> Thanks. I need to draw animation in real time. 

Real time? Maybe so, but certainly not at 1000 frames per second.

If you draw more frames than your system can display, you are simply
wasting resources.

If you draw more frames than a human observer can perceive, you are
simply wasting resources.

So... 10 frames per second might be a bit marginal for the average
observer, but adequate.
20 FPS would look smooth to most observers.
1000 FPS is pointless.

> But it seemed that I can use 
> fl_begin_offscreen(),fl_end_offscreen(),fl_copy_offscreen()  
> + Fl_Window to replace Fl_Double_Window.
> 
> By the way, why will Fl_Double_Window drawing be slower than 
> Fl_Window drawing? Is it due to fl_*_offscreen() call?

Yes, basically.

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