I want to create a function that finds the real space distance between two three-dimensional points. Each point is a list of three real numbers.
I can create functions dist1
, dist2
, and distCompiled1
and test them on example points myPoint1
and myPoint2
:
dist1[pt1_List, pt2_List] := Sqrt[(pt1[[1]] - pt2[[1]])^2 + (pt1[[2]] - pt2[[2]])^2
+ (pt1[[3]] - pt2[[3]])^2];
dist2[pt1_List, pt2_List] := Sqrt[(pt1 - pt2).(pt1 - pt2)];
distCompiled1 := Compile[{{pt1, _Real, 1}, {pt2, _Real, 1}},
Sqrt[(pt1 - pt2).(pt1 - pt2)]];
myPoint1 = {1.0, 2.0, 3.0};
myPoint2 = {4.0, 5.0, 6.0};
dist1[myPoint1, myPoint2]
dist2[myPoint1, myPoint2]
distCompiled1[myPoint1, myPoint2]
5.19615
5.19615
5.19615
So, dist1
, dist2
, and distCompiled1
all give the same result.
But, I will be computing distances between many millions of points, so I want to optimize my function for speed. If I calculate the distance between 1 million pairs of points, for example, I get the following results:
dist1[pt1_List, pt2_List] := Sqrt[(pt1[[1]] - pt2[[1]])^2 + (pt1[[2]] - pt2[[2]])^2
+ (pt1[[3]] - pt2[[3]])^2];
dist2[pt1_List, pt2_List] := Sqrt[(pt1 - pt2).(pt1 - pt2)];
distCompiled1 := Compile[{{pt1, _Real, 1}, {pt2, _Real, 1}},
Sqrt[(pt1 - pt2).(pt1 - pt2)]];
maxVal = 8.009469032;
numPts = 10^6;
SeedRandom[1234];
coords1 = Table[RandomReal[{0, maxVal}, 3], {numPts}];
SeedRandom[1235];
coords2 = Table[RandomReal[{0, maxVal}, 3], {numPts}];
(* Time the computations, and verify that the functions give the same results. *)
AbsoluteTiming[
dist1Result = Table[dist1[coords1[[i]], coords2[[i]]], {i, 1, numPts}];
]
AbsoluteTiming[
dist2Result = Table[dist2[coords1[[i]], coords2[[i]]], {i, 1, numPts}];
]
AbsoluteTiming[
distCompiled1Result = Table[distCompiled1[coords1[[i]], coords2[[i]]], {i, 1, numPts}];
]
(dist1Result == dist2Result) && (dist2Result == distCompiled1Result)
{6.725385, Null}
{5.900337, Null}
{30.191727, Null}
True