Initial analysis and plan for implementing find_alpha forward rule

Co-authored-by: yebai <[email protected]>

Author: Copilot

Project.toml

@@ -14,6 +14,7 @@ IrrationalConstants = "92d709cd-6900-40b7-9082-c6be49f344b6"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 LogExpFunctions = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
 MappedArrays = "dbb5928d-eab1-5f90-85c2-b9b0edb7c900"
+Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 Roots = "f2b01f46-fcfa-551c-844a-d8ac1e96c665"
@@ -26,7 +27,6 @@ DistributionsAD = "ced4e74d-a319-5a8a-b0ac-84af2272839c"
 EnzymeCore = "f151be2c-9106-41f4-ab19-57ee4f262869"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 LazyArrays = "5078a376-72f3-5289-bfd5-ec5146d43c02"
-Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6"
 ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267"
 Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
 
@@ -36,8 +36,8 @@ BijectorsEnzymeCoreExt = "EnzymeCore"
 BijectorsForwardDiffExt = "ForwardDiff"
 BijectorsLazyArraysExt = "LazyArrays"
 BijectorsMooncakeExt = "Mooncake"
-BijectorsReverseDiffExt = "ReverseDiff"
 BijectorsReverseDiffChainRulesExt = ["ChainRules", "ReverseDiff"]
+BijectorsReverseDiffExt = "ReverseDiff"
 BijectorsTrackerExt = "Tracker"
 
 [compat]