diff --git a/flang/include/flang/Decimal/binary-floating-point.h b/flang/include/flang/Decimal/binary-floating-point.h
--- a/flang/include/flang/Decimal/binary-floating-point.h
+++ b/flang/include/flang/Decimal/binary-floating-point.h
@@ -79,10 +79,26 @@
     return (raw_ & ((RawType{1} << (bits - 1)) - 1)) == 0;
   }
   constexpr bool IsNaN() const {
-    return BiasedExponent() == maxExponent && Significand() != 0;
+    auto expo{BiasedExponent()};
+    auto sig{Significand()};
+    if constexpr (bits == 80) { // x87
+      if (expo == maxExponent) {
+        return sig != (significandMask >> 1) + 1;
+      } else {
+        return expo != 0 && !(sig & (RawType{1} << (significandBits - 1)));
+        ;
+      }
+    } else {
+      return expo == maxExponent && sig != 0;
+    }
   }
   constexpr bool IsInfinite() const {
-    return BiasedExponent() == maxExponent && Significand() == 0;
+    if constexpr (bits == 80) { // x87
+      return BiasedExponent() == maxExponent &&
+          Significand() == ((significandMask >> 1) + 1);
+    } else {
+      return BiasedExponent() == maxExponent && Significand() == 0;
+    }
   }
   constexpr bool IsMaximalFiniteMagnitude() const {
     return BiasedExponent() == maxExponent - 1 &&
diff --git a/flang/include/flang/Evaluate/real.h b/flang/include/flang/Evaluate/real.h
--- a/flang/include/flang/Evaluate/real.h
+++ b/flang/include/flang/Evaluate/real.h
@@ -69,19 +69,60 @@
     return !IsNotANumber() && IsSignBitSet();
   }
   constexpr bool IsNotANumber() const {
-    return Exponent() == maxExponent && !GetSignificand().IsZero();
+    auto expo{Exponent()};
+    auto sig{GetSignificand()};
+    if constexpr (bits == 80) { // x87
+      // 7FFF8000000000000000 is Infinity, not NaN, on 80387 & later.
+      if (expo == maxExponent) {
+        return sig != Significand{}.IBSET(63);
+      } else {
+        return expo != 0 && !sig.BTEST(63);
+      }
+    } else {
+      return expo == maxExponent && !sig.IsZero();
+    }
   }
   constexpr bool IsQuietNaN() const {
-    return Exponent() == maxExponent &&
-        GetSignificand().BTEST(significandBits - 1);
+    auto expo{Exponent()};
+    auto sig{GetSignificand()};
+    if constexpr (bits == 80) { // x87
+      if (expo == maxExponent) {
+        return sig.IBITS(62, 2) == 3;
+      } else {
+        return expo != 0 && !sig.BTEST(63);
+      }
+    } else {
+      return expo == maxExponent && sig.BTEST(significandBits - 1);
+    }
   }
   constexpr bool IsSignalingNaN() const {
-    return IsNotANumber() && !GetSignificand().BTEST(significandBits - 1);
+    auto expo{Exponent()};
+    auto sig{GetSignificand()};
+    if constexpr (bits == 80) { // x87
+      return expo == maxExponent && sig != Significand{}.IBSET(63) &&
+          sig.IBITS(62, 2) != 3;
+    } else {
+      return expo == maxExponent && !sig.IsZero() &&
+          !sig.BTEST(significandBits - 1);
+    }
   }
   constexpr bool IsInfinite() const {
-    return Exponent() == maxExponent && GetSignificand().IsZero();
+    if constexpr (bits == 80) { // x87
+      // 7FFF8000000000000000 is Infinity, not NaN, on 80387 & later.
+      return Exponent() == maxExponent &&
+          GetSignificand() == Significand{}.IBSET(63);
+    } else {
+      return Exponent() == maxExponent && GetSignificand().IsZero();
+    }
+  }
+  constexpr bool IsFinite() const {
+    auto expo{Exponent()};
+    if constexpr (bits == 80) { // x87
+      return expo != maxExponent && (expo == 0 || GetSignificand().BTEST(63));
+    } else {
+      return expo != maxExponent;
+    }
   }
-  constexpr bool IsFinite() const { return Exponent() != maxExponent; }
   constexpr bool IsZero() const {
     return Exponent() == 0 && GetSignificand().IsZero();
   }
@@ -226,6 +267,10 @@
     if (negative) {
       infinity = infinity.IBSET(infinity.bits - 1);
     }
+    if constexpr (bits == 80) { // x87
+      // 7FFF8000000000000000 is Infinity, not NaN, on 80387 & later.
+      infinity.IBSET(63);
+    }
     return {infinity};
   }
 
diff --git a/flang/lib/Decimal/big-radix-floating-point.h b/flang/lib/Decimal/big-radix-floating-point.h
--- a/flang/lib/Decimal/big-radix-floating-point.h
+++ b/flang/lib/Decimal/big-radix-floating-point.h
@@ -348,11 +348,26 @@
   using Raw = typename Real::RawType;
   constexpr Raw SignBit() const { return Raw{isNegative_} << (Real::bits - 1); }
   constexpr Raw Infinity() const {
-    return (Raw{Real::maxExponent} << Real::significandBits) | SignBit();
+    Raw result{static_cast<Raw>(Real::maxExponent)};
+    result <<= Real::significandBits;
+    result |= SignBit();
+    if constexpr (Real::bits == 80) { // x87
+      result |= Raw{1} << 63;
+    }
+    return result;
   }
   constexpr Raw NaN(bool isQuiet = true) {
-    return (Raw{Real::maxExponent} << Real::significandBits) |
-        (Raw{1} << (Real::significandBits - (isQuiet ? 1 : 2))) | SignBit();
+    Raw result{Real::maxExponent};
+    result <<= Real::significandBits;
+    result |= SignBit();
+    if constexpr (Real::bits == 80) { // x87
+      result |= Raw{isQuiet ? 3u : 2u} << 62;
+    } else {
+      Raw quiet{isQuiet ? Raw{2} : Raw{1}};
+      quiet <<= Real::significandBits - 2;
+      result |= quiet;
+    }
+    return result;
   }
 
   Digit digit_[maxDigits]; // in little-endian order: digit_[0] is LSD
diff --git a/flang/lib/Decimal/decimal-to-binary.cpp b/flang/lib/Decimal/decimal-to-binary.cpp
--- a/flang/lib/Decimal/decimal-to-binary.cpp
+++ b/flang/lib/Decimal/decimal-to-binary.cpp
@@ -298,7 +298,11 @@
   if (expo >= Binary::maxExponent) {
     expo = Binary::maxExponent; // Inf
     flags |= Overflow;
-    fraction = 0;
+    if constexpr (Binary::bits == 80) { // x87
+      fraction = IntType{1} << 63;
+    } else {
+      fraction = 0;
+    }
   }
   using Raw = typename Binary::RawType;
   Raw raw = static_cast<Raw>(isNegative) << (Binary::bits - 1);
diff --git a/flang/test/Evaluate/folding03.f90 b/flang/test/Evaluate/folding03.f90
--- a/flang/test/Evaluate/folding03.f90
+++ b/flang/test/Evaluate/folding03.f90
@@ -138,7 +138,7 @@
   real(4), parameter :: r4_nan_add2 = r4_pinf + r4_ninf
   TEST_ISNAN(r4_nan_add2)
 
-  ! No warnings expected here (quite NaN propagation)
+  ! No warnings expected here (quiet NaN propagation)
   real(4), parameter :: r4_nan_sub3 = 0._4 - r4_nan
   TEST_ISNAN(r4_nan_sub3)
   real(4), parameter :: r4_nan_sub4 = r4_nan - r4_pmax
@@ -201,7 +201,7 @@
   real(4), parameter :: r4_nan_mult2 = 0._4*r4_ninf
   TEST_ISNAN(r4_nan_mult2)
 
-  ! No warnings expected here (quite NaN propagation)
+  ! No warnings expected here (quiet NaN propagation)
   real(4), parameter :: r4_nan_div6 = 0._4/r4_nan
   TEST_ISNAN(r4_nan_div6)
   real(4), parameter :: r4_nan_div7 = r4_nan/r4_nan
diff --git a/flang/unittests/Evaluate/real.cpp b/flang/unittests/Evaluate/real.cpp
--- a/flang/unittests/Evaluate/real.cpp
+++ b/flang/unittests/Evaluate/real.cpp
@@ -95,7 +95,15 @@
   int exponentBits{R::bits - significandBits - 1};
   std::uint64_t maxExponent{(std::uint64_t{1} << exponentBits) - 1};
   MATCH(nan.Exponent(), maxExponent)(desc);
-  R inf{Word{maxExponent}.SHIFTL(significandBits)};
+  Word infWord{Word{maxExponent}.SHIFTL(significandBits)};
+  Word negInfWord{
+      Word{maxExponent}.SHIFTL(significandBits).IOR(Word::MASKL(1))};
+  if constexpr (kind == 10) { // x87
+    infWord = infWord.IBSET(63);
+    negInfWord = negInfWord.IBSET(63);
+  }
+  R inf{infWord};
+  R negInf{negInfWord};
   TEST(!inf.IsNegative())(desc);
   TEST(!inf.IsNotANumber())(desc);
   TEST(inf.IsInfinite())(desc);
@@ -106,7 +114,6 @@
   TEST(minusZero.Compare(inf) == Relation::Less)(desc);
   TEST(nan.Compare(inf) == Relation::Unordered)(desc);
   TEST(inf.Compare(inf) == Relation::Equal)(desc);
-  R negInf{Word{maxExponent}.SHIFTL(significandBits).IOR(Word::MASKL(1))};
   TEST(negInf.IsNegative())(desc);
   TEST(!negInf.IsNotANumber())(desc);
   TEST(negInf.IsInfinite())(desc);