Why is LogLogPlot[] slower than Plot[]?

Question

I notice that LogLogPlot runs quite bit more slowly than ordinary Plot. For example, consider the following two blocks of code which plots (using Manipulate) a region in linear scale and log-linear scale. Notice how the log-log-scale manipulation runs more sluggishly. Is there a way to speed up the log-log-scale version?

Linear-scale code:

 Mhad = 0.93827;
 Table[Tooltip[(MxSq - M^2) x/(1 - x) /. M -> Mhad, "x=" <> ToString[x]], {x, 0, .9, .1}];
 Manipulate[Plot[{1/s (s - M^2) (s - MxSq) /. M -> Mhad, %},
   {MxSq, Mhad^2, 100}, PlotRange -> {{0, 100}, {0, 100}},
   AspectRatio -> 1, Filling -> {1 -> Axis}, Frame -> True,
   PlotStyle -> Table[If[i == 1,
     RGBColor[0, 0, 0],
     {RGBColor[1, 0, 0], Thickness[.001]}], {i, 1, 11}]], {{s, 50}, 1,
     400}]

Log-log-scale:

 Table[Tooltip[(MxSq - M^2) x/(1 - x) /. M -> Mhad, "x=" <> ToString[x]], {x, 0, .9, .1}];
 Manipulate[LogLogPlot[{1/s (s - M^2) (s - MxSq) /. M -> Mhad, %},
   {MxSq, Mhad^2, 100}, PlotRange -> {{.8, 100}, {.8, 100}},
   AspectRatio -> 1, Filling -> {1 -> Axis}, Frame -> True,
   PlotStyle -> Table[If[i == 1,
     RGBColor[0, 0, 0],
     {RGBColor[1, 0, 0], Thickness[.001]}], {i, 1, 11}]], {{s, 50}, 1,
     400}]

Answer 1

Using

Mhad = 0.93827;
t = Table[
   Tooltip[(MxSq - M^2) x/(1 - x) /. M -> Mhad,"x=" <> ToString[x]], {x, 0, .9, .1}];
With[{s = 50}, 
  p1=Plot[{1/s (s - M^2) (s - MxSq) /. M -> Mhad, t}, {MxSq, Mhad^2,100}, 
        PlotRange -> {{.8, 100}, {.8, 100}}, AspectRatio -> 1, 
        Filling -> {1 -> Axis}, Frame -> True, 
        PlotStyle -> 
          Table[If[i == 1, RGBColor[0, 0, 0],
                          {RGBColor[1, 0, 0], Thickness[.001]}], {i, 1,11}]]
 ] // AbsoluteTiming

and a similar one for LogLogPlot I find that Plot is about 2.7 times faster than LogLogPlot.

This is hardly surprising as the LogLogPlot is much more complex than the Plot. You can use FullForm to examine the underlying graphics primitives. The bytecount of these primitives have the following ratio:

(p2 // FullForm // ByteCount)/ (p1 // FullForm // ByteCount)

2.604100491

or take for instance the number of List instructions (all coordinates are in lists):

Cases[p1, List[___], Infinity] // Length

687

Cases[p2, List[___], Infinity] // Length

1202

It simply takes more coordinates to describe a curve accurately than for a set of straight lines.

One way to speed up the Manipulate may be the use of ControlActive, where you draw graphics with lower quality as long as the controls are being manipulated and a final one with higher quality if user-interaction stops.

Manipulate[
 ControlActive[
  LogLogPlot[{1/s (s - M^2) (s - MxSq) /. M -> Mhad}, {MxSq, Mhad^2,100}, 
              PlotRange -> {{.8, 100}, {.8, 100}}, AspectRatio -> 1, Frame -> True,
              PlotPoints -> 10, MaxRecursion -> 1],
  LogLogPlot[{1/s (s - M^2) (s - MxSq) /. M -> Mhad, t}, {MxSq, Mhad^2, 100}, 
              PlotRange -> {{.8, 100}, {.8, 100}}, 
              AspectRatio -> 1, Filling -> {1 -> Axis}, Frame -> True, 
              PlotStyle -> 
                 Table[If[i == 1, RGBColor[0, 0, 0], 
                                  {RGBColor[1, 0, 0], Thickness[.001]}], {i, 1, 11}]]
  ], 
  {{s, 50}, 1, 400}
]

which switches between

and

I changed PlotPoints, MaxRecursion, Filling and the plotting of the background curves. Adapt to your own needs.