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Plotting the sum of curves without recalculation


I want to plot several curves of functions which are hard to calculate on the same plot. I would like to add to the same plot the sum of these functions without recalculating them twice at each point.


So instead of


 Plot[{Sin[x], Sin[2 x], Sin[x] + Sin[2 x]}, {x, 0, 6 Pi}]

I would like somthing like


 Plot[{aa = Sin[x], bb = Sin[2 x], aa + bb}, {x, 0, 6 Pi}]

Answer




How about this:


f[x_] := Sin[x]
Plot[{#1, #2, #1 + #2} &[f[x], f[2 x]], {x, 0, 4}]

Strangely enough, this solution is slower that expected:


f[x_] := NIntegrate[Sin[1/y^2], {y, -x, x}] (*slow function*)

AbsoluteTiming[Plot[{f[x], f[2 x], f[x] + f[2 x]}, {x, 0, 1}]] (*naïve approach*)
AbsoluteTiming[Plot[{#1, #2, #1 + #2} &[f[x], f[2 x]], {x, 0, 1}]] (*my solution*)
AbsoluteTiming[Plot[With[{aa=f[x],bb=f[2x]},{aa,bb,aa+bb}],{x,0,1},Evaluated->False]] (*Szabolcs' comment*)


(*65.7*)
(*106.2*)
(*102.0*)

We do get a substantial improvement with memoization:


f[x_] := f[x] = NIntegrate[Sin[1/y^2], {y, -x, x}]
AbsoluteTiming[Plot[{f[x], f[2 x], f[x] + f[2 x]}, {x, 0, 1}]] (*naïve approach with memoization*)

(*40.5*)


Finally, the best approach so far is to follow BlacKow's suggestion and precalculate the function at discrete points:


f[x_] := NIntegrate[Sin[1/y^2], {y, -x, x}] (*two slow*)
g[x_] := NIntegrate[Cos[1/y^2], {y, -x, x}] (*functions*)

AbsoluteTiming[
  points = Range[0, 1, .01];
  F = f /@ points;
  G = g /@ points;
  ListPlot[{Transpose@{points, F}, Transpose@{points, G}, 

       Transpose@{points, F + G}}, Joined -> True]
]

(*3.4*)

though it can get tricky to choose the appropriate spacing.


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