Ok so, I have mentioned this build before but I am trying to figure out the numbers on paper. edit: I am really bored at work today

Build: Runic Echoes, Sorc Shoes, Liandries, Sunfire Cape, Frozen Heart, Abyssal Mask,

Runes.Not sure on primary but you have to have Coup de Grace so this could be Precision/resolve or Resolve/Precision. Either way make sure you have Presence of Mind and Coup De Grace in there. If you go precision primary then I would recommend Conqueror since it got buffed for double 10% true damage (amu passive). Otherwise take aftershock and whatever you want really.

So heres what goes into making it work:

2 components of Liandry's Torment: MADNESS: stacks (2% per stack, 10% max) 10% increased damage at max stacks TORMENT: Dealing ability damage applies a damage-over-time effect for 3 seconds that deals a total of 4.5% of target's maximum health bonus magic damage, with each tick of damage occurring every 0.5 seconds. This total damage is increased to 7.5% of target's maximum health against movement impaired units. (note torment does not work with item passives that deal magic damage, sunfire/thornmail, madness does increase their damage though)

Abyssal Mask : AURA - Nearby enemy champions take 15% more magic damage (325 range).

Sunfire Cape: IMMOLATE: Deals 26 − 43 (based on level) magic damage per second to nearby enemies.

Frozen heart: Cripple: Cripples nearby enemies' attack speed by 15% (700 range).

Coup de Grace Rune: Deal 7% increased damage to champions below 40% maximum health.

So I looked it up and the damage stacks multiplicately on all these sources.

Lets assume target is below 40% health you get the following damage modifiers 7% (coup de grace) 10% (madness stacks) 15% (abyssal aura)

Multiplicative figure like such, lets say 3 buffs 10% each, 1000 damage 10% = 1100, 10% more (1100 10%) = 1210. 10% more (1210 10%) = 1331 total damage. It doesnt matter what order you apply the percentages.

So sunfire cape damage gets 15% from abyssal aura, 10% from madness, and 7% once they drop below 40% health. Assuming max level your sunfire should do 43 10% 15% = 54.395 damage per sec, 58.20 once they drop below 40%

Liandry Torment damage. If you have frozen heart then they are considered movement impaired from the aura and get the 7.5% damage.

Lets assume 3000 health on enemy champ. 3000*7.5%=225 damage per sec. add in the damage multipliers though. 225 15% 10%=297.56 damage per sec, 318.39 once they drop below 40%

Amumu W: MAGIC DAMAGE PER HALF SECOND: 5 / 7.5 / 10 / 12.5 / 15 ( 0.5 / 0.625 / 0.75 / 0.875 / 1% ( 0.5% per 100 AP) (this last part is actually calculated per 1 ap, so .005% per 1 ap) of target's maximum health).

Assuming 3k health and runic liandry ap (80 75=155) and max level: 15 (1%( .005%155) target max health = 2.55775%3000) for total of 76.73) = 91.7325 damage per second

Add in the damage buffs, 10/15% =116.04 per sec, or 124.16 after the drop below 40%.

So to bring this all together on a 3k target, above 40% health you do 467.995 (468) damage per second and 500.75 (500) dps once they hit 40%

This also assumes you are in combat 5 seconds to get max stacks on liandry.

Does all this look sound? Did I do the math right?