Day 8 (#8)

Blown way through my time budget and there's a bunch of low hanging fruit (data dependencies in LCM, for example) but I'm late for work.

Co-authored-by: Smaug123 <patrick+github@patrickstevens.co.uk>
Reviewed-on: #8

AdventOfCode2023.FSharp/AdventOfCode2023.FSharp.Lib/Arithmetic.fs

@@ -0,0 +1,67 @@
+namespace AdventOfCode2023
+
+[<RequireQualifiedAccess>]
+module Arithmetic =
+
+    [<Struct>]
+    type EuclidResult<'a> =
+        {
+            Hcf : 'a
+            A : 'a
+            B : 'a
+        }
+
+    /// Compute floor(sqrt(i)).
+    let inline sqrt (i : ^a) =
+        if i <= LanguagePrimitives.GenericOne then
+            i
+        else
+            let rec go start =
+                let next = start + LanguagePrimitives.GenericOne
+                let sqr = next * next
+
+                if sqr < LanguagePrimitives.GenericZero then
+                    // Overflow attempted, so the sqrt is between start and next
+                    start
+                elif i < sqr then
+                    start
+                elif i = sqr then
+                    next
+                else
+                    go next
+
+            go LanguagePrimitives.GenericOne
+
+    /// Find Hcf, A, B s.t. A * a + B * b = Hcf, and Hcf is the highest common factor of a and b.
+    let inline euclideanAlgorithm (a : ^a) (b : ^a) : EuclidResult< ^a > =
+        let rec go rMin1 r sMin1 s tMin1 t =
+            if r = LanguagePrimitives.GenericZero then
+                {
+                    Hcf = rMin1
+                    A = sMin1
+                    B = tMin1
+                }
+            else
+                let newQ = rMin1 / r
+                go r (rMin1 - newQ * r) s (sMin1 - newQ * s) t (tMin1 - newQ * t)
+
+        let maxA = max a b
+        let minB = min a b
+
+        let result =
+            go
+                maxA
+                minB
+                LanguagePrimitives.GenericOne
+                LanguagePrimitives.GenericZero
+                LanguagePrimitives.GenericZero
+                LanguagePrimitives.GenericOne
+
+        if a = maxA then
+            result
+        else
+            {
+                Hcf = result.Hcf
+                A = result.B
+                B = result.A
+            }