diff --git a/llvm/lib/Analysis/BasicAliasAnalysis.cpp b/llvm/lib/Analysis/BasicAliasAnalysis.cpp
--- a/llvm/lib/Analysis/BasicAliasAnalysis.cpp
+++ b/llvm/lib/Analysis/BasicAliasAnalysis.cpp
@@ -360,8 +360,10 @@
   }
 
   LinearExpression mul(const APInt &Other, bool MulIsNSW) const {
-    return LinearExpression(Val, Scale * Other, Offset * Other,
-                            IsNSW && (Other.isOne() || MulIsNSW));
+    // The check for zero offset is necessary, because generally
+    // (X +nsw Y) *nsw Z does not imply (X *nsw Z) +nsw (Y *nsw Z).
+    bool NSW = IsNSW && (Other.isOne() || (MulIsNSW && Offset.isZero()));
+    return LinearExpression(Val, Scale * Other, Offset * Other, NSW);
   }
 };
 }
@@ -1249,12 +1251,14 @@
       CR = CR.intersectWith(
           ConstantRange::fromKnownBits(Known, /* Signed */ true),
           ConstantRange::Signed);
+      CR = Index.Val.evaluateWith(CR).sextOrTrunc(OffsetRange.getBitWidth());
 
       assert(OffsetRange.getBitWidth() == Scale.getBitWidth() &&
              "Bit widths are normalized to MaxPointerSize");
-      OffsetRange = OffsetRange.add(
-          Index.Val.evaluateWith(CR).sextOrTrunc(OffsetRange.getBitWidth())
-                                    .smul_fast(ConstantRange(Scale)));
+      if (Index.IsNSW)
+        OffsetRange = OffsetRange.add(CR.smul_sat(ConstantRange(Scale)));
+      else
+        OffsetRange = OffsetRange.add(CR.smul_fast(ConstantRange(Scale)));
     }
 
     // We now have accesses at two offsets from the same base:
diff --git a/llvm/test/Analysis/BasicAA/assume-index-positive.ll b/llvm/test/Analysis/BasicAA/assume-index-positive.ll
--- a/llvm/test/Analysis/BasicAA/assume-index-positive.ll
+++ b/llvm/test/Analysis/BasicAA/assume-index-positive.ll
@@ -145,12 +145,12 @@
   ret void
 }
 
-; TODO: Unlike the previous case, %ptr.neg and %ptr.shl can't alias, because
+; Unlike the previous case, %ptr.neg and %ptr.shl can't alias, because
 ; shl nsw of non-negative is non-negative.
 define void @shl_nsw_of_non_negative(i8* %ptr, i64 %a) {
 ; CHECK-LABEL: Function: shl_nsw_of_non_negative
 ; CHECK: NoAlias: i8* %ptr.a, i8* %ptr.neg
-; CHECK: MayAlias: i8* %ptr.neg, i8* %ptr.shl
+; CHECK: NoAlias: i8* %ptr.neg, i8* %ptr.shl
   %a.cmp = icmp sge i64 %a, 0
   call void @llvm.assume(i1 %a.cmp)
   %ptr.neg = getelementptr i8, i8* %ptr, i64 -2