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


/*REXX program calculates the total area of  (possibly overlapping)  circles.           */
parse arg box dig .                              /*obtain optional argument from the CL.*/
if box=='' | box==','  then box= 500             /*Not specified?  Then use the default.*/
if dig=='' | dig==','  then dig=  12             /* "      "         "   "   "     "    */
numeric digits dig                               /*have enough decimal digits for points*/
                       data = ' 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'
circles= words(data) % 3     /*     ══x══          ══y══       ══radius══     */
parse var data minX minY . 1 maxX maxY .              /*assign minimum & maximum values.*/
            do j=1  for circles;      _= j * 3  -  2  /*assign some circles with datum. */
            @x.j= word(data,_);    @y.j= word(data, _ + 1)
                                   @r.j= word(data,_+2);    @rr.j= @r.j**2
            minX= min(minX, @x.j - @r.j);   maxX= max(maxX, @x.j + @r.j)
            minY= min(minY, @y.j - @r.j);   maxY= max(maxY, @y.j + @r.j)
            end   /*j*/
dx= (maxX-minX) / box
dy= (maxY-minY) / box
#= 0                                                  /*count of sample points (so far).*/
      do   row=0  to box;          y= minY + row*dy   /*process each of the grid rows.  */
        do col=0  to box;          x= minX + col*dx   /*   "      "   "  "    "  column.*/
          do k=1  for circles                         /*now process each new circle.    */
          if (x - @x.k)**2 + (y - @y.k)**2 <= @rr.k  then  do;   #= # + 1;    leave;   end
          end   /*k*/
        end     /*col*/
      end       /*row*/
                                                      /*stick a fork in it, we're done. */
say 'Using ' box  " boxes (which have "  box**2  ' points)  and '  dig  " decimal digits,"
say 'the approximate area is: '       # * dx * dy