Index: llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
===================================================================
--- llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
+++ llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
@@ -680,9 +680,9 @@
     if (!A->getType()->isSingleValueType() || A->user_empty())
       return false;
 
-    // If the argument isn't overdefined, there's nothing to do. It should
-    // already be constant.
-    if (!Solver.getLatticeValueFor(A).isOverdefined()) {
+    // If the argument is Constant, there's nothing to do. IPSCCP will
+    // propagate the argument.
+    if (isConstant(Solver.getLatticeValueFor(A))) {
       LLVM_DEBUG(dbgs() << "FnSpecialization: Nothing to do, argument "
                         << A->getNameOrAsOperand()
                         << " is already constant?\n");
@@ -700,7 +700,7 @@
     // might be beneficial to take the occurrences into account in the cost
     // model, so we would need to find the unique constants.
     //
-    // TODO 2: this currently does not support constants, i.e. integer ranges.
+    // TODO 2: this currently does not support constant range arguments.
     //
     IsPartial = !getPossibleConstants(A, Constants);
     LLVM_DEBUG(dbgs() << "FnSpecialization: Found interesting argument "
Index: llvm/test/Transforms/FunctionSpecialization/function-specialization6.ll
===================================================================
--- /dev/null
+++ llvm/test/Transforms/FunctionSpecialization/function-specialization6.ll
@@ -0,0 +1,59 @@
+; RUN: opt -function-specialization -function-specialization-for-literal-constant -force-function-specialization -S < %s | FileCheck %s
+
+define i64 @main(i64 %x, i1 %flag) {
+;
+; CHECK-LABEL: @main(i64 %x, i1 %flag) {
+; CHECK:      entry:
+; CHECK-NEXT:   br i1 %flag, label %three, label %four
+; CHECK:      three:
+; CHECK-NEXT:   [[TMP0:%.+]] = call i64 @compute.1(i64 3, i64 (i64)* @plus)
+; CHECK-NEXT:   br label %merge
+; CHECK:      four:
+; CHECK-NEXT:   [[TMP1:%.+]] = call i64 @compute.2(i64 4, i64 (i64)* @plus)
+; CHECK-NEXT:   br label %merge
+; CHECK:      merge:
+; CHECK-NEXT:   [[TMP2:%.+]] = phi i64 [ [[TMP0]], %three ], [ [[TMP1]], %four ]
+; CHECK-NEXT:   ret i64 [[TMP2]]
+; CHECK-NEXT: }
+;
+entry:
+  br i1 %flag, label %three, label %four
+
+three:
+  %tmp0 = call i64 @compute(i64 3, i64 (i64)* @plus)
+  br label %merge
+
+four:
+  %tmp1 = call i64 @compute(i64 4, i64 (i64)* @plus)
+  br label %merge
+
+merge:
+  %tmp2 = phi i64 [ %tmp0, %three ], [ %tmp1, %four]
+  ret i64 %tmp2
+}
+
+; CHECK-NOT: define internal i64 @compute(
+;
+; CHECK-LABEL: define internal i64 @compute.1(i64 %x, i64 (i64)* %binop) {
+; CHECK-NEXT:  entry:
+; CHECK-NEXT:    [[TMP0:%.+]] = call i64 @plus(i64 3)
+; CHECK-NEXT:    ret i64 [[TMP0]]
+; CHECK-NEXT:  }
+;
+; CHECK-LABEL: define internal i64 @compute.2(i64 %x, i64 (i64)* %binop) {
+; CHECK-NEXT:  entry:
+; CHECK-NEXT:    [[TMP0:%.+]] = call i64 @plus(i64 4)
+; CHECK-NEXT:    ret i64 [[TMP0]]
+; CHECK-NEXT:  }
+;
+define internal i64 @compute(i64 %x, i64 (i64)* %binop) {
+entry:
+  %tmp0 = call i64 %binop(i64 %x)
+  ret i64 %tmp0
+}
+
+define internal i64 @plus(i64 %x) {
+entry:
+  %tmp0 = add i64 %x, 1
+  ret i64 %tmp0
+}