fix: "Enrich" (add more x points) when integrating

in order to get more numerical precision.
Note: not complete yet.

Value: [1e-3 to 3e-1]

packages/squiggle-lang/__tests__/Distributions/KlDivergence_test.res

@@ -69,7 +69,7 @@ describe("klDivergence: continuous -> continuous -> float", () => {
       2.0 ** 2.0 *.
       (10.0 ** 2.0 -. (10.0 +. 9.0) *. 10.0 +. (9.0 ** 2.0 +. 10.0 *. 9.0 +. 10.0 ** 2.0) /. 3.0)
     switch kl {
-    | Ok(kl') => kl'->expect->toBeSoCloseTo(analyticalKl, ~digits=1)
+    | Ok(kl') => kl'->expect->toBeSoCloseTo(analyticalKl, ~digits=3)
     | Error(err) => {
         Js.Console.log(DistributionTypes.Error.toString(err))
         raise(KlFailed)

packages/squiggle-lang/src/rescript/Distributions/PointSetDist/Continuous.res

@@ -272,10 +272,11 @@ module T = Dist({
     XYShape.Analysis.getVarianceDangerously(t, mean, Analysis.getMeanOfSquares)
 
   let klDivergence = (prediction: t, answer: t) => {
+    let enrichedAnswer = XYShape.PointwiseCombination.enrichXyShape(answer.xyShape) // answer.xyShape //
     let newShape = XYShape.PointwiseCombination.combineAlongSupportOfSecondArgument(
       PointSetDist_Scoring.KLDivergence.integrand,
       prediction.xyShape,
-      answer.xyShape,
+      enrichedAnswer,
     )
     let xyShapeToContinuous: XYShape.xyShape => t = xyShape => {
       xyShape: xyShape,

packages/squiggle-lang/src/rescript/MagicNumbers.res

@@ -12,6 +12,7 @@ module Epsilon = {
 module Environment = {
   let defaultXYPointLength = 1000
   let defaultSampleCount = 10000
+  let enrichmentFactor = 10
 }
 
 module OpCost = {

packages/squiggle-lang/src/rescript/Utility/XYShape.res

@@ -453,6 +453,44 @@ module PointwiseCombination = {
     T.filterOkYs(newXs, newYs)->Ok
   }
 
+  let enrichXyShape = (t: T.t): T.t => {
+    let length = E.A.length(t.xs)
+    Js.Console.log(length)
+    let points = switch length < MagicNumbers.Environment.defaultXYPointLength {
+    | true =>
+      Belt.Int.fromFloat(
+        Belt.Float.fromInt(
+          MagicNumbers.Environment.enrichmentFactor * MagicNumbers.Environment.defaultXYPointLength,
+        ) /.
+        Belt.Float.fromInt(length),
+      )
+    | false => MagicNumbers.Environment.enrichmentFactor
+    }
+
+    let getInBetween = (x1: float, x2: float): array<float> => {
+      let newPointsArray = Belt.Array.makeBy(points - 1, i => i)
+      // don't repeat the x2 point, it will be gotten in the next iteration.
+      let result = Js.Array.mapi((pos, i) =>
+        switch i {
+        | 0 => x1
+        | _ =>
+          x1 *.
+          (Belt.Float.fromInt(points) -. Belt.Float.fromInt(pos)) /.
+          Belt.Float.fromInt(points) +. x2 *. Belt.Float.fromInt(pos) /. Belt.Float.fromInt(points)
+        }
+      , newPointsArray)
+      result
+    }
+    let newXsUnflattened = Js.Array.mapi((x, i) =>
+      switch i < length - 1 {
+      | true => getInBetween(x, t.xs[i + 1])
+      | false => [x]
+      }
+    , t.xs)
+    let newXs = Belt.Array.concatMany(newXsUnflattened)
+    let newYs = E.A.fmap(x => XtoY.linear(x, t), newXs) //XtoY.linear(newXs)
+    {xs: newXs, ys: newYs}
+  }
   // This function is used for klDivergence
   let combineAlongSupportOfSecondArgument: (
     (float, float) => result<float, Operation.Error.t>,