For those who might want to use a flat damage bonus instead for Battlefield Blaster, here's the damage increase by damage die type:
- d2: 1.5 + 0.500 + 0.375 + 0.188 ... = 3.00 average, +100% damage
- d3: 2.0 + 0.667 + 0.222 + 0.074 ... = 3.00 average, +50% damage
- d4: 2.5 + 0.625 + 0.156 + 0.039 ... = 3.33 average, +33% damage
- d5: 3.0 + 0.600 + 0.120 + 0.024 ... = 3.75 average, +25% damage
- d6: 3.5 + 0.583 + 0.097 + 0.016 ... = 4.20 average, +20% damage
- d8: 4.5 + 0.562 + 0.070 + 0.009 ... = 5.14 average, +14% damage
- d10: 5.5 + 0.550 + 0.055 + 0.006 ... = 6.11 average, +11% damage
- d12: 6.5 + 0.542 + 0.045 + 0.004 ... = 7.09 average, +9% damage
- d20: 10.5 + 0.525 + 0.026 + 0.001 ... = 11.05 average, +5% damage
- d100: 50.5 + 0.505 + 0.005 ... = 51.01 average, +1% damage
If you roll 4d4 for Pestilence cantrip cast at 15th level, your average damage increases to 4d4+3.
If you roll 6d6 for Fireball, your average damage increases to 6d6+4.
If you roll 14d6 for Meteor Swarm, your average damage increases to 14d6+10.
If you roll 8d8 for Cone of Cold, your average damage increases to 8d8+5.
If you roll 6d10 for Blade Barrier, your average damage increases to 6d10+4.
So as a rule of thumb instead of doing the math, you get +1 damage per damage die rolled. It is a little on the generous side, but much simpler to handle in combat to save time.