## Simulate dice rolls

For the discussion of math. Duh.

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elminster
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### Simulate dice rolls

I'm working some source code for a game, doing repeated rolls for large numbers of dice is cpu time consuming; so I was wondering if there was a way to simulate it using less cpu time. It doesn't have to be as accurate, but anything reasonable would do.
I was thinking, use an approximation of a Gaussian distribution, the mean and standard deviations if the dice values to come up with a similar sort of thing. Problem is, I wouldn't know how to go about it.
The person here:http://www.protonfish.com/random.shtml suggests:

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`function rnd_snd() {  return (Math.random()*2-1)+(Math.random()*2-1)+(Math.random()*2-1);} function rnd(mean, stdev) {  return Math.round(rnd_snd()*stdev+mean);} `
Although I've got no idea how to go about proving the quality of the values that would generate. Say for example, I was trying to get a random number for 20D6 (i.e. 20 six-sided dice rolls added together) how similar would something like that be to the theoretical ideal results and is there a better (Ideally using as little resources as possible) way to do it?

quintopia
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### Re: Simulate dice rolls

For enough dice, the approximation to the normal distribution is reasonable. I have found on the internet at one point a way to compute random numbers from a standard normal using only uniform random variables, but I can't find it right now. Just google it for a while.

evilbeanfiend
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### Re: Simulate dice rolls

i have to say this sounds like premature optimization to me, is your prng really your bottleneck? have you profiled it properly?
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NathanielJ
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### Re: Simulate dice rolls

quintopia wrote:For enough dice, the approximation to the normal distribution is reasonable. I have found on the internet at one point a way to compute random numbers from a standard normal using only uniform random variables, but I can't find it right now. Just google it for a while.

Actually, the method used in the original post uses one form of this. If you sum a whole bunch of uniform random variables, it starts to look normal. The code in the OP doesn't truly find values from a normal, but rather sums 3 uniform random variables and pretends that the result is normal.
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quintopia
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### Re: Simulate dice rolls

I could have sworn the method I saw got as precise as the underlying data types with only a constant number of uniform random numbers. I found something involving building from exponentials. Is that how it's done?

EDIT: Found the note: http://www.taygeta.com/random/gaussian.html

tehtmi
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### Re: Simulate dice rolls

For the method of adding three uniform random variables (mentioned by the OP), here is an illustration of how good of an approximation to the normal you get:

http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/

If all you are worried about is minimizing the number of calls to your psuedo-random number generator, you could do something like this:

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`function roll(int dice, int sides) {  r = random_integer(sides ** dice); // a number in {0,...,sides ** dice - 1}  sum = 0;  while(r > 0) {    sum += r % sides;    r  /= sides;  }  return sum + dice;}`

...just be sure not to overflow with (sides ** dice).

elminster
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### Re: Simulate dice rolls

Well the problem was that the original coder done it in such a way that a certain monster would have 15000d4 health points, so 15000 calls loops inside it (I'll probably change this to 150d400 or a very basic formulae for the time being). This would even lag the intel Q9400 based server it's running on (A high powered cpu). That particular monster is a more unique case (Rarely comes into the game), but the current dice function is used by quite a few things.

I did actually run a profiler on it (VTune) and it does indeed show it up as one of the most time consuming functions. So I was looking for a way to preserve the numbers generated but using less resources. The other methods probably use up more resources compared to a low number of throws, but I was looking for something more efficient for the higher numbers.
The current loop is:

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`for (i = 1; i <= _throw; i++)      ret += (rand() % range) + 1;`
_throw being number of throws, range being the number of sides, ret being the returned value, rand being the random number generator. It's basically looping for the number of throws and accumulating each value, which would between 1 and the number of sides.

I didn't put this in the coding forum because it's more about the maths. I could implement or find the code for any mathematical function, but I don't know the maths involved nor how to calculate the accuracy of the results (Mathematically proving I mean, since just generating numbers only gives a certain amount of accuracy)

GreedyAlgorithm
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### Re: Simulate dice rolls

Box-Muller transform. Pseudo-code:

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`roll(dice, sides)  mean = (sides+1)/2.0;  variance = (sides+1)*(2*sides+1)/6.0 - mean*mean;  mean /= dice;  variance /= dice;  sum = floor(normalboxmuller(mean,variance) + 0.5); //add the 0.5 so that e.g. values from 34.5 to 35.5 are counted as 35  return sum;normalboxmuller(mean, variance)  X = random(); //uniform (0,1]  Y = random();  N = sqrt(-2 * ln(X)) * cos(2 * pi * Y);  N = N * sqrt(variance) + mean;  return N;`

You'll want to truncate roll, and maybe only use it if dice>10 or something. You can get slightly more accurate by changing the "+ 0.5" bit to depend on how far into the tail of the distribution you are, but it's really not worth it. This'll be quite accurate as long as you don't call it for 2d10 or something similarly small.
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Yakk
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### Re: Simulate dice rolls

Hmm. A fast XdY function could be fun.

Here is an attempt:
Use a binomial distribution trick to calculate the number of X dice with Y faces that land on 1.
Then eliminate that many dice from your pool. Do the same for the number of X' dice with Y-1 faces that land on 1 -- this is the number of 2s you get.
Repeat until you eat all of the faces (ie, when you have X'''''..''' dice with 1 face, all of them land on 1, so it is guaranteed to terminate).

Now you can precalculate the exact binomial distributions and store them in a table of dimension X x Y x Y.

If the largest dice you use are d20s, and you use them in clumps of 100 dice, that's a table with 40000 entries, each of them a probability of (X dice with Y sides) having (exactly Q dice landing on a 1).

For a given XdY roll, you break X down into chunks of 100. Do a RNG. Accumulate up the table until you hit the threshold. Repeat for each face of the die.

This gives you mathematically perfect results in O(Y) time for up to 100 dice being rolled.

For beyond 100 dice... you could just scale the result (so 15000d4 is 150 * 100d4 ) naively, or you could do a double-scaling of both magnitude and variance 15000d4 = {[(100d4 - 100*2.5)*SD_scale_factor] + 100*2.5} * 150. Then fuzzy it with a random +/- (SD_scale_factor * 150)/2 factor thrown on top (so you don't get discreet pumps every SD_scale_factor * 150 units).

And for lower amounts (like 1d4), you just roll 1d4.

In C++, I'd implement this as a die roll description object. Ie, die( 4 ) would be a 1d4, unrolled.

Calling .roll() would do work.

die(4)^100 might be 100d4 (the interior term rolled 100 times, and added).
die(4)+50 would be 1d4+50.
die(4)*50 would be 50*1d4.

( 2*die(4)+die(8)+2 )^10 is 10d4 *2 + 10d8 + 20.

The object would have a collection of (die size) (die count) and (multiplier), and a constant factor (or the d1) term.

^ number scales the die count of the term (and the constant term).
* and / scales the multipliers (and the constant term).
+ adds two sets of terms.

.roll() then runs something like the above, doing different things for low die-totals than huge die totals, in order to generate a reasonably fast result.

If I was really crazy, I'd extend it to support (1d4)d6. (not that tricky, but probably not worth it).

The best part would be that the 'using' code would now look more natural, and I could change how dice are rolled willy-nilly without ever touching client code every again (by modifying how .roll() works).

But I like coding this kind of thing.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

LaserGuy
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### Re: Simulate dice rolls

If you're using a relatively friendly language, it probably has a gaussian random number generator as well as a uniform generator. Then all you need to do is calculate the mean and variance for NdK dice, and throw those values into the gaussian RNG algorithm. This should give for one calculation any arbitrarily large number of dice. You will need to set some threshold (4 dice is probably okay) for the point where you'll need to calculate each roll individually again.

The variance on 15000d4 is pretty small, by the way. For all practical purposes, this monster will have 37500 health.