if(maxtime > mintime || maxdps > mindps)
{
+ // Constraints:
+
+ // damage we have right now
mindamage = mindps * mintime;
- maxdamage = mindamage + d;
-
- // interval [mintime, maxtime] * [mindps, maxdps]
- // intersected with
- // [mindamage, maxdamage]
- // maximum of this!
-
- if(maxdamage >= maxtime * maxdps)
- {
- totaltime = maxtime;
- totaldamage = maxtime * maxdps;
- // this branch increases totaldamage if either t > mintime, or dps > mindps
- }
- else
- {
- // maxdamage is inside the interval!
- // first, try to use mindps; only if this fails, increase dps as needed
- totaltime = min(maxdamage / mindps, maxtime); // maxdamage / mindps >= mindamage / mindps = mintime
- totaldamage = maxdamage;
- // can totaldamage / totaltime be >= maxdps?
- // max(mindps, maxdamage / maxtime) >= maxdps?
- // we know maxdamage < maxtime * maxdps
- // so it cannot be
-
- // this branch ALWAYS increases totaldamage, but requires maxdamage < maxtime * maxdps
- }
+ // damage we want to get
+ maxdamage = mindamage + d;
- // total conditions for increasing:
- // maxtime > mintime OR maxdps > mindps OR maxtime * maxdps > maxdamage
- // however:
- // if maxtime = mintime, maxdps = mindps
- // then:
- // maxdamage = mindamage + d
- // mindamage = mindps * mintime = maxdps * maxtime < maxdamage!
- // so the last condition is not needed
+ // but we can't exceed maxtime * maxdps!
+ totaldamage = min(maxdamage, maxtime * maxdps);
+
+ // LEMMA:
+ // Look at:
+ // totaldamage = min(mindamage + d, maxdamage * maxdps)
+ // We see:
+ // totaldamage <= maxdamage * maxdps
+ // ==> totaldamage / maxdps <= maxtime.
+ // We also see:
+ // totaldamage / mindps = min(mindamage / mindps + d, maxtime * maxdps / mindps)
+ // >= min(mintime, maxtime)
+ // ==> totaldamage / maxdps >= mintime.
+
+ /*
+ // how long do we damage then?
+ // at least as long as before
+ // but, never exceed maxdps
+ totaltime = max(mintime, totaldamage / maxdps); // always <= maxtime due to lemma
+ */
+
+ // alternate:
+ // at most as long as maximum allowed
+ // but, never below mindps
+ totaltime = min(maxtime, totaldamage / mindps); // always >= mintime due to lemma
+
+ // assuming t > mintime, dps > mindps:
+ // we get d = t * dps = maxtime * maxdps
+ // totaldamage = min(maxdamage, maxtime * maxdps) = min(... + d, maxtime * maxdps) = maxtime * maxdps
+ // totaldamage / maxdps = maxtime
+ // totaldamage / mindps > totaldamage / maxdps = maxtime
+ // FROM THIS:
+ // a) totaltime = max(mintime, maxtime) = maxtime
+ // b) totaltime = min(maxtime, totaldamage / maxdps) = maxtime
+
+ // assuming t <= mintime:
+ // we get maxtime = mintime
+ // a) totaltime = max(mintime, ...) >= mintime, also totaltime <= maxtime by the lemma, therefore totaltime = mintime = maxtime
+ // b) totaltime = min(maxtime, ...) <= maxtime, also totaltime >= mintime by the lemma, therefore totaltime = mintime = maxtime
+
+ // assuming dps <= mindps:
+ // we get mindps = maxdps.
+ // With this, the lemma says that mintime <= totaldamage / mindps = totaldamage / maxdps <= maxtime.
+ // a) totaltime = max(mintime, totaldamage / maxdps) = totaldamage / maxdps
+ // b) totaltime = min(maxtime, totaldamage / mindps) = totaldamage / maxdps
e.fire_damagepersec = totaldamage / totaltime;
e.fire_endtime = time + totaltime;