diff --git a/llvm/include/llvm/Support/MathExtras.h b/llvm/include/llvm/Support/MathExtras.h
index 05d87e176dec1..5bcefe4b6c361 100644
--- a/llvm/include/llvm/Support/MathExtras.h
+++ b/llvm/include/llvm/Support/MathExtras.h
@@ -385,6 +385,8 @@ inline uint64_t PowerOf2Ceil(uint64_t A) {
 ///   alignTo(~0LL, 8) = 0
 ///   alignTo(321, 255) = 510
 /// \endcode
+///
+/// May overflow.
 inline uint64_t alignTo(uint64_t Value, uint64_t Align) {
   assert(Align != 0u && "Align can't be 0.");
   return (Value + Align - 1) / Align * Align;
@@ -424,33 +426,38 @@ template <uint64_t Align> constexpr inline uint64_t alignTo(uint64_t Value) {
   return (Value + Align - 1) / Align * Align;
 }
 
-/// Returns the integer ceil(Numerator / Denominator). Unsigned integer version.
+/// Returns the integer ceil(Numerator / Denominator). Unsigned version.
+/// Guaranteed to never overflow.
 inline uint64_t divideCeil(uint64_t Numerator, uint64_t Denominator) {
-  return alignTo(Numerator, Denominator) / Denominator;
+  assert(Denominator && "Division by zero");
+  uint64_t Bias = (Numerator != 0);
+  return (Numerator - Bias) / Denominator + Bias;
 }
 
-/// Returns the integer ceil(Numerator / Denominator). Signed integer version.
+/// Returns the integer ceil(Numerator / Denominator). Signed version.
+/// Guaranteed to never overflow.
 inline int64_t divideCeilSigned(int64_t Numerator, int64_t Denominator) {
   assert(Denominator && "Division by zero");
   if (!Numerator)
     return 0;
   // C's integer division rounds towards 0.
-  int64_t X = (Denominator > 0) ? -1 : 1;
-  bool SameSign = (Numerator > 0) == (Denominator > 0);
-  return SameSign ? ((Numerator + X) / Denominator) + 1
+  int64_t Bias = (Denominator >= 0 ? 1 : -1);
+  bool SameSign = (Numerator >= 0) == (Denominator >= 0);
+  return SameSign ? (Numerator - Bias) / Denominator + 1
                   : Numerator / Denominator;
 }
 
-/// Returns the integer floor(Numerator / Denominator). Signed integer version.
+/// Returns the integer floor(Numerator / Denominator). Signed version.
+/// Guaranteed to never overflow.
 inline int64_t divideFloorSigned(int64_t Numerator, int64_t Denominator) {
   assert(Denominator && "Division by zero");
   if (!Numerator)
     return 0;
   // C's integer division rounds towards 0.
-  int64_t X = (Denominator > 0) ? -1 : 1;
-  bool SameSign = (Numerator > 0) == (Denominator > 0);
+  int64_t Bias = Denominator >= 0 ? -1 : 1;
+  bool SameSign = (Numerator >= 0) == (Denominator >= 0);
   return SameSign ? Numerator / Denominator
-                  : -((-Numerator + X) / Denominator) - 1;
+                  : (Numerator - Bias) / Denominator - 1;
 }
 
 /// Returns the remainder of the Euclidean division of LHS by RHS. Result is
@@ -461,9 +468,12 @@ inline int64_t mod(int64_t Numerator, int64_t Denominator) {
   return Mod < 0 ? Mod + Denominator : Mod;
 }
 
-/// Returns the integer nearest(Numerator / Denominator).
+/// Returns (Numerator / Denominator) rounded by round-half-up. Guaranteed to
+/// never overflow.
 inline uint64_t divideNearest(uint64_t Numerator, uint64_t Denominator) {
-  return (Numerator + (Denominator / 2)) / Denominator;
+  assert(Denominator && "Division by zero");
+  uint64_t Mod = Numerator % Denominator;
+  return (Numerator / Denominator) + (Mod > (Denominator - 1) / 2);
 }
 
 /// Returns the largest uint64_t less than or equal to \p Value and is
diff --git a/llvm/unittests/Support/MathExtrasTest.cpp b/llvm/unittests/Support/MathExtrasTest.cpp
index bcccb963c96ae..bd09bab9be004 100644
--- a/llvm/unittests/Support/MathExtrasTest.cpp
+++ b/llvm/unittests/Support/MathExtrasTest.cpp
@@ -459,15 +459,37 @@ TEST(MathExtras, DivideNearest) {
   EXPECT_EQ(divideNearest(14, 3), 5u);
   EXPECT_EQ(divideNearest(15, 3), 5u);
   EXPECT_EQ(divideNearest(0, 3), 0u);
+  EXPECT_EQ(divideNearest(5, 4), 1u);
+  EXPECT_EQ(divideNearest(6, 4), 2u);
+  EXPECT_EQ(divideNearest(3, 1), 3u);
+  EXPECT_EQ(divideNearest(3, 6), 1u);
+  EXPECT_EQ(divideNearest(3, 7), 0u);
   EXPECT_EQ(divideNearest(std::numeric_limits<uint32_t>::max(), 2),
-            2147483648u);
+            std::numeric_limits<uint32_t>::max() / 2 + 1);
+  EXPECT_EQ(divideNearest(std::numeric_limits<uint64_t>::max(), 2),
+            std::numeric_limits<uint64_t>::max() / 2 + 1);
+  EXPECT_EQ(divideNearest(std::numeric_limits<uint64_t>::max(), 1),
+            std::numeric_limits<uint64_t>::max());
+  EXPECT_EQ(divideNearest(std::numeric_limits<uint64_t>::max() - 1,
+                          std::numeric_limits<uint64_t>::max()),
+            1u);
 }
 
 TEST(MathExtras, DivideCeil) {
   EXPECT_EQ(divideCeil(14, 3), 5u);
   EXPECT_EQ(divideCeil(15, 3), 5u);
   EXPECT_EQ(divideCeil(0, 3), 0u);
-  EXPECT_EQ(divideCeil(std::numeric_limits<uint32_t>::max(), 2), 2147483648u);
+  EXPECT_EQ(divideCeil(5, 4), 2u);
+  EXPECT_EQ(divideCeil(6, 4), 2u);
+  EXPECT_EQ(divideCeil(3, 1), 3u);
+  EXPECT_EQ(divideCeil(3, 6), 1u);
+  EXPECT_EQ(divideCeil(3, 7), 1u);
+  EXPECT_EQ(divideCeil(std::numeric_limits<uint32_t>::max(), 2),
+            std::numeric_limits<uint32_t>::max() / 2 + 1);
+  EXPECT_EQ(divideCeil(std::numeric_limits<uint64_t>::max(), 2),
+            std::numeric_limits<uint64_t>::max() / 2 + 1);
+  EXPECT_EQ(divideCeil(std::numeric_limits<uint64_t>::max(), 1),
+            std::numeric_limits<uint64_t>::max());
 
   EXPECT_EQ(divideCeilSigned(14, 3), 5);
   EXPECT_EQ(divideCeilSigned(15, 3), 5);
@@ -479,8 +501,14 @@ TEST(MathExtras, DivideCeil) {
   EXPECT_EQ(divideCeilSigned(0, -3), 0);
   EXPECT_EQ(divideCeilSigned(std::numeric_limits<int32_t>::max(), 2),
             std::numeric_limits<int32_t>::max() / 2 + 1);
+  EXPECT_EQ(divideCeilSigned(std::numeric_limits<int64_t>::max(), 2),
+            std::numeric_limits<int64_t>::max() / 2 + 1);
   EXPECT_EQ(divideCeilSigned(std::numeric_limits<int32_t>::max(), -2),
             std::numeric_limits<int32_t>::min() / 2 + 1);
+  EXPECT_EQ(divideCeilSigned(std::numeric_limits<int64_t>::max(), -2),
+            std::numeric_limits<int64_t>::min() / 2 + 1);
+  EXPECT_EQ(divideCeilSigned(std::numeric_limits<int64_t>::min(), 1),
+            std::numeric_limits<int64_t>::min());
 }
 
 TEST(MathExtras, DivideFloorSigned) {
@@ -494,8 +522,14 @@ TEST(MathExtras, DivideFloorSigned) {
   EXPECT_EQ(divideFloorSigned(0, -3), 0);
   EXPECT_EQ(divideFloorSigned(std::numeric_limits<int32_t>::max(), 2),
             std::numeric_limits<int32_t>::max() / 2);
+  EXPECT_EQ(divideFloorSigned(std::numeric_limits<int64_t>::max(), 2),
+            std::numeric_limits<int64_t>::max() / 2);
   EXPECT_EQ(divideFloorSigned(std::numeric_limits<int32_t>::max(), -2),
             std::numeric_limits<int32_t>::min() / 2);
+  EXPECT_EQ(divideFloorSigned(std::numeric_limits<int64_t>::max(), -2),
+            std::numeric_limits<int64_t>::min() / 2);
+  EXPECT_EQ(divideFloorSigned(std::numeric_limits<int64_t>::min(), 1),
+            std::numeric_limits<int64_t>::min());
 }
 
 TEST(MathExtras, Mod) {