Pet attack formula... mostly solved

Ok, I don't have much time to work on this the next week or so, but I think I'm making some real progress on the formula. Maybe someone else can figure it out from here. I think the only things I'm off on still, is my gain per level is slightly too low and some rounding vs truncating errors.

Using these dividers on attack:
1 1.00
2 1.11
3 1.22
4 1.33
5 1.44
6 1.56
7 1.67
8 1.78
9 1.89
10 2.00

It should be possible to remove the differences in attack per level.

Then, by multiplying that number by attacks per second, it seems to create a base rate. For example a Sharptooth Wolfling has an attack stat of 5 with an attack rate of .8 per second. A Pollen Scorpion has an attack stat of 6 with an attack rate of .6 per second. Using data I collected, here's the base attacks for each over the first 22 levels (2-23):

Level	Wolf	Scorpion
2	21	30
3	27	39
4	34	49
5	41	59
6	48	70
7	56	81
8	64	93
9	73	105
10	81	118
11	91	131
12	100	145
13	110	159
14	120	174
15	131	189
16	142	205
17	153	221
18	165	238
19	177	255
20	189	273
21	202	292
22	215	310
23	228	330

When doing what I described above to their attacks here's what happens:

Level	Wolfling	Scorpion
2	11.57	11.63
3	15.04	14.95
4	18.9	18.83
5	22.76	22.71
6	27	26.58
7	31.24	31.02
8	35.87	35.45
9	40.5	40.43
10	45.51	44.86
11	50.53	50.4
12	55.93	55.38
13	61.33	60.92
14	67.11	66.46
15	72.9	72.55
16	79.07	78.65
17	85.24	84.74
18	91.8	91.38
19	98.36	98.03
20	105.3	104.68
21	112.63	111.88
22	119.57	119.08
23	127.29	126.28

Look how close they are. Infact, here's the midpoints, which I'll be using for the rest of this:

8.4
11.6
15
18.87
22.73
26.79
31.13
35.66
40.47
45.19
50.46
55.66
61.13
66.79
72.73
78.86
84.99
91.59
98.19
104.99
112.25
119.32
126.78

See the pattern? If not, check the increase per level, each time it goes up by level*.2. Then look at what's left. It starts at 5 and adds to the previous level giving you:

Level	Gain	Level+Previous	Base=5
1	3	8	
2	3.2	11.2	
3	3.4	14.6	
4	3.6	18.2	
5	3.8	22	
6	4	26	
7	4.2	30.2	
8	4.4	34.6	
9	4.6	39.2	
10	4.8	44	
11	5	49	
12	5.2	54.2	
13	5.4	59.6	
14	5.6	65.2	
15	5.8	71	
16	6	77	
17	6.2	83.2	
18	6.4	89.6	
19	6.6	96.2	
20	6.8	103	
21	7	110	
22	7.2	117.2	
23	7.4	124.6	
24	7.6	132.2	
25	7.8	140	
26	8	148	
27	8.2	156.2	
28	8.4	164.6	
29	8.6	173.2	
30	8.8	182	
31	9	191	
32	9.2	200.2	
33	9.4	209.6	
34	9.6	219.2	
35	9.8	229	
36	10	239	
37	10.2	249.2	
38	10.4	259.6	
39	10.6	270.2	
40	10.8	281	
41	11	292	
42	11.2	303.2	
43	11.4	314.6	
44	11.6	326.2	
45	11.8	338	
46	12	350	
47	12.2	362.2	
48	12.4	374.6	
49	12.6	387.2	
50	12.8	400	
51	13	413	
52	13.2	426.2	
53	13.4	439.6	
54	13.6	453.2	
55	13.8	467	
56	14	481	
57	14.2	495.2	
58	14.4	509.6	
59	14.6	524.2	
60	14.8	539	
61	15	554	
62	15.2	569.2	
63	15.4	584.6	
64	15.6	600.2	
65	15.8	616	
66	16	632	
67	16.2	648.2	
68	16.4	664.6	
69	16.6	681.2	
70	16.8	698	
71	17	715	
72	17.2	732.2	
73	17.4	749.6	
74	17.6	767.2	
75	17.8	785	
76	18	803	
77	18.2	821.2	
78	18.4	839.6	
79	18.6	858.2	
80	18.8	877	
81	19	896	
82	19.2	915.2	
83	19.4	934.6	
84	19.6	954.2	
85	19.8	974	
86	20	994	
87	20.2	1014.2	
88	20.4	1034.6	
89	20.6	1055.2	
90	20.8	1076	
91	21	1097	
92	21.2	1118.2	
93	21.4	1139.6	
94	21.6	1161.2	
95	21.8	1183	
96	22	1205	
97	22.2	1227.2	
98	22.4	1249.6	
99	22.6	1272.2	
100	22.8	1295	
101	23	1318	
102	23.2	1341.2	
103	23.4	1364.6	
104	23.6	1388.2	
105	23.8	1412	

Compare those to the previous set of numbers and look how close they are:

Level	Actual	Calculated
2	11.6	11.2
3	15	14.6
4	18.87	18.2
5	22.73	22
6	26.79	26
7	31.13	30.2
8	35.66	34.6
9	40.47	39.2
10	45.19	44
11	50.46	49
12	55.66	54.2
13	61.13	59.6
14	66.79	65.2
15	72.73	71
16	78.86	77
17	84.99	83.2
18	91.59	89.6
19	98.19	96.2
20	104.99	103
21	112.25	110
22	119.32	117.2
23	126.78	124.6

Therefore, it's not accurate, but I think it's safe to say that long string of numbers above is the base attack for pets.

Now, with that base attack, divide it by the attack/sec and multiply by the attack tier (those 10 numbers above that ranged from 1 to 2) or the formula (1+((Tier-1)/9) to get the expected pet attack for a level. Remember, when testing this pick things with low tame levels as it's not accounting for that reduction higher tame level pets take. And don't forget about the 20% attack bonus from loyalty if applicable.

Lets verify it from my pet bag:
A Herc at 96 is 2989 attack. Hercs have .8/sec and an attack of 7. Meaning 1205/.8*1.67=2515. There's still the attack bonus to account for though, 2515*1.2=3018. As you can see, I'm close.
A Sharptooth Wolfling at 23 is 228 attack without a loyalty bonus. It has an attack of .8/sec with an attack of 5. That's 124.6/.8*1.44=224.28. Again, close.
My Pollen Scorpion at 57 with a loyalty bonus is 1546 attack. Scorpions have an attack of 6 with an attack/sec of .6. So we want 495.2/.6*1.56*1.2 that's 1545.024 attack.

And for good measure, although I don't have one, according to ecatomb a Jaden Qingfu is 2137 attack at 90 with an attack/sec of .8. Using my pet database (or growth chart), I know a Qingfu has an attack of 4 (actually, I think I got this value from the stickied spreadsheet originally). So we're looking at 1076/.8*1.33*1.2 which equals... 2146.62. Once again, I'm pretty close.

Oh, and if you're curious about what this means for a nix since if you try it you'll quickly realize it's way past 10... it's attack is 21. So much for assuming the scale stopped at 10.