Total circles area

You are encouraged to solve this task according to the task description, using any language you may know.
 Example circles
 Example circles filtered
Given some partially overlapping circles on the plane, compute and show the total area covered by them, with four or six (or a little more) decimal digits of precision. The area covered by two or more disks needs to be counted only once.

One point of this Task is also to compare and discuss the relative merits of various solution strategies, their performance, precision and simplicity. This means keeping both slower and faster solutions for a language (like C) is welcome.

To allow a better comparison of the different implementations, solve the problem with this standard dataset, each line contains the x and y coordinates of the centers of the disks and their radii   (11 disks are fully contained inside other disks):

         xc              yc          radius
     1.6417233788   1.6121789534   0.0848270516
    -1.4944608174   1.2077959613   1.1039549836
     0.6110294452  -0.6907087527   0.9089162485
     0.3844862411   0.2923344616   0.2375743054
    -0.2495892950  -0.3832854473   1.0845181219
     1.7813504266   1.6178237031   0.8162655711
    -0.1985249206  -0.8343333301   0.0538864941
    -1.7011985145  -0.1263820964   0.4776976918
    -0.4319462812   1.4104420482   0.7886291537
     0.2178372997  -0.9499557344   0.0357871187
    -0.6294854565  -1.3078893852   0.7653357688
     1.7952608455   0.6281269104   0.2727652452
     1.4168575317   1.0683357171   1.1016025378
     1.4637371396   0.9463877418   1.1846214562
    -0.5263668798   1.7315156631   1.4428514068
    -1.2197352481   0.9144146579   1.0727263474
    -0.1389358881   0.1092805780   0.7350208828
     1.5293954595   0.0030278255   1.2472867347
    -0.5258728625   1.3782633069   1.3495508831
    -0.1403562064   0.2437382535   1.3804956588
     0.8055826339  -0.0482092025   0.3327165165
    -0.6311979224   0.7184578971   0.2491045282
     1.4685857879  -0.8347049536   1.3670667538
    -0.6855727502   1.6465021616   1.0593087096
     0.0152957411   0.0638919221   0.9771215985

The result is   21.56503660... .



Related task

   Circles of given radius through two points.


See also

 http://www.reddit.com/r/dailyprogrammer/comments/zff9o/9062012_challenge_96_difficult_water_droplets/
 http://stackoverflow.com/a/1667789/10562


import sequtils
 
type Circle = tuple[x, y, r: float]
 
const circles: seq[Circle] = @[
  ( 1.6417233788,  1.6121789534, 0.0848270516),
  (-1.4944608174,  1.2077959613, 1.1039549836),
  ( 0.6110294452, -0.6907087527, 0.9089162485),
  ( 0.3844862411,  0.2923344616, 0.2375743054),
  (-0.2495892950, -0.3832854473, 1.0845181219),
  ( 1.7813504266,  1.6178237031, 0.8162655711),
  (-0.1985249206, -0.8343333301, 0.0538864941),
  (-1.7011985145, -0.1263820964, 0.4776976918),
  (-0.4319462812,  1.4104420482, 0.7886291537),
  ( 0.2178372997, -0.9499557344, 0.0357871187),
  (-0.6294854565, -1.3078893852, 0.7653357688),
  ( 1.7952608455,  0.6281269104, 0.2727652452),
  ( 1.4168575317,  1.0683357171, 1.1016025378),
  ( 1.4637371396,  0.9463877418, 1.1846214562),
  (-0.5263668798,  1.7315156631, 1.4428514068),
  (-1.2197352481,  0.9144146579, 1.0727263474),
  (-0.1389358881,  0.1092805780, 0.7350208828),
  ( 1.5293954595,  0.0030278255, 1.2472867347),
  (-0.5258728625,  1.3782633069, 1.3495508831),
  (-0.1403562064,  0.2437382535, 1.3804956588),
  ( 0.8055826339, -0.0482092025, 0.3327165165),
  (-0.6311979224,  0.7184578971, 0.2491045282),
  ( 1.4685857879, -0.8347049536, 1.3670667538),
  (-0.6855727502,  1.6465021616, 1.0593087096),
  ( 0.0152957411,  0.0638919221, 0.9771215985)]
 
template sqr(x: SomeNumber): SomeNumber = x * x
 
let xMin = min circles.mapIt(it.x - it.r)
let xMax = max circles.mapIt(it.x + it.r)
let yMin = min circles.mapIt(it.y - it.r)
let yMax = max circles.mapIt(it.y + it.r)
 
const boxSide = 500
 
let dx = (xMax - xMin) / boxSide
let dy = (yMax - yMin) / boxSide
 
var count = 0
 
for r in 0 ..< boxSide:
  let y = yMin + float(r) * dy
  for c in 0 ..< boxSide:
    let x = xMin + float(c) * dx
    for circle in circles:
      if sqr(x - circle.x) + sqr(y - circle.y) <= sqr(circle.r):
        inc count
        break
 
echo "Approximated area: ", float(count) * dx * dy