preroll: Use fewer floating-point operations when calculating variance

1 files changed, 11 insertions, 8 deletions

diff --git a/preroll.go b/preroll.go
index ae9d5a3..ab3d0a0 100644
--- a/preroll.go
+++ b/preroll.go
@@ -1,4 +1,4 @@
-/* Copyright (C) 2011, 2013-2014, 2017 Luke Shumaker <lukeshu@sbcglobal.net> */
+/* Copyright (C) 2011, 2013-2014, 2017, 2019 Luke Shumaker <lukeshu@sbcglobal.net> */
 package main
 
 import (
@@ -35,14 +35,17 @@ func rollSpec(spec string) (mean float64, variance float64) {
 			dice = 1
 		}
 
-		// mean is E(X)
+		// - Let 'd' be the die_size
+		// - Let 'X' be the the uniform random variable that is the result of a 1d${d} roll
+		// - E(X)   = (1+d)/2
 		mean = float64(1 + die_size)/2.0
-
-		// sum is Σi² for i=1..die_size
-		sum := (die_size*(die_size+1)*(2*die_size+1))/6
-		// mean2 is E(X²)
-		mean2 := float64(sum)/float64(die_size)
-		variance = mean2-(mean*mean)
+		// - Var(X) = E(X²)-E(X)²
+		// - E(X²)  = (Σi² for i=1..d) / d
+		//          = ¹/₆d(d+1)(2d+1)  / d
+		//          =    d(d+1)(2d+1)  / (6d)
+		// ∴ Var(X) = (d(d+1)(2d+1) / (6d)) - ((1+d)/2)²
+		//          = ¹/₁₂(d+1)(d-1)
+		variance = float64((die_size+1)*(die_size-1))/12.0
 
 		mean *= float64(dice)
 		variance *= float64(dice)