See for yourself:

http://rm.endoftheinternet.org/~nexuiz/falloff/

The falloff curves are:

• 0: like Quake, dir+randomvec*spread; 1: same without varying velocity
  
  
• 2: solid circle
  
  
• 3: gaussian falloff, varies velocity like 0; 4: same without varying velocity
  
  
• 5: linear falloff
  
  
• 6: (1-r^2) falloff
  
  
• 7: (1-r)(2-r) falloff
  
  

They are all coded so their standard deviation matches among them, so changing them shouldn't change gameplay much.

two things come to mind:
road salt? (haarghhnnn)
nice work there!

So which one will become default?

Not sure. I prefer 5 and 7.

I vote for 5 and 7 on that order. Good stuff.

I vote for 7 too, thats how i imagine a real shotgun to operate: The graph for 5 and 7 looks cool but I think 5 has a little bit too uniform spread in practice based on the decal-picture. Maybe its just me.

I prefer 7.

Let's say 7 has the simpler implementation:

dx = random() * 2 * M_PI;
dy = sin(dx);
dx = cos(dx);
// radius in our dist function
r = random();
r = 1 - sqrt(r);
r = 1 - sqrt(r);
return normalize(forward + (v1 * dx + v2 * dy) * r * sigma);

compared to 5:

dx = random() * 2 * M_PI;
dy = sin(dx);
dx = cos(dx);
// radius in our dist function
r = random();
r = solve_cubic_abcd(-2, 3, 0, -r) * '0 1 0';
return normalize(forward + (v1 * dx + v2 * dy) * r * sigma);

because, the cubic root solving is anything but simple compared to two square roots

So, 7 should be the faster one.

I'll now order them in speed of the code:
0 (277k/sec)
1 (237k/sec)
2 (165k/sec)
6, 7 (158k/sec)
5 (110k/sec)
3, 4 (80k/sec)

Well then with those performance values its looks like 7 is your best option. Nothing else with as realistic looking and unspammy a pattern is faster.

I think personally a gaussian is probably a good approximation of the sum of the different random parts in a weapon

Yes, that is what gaussian normally models.

However, it is bad for gameplay that a projectile can end up anywhere - just with low probability. It's better especially in team games if a bullet can only hit a certain maximum distance around the crosshair (so you can AVOID hitting someone) - which is why I suggest the (2-r)(1-r) spread instead of gaussian.

Other than that, the functions for gaussian and (2-r)(1-r) are very similar at higher distances, and dissimilar at lower distances.