diff --git a/llvm/lib/Transforms/Scalar/ConstraintElimination.cpp b/llvm/lib/Transforms/Scalar/ConstraintElimination.cpp
index 8f09569d0d9cc..7b672e89b67aa 100644
--- a/llvm/lib/Transforms/Scalar/ConstraintElimination.cpp
+++ b/llvm/lib/Transforms/Scalar/ConstraintElimination.cpp
@@ -1061,11 +1061,16 @@ void State::addInfoFor(BasicBlock &BB) {
           FactOrCheck::getCheck(DT.getNode(&BB), cast<CallInst>(&I)));
       break;
     // Enqueue the intrinsics to add extra info.
-    case Intrinsic::abs:
     case Intrinsic::umin:
     case Intrinsic::umax:
     case Intrinsic::smin:
     case Intrinsic::smax:
+      // TODO: Check if it is possible to instead only added the min/max facts
+      // when simplifying uses of the min/max intrinsics.
+      if (!isGuaranteedNotToBePoison(&I))
+        break;
+      [[fallthrough]];
+    case Intrinsic::abs:
       WorkList.push_back(FactOrCheck::getInstFact(DT.getNode(&BB), &I));
       break;
     }
diff --git a/llvm/test/Transforms/ConstraintElimination/minmax.ll b/llvm/test/Transforms/ConstraintElimination/minmax.ll
index a31cf6845ad67..82b932f14c4ff 100644
--- a/llvm/test/Transforms/ConstraintElimination/minmax.ll
+++ b/llvm/test/Transforms/ConstraintElimination/minmax.ll
@@ -306,7 +306,9 @@ define i1 @smin_branchless(i32 %x, i32 %y) {
 ; CHECK-SAME: (i32 [[X:%.*]], i32 [[Y:%.*]]) {
 ; CHECK-NEXT:  entry:
 ; CHECK-NEXT:    [[MIN:%.*]] = call i32 @llvm.smin.i32(i32 [[X]], i32 [[Y]])
-; CHECK-NEXT:    [[RET:%.*]] = xor i1 true, false
+; CHECK-NEXT:    [[CMP1:%.*]] = icmp sle i32 [[MIN]], [[X]]
+; CHECK-NEXT:    [[CMP2:%.*]] = icmp sgt i32 [[MIN]], [[X]]
+; CHECK-NEXT:    [[RET:%.*]] = xor i1 [[CMP1]], [[CMP2]]
 ; CHECK-NEXT:    ret i1 [[RET]]
 ;
 entry:
diff --git a/llvm/test/Transforms/ConstraintElimination/umin-result-may-be-poison.ll b/llvm/test/Transforms/ConstraintElimination/umin-result-may-be-poison.ll
new file mode 100644
index 0000000000000..6d1d95ec4fdba
--- /dev/null
+++ b/llvm/test/Transforms/ConstraintElimination/umin-result-may-be-poison.ll
@@ -0,0 +1,62 @@
+; NOTE: Assertions have been autogenerated by utils/update_test_checks.py UTC_ARGS: --version 4
+; RUN: opt -p constraint-elimination -S %s | FileCheck %s
+
+; Tests for https://github.com/llvm/llvm-project/issues/78621.
+
+define i1 @umin_not_used(i32 %arg) {
+; CHECK-LABEL: define i1 @umin_not_used(
+; CHECK-SAME: i32 [[ARG:%.*]]) {
+; CHECK-NEXT:    [[ICMP:%.*]] = icmp slt i32 [[ARG]], 0
+; CHECK-NEXT:    [[SHL:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    [[TMP1:%.*]] = call i32 @llvm.umin.i32(i32 [[SHL]], i32 80)
+; CHECK-NEXT:    [[CMP2:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    ret i1 [[ICMP]]
+;
+  %icmp = icmp slt i32 %arg, 0
+  %shl = shl nuw nsw i32 %arg, 3
+  call i32 @llvm.umin.i32(i32 %shl, i32 80)
+  %cmp2 = shl nuw nsw i32 %arg, 3
+  ret i1 %icmp
+}
+
+define i1 @umin_poison_is_UB_via_call(i32 %arg) {
+; CHECK-LABEL: define i1 @umin_poison_is_UB_via_call(
+; CHECK-SAME: i32 [[ARG:%.*]]) {
+; CHECK-NEXT:    [[SHL:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    [[MIN:%.*]] = call i32 @llvm.umin.i32(i32 [[SHL]], i32 80)
+; CHECK-NEXT:    call void @noundef(i32 noundef [[MIN]])
+; CHECK-NEXT:    [[CMP2:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    ret i1 false
+;
+  %icmp = icmp slt i32 %arg, 0
+  %shl = shl nuw nsw i32 %arg, 3
+  %min = call i32 @llvm.umin.i32(i32 %shl, i32 80)
+  call void @noundef(i32 noundef %min)
+  %cmp2 = shl nuw nsw i32 %arg, 3
+  ret i1 %icmp
+}
+
+define i1 @umin_poison_call_before_UB(i32 %arg) {
+; CHECK-LABEL: define i1 @umin_poison_call_before_UB(
+; CHECK-SAME: i32 [[ARG:%.*]]) {
+; CHECK-NEXT:    [[ICMP:%.*]] = icmp slt i32 [[ARG]], 0
+; CHECK-NEXT:    [[SHL:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    [[MIN:%.*]] = call i32 @llvm.umin.i32(i32 [[SHL]], i32 80)
+; CHECK-NEXT:    call void @fn()
+; CHECK-NEXT:    call void @noundef(i32 noundef [[MIN]])
+; CHECK-NEXT:    [[CMP2:%.*]] = shl nuw nsw i32 [[ARG]], 3
+; CHECK-NEXT:    ret i1 [[ICMP]]
+;
+  %icmp = icmp slt i32 %arg, 0
+  %shl = shl nuw nsw i32 %arg, 3
+  %min = call i32 @llvm.umin.i32(i32 %shl, i32 80)
+  call void @fn()
+  call void @noundef(i32 noundef %min)
+  %cmp2 = shl nuw nsw i32 %arg, 3
+  ret i1 %icmp
+}
+
+declare i32 @llvm.umin.i32(i32, i32) #0
+
+declare void @noundef(i32 noundef)
+declare void @fn()