[flamefury]

Sometime shortly after that the formula was updated to divide the sum of skill levels by 8, becauseis exactly how it works, and that was indeed way overpowered .

"Working like a normal unresistable spell" translates to "spell power is increased by floor(caster's Spirit / 4)" + "average number of hits = spell level + spell level * accuracy %" + "target's magic evasion is ignored", so the formula for Ultima looks more like:

spell power = 25 + floor(caster's Spirit / 4) + floor(sum(caster's skill levels) / [skill levels are 0..15, not 1..16]

and then that spell power gets passed in to the standard routine for unresistable spell damage:

average damage = (spell power * (1 + random(0..spell power)) + spell power * 0.05) * (spell level * accuracy %)

So in the PSP formula, it looks like the base damage value incorporates most of the damage calculation, and the average damage is basically the base damage * 1.05, but in the NES version, the spell power is part of a larger calculation, and the average damage is basically (spell power * 1.55) * number of hits.

The damage functions scale very differently (PSP is quadratic in skill levels vs. linear in ours), but in general, the PSP version appears to be more powerful than our current version for low levels of Ultima (where the +100 dominates) regardless of the caster's overall power and when the caster has at least 50% of their maximum power (where the skill level squared dominates) regardless of Ultima's level.

I'm still not sure how you're coming to the conclusion that PSP version is more powerful at lower levels. Reminder that the +100 is not multiplied by Ultima's level. It's only relevant for very low level sums.

Even against a 25 power spell with no bells or whistles (other than always having 2*spell level hits), you would need a sizeable amount of skill+magic levels to overpower it decisively:



EDIT: Also, later releases also only use magic "evasion" though it's called defense. There's no way to lower the damage from a magic hit outside of element resist, you can only avoid magic hits to reduce damage from the spell cast overall, but each hit is more or less set on how much it will deal.

[abw]

I'm still not sure how you're coming to the conclusion that PSP version is more powerful at lower levels. Reminder that the +100 is not multiplied by Ultima's level. It's only relevant for very low level sums.

Try using the entire damage formulae, including critical hits and random boni. Assuming I've got enough mental energy left to do this correctly, the average damage from the PSP version based on https://finalfantasy.fandom.com/wiki/Ultima_(Final_Fantasy_II) should be:

Code: [Select]

<Average Damage> = (<Ultima Level> * (<Tier> + floor(<Tier> / 2)) ^ 2 + 100) * 1.05 + 0.5 * ((<Ultima Level> * (<Tier> + floor(<Tier> / 2)) ^ 2 + 100) % 256)
giving (rounded to the nearest integer):

Ultima Level / Tier	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	155	157	169	180	211	231	281	310	378	289	376	424	529	587	583	649	792
2	155	158	183	205	267	307	278	337	473	551	597	693	903	890	1138	1143	1429
3	155	160	197	229	322	383	404	492	569	685	817	961	1150	1322	1566	1638	1937
4	155	161	211	254	378	331	529	647	792	819	1166	1230	1524	1625	1993	2260	2574
5	155	163	225	279	306	407	655	674	887	1081	1387	1499	1898	2057	2421	2754	3211
6	155	164	239	304	362	483	652	829	1110	1215	1608	1768	2272	2488	2976	3248	3848
7	155	166	253	329	418	559	778	856	1205	1349	1828	2037	2518	2792	3404	3742	4357
8	155	167	267	353	473	635	903	1011	1429	1611	2049	2305	2893	3223	3831	4237	4993
9	155	169	281	378	529	583	901	1166	1524	1745	2270	2574	3267	3527	4387	4731	5630
10	155	171	295	275	585	659	1027	1193	1619	2007	2491	2843	3513	3959	4815	5225	6267
11	155	172	308	300	641	734	1152	1348	1842	2140	2711	3112	3887	4390	5242	5719	6776
12	155	174	322	325	569	810	1150	1375	1937	2274	3060	3381	4261	4694	5670	6213	7413
13	155	175	336	349	624	886	1275	1530	2161	2536	3281	3649	4636	5125	6225	6836	8049
14	155	177	350	374	680	834	1401	1685	2256	2670	3502	3918	4882	5429	6653	7330	8686
15	155	178	364	399	736	910	1398	1712	2479	2804	3722	4187	5256	5860	7080	7824	9195
16	155	180	378	424	792	986	1524	1867	2574	3066	3943	4456	5630	6292	7636	8318	9832

Based on those numbers, however, I'm questioning whether the linked formula is correct, as the damage growth from one tier to the next is pretty erratic and there are several places where gaining a tier actually reduces the average damage (e.g. at tier 9, Ultima 1 does about 90 damage less than at tier 8 since tier 9's maximum random damage = ((9+4)^2+100)%256 = 13 vs. tier 8's ((8+4)^2+100)%256 = 244).

Our version takes an extra variable (caster's Spirit), so to try to keep the comparison somewhat even, let's define a tier for our version as a percentage of the caster's combined skill + stat power out of a total of 16, so e.g. a caster with a total of 24 * 16 = 384 skill points and 99 Spirit has a tier of 16, a caster with 192 skill points and 50 Spirit has a tier of 8, etc. It's not entirely a fair comparison since Spirit is both easier to raise and has a larger marginal effect on average damage than do skill levels, but it's good enough for hand-waving. In that case, the average damage from our version should be:

Code: [Select]

<Average Damage> = (<Ultima Level> * (1.5 + min(0.5, <Tier> / 16))) * (25 + floor(99 * <Tier> / 16 / 4) + floor(24 * (<Tier> - 1) / 8)) * 1.55
giving (rounded to the nearest integer):

Ultima Level / Tier	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	58	63	78	92	109	124	142	159	180	192	208	223	236	251	264	279	291
2	116	126	156	183	217	247	285	318	360	384	415	446	471	502	527	558	583
3	174	189	234	275	326	371	427	477	539	577	623	670	707	753	791	837	874
4	233	252	312	366	434	494	570	637	719	769	831	893	942	1004	1054	1116	1166
5	291	315	390	458	543	618	712	796	899	961	1039	1116	1178	1256	1318	1395	1457
6	349	378	468	549	651	742	854	955	1079	1153	1246	1339	1414	1507	1581	1674	1748
7	407	441	547	641	760	865	997	1114	1259	1345	1454	1562	1649	1758	1845	1953	2040
8	465	504	625	732	868	989	1139	1273	1438	1538	1662	1786	1885	2009	2108	2232	2331
9	523	567	703	824	977	1113	1282	1432	1618	1730	1869	2009	2120	2260	2372	2511	2623
10	581	630	781	915	1085	1236	1424	1592	1798	1922	2077	2232	2356	2511	2635	2790	2914
11	639	693	859	1007	1194	1360	1566	1751	1978	2114	2285	2455	2592	2762	2899	3069	3205
12	698	756	937	1099	1302	1483	1709	1910	2158	2306	2492	2678	2827	3013	3162	3348	3497
13	756	819	1015	1190	1411	1607	1851	2069	2337	2499	2700	2902	3063	3264	3426	3627	3788
14	814	882	1093	1282	1519	1731	1994	2228	2517	2691	2908	3125	3298	3515	3689	3906	4080
15	872	945	1171	1373	1628	1854	2136	2387	2697	2883	3116	3348	3534	3767	3953	4185	4371
16	930	1008	1249	1465	1736	1978	2279	2547	2877	3075	3323	3571	3770	4018	4216	4464	4662

The damage growth here is much less erratic, plus if we drop the tier simplification and look at the base skill levels and Spirit stat, we get damage growth with every 8 skill levels (vs. 24 for PSP) and every 4 points of Spirit; furthermore, every point of Spirit increases accuracy by 1% (up to a cap of 50% since Ultima has a base accuracy of 50%, but Spirit beyond 50 is still useful for counteracting accuracy reduction due to equipment penalties), making extra hits more likely and raising the average damage.

As for expected player stats/skill levels, our version of Ultima is the same in both the Bug Fix and Restored patches, but Restored's passive skill growth makes it pretty easy to have level 7+ in all weapon skills and level 9+ in all spell skills by the endgame, plus whatever levels you get from active usage. Give Ultima to somebody with a decent Spirit stat, and they'll likely be dealing damage around the tier 10 level with Ultima 8+, at which point the PSP version deals about 400 more damage (~20%). Of course, there are only about a dozen monsters in the entire game that have high enough max HP for that 400 damage to make a difference .

[flamefury]

Try using the entire damage formulae, including critical hits and random boni. Assuming I've got enough mental energy left to do this correctly, the average damage from the PSP version based on https://finalfantasy.fandom.com/wiki/Ultima_(Final_Fantasy_II) should be:

Code: [Select]

<Average Damage> = (<Ultima Level> * (<Tier> + floor(<Tier> / 2)) ^ 2 + 100) * 1.05 + 0.5 * ((<Ultima Level> * (<Tier> + floor(<Tier> / 2)) ^ 2 + 100) % 256)
giving (rounded to the nearest integer):

Ultima Level / Tier	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	155	157	169	180	211	231	281	310	378	289	376	424	529	587	583	649	792
2	155	158	183	205	267	307	278	337	473	551	597	693	903	890	1138	1143	1429
3	155	160	197	229	322	383	404	492	569	685	817	961	1150	1322	1566	1638	1937
4	155	161	211	254	378	331	529	647	792	819	1166	1230	1524	1625	1993	2260	2574
5	155	163	225	279	306	407	655	674	887	1081	1387	1499	1898	2057	2421	2754	3211
6	155	164	239	304	362	483	652	829	1110	1215	1608	1768	2272	2488	2976	3248	3848
7	155	166	253	329	418	559	778	856	1205	1349	1828	2037	2518	2792	3404	3742	4357
8	155	167	267	353	473	635	903	1011	1429	1611	2049	2305	2893	3223	3831	4237	4993
9	155	169	281	378	529	583	901	1166	1524	1745	2270	2574	3267	3527	4387	4731	5630
10	155	171	295	275	585	659	1027	1193	1619	2007	2491	2843	3513	3959	4815	5225	6267
11	155	172	308	300	641	734	1152	1348	1842	2140	2711	3112	3887	4390	5242	5719	6776
12	155	174	322	325	569	810	1150	1375	1937	2274	3060	3381	4261	4694	5670	6213	7413
13	155	175	336	349	624	886	1275	1530	2161	2536	3281	3649	4636	5125	6225	6836	8049
14	155	177	350	374	680	834	1401	1685	2256	2670	3502	3918	4882	5429	6653	7330	8686
15	155	178	364	399	736	910	1398	1712	2479	2804	3722	4187	5256	5860	7080	7824	9195
16	155	180	378	424	792	986	1524	1867	2574	3066	3943	4456	5630	6292	7636	8318	9832

Based on those numbers, however, I'm questioning whether the linked formula is correct, as the damage growth from one tier to the next is pretty erratic and there are several places where gaining a tier actually reduces the average damage (e.g. at tier 9, Ultima 1 does about 90 damage less than at tier 8 since tier 9's maximum random damage = ((9+4)^2+100)%256 = 13 vs. tier 8's ((8+4)^2+100)%256 = 244).

Our version takes an extra variable (caster's Spirit), so to try to keep the comparison somewhat even, let's define a tier for our version as a percentage of the caster's combined skill + stat power out of a total of 16, so e.g. a caster with a total of 24 * 16 = 384 skill points and 99 Spirit has a tier of 16, a caster with 192 skill points and 50 Spirit has a tier of 8, etc. It's not entirely a fair comparison since Spirit is both easier to raise and has a larger marginal effect on average damage than do skill levels, but it's good enough for hand-waving. In that case, the average damage from our version should be:

Code: [Select]

<Average Damage> = (<Ultima Level> * (1.5 + min(0.5, <Tier> / 16))) * (25 + floor(99 * <Tier> / 16 / 4) + floor(24 * (<Tier> - 1) / 8)) * 1.55
giving (rounded to the nearest integer):

Ultima Level / Tier	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	58	63	78	92	109	124	142	159	180	192	208	223	236	251	264	279	291
2	116	126	156	183	217	247	285	318	360	384	415	446	471	502	527	558	583
3	174	189	234	275	326	371	427	477	539	577	623	670	707	753	791	837	874
4	233	252	312	366	434	494	570	637	719	769	831	893	942	1004	1054	1116	1166
5	291	315	390	458	543	618	712	796	899	961	1039	1116	1178	1256	1318	1395	1457
6	349	378	468	549	651	742	854	955	1079	1153	1246	1339	1414	1507	1581	1674	1748
7	407	441	547	641	760	865	997	1114	1259	1345	1454	1562	1649	1758	1845	1953	2040
8	465	504	625	732	868	989	1139	1273	1438	1538	1662	1786	1885	2009	2108	2232	2331
9	523	567	703	824	977	1113	1282	1432	1618	1730	1869	2009	2120	2260	2372	2511	2623
10	581	630	781	915	1085	1236	1424	1592	1798	1922	2077	2232	2356	2511	2635	2790	2914
11	639	693	859	1007	1194	1360	1566	1751	1978	2114	2285	2455	2592	2762	2899	3069	3205
12	698	756	937	1099	1302	1483	1709	1910	2158	2306	2492	2678	2827	3013	3162	3348	3497
13	756	819	1015	1190	1411	1607	1851	2069	2337	2499	2700	2902	3063	3264	3426	3627	3788
14	814	882	1093	1282	1519	1731	1994	2228	2517	2691	2908	3125	3298	3515	3689	3906	4080
15	872	945	1171	1373	1628	1854	2136	2387	2697	2883	3116	3348	3534	3767	3953	4185	4371
16	930	1008	1249	1465	1736	1978	2279	2547	2877	3075	3323	3571	3770	4018	4216	4464	4662

The damage growth here is much less erratic, plus if we drop the tier simplification and look at the base skill levels and Spirit stat, we get damage growth with every 8 skill levels (vs. 24 for PSP) and every 4 points of Spirit; furthermore, every point of Spirit increases accuracy by 1% (up to a cap of 50% since Ultima has a base accuracy of 50%, but Spirit beyond 50 is still useful for counteracting accuracy reduction due to equipment penalties), making extra hits more likely and raising the average damage.

As for expected player stats/skill levels, our version of Ultima is the same in both the Bug Fix and Restored patches, but Restored's passive skill growth makes it pretty easy to have level 7+ in all weapon skills and level 9+ in all spell skills by the endgame, plus whatever levels you get from active usage. Give Ultima to somebody with a decent Spirit stat, and they'll likely be dealing damage around the tier 10 level with Ultima 8+, at which point the PSP version deals about 400 more damage (~20%). Of course, there are only about a dozen monsters in the entire game that have high enough max HP for that 400 damage to make a difference .

EDIT: Whoops I was asleep, my other calc here was wrong.

Just to visualize the times when tier increases lower damage (blue beside orange) and Ultima level increases lower damage (green ontop of red):


Re the linked Ultima formula being wrong, it's definitely possible I made a mistake, but I'm fairly confident my hacking was right. You can double-check my work if you like (don't mind some of the comments around the 0~99 and random bonus roll, I made them before I deciphered what it was doing):
https://pastebin.com/rie8YhB6

Or if you want to hack it yourself, the addresses are 0x0893298c (most likely the generic spellcast function) and 08932d08 (random damage and critical hit check function) in ULUS10263.

The thing causing the damage drops on some Ultima levels and some tier jumps is the mod 256 random bonus. But I'm near certain that this part I understood correctly.

Code: [Select]

andi a0,s3,0xFF     # s3 = base damage, get the lowest bits of damage, save into a0
pos_08932D74:
jal z_un_088a6470  # Rolls 0~a0(=damage % 256) for random damage bonus, outputs into v0
andi s4,s3,0xFFFF   # Copies base damage into s4
andi v0,v0,0xFF     # Not necessary, but capping v0 to 255
addu v0,s4,v0       # Adding into s4 for running sum of damage

EDIT2: Now that I look at this, I understand what's happening. This is the same function that controls damage hits, Ultima is coded as a single damage hit. However, everything else in the game would be capped at 255 power (eg, normal attacks, other spells, etc.) and the random bonus for each of their individual hits has a range of Power to 2*Power. That's this randomized bonus.

This breaks on Ultima because it's a single hit that can go beyond 255, causing its damage to fluctuate around even as its level and tier increases.

[abw]

Ha, that's actually a bug from the NES version (which we've fixed) that has persisted through to the PSP version . In the NES version, it can be triggered by physical attacks with weapons that deal bonus elemental or family damage and push the base damage per hit above 255; just as in the PSP version, the random bonus only operates on the low 8 bits rather than the full 16 bits. Fixing the PSP version's random bonus would increase its average damage and push it even further beyond our version's.

I'm leaning towards keeping our version as-is for now, but the PSP formula is definitely a great find and something to keep in mind if/when rebalancing the game.

[flamefury]

Yeah you'd probably have to make up another randomization function just for Ultima. I doubt they ever wanted it to randomly do (base damage ~ 2 * base damage) because that would be completely insane in terms of damage output.

Maybe something more like a randomization cap that's determined as a function of Ultima level and the level sum, starting at 0 when level sum tier is 0 and Ultima 1 and then capping out at 255 (or 510 or something like that) when the level sum tier is capped and Ultima 16.

[Mirage]

On the subject of Ultima, would it be possible to break up the Ultima-level-dependent part of the damage formula into individual hits and then tack on the base 100 damage at the end? I'm curious on how having multiple lower-damage hits would affect the randomization and critical damage parts of the formula.