How much health does it take for Atma's Impaler to become cost effective?

Question

The item Atma's Impaler has a unique passive as follows:

UNIQUE: 2% of your maximum health is granted as bonus attack damage.

The item's combine cost is 825, and the total item costs 2355. How much health do I need to make Atma's Impaler a cost effective +attack damage item? That is to say, how much health do I need in order to gain more bonus attack damage than if I had spent the gold on +AD items alone?

I'm trying to decide what to do when situations merit the armor bonus the item provides, but I still need to build damage.

Answer

I'm not sure what the true cost of a single attack damage is, but here are some points of comparison:

Long Sword: 415g, 10ad = 41.5g/ad
BF Sword: 1650g, 45ad = 36.7g/ad

So, while the reality is more complicated, you could say that one point of attack damage costs around 39g. Here are the values of the Atmas passive at different health levels:

1000 hp: 2355g, 20 ad = 117.75g/ad
1250 hp: 2355g, 25 ad = 94.20g/ad
1500 hp: 2355g, 30 ad = 78.50g/ad
1750 hp: 2355g, 35 ad = 67.29g/ad
2000 hp: 2355g, 40 ad = 58.88g/ad
2250 hp: 2355g, 45 ad = 52.33g/ad
2500 hp: 2355g, 50 ad = 47.10g/ad
2750 hp: 2355g, 55 ad = 42.82g/ad
3000 hp: 2355g, 60 ad = 39.25g/ad
3250 hp: 2355g, 65 ad = 36.23g/ad
3500 hp: 2355g, 70 ad = 33.64g/ad

So at somewhere around 3000 hp it becomes worth it for the damage alone.

Here's the same table considering only the cost of the combine, rather than the total cost of the item:

1000 hp: 825g, 20 ad = 41.25g/ad
1250 hp: 825g, 25 ad = 33.00g/ad
1500 hp: 825g, 30 ad = 27.50g/ad
1750 hp: 825g, 35 ad = 23.57g/ad
2000 hp: 825g, 40 ad = 20.62g/ad
2250 hp: 825g, 45 ad = 18.33g/ad
2500 hp: 825g, 50 ad = 16.50g/ad
2750 hp: 825g, 55 ad = 15.00g/ad
3000 hp: 825g, 60 ad = 13.75g/ad
3250 hp: 825g, 65 ad = 12.69g/ad
3500 hp: 825g, 70 ad = 11.79g/ad

So if you already have the two base items, you only need around 1100 health to make the combine worthwhile.

The wiki page says:

The amount of health required to be as effective as the B. F. Sword is 1125, when taking both armor and crit hit chance into account.

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