Calculate Dice Distributions

Syntax for Specifying Dice

You can combine the following symbols in order to create interesting distributions for games and statistical questions.

• 20 Numeric Constants.
• d6 Fair Dice. Uniform distribution from 1 until 6, 20 and 100 inclusive respectively.
• * Multiplication. Example: d6*d6 could be 36 if we roll a 6 two times, but it can never be 35. 2*d6 can never be 11, while 2d6 can be.
• + - Addition and Substraction. Example: d6+d6 is the same as often denoted as 2d6. d20-d20 is the difference between two 20-sided dice.
• x Multiaddition. Example: 2xd6 is the same as often denoted as 2d6. It is rolling two dice. d2xd6 is saying, we roll a d2, either giving 1 or 2 and then rolling a d6 that many times adding up the results. This operator is left-assosiative. When two factors like Constants (6) or Dice (d20) follow each other without an operator between, a x is automatically inserted between them. That means 3d20 = 3xd20 = d20+d20+d20. For Constants: a x b = a * b (e.g. 5 x 3 = 5 * 3)
• / Division. Values are divided and rounded to the next integer. For example d6/2 = d3 . The value range of d6/3 however is [0,1,2]. Also left-associative.
• max(x, y, ...) The maximum of some comma-separated values. For example max(d6,d6) is throwing 2 dice and just keeping the highest.
• min(x, y, ...) The minimum of some comma-separated values. For example min(d6,d6) is throwing 2 dice and just keeping the lowest.
• abs(x) The absolute value of x. For a probability distribution, the probability of each negative value is added to the probability of the corresponding positive value. Example: for abs(d6-3) the probability of 2 is 1/3, because for d6-3 the probability of 2 is 1/6 and of -2 is also 1/6. 1/6 + 1/6 = 1/3.

Details

• Integers All values the dice can take are limited to integers. Even the divide operator rounds everything to integers. Also inputting non-integers will return errors and not compute a distribution.
• Fractions Probabilities are computed as mathematically accurate fractions with infinite precision. You can however see them as decimal values as well in the table and the chart. This design decision allows for accuracy, though sacrificing a lot of performance. It's a tradeoff. Input e.g. 40d6 and hover over the graph to see how precise these fractions can get.
• Precedence The operators in order by their precedence (highest to lowest): x, *, /, +/-. You can use Brackets to explicitly control the order of evaluation if you are not sure.

Performance Limitations and Background

In general, if the number of different values the distribution has, exceeds ca. 1000 you might have to wait longer than 1-3 seconds until the computation is complete. Computation could be roughly 100x faster if we used only floats for probabilities, but I chose to use fractions with infinite precision to keep everything mathematically accurate. Also for highly convoluted distributions even 64-floats will become 0.0 very fast and do not tell us anything.
Rolling dice (even 1000 at once) is pretty cheap though even for very complex distributions and can be usually done in a couple of milliseconds in the webbrowsers worker thread that is using Web Assembly.

Rust and Web Assembly

This project uses theRust Dices Library that I wrote with the goal in mind to make it compile to WebAssembly. The web-app you see spins up a separate worker thread which interops with WebAssembly. In this way the main thread is not blocked, even during heavy computations.