The Dice Roller included in this app is quite advanced, especially if you spend some time to type and save formulas that are specific to your game. This is an implementation of the Dart Dice Parser and the text below is taken from their support page:

Dice Notation 

Examples: 

• 2d20 #cf #cs
  • roll 2d20, result will include counts of critical successes (20) and failures (1)
• advantage
  • 2d20-L -- drop lowest
  • 2d20k, 2d20kh -- keep highest
• disadvantage
  • 2d20-H -- drop highest
  • 2d20-kl -- keep lowest
• (2d10+3d20)-L3 -- roll 2d10 and 3d20, combine the two results lists, and drop lowest 3 results
• 20d10-<3->8# -- roll 20 d10, drop any less than 3 or greater than 8 and count the number of remaining dice

Supported notation 

• 2d6 -- roll 2 dice of 6 sides
• special dice variations:
  • 4dF -- roll 4 fudge dice (sides: [-1, -1, 0, 0, 1, 1])
  • 1d% -- roll 1 percentile dice (equivalent to 1d100)
  • 1D66 -- roll 1 D66, aka 1d6*10 + 1d6
    • NOTE: you must use uppercase D66, lowercase d66 will be interpreted as a 66-sided die
  • 2d[2,3,5,7]-- roll 2 dice with values [2,3,5,7]
• exploding dice
  • 4d6! -- roll 4 6-sided dice, explode if max (6) is rolled (re-roll and include in results)
    • 4d6 !=5 or 4d6!5 -- explode a roll if equal to 5
    • 4d6 !>=4 - explode if >= 4
    • 4d6 !<=2 - explode if <=2
    • 4d6 !>5 - explode if > 5
    • 4d6 !<2 - explode if <2
    • To explode only once, use syntax !o
      • 4d6 !o<5
• compounding dice (Shadowrun, L5R, etc). Similar to exploding, but the additional rolls for each dice are added together as a single "roll". The original roll is replaced by the sum of it and any additional rolls.
  • 5d6 !! -- roll 5 6-sided dice, compound
    • 5d6 !!=5 or 5d6!5 -- compound a roll if equal to 5
    • 5d6 !!>=4 - compound if >= 4
    • 5d6 !!<=4 - compound if <= 4
    • 5d6 !!>5 - compound if > 5
    • 5d6 !!<3 - compound if < 3
    • To compound only once, use syntax !!o
      • 5d6 !!o<2
• re-rolling dice:
  • 4d4 r2 -- roll 4d4, re-roll any result = 2
  • 4d4 r=2 -- roll 4d4, re-roll any result = 2
  • 4d4 r<=2 -- roll 4d4, re-roll any <= 2
  • 4d4 r>=3 -- roll 4d4, re-roll any >= 3
  • 4d4 r<2 -- roll 4d4, re-roll any < 2
  • 4d4 r>3 -- roll 4d4, re-roll any > 3
  • To reroll only once, use syntax ro
    • 4d4 ro<2
• keeping dice:
  • 3d20 k 2 -- roll 3d20, keep 2 highest
  • 3d20 kh 2 -- roll 3d20, keep 2 highest
  • 3d20 kl 2 -- roll 3d20, keep 2 lowest
• dropping dice:
  • 4d6 -H -- roll 4d6, drop 1 highest
  • 4d6 -L -- roll 4d6, drop 1 lowest
  • 4d6 -H2 -- roll 4d6, drop 2 highest
  • 4d6 -L2 -- roll 4d6, drop 2 lowest
  • 4d6 ->5 -- roll 4d6, drop any results > 5
  • 4d6 -<2 -- roll 4d6, drop any results < 2
  • 4d6 ->=5 -- roll 4d6, drop any results >= 5
  • 4d6 -<=2 -- roll 4d6, drop any results <= 2
  • 4d6 -=1 -- roll 4d6, drop any results equal to 1
  • NOTE: the drop operators have higher precedence than the arithmetic operators; 4d10-L2+2 is equivalent to (4d10-L2)+2
  • NOTE: drop is not subtraction.
    • 4d6 - 3 -- roll 4d6, subtract 3
    • 4d6 - 2d6 -- roll 4d6, subtract the result of rolling 2d6
• cap/clamp:
  • 4d20 C<5 -- roll 4d20, change any value < 5 to 5
  • 4d20 C>15 -- roll 4d20, change any value > 15 to 15
• scoring dice rolls:
  • counting:
    • 4d6 # -- how many results?
      • For example, you might use this to count # of dice above a target. (5d10 -<6)# -- roll 5 d10, drop any less than 6, count results
    • 4d6 #>3 -- roll 4d6, count any > 3
    • 4d6 #<3 -- roll 4d6, count any < 3
    • 4d6 #>=5 -- roll 4d6, count any >= 5
    • 4d6 #<=2 -- roll 4d6, count any <= 2
    • 4d6 #=5 -- roll 4d6, count any equal to 5
  • successes and failures
    • A normal count operation # discards the rolled dice and changes the result to be the count
      • For example, 2d6#<=3 rolls [3,4] then counts which results are <=3 , returning [1]
    • But, sometimes you want to be able to count successes/failures without discarding the dice rolls. In this case, use modifiers #s, #f, #cs, #cf to add metadata to the results.
      • 6d6 #f<=2 #s>=5 #cs6 #cf1 -- roll 6d6, count results <= 2 as failures, >= 5 as successes, =6 as critical successes, =1 as critical failures
        • The above returns a result like: RollSummary(total: 22, results: [6, 2, 1, 5, 3, 5], metadata: {rolled: [6, 2, 1, 5, 3, 5], score: {successes: [6, 5, 5], failures: [2, 1], critSuccesses: [6], critFailures: [1]}})
    • NOTE: order matters
      • 2d20 kh #cf #cs -- roll 2d20, keep the highest, count critical successes & failures. If this rolled [1,18], the 1 is dropped and the result metadata won't record a critical failure. If that's not the behavior you want, move the counts prior to the drop (2d20 #cf #cs kh).
• arithmetic operations
  • parenthesis to force a certain order of operations
  • addition is a little special -- could be a sum of ints, or it can be used to aggregate results of multiple dice rolls
    • Addition of integers is the usual sum
      • 4+5
      • 2d6 + 1
    • Addition of roll results combines the results (use parens to ensure the order of operations is what you desire)
      • (5d6+5d10)-L2 -- roll 5d6 and 5d10, and from aggregate results drop the lowest 2.
      • 5d6+5d10-L2 -- roll 5d6 and 5d10, and from only the 5d10 results drop the lowest 2. equivalent to 5d6+(5d10-L2)
  • * for multiplication
  • - for subtraction
  • numbers must be integers
  • division is not supported.