Resolute Technique - break even point (Chance to hit vs Critical Strike)

*I didn't post this in the "Beta Passive Tree/Keystone Feedback" sub-forum because this thread doesn't contain something I would consider as feedback, but the forum is kind of a clusterfuck, so moderators: feel free to move this thread.
*English is not my native language so I would be very thankful for any kind of errors I make
*contains simple math

With RT you trade your chance to deal critical strike for your chance to miss. So to compare a build with RT to one which doesn't use it you have to compare the loss of damage due to missed hits to the increase in damage due to critical strikes.
I will calculate the required "chance to hit", given a certain value of "critical strike chance/critical strike multiplier", at which the expected damage without RT exceeds damage with RT.

From the mechanicas thread we can take that there are three important values to calculate damage:
- Chance to hit/Hit percantage ("Hit")
- Critical Strike Chance ("CS")
- Critical Strike Damage Multiplier ("CM")

When you strike an enemy you first check if it hits (possibility to hit=Hit), afterwards you check if the strike is a crit AND the enemy checks if he can evade the crit, making it a normal hit (possibilty to crit=CS*Hit).
A normal hit deals your regular damage while a critical strike deals 150% of your damage, further increased by your critical strike damage multiplier (resulting in 1.5*(1+CM)*damage).

So to calculate the Expected Damagevalue of an attack without RT we get:
Hit*(1-CS) (Attack hit but didn't crit)
+Hit*CS*(1-Hit) (Attack hit, did crit but the crit was evaded)
+Hit*CS*Hit*1.5*(1+CM) (Attack hit, did crit and the crit hit)

EV(Damage)=Damage*[Hit*(1-CS)+Hit*CS*(1-Hit)+Hit^2*CS*1.5*(1+CM)]=...=Damage*[Hit^2*1/2*CS*(1+3*CM)+Hit]

We now compare that to an attack with RT. In this case Hit=1 (100%), Critical Strike Chance=0 (0%). And we get something way simpler:

EV(Damage, Resolut Technique)=Damage

As you can see the expected damage is equal in both cases when
Hit^2*1/2*CS*(1+3*CM)+Hit=1

As your chance to deal crits/your crit multiplier are pretty much static, while your chance to hit is a lot more flexible we will now transpose (is that the right english word?) the equation for "Hit", resulting in:

Hit=(sqrt(1+2*CS*(1+3*CM))-1)/(CS*(1+3*CM)), with sqrt(..) declaring the square root.

With this we can calculate the minimum chance to hit ("Hit") at which an attack with RT and without it deal equal damage, given a certain value of "critical strike chance/critical strike multiplier".
So if our chance to hit exceeds this value we can expect to deal more damage without RT, while we should take RT if we are bellow that value.

*If someone knows a way to make tables in this forum, tell me^^

The following "table" contains some combinations of critical strike chance ("CS") and critical strike damage modifier ("CM") and the required chance to hit ("Hit") to deal equal damage with and without RT.
Example: CS=5%, CM=0% (your Critical strikes deal the usual 150% damage). Your chance to hit has to be about 97.6%. With a higher chance to hit RT would decrease your damage, while at a lower chance (read: almost always) you want to take RT to increase your average damage.

CS | CM| Hit
5% 0% 97.6%
6% 0% 97.2%
7% 0% 96.7%
10% 0% 95.4%
20% 0% 91.6%
30% 0% 88.3%
40% 0% 85.4%
95% 0% 74.0%
5% 40% 95.0%
6% 40% 94.1%
7% 40% 93.2%
10% 40% 90.9%
20% 40% 84.3%
30% 40% 79.3%
40% 40% 75.2%
95% 40% 61.1%

Taking away critical strike also takes away anything related to crit, such as:
- elemental status alignements such as ignite and shock, resulting in more damage if you do that kind of elemental damage
- your spells also lose the chance to crit but didn't miss in the first place
- the "surgeon's" mod for flasks (flask gains 1 charge on crit)
So even if you don't have the exact chance to hit needed you might still skip RT to get those bonuses.

Critical Strike="CS"
Critical damage multiplier="CM"
Chance to hit=Hit

With this:

Hit=(sqrt(1+2*CS*(1+3*CM))-1)/(CS*(1+3*CM))

You can calculate your required chance to hit at which you deal more damage without RT instead of taking it. As soon as your hitchance exceeds that value NEVER take RT.

As you might have noticed it takes a significant amount of critchance,critmultiplier and hitchance to not get a bonus from RT, a usual build with 10%Critchance and 40%Crit Multiplier will still need a hitchance around 90% to be more efficient than a build with RT.

So I guess if you want a TL;DR which doesnt involve math I might as well give you the answer which most of you already suspected:
Take RT when you are near it in the passiv skill tree or don't have to spend many points to get there to increase your attack damage significantly.
Do not takte RT if you rely heavily on spells, elemental status alignements or want to use things that trigger on crits like the suregons mod on Flasks or knockback on bows.

In the end I guess my "conlusion" is kinda lame, as I don't tell you anything which you didn't already know, but hey that's what math is there for: support obvious facts with numbers, so you can say "DUH". Which brings us to the

real TL;DR: Thank you captain obvious!