diff --git a/src/Bridges/Constraint/Constraint.jl b/src/Bridges/Constraint/Constraint.jl
index 8b67cf9904..996934c359 100644
--- a/src/Bridges/Constraint/Constraint.jl
+++ b/src/Bridges/Constraint/Constraint.jl
@@ -29,6 +29,12 @@ Add all bridges defined in the `Bridges.Constraint` submodule to `model`.
 The coefficient type used is `T`.
 """
 function add_all_bridges(model, ::Type{T}) where {T}
+    if T <: AbstractFloat
+        # Out of order: put these bridges at the start for backwards
+        # compatibility.
+        MOI.Bridges.add_bridge(model, GreaterToIntervalBridge{T})
+        MOI.Bridges.add_bridge(model, LessToIntervalBridge{T})
+    end
     MOI.Bridges.add_bridge(model, AllDifferentToCountDistinctBridge{T})
     MOI.Bridges.add_bridge(model, BinPackingToMILPBridge{T})
     MOI.Bridges.add_bridge(model, CircuitToMILPBridge{T})
@@ -48,9 +54,6 @@ function add_all_bridges(model, ::Type{T}) where {T}
     MOI.Bridges.add_bridge(model, GeoMeanBridge{T})
     MOI.Bridges.add_bridge(model, GeoMeanToPowerBridge{T})
     MOI.Bridges.add_bridge(model, GeoMeantoRelEntrBridge{T})
-    if T <: AbstractFloat
-        MOI.Bridges.add_bridge(model, GreaterToIntervalBridge{T})
-    end
     MOI.Bridges.add_bridge(model, GreaterToLessBridge{T})
     MOI.Bridges.add_bridge(model, HermitianToSymmetricPSDBridge{T})
     MOI.Bridges.add_bridge(model, IndicatorActiveOnFalseBridge{T})
@@ -65,9 +68,6 @@ function add_all_bridges(model, ::Type{T}) where {T}
     MOI.Bridges.add_bridge(model, InequalityToComplementsBridge{T})
     MOI.Bridges.add_bridge(model, IntegerToZeroOneBridge{T})
     MOI.Bridges.add_bridge(model, LessToGreaterBridge{T})
-    if T <: AbstractFloat
-        MOI.Bridges.add_bridge(model, LessToIntervalBridge{T})
-    end
     MOI.Bridges.add_bridge(model, LogDetBridge{T})
     MOI.Bridges.add_bridge(model, NonnegToNonposBridge{T})
     MOI.Bridges.add_bridge(model, NonposToNonnegBridge{T})
diff --git a/src/Bridges/Variable/bridges/RSOCtoPSDBridge.jl b/src/Bridges/Variable/bridges/RSOCtoPSDBridge.jl
index cc3b30ca86..a58464abc1 100644
--- a/src/Bridges/Variable/bridges/RSOCtoPSDBridge.jl
+++ b/src/Bridges/Variable/bridges/RSOCtoPSDBridge.jl
@@ -56,6 +56,8 @@ end
 const RSOCtoPSD{T,OT<:MOI.ModelLike} =
     SingleBridgeOptimizer{RSOCtoPSDBridge{T},OT}
 
+MOI.Bridges.bridging_cost(::Type{<:RSOCtoPSDBridge}) = 1.5
+
 function bridge_constrained_variable(
     ::Type{RSOCtoPSDBridge{T}},
     model::MOI.ModelLike,
diff --git a/test/Bridges/lazy_bridge_optimizer.jl b/test/Bridges/lazy_bridge_optimizer.jl
index f6b4ebb4e1..1d629f13d2 100644
--- a/test/Bridges/lazy_bridge_optimizer.jl
+++ b/test/Bridges/lazy_bridge_optimizer.jl
@@ -1041,13 +1041,13 @@ Objective function of type `MOI.ScalarQuadraticFunction{$T}` is not supported an
             """
 Bridge graph with 2 variable nodes, 5 constraint nodes and 2 objective nodes.
  [1] constrained variables in `MOI.Reals` are bridged (distance 1) by $(MOI.Bridges.Variable.FreeBridge{T}).
- [2] constrained variables in `MOI.RotatedSecondOrderCone` are bridged (distance 2) by $(MOI.Bridges.Variable.RSOCtoPSDBridge{T}).
- (1) `MOI.ScalarQuadraticFunction{$T}`-in-`MOI.GreaterThan{$T}` constraints are bridged (distance 5) by $(MOI.Bridges.Constraint.QuadtoSOCBridge{T}).
- (2) `MOI.VectorAffineFunction{$T}`-in-`MOI.RotatedSecondOrderCone` constraints are bridged (distance 4) by $(MOI.Bridges.Constraint.VectorSlackBridge{T,MOI.VectorAffineFunction{T},MOI.RotatedSecondOrderCone}).
+ [2] constrained variables in `MOI.RotatedSecondOrderCone` are bridged (distance 2.5) by $(MOI.Bridges.Variable.RSOCtoPSDBridge{T}).
+ (1) `MOI.ScalarQuadraticFunction{$T}`-in-`MOI.GreaterThan{$T}` constraints are bridged (distance 5.5) by $(MOI.Bridges.Constraint.QuadtoSOCBridge{T}).
+ (2) `MOI.VectorAffineFunction{$T}`-in-`MOI.RotatedSecondOrderCone` constraints are bridged (distance 4.5) by $(MOI.Bridges.Constraint.VectorSlackBridge{T,MOI.VectorAffineFunction{T},MOI.RotatedSecondOrderCone}).
  (3) `MOI.VariableIndex`-in-`MOI.EqualTo{$T}` constraints are bridged (distance 1) by $(MOI.Bridges.Constraint.ScalarFunctionizeBridge{T,MOI.EqualTo{T}}).
  (4) `MOI.VectorAffineFunction{$T}`-in-`MOI.Zeros` constraints are bridged (distance 1) by $(MOI.Bridges.Constraint.ScalarizeBridge{T,MOI.ScalarAffineFunction{T},MOI.EqualTo{T}}).
- (5) `MOI.ScalarQuadraticFunction{$T}`-in-`MOI.LessThan{$T}` constraints are bridged (distance 5) by $(MOI.Bridges.Constraint.QuadtoSOCBridge{T}).
- |1| objective function of type `MOI.ScalarQuadraticFunction{$T}` is bridged (distance 13) by $(MOI.Bridges.Objective.SlackBridge{T,MOI.ScalarQuadraticFunction{T},MOI.ScalarQuadraticFunction{T}}).
+ (5) `MOI.ScalarQuadraticFunction{$T}`-in-`MOI.LessThan{$T}` constraints are bridged (distance 5.5) by $(MOI.Bridges.Constraint.QuadtoSOCBridge{T}).
+ |1| objective function of type `MOI.ScalarQuadraticFunction{$T}` is bridged (distance 14) by $(MOI.Bridges.Objective.SlackBridge{T,MOI.ScalarQuadraticFunction{T},MOI.ScalarQuadraticFunction{T}}).
  |2| objective function of type `MOI.VariableIndex` is bridged (distance 1) by $(MOI.Bridges.Objective.FunctionizeBridge{T,MOI.VariableIndex}).
 """,
         )