[APFloat] Fix FP remainder operation

[ekatz]

Summary

Fix IEEEFloat::remainder() function.

Fix PR3359.

[ekatz]

Rebased to master.

[ekatz]

Ping

[ekatz]

Ping

[hfinkel]

Thanks for posting patches to fix problems with the APFloat implementation. FWIW, this code is hard to review, from my perspective, because it lacks comments (both your new code and the code it is replacing). As you clearly have gone through this and understand what it's doing, could you add a bunch of comments explaining what's going on? I think that would be very helpful for everyone else.

[arsenm]

llvm/unittests/ADT/APFloatTest.cpp

3418

These should be swapped due to a gtestism (same for the rest, it should be expected, actual)

[ekatz]

Added comments and fix tests to meet gtestism.

[ekatz]

Ping

[ekatz]

Ping

[efriedma]

I'd like to see more test coverage for denormal denominators, and denominators close to the max finite float.

llvm/lib/Support/APFloat.cpp

1823

Can we return early if rhs.isFinite() is false? The following code might work for rhs==infinity, but it makes everything more complicated to reason about.

1825

If the add is not opOK, that means the add returned opOverflow, so the value is already less than twice rhs? I guess that's right, but it would be nice to spell it out.

1842

Is the special case for compare(P) == cmpEqual necessary?

1913

If this check fails, that means P2 is denormal?

Can you make this work for denormal cases by just fiddling with the exponent of P2, instead of doing a proper divide? The resulting APFloat might not be properly normalized, but that doesn't matter here.

[ekatz]

Simplified logics according to the notes.
Also added more extensive tests.

[ekatz]

@efriedma , I updated the files according to your requests.

[efriedma]

llvm/lib/Support/APFloat.cpp

1818

The opDivByZero thing is sort of confusing. Maybe just something like if (!isFinite() || !rhs.isFinite()) return remainderSpecials(rhs);?

1886

Are all the subtractions here guaranteed to produce exact results? If they are, would it make sense to assert that?

[ekatz]

llvm/lib/Support/APFloat.cpp

1818

I thought about that, but I just used the same convention as in addOrSubtractSpecials.

If we were to add this if statement, then we are actually handling specials outside the function that supposed to do exactly that. And that alone makes even less sense than the strange opStatus (which is commented, and used only internally).

It would have made more sense to add a new opStatus just for that case, but it seems a bit too much, I guess (at least for those internal functions).

Which is why I decided to take this approach.

[ekatz]

Added asserts for the opStatus returned by the operations.

[efriedma]

LGTM

llvm/include/llvm/ADT/APFloat.h

private:
 
   /// \name Arithmetic on special values.
   /// @{
 
   opStatus addOrSubtractSpecials(const IEEEFloat &, bool subtract);
   opStatus divideSpecials(const IEEEFloat &);
   opStatus multiplySpecials(const IEEEFloat &);
   opStatus modSpecials(const IEEEFloat &);
+  opStatus remainderSpecials(const IEEEFloat&);
 
   /// @}
 
   /// \name Miscellany
   /// @{
 
   bool convertFromStringSpecials(StringRef str);
   opStatus normalize(roundingMode, lostFraction);

llvm/lib/Support/APFloat.cpp

case PackCategoriesIntoKey(fcZero, fcZero):
     makeNaN();
     return opInvalidOp;
 
   case PackCategoriesIntoKey(fcNormal, fcNormal):
     return opOK;
   }
 }
 
+IEEEFloat::opStatus IEEEFloat::remainderSpecials(const IEEEFloat &rhs) {
+  switch (PackCategoriesIntoKey(category, rhs.category)) {
+  default:
+    llvm_unreachable(nullptr);
+
+  case PackCategoriesIntoKey(fcZero, fcNaN):
+  case PackCategoriesIntoKey(fcNormal, fcNaN):
+  case PackCategoriesIntoKey(fcInfinity, fcNaN):
+    assign(rhs);
+    LLVM_FALLTHROUGH;
+  case PackCategoriesIntoKey(fcNaN, fcZero):
+  case PackCategoriesIntoKey(fcNaN, fcNormal):
+  case PackCategoriesIntoKey(fcNaN, fcInfinity):
+  case PackCategoriesIntoKey(fcNaN, fcNaN):
+    if (isSignaling()) {
+      makeQuiet();
+      return opInvalidOp;
+    }
+    return rhs.isSignaling() ? opInvalidOp : opOK;
+
+  case PackCategoriesIntoKey(fcZero, fcInfinity):
+  case PackCategoriesIntoKey(fcZero, fcNormal):
+  case PackCategoriesIntoKey(fcNormal, fcInfinity):
+    return opOK;
+
+  case PackCategoriesIntoKey(fcNormal, fcZero):
+  case PackCategoriesIntoKey(fcInfinity, fcZero):
+  case PackCategoriesIntoKey(fcInfinity, fcNormal):
+  case PackCategoriesIntoKey(fcInfinity, fcInfinity):
+  case PackCategoriesIntoKey(fcZero, fcZero):
+    makeNaN();
+    return opInvalidOp;
+
+  case PackCategoriesIntoKey(fcNormal, fcNormal):
+    return opDivByZero; // fake status, indicating this is not a special case
+  }
+}
+
 /* Change sign.  */
 void IEEEFloat::changeSign() {
   /* Look mummy, this one's easy.  */
   sign = !sign;
 }
 
 /* Normalized addition or subtraction.  */
 IEEEFloat::opStatus IEEEFloat::addOrSubtract(const IEEEFloat &rhs,
if (isFiniteNonZero()) {
     fs = normalize(rounding_mode, lost_fraction);
     if (lost_fraction != lfExactlyZero)
       fs = (opStatus) (fs | opInexact);
   }
 
   return fs;
 }
 
-/* Normalized remainder.  This is not currently correct in all cases.  */
+/* Normalized remainder.  */
 IEEEFloat::opStatus IEEEFloat::remainder(const IEEEFloat &rhs) {
   opStatus fs;
-  IEEEFloat V = *this;
   unsigned int origSign = sign;
 
-  fs = V.divide(rhs, rmNearestTiesToEven);
-  if (fs == opDivByZero)
+  // First handle the special cases.
+  fs = remainderSpecials(rhs);
+  if (fs != opDivByZero)
     return fs;
 
-  int parts = partCount();
-  integerPart *x = new integerPart[parts];
-  bool ignored;
-  fs = V.convertToInteger(makeMutableArrayRef(x, parts),
-                          parts * integerPartWidth, true, rmNearestTiesToEven,
-                          &ignored);
-  if (fs == opInvalidOp) {
-    delete[] x;
-    return fs;
+  fs = opOK;
+
+  // Make sure the current value is less than twice the denom. If the addition
+  // did not succeed (an overflow has happened), which means that the finite
+  // value we currently posses must be less than twice the denom (as we are
+  // using the same semantics).
+  IEEEFloat P2 = rhs;
+  if (P2.add(rhs, rmNearestTiesToEven) == opOK) {
+    fs = mod(P2);
+    assert(fs == opOK);
   }
 
-  fs = V.convertFromZeroExtendedInteger(x, parts * integerPartWidth, true,
-                                        rmNearestTiesToEven);
-  assert(fs==opOK);   // should always work
+  // Lets work with absolute numbers.
+  IEEEFloat P = rhs;
+  P.sign = false;
+  sign = false;
 
-  fs = V.multiply(rhs, rmNearestTiesToEven);
-  assert(fs==opOK || fs==opInexact);   // should not overflow or underflow
+  //
+  // To calculate the remainder we use the following scheme.
+  //
+  // The remainder is defained as follows:
+  //
+  // remainder = numer - rquot * denom = x - r * p
+  //
+  // Where r is the result of: x/p, rounded toward the nearest integral value
+  // (with halfway cases rounded toward the even number).
+  //
+  // Currently, (after x mod 2p):
+  // r is the number of 2p's present inside x, which is inherently, an even
+  // number of p's.
+  //
+  // We may split the remaining calculation into 4 options:
+  // - if x < 0.5p then we round to the nearest number with is 0, and are done.
+  // - if x == 0.5p then we round to the nearest even number which is 0, and we
+  //   are done as well.
+  // - if 0.5p < x < p then we round to nearest number which is 1, and we have
+  //   to subtract 1p at least once.
+  // - if x >= p then we must subtract p at least once, as x must be a
+  //   remainder.
+  //
+  // By now, we were done, or we added 1 to r, which in turn, now an odd number.
+  //
+  // We can now split the remaining calculation to the following 3 options:
+  // - if x < 0.5p then we round to the nearest number with is 0, and are done.
+  // - if x == 0.5p then we round to the nearest even number. As r is odd, we
+  //   must round up to the next even number. so we must subtract p once more.
+  // - if x > 0.5p (and inherently x < p) then we must round r up to the next
+  //   integral, and subtract p once more.
+  //
 
-  fs = subtract(V, rmNearestTiesToEven);
-  assert(fs==opOK || fs==opInexact);   // likewise
+  // Extend the semantics to prevent an overflow/underflow or inexact result.
+  bool losesInfo;
+  fltSemantics extendedSemantics = *semantics;
+  extendedSemantics.maxExponent++;
+  extendedSemantics.minExponent--;
+  extendedSemantics.precision += 2;
+
+  IEEEFloat VEx = *this;
+  fs = VEx.convert(extendedSemantics, rmNearestTiesToEven, &losesInfo);
+  assert(fs == opOK && !losesInfo);
+  IEEEFloat PEx = P;
+  fs = PEx.convert(extendedSemantics, rmNearestTiesToEven, &losesInfo);
+  assert(fs == opOK && !losesInfo);
+
+  // It is simpler to work with 2x instead of 0.5p, and we do not need to lose
+  // any fraction.
+  fs = VEx.add(VEx, rmNearestTiesToEven);
+  assert(fs == opOK);
+
+  if (VEx.compare(PEx) == cmpGreaterThan) {
+    fs = subtract(P, rmNearestTiesToEven);
+    assert(fs == opOK);
+
+    // Make VEx = this.add(this), but because we have different semantics, we do
+    // not want to `convert` again, so we just subtract PEx twice (which equals
+    // to the desired value).
+    fs = VEx.subtract(PEx, rmNearestTiesToEven);
+    assert(fs == opOK);
+    fs = VEx.subtract(PEx, rmNearestTiesToEven);
+    assert(fs == opOK);
+
+    cmpResult result = VEx.compare(PEx);
+    if (result == cmpGreaterThan || result == cmpEqual) {
+      fs = subtract(P, rmNearestTiesToEven);
+      assert(fs == opOK);
+    }
+  }
 
   if (isZero())
     sign = origSign;    // IEEE754 requires this
-  delete[] x;
+  else
+    sign ^= origSign;
   return fs;
 }
 
 /* Normalized llvm frem (C fmod). */
 IEEEFloat::opStatus IEEEFloat::mod(const IEEEFloat &rhs) {
   opStatus fs;
   fs = modSpecials(rhs);
   unsigned int origSign = sign;
 

llvm/unittests/ADT/APFloatTest.cpp

TEST(APFloatTest, mod) {
     APFloat f1(APFloat::IEEEdouble(), "-4.0");
     APFloat f2(APFloat::IEEEdouble(), "2.0");
     APFloat expected(APFloat::IEEEdouble(), "-0.0");
     EXPECT_EQ(f1.mod(f2), APFloat::opOK);
     EXPECT_TRUE(f1.bitwiseIsEqual(expected));
   }
 }
 
+TEST(APFloatTest, remainder) {
+  // Test Special Cases against each other and normal values.
+
+  APFloat PInf = APFloat::getInf(APFloat::IEEEsingle(), false);
+  APFloat MInf = APFloat::getInf(APFloat::IEEEsingle(), true);
+  APFloat PZero = APFloat::getZero(APFloat::IEEEsingle(), false);
+  APFloat MZero = APFloat::getZero(APFloat::IEEEsingle(), true);
+  APFloat QNaN = APFloat::getNaN(APFloat::IEEEsingle(), false);
+  APFloat SNaN = APFloat(APFloat::IEEEsingle(), "snan123");
+  APFloat PNormalValue = APFloat(APFloat::IEEEsingle(), "0x1p+0");
+  APFloat MNormalValue = APFloat(APFloat::IEEEsingle(), "-0x1p+0");
+  APFloat PLargestValue = APFloat::getLargest(APFloat::IEEEsingle(), false);
+  APFloat MLargestValue = APFloat::getLargest(APFloat::IEEEsingle(), true);
+  APFloat PSmallestValue = APFloat::getSmallest(APFloat::IEEEsingle(), false);
+  APFloat MSmallestValue = APFloat::getSmallest(APFloat::IEEEsingle(), true);
+  APFloat PSmallestNormalized =
+      APFloat::getSmallestNormalized(APFloat::IEEEsingle(), false);
+  APFloat MSmallestNormalized =
+      APFloat::getSmallestNormalized(APFloat::IEEEsingle(), true);
+
+  APFloat PVal1(APFloat::IEEEsingle(), "0x1.fffffep+126");
+  APFloat MVal1(APFloat::IEEEsingle(), "-0x1.fffffep+126");
+  APFloat PVal2(APFloat::IEEEsingle(), "0x1.fffffep-126");
+  APFloat MVal2(APFloat::IEEEsingle(), "-0x1.fffffep-126");
+  APFloat PVal3(APFloat::IEEEsingle(), "0x1p-125");
+  APFloat MVal3(APFloat::IEEEsingle(), "-0x1p-125");
+  APFloat PVal4(APFloat::IEEEsingle(), "0x1p+127");
+  APFloat MVal4(APFloat::IEEEsingle(), "-0x1p+127");
+  APFloat PVal5(APFloat::IEEEsingle(), "1.5");
+  APFloat MVal5(APFloat::IEEEsingle(), "-1.5");
+  APFloat PVal6(APFloat::IEEEsingle(), "1");
+  APFloat MVal6(APFloat::IEEEsingle(), "-1");
+
+  struct {
+    APFloat x;
+    APFloat y;
+    const char *result;
+    int status;
+    int category;
+  } SpecialCaseTests[] = {
+    { PInf, PInf, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MInf, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PInf, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, PNormalValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MNormalValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, PLargestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MLargestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, PSmallestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MSmallestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, PSmallestNormalized, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PInf, MSmallestNormalized, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PInf, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MInf, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MInf, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PNormalValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MNormalValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PLargestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MLargestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PSmallestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MSmallestValue, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, PSmallestNormalized, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MInf, MSmallestNormalized, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PZero, PInf, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, MInf, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PZero, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PZero, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PZero, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PZero, PNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, MNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, PLargestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, MLargestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, PSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, MSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, PSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PZero, MSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, PInf, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, MInf, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MZero, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MZero, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MZero, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MZero, PNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, MNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, PLargestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, MLargestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, PSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, MSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, PSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MZero, MSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { QNaN, PInf, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MInf, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, PZero, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MZero, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, SNaN, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { QNaN, PNormalValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MNormalValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, PLargestValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MLargestValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, PSmallestValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MSmallestValue, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, PSmallestNormalized, "nan", APFloat::opOK, APFloat::fcNaN },
+    { QNaN, MSmallestNormalized, "nan", APFloat::opOK, APFloat::fcNaN },
+    { SNaN, PInf, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MInf, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, PZero, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MZero, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, QNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, PNormalValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MNormalValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, PLargestValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MLargestValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, PSmallestValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MSmallestValue, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, PSmallestNormalized, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { SNaN, MSmallestNormalized, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PNormalValue, PInf, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PNormalValue, MInf, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PNormalValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PNormalValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PNormalValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PNormalValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PNormalValue, PNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PNormalValue, MNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PNormalValue, PLargestValue, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PNormalValue, MLargestValue, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PNormalValue, PSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PNormalValue, MSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PNormalValue, PSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PNormalValue, MSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, PInf, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MNormalValue, MInf, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MNormalValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MNormalValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MNormalValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MNormalValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MNormalValue, PNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, MNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, PLargestValue, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MNormalValue, MLargestValue, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MNormalValue, PSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, MSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, PSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MNormalValue, MSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, PInf, "0x1.fffffep+127", APFloat::opOK, APFloat::fcNormal },
+    { PLargestValue, MInf, "0x1.fffffep+127", APFloat::opOK, APFloat::fcNormal },
+    { PLargestValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PLargestValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PLargestValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PLargestValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PLargestValue, PNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, MNormalValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, PLargestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, MLargestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, PSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, MSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, PSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PLargestValue, MSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, PInf, "-0x1.fffffep+127", APFloat::opOK, APFloat::fcNormal },
+    { MLargestValue, MInf, "-0x1.fffffep+127", APFloat::opOK, APFloat::fcNormal },
+    { MLargestValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MLargestValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MLargestValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MLargestValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MLargestValue, PNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, MNormalValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, PLargestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, MLargestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, PSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, MSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, PSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MLargestValue, MSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestValue, PInf, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, MInf, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PSmallestValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestValue, PNormalValue, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, MNormalValue, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, PLargestValue, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, MLargestValue, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, PSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestValue, MSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestValue, PSmallestNormalized, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestValue, MSmallestNormalized, "0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, PInf, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, MInf, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestValue, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestValue, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MSmallestValue, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestValue, PNormalValue, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, MNormalValue, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, PLargestValue, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, MLargestValue, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, PSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestValue, MSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestValue, PSmallestNormalized, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestValue, MSmallestNormalized, "-0x1p-149", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, PInf, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, MInf, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestNormalized, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestNormalized, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { PSmallestNormalized, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { PSmallestNormalized, PNormalValue, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, MNormalValue, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, PLargestValue, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, MLargestValue, "0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { PSmallestNormalized, PSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestNormalized, MSmallestValue, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestNormalized, PSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PSmallestNormalized, MSmallestNormalized, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestNormalized, PInf, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, MInf, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, PZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestNormalized, MZero, "nan", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestNormalized, QNaN, "nan", APFloat::opOK, APFloat::fcNaN },
+    { MSmallestNormalized, SNaN, "nan123", APFloat::opInvalidOp, APFloat::fcNaN },
+    { MSmallestNormalized, PNormalValue, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, MNormalValue, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, PLargestValue, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, MLargestValue, "-0x1p-126", APFloat::opOK, APFloat::fcNormal },
+    { MSmallestNormalized, PSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestNormalized, MSmallestValue, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestNormalized, PSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MSmallestNormalized, MSmallestNormalized, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+
+    { PVal1, PVal1, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, MVal1, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, PVal2, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, MVal2, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, PVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, MVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, PVal4, "-0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { PVal1, MVal4, "-0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { PVal1, PVal5, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, MVal5, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, PVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal1, MVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, PVal1, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, MVal1, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, PVal2, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, MVal2, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, PVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, MVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, PVal4, "0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { MVal1, MVal4, "0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { MVal1, PVal5, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, MVal5, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, PVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal1, MVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal2, PVal1, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, MVal1, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, PVal2, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal2, MVal2, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal2, PVal3, "-0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, MVal3, "-0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, PVal4, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, MVal4, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, PVal5, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, MVal5, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, PVal6, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal2, MVal6, "0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, PVal1, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, MVal1, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, PVal2, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal2, MVal2, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal2, PVal3, "0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, MVal3, "0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, PVal4, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, MVal4, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, PVal5, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, MVal5, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, PVal6, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal2, MVal6, "-0x1.fffffep-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, PVal1, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, MVal1, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, PVal2, "0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, MVal2, "0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, PVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal3, MVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal3, PVal4, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, MVal4, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, PVal5, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, MVal5, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, PVal6, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal3, MVal6, "0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, PVal1, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, MVal1, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, PVal2, "-0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, MVal2, "-0x0.000002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, PVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal3, MVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal3, PVal4, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, MVal4, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, PVal5, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, MVal5, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, PVal6, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { MVal3, MVal6, "-0x1p-125", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, PVal1, "0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, MVal1, "0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, PVal2, "0x0.002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, MVal2, "0x0.002p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, PVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal4, MVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal4, PVal4, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal4, MVal4, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal4, PVal5, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, MVal5, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal4, PVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal4, MVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, PVal1, "-0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, MVal1, "-0x1p+103", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, PVal2, "-0x0.002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, MVal2, "-0x0.002p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, PVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, MVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, PVal4, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, MVal4, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, PVal5, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, MVal5, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal4, PVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal4, MVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal5, PVal1, "1.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, MVal1, "1.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, PVal2, "0x0.00006p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, MVal2, "0x0.00006p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, PVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal5, MVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal5, PVal4, "1.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, MVal4, "1.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, PVal5, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal5, MVal5, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal5, PVal6, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal5, MVal6, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, PVal1, "-1.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, MVal1, "-1.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, PVal2, "-0x0.00006p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, MVal2, "-0x0.00006p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, PVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal5, MVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal5, PVal4, "-1.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, MVal4, "-1.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, PVal5, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal5, MVal5, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal5, PVal6, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal5, MVal6, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, PVal1, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, MVal1, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, PVal2, "0x0.00004p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, MVal2, "0x0.00004p-126", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, PVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal6, MVal3, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal6, PVal4, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, MVal4, "0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, PVal5, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, MVal5, "-0.5", APFloat::opOK, APFloat::fcNormal },
+    { PVal6, PVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { PVal6, MVal6, "0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal6, PVal1, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, MVal1, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, PVal2, "-0x0.00004p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, MVal2, "-0x0.00004p-126", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, PVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal6, MVal3, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal6, PVal4, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, MVal4, "-0x1p+0", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, PVal5, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, MVal5, "0.5", APFloat::opOK, APFloat::fcNormal },
+    { MVal6, PVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+    { MVal6, MVal6, "-0x0p+0", APFloat::opOK, APFloat::fcZero },
+  };
+
+  for (size_t i = 0; i < array_lengthof(SpecialCaseTests); ++i) {
+    APFloat x(SpecialCaseTests[i].x);
+    APFloat y(SpecialCaseTests[i].y);
+    APFloat::opStatus status = x.remainder(y);
+
+    APFloat result(x.getSemantics(), SpecialCaseTests[i].result);
+
+    EXPECT_TRUE(result.bitwiseIsEqual(x));
+    EXPECT_EQ(SpecialCaseTests[i].status, (int)status);
+    EXPECT_EQ(SpecialCaseTests[i].category, (int)x.getCategory());
+  }
+
+  {
+    APFloat f1(APFloat::IEEEdouble(), "0x1.3333333333333p-2"); // 0.3
+    APFloat f2(APFloat::IEEEdouble(), "0x1.47ae147ae147bp-7"); // 0.01
+    APFloat expected(APFloat::IEEEdouble(), "-0x1.4p-56");
+    EXPECT_EQ(APFloat::opOK, f1.remainder(f2));
+    EXPECT_TRUE(f1.bitwiseIsEqual(expected));
+  }
+  {
+    APFloat f1(APFloat::IEEEdouble(), "0x1p64"); // 1.8446744073709552e19
+    APFloat f2(APFloat::IEEEdouble(), "1.5");
+    APFloat expected(APFloat::IEEEdouble(), "-0.5");
+    EXPECT_EQ(APFloat::opOK, f1.remainder(f2));
+    EXPECT_TRUE(f1.bitwiseIsEqual(expected));
+  }
+  {
+    APFloat f1(APFloat::IEEEdouble(), "0x1p1000");
+    APFloat f2(APFloat::IEEEdouble(), "0x1p-1000");
+    APFloat expected(APFloat::IEEEdouble(), "0.0");
+    EXPECT_EQ(APFloat::opOK, f1.remainder(f2));
+    EXPECT_TRUE(f1.bitwiseIsEqual(expected));
+  }
+  {
+    APFloat f1 = APFloat::getInf(APFloat::IEEEdouble(), false);
+    APFloat f2(APFloat::IEEEdouble(), "1.0");
+    EXPECT_EQ(f1.remainder(f2), APFloat::opInvalidOp);
+    EXPECT_TRUE(f1.isNaN());
+  }
+  {
+    APFloat f1(APFloat::IEEEdouble(), "-4.0");
+    APFloat f2(APFloat::IEEEdouble(), "-2.0");
+    APFloat expected(APFloat::IEEEdouble(), "-0.0");
+    EXPECT_EQ(APFloat::opOK, f1.remainder(f2));
+    EXPECT_TRUE(f1.bitwiseIsEqual(expected));
+  }
+  {
+    APFloat f1(APFloat::IEEEdouble(), "-4.0");
+    APFloat f2(APFloat::IEEEdouble(), "2.0");
+    APFloat expected(APFloat::IEEEdouble(), "-0.0");
+    EXPECT_EQ(APFloat::opOK, f1.remainder(f2));
+    EXPECT_TRUE(f1.bitwiseIsEqual(expected));
+  }
+}
+
 TEST(APFloatTest, PPCDoubleDoubleAddSpecial) {
   using DataType = std::tuple<uint64_t, uint64_t, uint64_t, uint64_t,
                               APFloat::fltCategory, APFloat::roundingMode>;
   DataType Data[] = {
       // (1 + 0) + (-1 + 0) = fcZero
       std::make_tuple(0x3ff0000000000000ull, 0, 0xbff0000000000000ull, 0,
                       APFloat::fcZero, APFloat::rmNearestTiesToEven),
       // LDBL_MAX + (1.1 >> (1023 - 106) + 0)) = fcInfinity