601.2g The player pays the total cost in any order. Partial payments are not allowed. Unpayable costs can’t be paid. Example: You cast Altar’s Reap, which costs {1}{B} and has an additional cost of sacrificing a creature. You sacrifice Thunderscape Familiar, whose effect makes your black spells cost {1} less to cast. Because a spell’s total cost is “locked in” before payments are actually made, you pay {B}, not {1}{B}, even though you’re sacrificing the amiliar.

601.2h Once the steps described in 601.2a–g are completed, the spell becomes cast. Any abilities that trigger when a spell is cast or put onto the stack trigger at this time. If the spell’s controller had priority before casting it, he or she gets priority.

So you start with 3 untapped pyromancers, cast the spell and pay the additional cost of tapping 3 red creatures (9 damage), then the pyromancers ability to untap triggers, you start resolving and you untap them and you may tap them for additional 3 damage. Without additional untap there are 12 damage as maximum possible.