Today I was playing with Peter de Jong attractor.

At the bottom of the page I've linked there are beautiful examples like:

My attempts are not so great:

It is around 10^5 points. For more than 5*10^5 my 4GB RAM gives up. How can I achieve such smooth result on standard/oldish pc?

Here is the piece of code to play with:

Points generator:

fr2 = Compile[{{p, _Real, 1}, {n, _Integer}, {a, _Real, 1}, {b, _Real, 1},
               {c, _Real, 1}},
   NestList[
    { Sin[a[[1]] #[[2]]] - Cos[a[[2]] #[[1]]],
      Sin[b[[1]] #[[1]]] - Cos[b[[2]] #[[2]]],
      Sin[c[[1]] #[[1]]] - Cos[c[[2]] #[[1]]]
      } &, p, n]];

Interactive toy:

For printed color image I've used approach introduced by Szabolcs in antialiasing3D.

DynamicModule[{a, b, c, n, type, fig, controls, print},
 Deploy@Dynamic[Refresh[
    Panel@Grid[{{fig, controls}}, Spacings -> 2],
    None]],
 
 Initialization :> (
   type = 1;
   n = 100000;
   {a, b, c} = {{1.4, -2.3}, {2.4, -2.1}, {2.41, 1.64}};
   
   controls = Column[{
      Slider2D[Dynamic@a, {-Pi, Pi, .01}], Dynamic@a,
      Slider2D[Dynamic@b, {-Pi, Pi, .01}], Dynamic@b,
      Slider2D[Dynamic@c, {-Pi, Pi, .01}], Dynamic@c,
      Button["Print", print[], Method -> "Queued"] }];
   
   fig = Graphics[{White, AbsolutePointSize@1, 
      Dynamic[
       Point@fr2[{.0, .0, .0}, ControlActive[2 10^4, 10^5], a, b, c][[;; , ;; 2]]]
      }, ImageSize -> {1, 1} 500, Background -> Black, AspectRatio -> Automatic];
   
   print[] := With[{t = 3, pointsize = 1, pts = 10^5, res = 72},
     Composition[
       CreateDocument,
       ImageResize[Rasterize[#, "Image", ImageResolution -> t res], Scaled[1/t]] &,
       Graphics[{AbsolutePointSize@pointsize, 
          Riffle[Hue@Rescale[#, {-2, 2}, {0, 1}] & /@ #[[;; , 3]], 
             Point /@ #[[;; , ;; 2]]] &@#
          }, ImageSize -> 800, Background -> Black] &
       ][fr2[{.0, .0, .0}, pts, a, b, c]]];
   )]