Problem 1301. RISK Calculator - Large Armies, High Accuracy, Fast

This Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100. [ Attack >= 2 and Defense >=1 ].

Related to Cody 1260 RISK Board Game Battle Simulation

Link to official Risk Rules

Simplified explanation of the dice play:

Attacker with 2 armies will throw one die.
Attacker with 3 armies will throw two die.
Attacker with 4 or more armies will throw three die.

Defense with 1 army will use one die.
Defense with 2 or more armies will throw 2 die.

The attacker High is compared to the Defender High.
If Attacker High > Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.
If the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. 
If Attack > Defense then Defense loses an army otherwise Attack loses an army.
Attack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose). 

Input: a,d where a is number of attacking armies and d is number defending

Output: pwin, the probability of the Attacker Winning

Accuracy: Accurate to +/- 1e-6

Scoring: Time (msec) to solve 10 Battle Scenarios

Risk Calculator