Index: lib/Support/APInt.cpp
===================================================================
--- lib/Support/APInt.cpp
+++ lib/Support/APInt.cpp
@@ -482,10 +482,14 @@
   APInt tmp(sufficient, StringRef(p, slen), radix);
 
   // Compute how many bits are required. If the log is infinite, assume we need
-  // just bit.
+  // just bit. If the log is exact and value is negative, then the value is
+  // MinSignedValue with (log + 1) bits.
   unsigned log = tmp.logBase2();
+  int32_t exactLog = tmp.exactLogBase2();
   if (log == (unsigned)-1) {
     return isNegative + 1;
+  } else if (isNegative && exactLog > 0) {
+    return isNegative + log;
   } else {
     return isNegative + log + 1;
   }
Index: unittests/ADT/APIntTest.cpp
===================================================================
--- unittests/ADT/APIntTest.cpp
+++ unittests/ADT/APIntTest.cpp
@@ -1263,8 +1263,10 @@
   EXPECT_EQ(6U, APInt::getBitsNeeded("-19", 10));
   EXPECT_EQ(6U, APInt::getBitsNeeded("-20", 10));
 
-  // TODO: INT_MIN cases need 1 less bit (PR40897)
-  EXPECT_EQ(9U, APInt::getBitsNeeded("-128", 10));
+  EXPECT_EQ(8U, APInt::getBitsNeeded("-127", 10));
+  EXPECT_EQ(8U, APInt::getBitsNeeded("-128", 10));
+  EXPECT_EQ(9U, APInt::getBitsNeeded("-255", 10));
+  EXPECT_EQ(9U, APInt::getBitsNeeded("-256", 10));
 }
 
 TEST(APIntTest, StringBitsNeeded16) {