The comparison between the kinetic energy delivery of a sling and a .44 magnum hold true, so long as the slingstone is large enough. The kinetic energy of a 16 gram .44 round is 1036.8 Joules. For example, it is possible to match the kinetic energy of a .44 magnum with an 8 kg stone thrown 26.45 meters, or a 4 kg stone thrown 52.9 meters. It becomes increasingly difficult with smaller stones. For example, a 125 gram stone must be thrown hard enough to reach 1692.73 meters (about a mile). These estimations are made, assuming a 45° launch angle, without taking wind resistance into account, using:

velocity = √(distance·9.8/sin(2·angle))

Joules = kg·m/s²/2

The following are some example throwing distances and slingstone masses that would require the same energy input as a .44 magnum.

 Meters     Kg 
13224.49  0.016
 3385.47  0.063
 1692.73  0.125
  846.37  0.250
  423.18  0.500
  211.59  1.000
  105.80  2.000
   52.90  4.000
   26.45  8.000
   13.22 16.000
    6.61 32.000

Spreadsheet: velocity based on distance.xlsx