Nourish v Regrowth

With the changes in 3.1, it’s interesting to examine the case of Nourish v. Regrowth.  I didn’t get into it with my last post on resto, because I got a bit too deep into lifebloom numbers.  I’d like to look at these two spells now.

Of utmost importance to me here is the direct heal portion.  Specifically, I think Blizzard has positioned these two spells in competing niches within the druid healing repertoire.  I’d like to examine the numbers and see if I am correct.

Nourish is the easier of the two to examine.  Let me look, first, purely at its healing output.  Nourish is a 1.5 sec. cast spell which heals for

(1 – crit) * M * (base + coeff * SP) + crit * 1.5 * 1.3 * M * (base + coeff * SP)

= (base + coeff * SP) * ((1-crit) * M + crit * 1.95 * M)

= M * (base + coeff * SP) * ((1 – crit) + crit * 1.5)

= M * (base + coeff * SP) * (1 + .95 * crit)

The crit value is based on the 1.5x multiplier critical heals receive plus the bonus from the Living Seed talent.

M is similar to Lifebloom (see my last post) but without Genesis.  That puts it at (1 + (.04 + .1)) *  1.06 = 1.208.  However, if you have any of the main druid HoTs on the target, it heals for an additive .2 before the ToL aura more.  We’ll assume that is the case, ao M = 1.42.  Additionally, Nourish has a new glyph incoming which increases M by .06 for each of your HoTs on the target.  Let’s assume that lifebloom stacks only count once.  That would mean Wild Growth, Rejuvenation, Regrowth, and Lifebloom in the best case scenario.  That increase M as above, so M = 1.675

base = 2035.

coeff = .6611

So, the equation thus far is 1.675 * (2035 + .6611 * SP) * (1 + .95 * crit) = (3408.6 + 1.107 * SP) * (1 + .95 * crit)

A two variable function is not optimal, but we’ll leave it like this for the moment.  One thing to note is that Nourish will, in the best case, scale well with both spell power and crit.  Keep in mind that crits will proc Nature’s Grace, reducing the cast time of Nourish to 1 sec and further increasing it’s output.

Let’s turn to Regrowth.

Regrowth’s direct heal is based on an equation very similar to Nourish:

(1 – crit) * M * (base + coeff * SP) + crit * 1.5 * 1.3 * M * (base + coeff * SP)

= (base + coeff * SP) * ((1-crit) * M + crit * 1.95 * M)

= M * (base + coeff * SP) * ((1 – crit) + crit * 1.5)

= M * (base + coeff * SP) * (1 + .95 * crit)

Crit value functions the same as Nourish.

M is, in the worst case, the same as LB: Regrowth benefits from Genesis, MS, GoN, and ToL Aura.  That leads to M = (1 + (.05 + .04 + .1)) * 1.06 = 1.261.  However, with the glyph, regrowth heals for 20% more if the target has regrowth.  As we already assumed the best case for Nourish, let’s do so for regrowth as well, which means that M = 1.473.

base = 2364

coeff = .6468

Regrowth’s avg direct heal is 1.473 * (2364 + .6468 * SP) * (1 + .95 * crit) = (3482.2 + .952 * SP) * (1 + .95 * crit)

From the above, we see that glyphed Nourish, in the best case, will, with 0 stats, be inferior to glyphed regrowth in the best case.  However, do to higher scaling coefficients, Nourish will eventually pull ahead.  Since the critical hit component is the same for both spells, we’ll ignore that in finding the SP at which Nourish overcomes Regrowth:

3482.2 + .952 * SP = 3408.6 + 1.107 * SP

73.6 = .155 * SP

SP = 474.84

So for any resto druid, Nourish will beat out Regrowth in healing output at any reasonable gear level.  Remember, though, that this is in the absolute best case that you have all 4 major hots on the target.  Having fewer increases the SP needed for Nourish to catch up.  While the base coefficient on Nourish is slightly higher than Regrowth’s, Nourish doesn’t benefit from Genesis.  That implies that, first, that when no HoTs are on the target, Nourish’s heal will be strictly inferior to Regrowth’s.

We have so far not accounted for the Regrowth HoT.  Let’s do that now.

Let’s first look at a single tick of regrowth:

Healing per tick = M * (base + coeff * SP)

base = 111.7

coeff = .2256

M is a bit interesting.  According to wowwiki, “The Regrowth Glyph increases both the Direct Heal and the HoT by 20% if there was already a regrowth HoT on the target when casted.”  That implies that the HoT receives the  same M as the direct heal in both cases.  In the best case, then, M = 1.473, or M = 1.261 if no HoT was present.

That gives a best case per tick healing value of:

1.473 * (111.7 + .2256 * SP) = 164.53 + .3323 * SP

In order to assess how this impacts Regrowth’s overall healing, we need to examine how regrowth might be used.  The perhaps worst case scenario is casting regrowth on a target on whom it hasn’t been cast before.  That is, let’s assume a target who is newly in need of healing and completely lacks for any of your HoTs.  We know that in this case, Regrowth is strictly superior to Nourish in terms of absolute healing.  In order to provide the advantage to Nourish, we would need to first supply a HoT, such as rejuvenation.  Having that in place would tend to put Nourish ahead of Regrowth, in the case that Regrowth is not present.  We’re ignoring, however, the HoT component of Regrowth in that case, and that needs to be watched.  In a situation of heavy raid damage, that HoT component may just be worth taking the time to cast Regrowth.  Consider that, talented, that HoT lasts for a full 27 seconds.

Using the equation above, we know the regrowth HoT will heal for

1.261 * (111.7 + .2256 * SP) = 140.85 + .284 * SP per tick if regrowth is not present.  That amounts to 46.95 + .095 * SP hps.  While that doesn’t compare to nourish or regrowth spam, it is essentially free.  That is, Regrowth provides a Nourish level heal, though not quite the HPS, and applies this additional buffer.

I think examination of HPS is probably warranted at this point.  We know that the nominal value of Nourish exceeds Regrowth in the best case, and that it has a faster cast time.  That implies that, in the best case, Nourish is naively the highest HPS cast.  Additionally, it is cheaper than Regrowth (even accounting for the ToL reduction).  If you are simply spam-casting a single healing spell, once hots are refreshed, Nourish is hands down the best choice for throughput.  However, what about the case where a single tick of Regrowth is allowed to occur before it is recast?  This is the situation where the spell is used as a semi-irregular top-off heal.  For instance, you cast all your HoTs on the OT and then look elsewhere.  Every once in a while you recast HoTs, stack LB, and top off their health.  Should you use Nourish or Regrowth in this case?

First, total healing throughput is less of a concern…that is, cast time is not an issue, since this isn’t repeated casting and both casts finish in the allotted time.  Rather, this is a matter of total healing done.  To find regrowth’s avg heal, considering direct heal + 1 tick, we simply add the two together:

(3482.2 + .952 * SP) * (1 + .95 * crit) + 164.53 + .3323 * SP

We can then compare to Nourish.  Since we’re basically calling this a best case scenario, we’ll assume Nourish with all HoTs:

(3408.6 + 1.107 * SP) * (1 + .95 * crit)

What falls out of this is that critical strike chance plays the role of determining which is the better choice.  In the case where crit is 0, Regrowth is superior nominal healing: it’s scaling from SP will be strictly superior, and it will have an advantage in base amount healed.  Beyond 0% crit, the coefficient on Nourish’s SP component will need to be greater than the direct heal + tick coefficient for Regrowth at the same crit value.  To find that point, we can ignore constant values (the base healing value for each) and assume SP is 1.  That leaves us strictly with a comparison of coefficients:

.952 * (1 + .95 * crit) + .3323 = 1.107 * (1 + .95 * crit)

.952 + .9044 * crit + .3323 = 1.107 + 1.051 * crit

1.284 + .9044 * crit = 1.107 + 1.051 * crit

.177 = .1466 * crit

crit = 1.207

So, at a critical strike chance of 120%, Nourish will begin to scale faster with SP than Regrowth.  As this is an impossible number to achieve, in the event that regrowth exists on the target and you will be able to allow it to tick at least once before you would either cast regrowth or nourish, casting regrowth is always superior.  The difference should grow even greater in the case that you do not have WG on the target.

Assuming that you have both the regrowth and the nourish glyph, there exist two potential spaces where Nourish is the better cast option compared to regrowth: in the case where the Regrowth glyph does not apply, i.e. Regrowth is not on the target, Nourish potentially a better choice, particularly when the target needs additional health as rapidly as possible.  The other case is when all possible HoTs are on the target and they need maximum additional throughput from you.  In all cases between these two extremes, regrowth is the better option.

Keep in mind that this implies that if you are casting Rejuvenation on everyone, perhaps to generate the revitalize proc, Nourishing a target with Rejuvenation but no Regrowth is almost definitely better.  If the target is not taking sufficient damage to warrant anything more than rejuv and simply needs topped off, a regrowth will be slower, less efficient, and substantial overkill compared to Nourish.

Of course, if that is the case, a lifebloom may be a better alternative than Nourish, depending on the state of their health and your prediction of incoming damage.