing

w2 = RandomComplex[-20 I, {10000000}];
Exp[w2]; // Timing

Results in 2.4 and 0.4 sec respectively. What mystifies me is that the involved numbers are not in a numerically "interesting" range.

The transition to slower times starts gradually at about -500 I and asymptotically stabilizes by about -10000 I. Past these ranges the timings are stable:

w = RandomComplex[-I, {1000000}];
t = Table[{10^n, Timing[Exp[10^n w]][[1]]}, {n, 1.5, 5, .01}];
ListLogLinearPlot[t, AxesLabel -> {"Scaling", "Time"}]

Any ideas what might be happening here? Any simple fixes to get faster performance with high scaling factors?

Comments

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