Embers skill reduces cast speed based on number of extra projectiles. It is not reading the proper number of projectiles. Applying the base cast rate of fireball (1.4333), the game is believing I have 3 to 4 more projectiles than I am actually producing. It is mistakenly picking up Arcane Divergence and Cinder Song as having double the projectiles that they are actually giving (because embers reduces projectiles by 50%).
The description of the skill does not indicate this interaction and instead indicates that it reads the extra projectile count, which in the above example is 6 (because 7 projectiles in total). It is reading the incorrect number, or the tooltip does not properly reflect. To repeat, the tooltip states it is based on the number of extra projectiles. (2+4+2)/2+2 = 6
Math:
1.4333 base cast
96% increased cast speed (regular, fire spell, skill tree increased cast additively)
Less 10% from Skira’s Gambit 2/3
7 projectiles would indicate -60% cast speed, per wording “…has reduced cast speed for each extra projectile”, which would indicate the first one is free, there are 6 projectiles in addition to the first.
Formula below based on: Skill Speed - Official Last Epoch Wiki
Update Cast per Sec = 1.4333 * (1+.96-.1-.6) = 1.806
There is a mention in the tooltip that it applies multiplicatively with other modifiers.
If true, that would be 1.4333 * (1+.96-.1) * (1-.6) = 1.0664
6 tests at 60 seconds per test, consistent 77 casts.
77/60 = 1.283
This is definitely slower than the additive approach and faster than the unlikely multiplicative approach.
If I take the above calculation and figure 3 extra projectiles (as in it is misreading Arcane Divergence and Cinder Song), that would be 1.4333 * (1+.96-.1-.9) = 1.376
The above is close.
If I assume it also is not reading Firespray and thus adding 4 projectiles beyond the actual count, that yields 1.23 cast per second, which is also fairly close.
It seems clear this is pulling the wrong number, in lieu of using the number of extra projectiles it is not realizing that Embers is removing half of those extra projectiles.