[APInt] Add support for dividing or remainder by a uint64_t or int64_t.

[craig.topper]

Summary

This patch adds udiv/sdiv/urem/srem/udivrem/sdivrem methods that can divide by a uint64_t. This makes division consistent with all the other arithmetic operations.

This modifies the interface of the divide helper method to work on raw arrays instead of APInts. This way we can pass the uint64_t in for the RHS without wrapping it in an APInt. This required moving all the Quotient and Remainder allocation handling up to the callers. For udiv/urem this was as simple as just creating the Quotient/Remainder with the right size when they were declared. For udivrem we have to rely on reallocate not changing the contents of the variable LHS or RHS is aliased with the Quotient or Remainder APInts. We also have to zero the upper bits of Remainder and Quotient that divide doesn't write to if lhsWords/rhsWords is smaller than the width.

I've update the toString method to use the new udivrem.

[craig.topper]

Forgot to mention, we should look into merging divide and tcDivide in the future.

[RKSimon]

Add extra tests?

lib/Support/APInt.cpp

1568

braces

[craig.topper]

Add tests. Remove braces.

[RKSimon]

LGTM

include/llvm/ADT/APInt.h

private:
   void fromString(unsigned numBits, StringRef str, uint8_t radix);
 
   /// \brief An internal division function for dividing APInts.
   ///
   /// This is used by the toString method to divide by the radix. It simply
   /// provides a more convenient form of divide for internal use since KnuthDiv
   /// has specific constraints on its inputs. If those constraints are not met
   /// then it provides a simpler form of divide.
-  static void divide(const APInt &LHS, unsigned lhsWords, const APInt &RHS,
-                     unsigned rhsWords, APInt *Quotient, APInt *Remainder);
+  static void divide(const WordType *LHS, unsigned lhsWords,
+                     const WordType *RHS, unsigned rhsWords, WordType *Quotient,
+                     WordType *Remainder);
 
   /// out-of-line slow case for inline constructor
   void initSlowCase(uint64_t val, bool isSigned);
 
   /// shared code between two array constructors
   void initFromArray(ArrayRef<uint64_t> array);
 
   /// out-of-line slow case for inline copy constructor
public:
 
   /// \brief Unsigned division operation.
   ///
   /// Perform an unsigned divide operation on this APInt by RHS. Both this and
   /// RHS are treated as unsigned quantities for purposes of this division.
   ///
   /// \returns a new APInt value containing the division result
   APInt udiv(const APInt &RHS) const;
+  APInt udiv(uint64_t RHS) const;
 
   /// \brief Signed division function for APInt.
   ///
   /// Signed divide this APInt by APInt RHS.
   APInt sdiv(const APInt &RHS) const;
+  APInt sdiv(int64_t RHS) const;
 
   /// \brief Unsigned remainder operation.
   ///
   /// Perform an unsigned remainder operation on this APInt with RHS being the
   /// divisor. Both this and RHS are treated as unsigned quantities for purposes
   /// of this operation. Note that this is a true remainder operation and not a
   /// modulo operation because the sign follows the sign of the dividend which
   /// is *this.
   ///
   /// \returns a new APInt value containing the remainder result
   APInt urem(const APInt &RHS) const;
+  uint64_t urem(uint64_t RHS) const;
 
   /// \brief Function for signed remainder operation.
   ///
   /// Signed remainder operation on APInt.
   APInt srem(const APInt &RHS) const;
+  int64_t srem(int64_t RHS) const;
 
   /// \brief Dual division/remainder interface.
   ///
   /// Sometimes it is convenient to divide two APInt values and obtain both the
   /// quotient and remainder. This function does both operations in the same
   /// computation making it a little more efficient. The pair of input arguments
   /// may overlap with the pair of output arguments. It is safe to call
   /// udivrem(X, Y, X, Y), for example.
   static void udivrem(const APInt &LHS, const APInt &RHS, APInt &Quotient,
                       APInt &Remainder);
+  static void udivrem(const APInt &LHS, uint64_t RHS, APInt &Quotient,
+                      uint64_t &Remainder);
 
   static void sdivrem(const APInt &LHS, const APInt &RHS, APInt &Quotient,
                       APInt &Remainder);
+  static void sdivrem(const APInt &LHS, int64_t RHS, APInt &Quotient,
+                      int64_t &Remainder);
 
   // Operations that return overflow indicators.
   APInt sadd_ov(const APInt &RHS, bool &Overflow) const;
   APInt uadd_ov(const APInt &RHS, bool &Overflow) const;
   APInt ssub_ov(const APInt &RHS, bool &Overflow) const;
   APInt usub_ov(const APInt &RHS, bool &Overflow) const;
   APInt sdiv_ov(const APInt &RHS, bool &Overflow) const;
   APInt smul_ov(const APInt &RHS, bool &Overflow) const;

lib/Support/APInt.cpp

if (shift) {
         DEBUG(dbgs() << " " << r[i]);
       }
     }
     DEBUG(dbgs() << '\n');
   }
   DEBUG(dbgs() << '\n');
 }
 
-void APInt::divide(const APInt &LHS, unsigned lhsWords, const APInt &RHS,
-                   unsigned rhsWords, APInt *Quotient, APInt *Remainder) {
+void APInt::divide(const WordType *LHS, unsigned lhsWords, const WordType *RHS,
+                   unsigned rhsWords, WordType *Quotient, WordType *Remainder) {
   assert(lhsWords >= rhsWords && "Fractional result");
 
   // First, compose the values into an array of 32-bit words instead of
   // 64-bit words. This is a necessity of both the "short division" algorithm
   // and the Knuth "classical algorithm" which requires there to be native
   // operations for +, -, and * on an m bit value with an m*2 bit result. We
   // can't use 64-bit operands here because we don't have native results of
   // 128-bits. Furthermore, casting the 64-bit values to 32-bit values won't
if ((Remainder?4:3)*n+2*m+1 <= 128) {
     Q = new uint32_t[m+n];
     if (Remainder)
       R = new uint32_t[n];
   }
 
   // Initialize the dividend
   memset(U, 0, (m+n+1)*sizeof(uint32_t));
   for (unsigned i = 0; i < lhsWords; ++i) {
-    uint64_t tmp = LHS.getRawData()[i];
+    uint64_t tmp = LHS[i];
     U[i * 2] = Lo_32(tmp);
     U[i * 2 + 1] = Hi_32(tmp);
   }
   U[m+n] = 0; // this extra word is for "spill" in the Knuth algorithm.
 
   // Initialize the divisor
   memset(V, 0, (n)*sizeof(uint32_t));
   for (unsigned i = 0; i < rhsWords; ++i) {
-    uint64_t tmp = RHS.getRawData()[i];
+    uint64_t tmp = RHS[i];
     V[i * 2] = Lo_32(tmp);
     V[i * 2 + 1] = Hi_32(tmp);
   }
 
   // initialize the quotient and remainder
   memset(Q, 0, (m+n) * sizeof(uint32_t));
   if (Remainder)
     memset(R, 0, n * sizeof(uint32_t));
void APInt::divide(const WordType *LHS, unsigned lhsWords, const WordType *RHS,
   } else {
     // Now we're ready to invoke the Knuth classical divide algorithm. In this
     // case n > 1.
     KnuthDiv(U, V, Q, R, m, n);
   }
 
   // If the caller wants the quotient
   if (Quotient) {
-    // Set up the Quotient value's memory.
-    Quotient->reallocate(LHS.BitWidth);
-    // Clear out any previous bits.
-    Quotient->clearAllBits();
-
-    // The quotient is in Q. Reconstitute the quotient into Quotient's low
-    // order words.
-    // This case is currently dead as all users of divide() handle trivial cases
-    // earlier.
-    if (lhsWords == 1) {
-      uint64_t tmp = Make_64(Q[1], Q[0]);
-      if (Quotient->isSingleWord())
-        Quotient->U.VAL = tmp;
-      else
-        Quotient->U.pVal[0] = tmp;
-    } else {
-      assert(!Quotient->isSingleWord() && "Quotient APInt not large enough");
-      for (unsigned i = 0; i < lhsWords; ++i)
-        Quotient->U.pVal[i] = Make_64(Q[i*2+1], Q[i*2]);
-    }
+      Quotient[i] = Make_64(Q[i*2+1], Q[i*2]);
   }
 
   // If the caller wants the remainder
   if (Remainder) {
-    // Set up the Remainder value's memory.
-    Remainder->reallocate(RHS.BitWidth);
-    // Clear out any previous bits.
-    Remainder->clearAllBits();
-
-    // The remainder is in R. Reconstitute the remainder into Remainder's low
-    // order words.
-    if (rhsWords == 1) {
-      uint64_t tmp = Make_64(R[1], R[0]);
-      if (Remainder->isSingleWord())
-        Remainder->U.VAL = tmp;
-      else
-        Remainder->U.pVal[0] = tmp;
-    } else {
-      assert(!Remainder->isSingleWord() && "Remainder APInt not large enough");
-      for (unsigned i = 0; i < rhsWords; ++i)
-        Remainder->U.pVal[i] = Make_64(R[i*2+1], R[i*2]);
-    }
+      Remainder[i] = Make_64(R[i*2+1], R[i*2]);
   }
 
   // Clean up the memory we allocated.
   if (U != &SPACE[0]) {
     delete [] U;
     delete [] V;
     delete [] Q;
     delete [] R;
   }
 }
 
-APInt APInt::udiv(const APInt& RHS) const {
+APInt APInt::udiv(const APInt &RHS) const {
   assert(BitWidth == RHS.BitWidth && "Bit widths must be the same");
 
   // First, deal with the easy case
   if (isSingleWord()) {
     assert(RHS.U.VAL != 0 && "Divide by zero?");
     return APInt(BitWidth, U.VAL / RHS.U.VAL);
   }
 
APInt APInt::udiv(const APInt &RHS) const {
   if (*this == RHS)
     // X / X ===> 1
     return APInt(BitWidth, 1);
   if (lhsWords == 1) // rhsWords is 1 if lhsWords is 1.
     // All high words are zero, just use native divide
     return APInt(BitWidth, this->U.pVal[0] / RHS.U.pVal[0]);
 
   // We have to compute it the hard way. Invoke the Knuth divide algorithm.
-  APInt Quotient; // to hold result.
-  divide(*this, lhsWords, RHS, rhsWords, &Quotient, nullptr);
+  APInt Quotient(BitWidth, 0); // to hold result.
+  divide(U.pVal, lhsWords, RHS.U.pVal, rhsWords, Quotient.U.pVal, nullptr);
+  return Quotient;
+}
+
+APInt APInt::udiv(uint64_t RHS) const {
+  assert(RHS != 0 && "Divide by zero?");
+
+  // First, deal with the easy case
+  if (isSingleWord())
+    return APInt(BitWidth, U.VAL / RHS);
+
+  // Get some facts about the LHS words.
+  unsigned lhsWords = getNumWords(getActiveBits());
+
+  // Deal with some degenerate cases
+  if (!lhsWords)
+    // 0 / X ===> 0
+    return APInt(BitWidth, 0);
+  if (RHS == 1)
+    // X / 1 ===> X
+    return *this;
+  if (this->ult(RHS))
+    // X / Y ===> 0, iff X < Y
+    return APInt(BitWidth, 0);
+  if (*this == RHS)
+    // X / X ===> 1
+    return APInt(BitWidth, 1);
+  if (lhsWords == 1) // rhsWords is 1 if lhsWords is 1.
+    // All high words are zero, just use native divide
+    return APInt(BitWidth, this->U.pVal[0] / RHS);
+
+  // We have to compute it the hard way. Invoke the Knuth divide algorithm.
+  APInt Quotient(BitWidth, 0); // to hold result.
+  divide(U.pVal, lhsWords, &RHS, 1, Quotient.U.pVal, nullptr);
   return Quotient;
 }
 
 APInt APInt::sdiv(const APInt &RHS) const {
   if (isNegative()) {
     if (RHS.isNegative())
       return (-(*this)).udiv(-RHS);
     return -((-(*this)).udiv(RHS));
   }
   if (RHS.isNegative())
     return -(this->udiv(-RHS));
   return this->udiv(RHS);
 }
 
-APInt APInt::urem(const APInt& RHS) const {
+APInt APInt::sdiv(int64_t RHS) const {
+  if (isNegative()) {
+    if (RHS < 0)
+      return (-(*this)).udiv(-RHS);
+    return -((-(*this)).udiv(RHS));
+  }
+  if (RHS < 0)
+    return -(this->udiv(-RHS));
+  return this->udiv(RHS);
+}
+
+APInt APInt::urem(const APInt &RHS) const {
   assert(BitWidth == RHS.BitWidth && "Bit widths must be the same");
   if (isSingleWord()) {
     assert(RHS.U.VAL != 0 && "Remainder by zero?");
     return APInt(BitWidth, U.VAL % RHS.U.VAL);
   }
 
   // Get some facts about the LHS
   unsigned lhsWords = getNumWords(getActiveBits());
APInt APInt::urem(const APInt &RHS) const {
   if (*this == RHS)
     // X % X == 0;
     return APInt(BitWidth, 0);
   if (lhsWords == 1)
     // All high words are zero, just use native remainder
     return APInt(BitWidth, U.pVal[0] % RHS.U.pVal[0]);
 
   // We have to compute it the hard way. Invoke the Knuth divide algorithm.
-  APInt Remainder;
-  divide(*this, lhsWords, RHS, rhsWords, nullptr, &Remainder);
+  APInt Remainder(BitWidth, 0);
+  divide(U.pVal, lhsWords, RHS.U.pVal, rhsWords, nullptr, Remainder.U.pVal);
+  return Remainder;
+}
+
+uint64_t APInt::urem(uint64_t RHS) const {
+  assert(RHS != 0 && "Remainder by zero?");
+
+  if (isSingleWord())
+    return U.VAL % RHS;
+
+  // Get some facts about the LHS
+  unsigned lhsWords = getNumWords(getActiveBits());
+
+  // Check the degenerate cases
+  if (lhsWords == 0)
+    // 0 % Y ===> 0
+    return 0;
+  if (RHS == 1)
+    // X % 1 ===> 0
+    return 0;
+  if (this->ult(RHS))
+    // X % Y ===> X, iff X < Y
+    return getZExtValue();
+  if (*this == RHS)
+    // X % X == 0;
+    return 0;
+  if (lhsWords == 1)
+    // All high words are zero, just use native remainder
+    return U.pVal[0] % RHS;
+
+  // We have to compute it the hard way. Invoke the Knuth divide algorithm.
+  uint64_t Remainder;
+  divide(U.pVal, lhsWords, &RHS, 1, nullptr, &Remainder);
   return Remainder;
 }
 
 APInt APInt::srem(const APInt &RHS) const {
   if (isNegative()) {
     if (RHS.isNegative())
       return -((-(*this)).urem(-RHS));
     return -((-(*this)).urem(RHS));
   }
   if (RHS.isNegative())
     return this->urem(-RHS);
   return this->urem(RHS);
 }
 
+int64_t APInt::srem(int64_t RHS) const {
+  if (isNegative()) {
+    if (RHS < 0)
+      return -((-(*this)).urem(-RHS));
+    return -((-(*this)).urem(RHS));
+  }
+  if (RHS < 0)
+    return this->urem(-RHS);
+  return this->urem(RHS);
+}
+
 void APInt::udivrem(const APInt &LHS, const APInt &RHS,
                     APInt &Quotient, APInt &Remainder) {
   assert(LHS.BitWidth == RHS.BitWidth && "Bit widths must be the same");
   unsigned BitWidth = LHS.BitWidth;
 
   // First, deal with the easy case
   if (LHS.isSingleWord()) {
     assert(RHS.U.VAL != 0 && "Divide by zero?");
void APInt::udivrem(const APInt &LHS, const APInt &RHS,
   }
 
   if (LHS == RHS) {
     Quotient  = 1;              // X / X ===> 1
     Remainder = 0;              // X % X ===> 0;
     return;
   }
 
+  // Make sure there is enough space to hold the results.
+  // NOTE: This assumes that reallocate won't affect any bits if it doesn't
+  // change the size. This is necessary if Quotient or Remainder is aliased
+  // with LHS or RHS.
+  Quotient.reallocate(BitWidth);
+  Remainder.reallocate(BitWidth);
+
   if (lhsWords == 1) { // rhsWords is 1 if lhsWords is 1.
     // There is only one word to consider so use the native versions.
     uint64_t lhsValue = LHS.U.pVal[0];
     uint64_t rhsValue = RHS.U.pVal[0];
-    // Make sure there is enough space to hold the results.
-    Quotient.reallocate(BitWidth);
-    Remainder.reallocate(BitWidth);
     Quotient = lhsValue / rhsValue;
     Remainder = lhsValue % rhsValue;
     return;
   }
 
   // Okay, lets do it the long way
-  divide(LHS, lhsWords, RHS, rhsWords, &Quotient, &Remainder);
+  divide(LHS.U.pVal, lhsWords, RHS.U.pVal, rhsWords, Quotient.U.pVal,
+         Remainder.U.pVal);
+  // Clear the rest of the Quotient and Remainder.
+  std::memset(Quotient.U.pVal + lhsWords, 0,
+              (getNumWords(BitWidth) - lhsWords) * APINT_WORD_SIZE);
+  std::memset(Remainder.U.pVal + rhsWords, 0,
+              (getNumWords(BitWidth) - rhsWords) * APINT_WORD_SIZE);
+}
+
+void APInt::udivrem(const APInt &LHS, uint64_t RHS, APInt &Quotient,
+                    uint64_t &Remainder) {
+  assert(RHS != 0 && "Divide by zero?");
+  unsigned BitWidth = LHS.BitWidth;
+
+  // First, deal with the easy case
+  if (LHS.isSingleWord()) {
+    uint64_t QuotVal = LHS.U.VAL / RHS;
+    Remainder = LHS.U.VAL % RHS;
+    Quotient = APInt(BitWidth, QuotVal);
+    return;
+  }
+
+  // Get some size facts about the dividend and divisor
+  unsigned lhsWords = getNumWords(LHS.getActiveBits());
+
+  // Check the degenerate cases
+  if (lhsWords == 0) {
+    Quotient = 0;                // 0 / Y ===> 0
+    Remainder = 0;               // 0 % Y ===> 0
+    return;
+  }
+
+  if (RHS == 1) {
+    Quotient = LHS;             // X / 1 ===> X
+    Remainder = 0;              // X % 1 ===> 0
+  }
+
+  if (LHS.ult(RHS)) {
+    Remainder = LHS.getZExtValue(); // X % Y ===> X, iff X < Y
+    Quotient = 0;                   // X / Y ===> 0, iff X < Y
+    return;
+  }
+
+  if (LHS == RHS) {
+    Quotient  = 1;              // X / X ===> 1
+    Remainder = 0;              // X % X ===> 0;
+    return;
+  }
+
+  // Make sure there is enough space to hold the results.
+  // NOTE: This assumes that reallocate won't affect any bits if it doesn't
+  // change the size. This is necessary if Quotient is aliased with LHS.
+  Quotient.reallocate(BitWidth);
+
+  if (lhsWords == 1) { // rhsWords is 1 if lhsWords is 1.
+    // There is only one word to consider so use the native versions.
+    uint64_t lhsValue = LHS.U.pVal[0];
+    Quotient = lhsValue / RHS;
+    Remainder = lhsValue % RHS;
+    return;
+  }
+
+  // Okay, lets do it the long way
+  divide(LHS.U.pVal, lhsWords, &RHS, 1, Quotient.U.pVal, &Remainder);
+  // Clear the rest of the Quotient.
+  std::memset(Quotient.U.pVal + lhsWords, 0,
+              (getNumWords(BitWidth) - lhsWords) * APINT_WORD_SIZE);
 }
 
 void APInt::sdivrem(const APInt &LHS, const APInt &RHS,
                     APInt &Quotient, APInt &Remainder) {
   if (LHS.isNegative()) {
     if (RHS.isNegative())
       APInt::udivrem(-LHS, -RHS, Quotient, Remainder);
     else {
       APInt::udivrem(-LHS, RHS, Quotient, Remainder);
       Quotient.negate();
     }
     Remainder.negate();
   } else if (RHS.isNegative()) {
     APInt::udivrem(LHS, -RHS, Quotient, Remainder);
     Quotient.negate();
   } else {
     APInt::udivrem(LHS, RHS, Quotient, Remainder);
   }
 }
 
+void APInt::sdivrem(const APInt &LHS, int64_t RHS,
+                    APInt &Quotient, int64_t &Remainder) {
+  uint64_t R = Remainder;
+  if (LHS.isNegative()) {
+    if (RHS < 0)
+      APInt::udivrem(-LHS, -RHS, Quotient, R);
+    else {
+      APInt::udivrem(-LHS, RHS, Quotient, R);
+      Quotient.negate();
+    }
+    R = -R;
+  } else if (RHS < 0) {
+    APInt::udivrem(LHS, -RHS, Quotient, R);
+    Quotient.negate();
+  } else {
+    APInt::udivrem(LHS, RHS, Quotient, R);
+  }
+  Remainder = R;
+}
+
 APInt APInt::sadd_ov(const APInt &RHS, bool &Overflow) const {
   APInt Res = *this+RHS;
   Overflow = isNonNegative() == RHS.isNonNegative() &&
              Res.isNonNegative() != isNonNegative();
   return Res;
 }
 
 APInt APInt::uadd_ov(const APInt &RHS, bool &Overflow) const {
if (Radix == 2 || Radix == 8 || Radix == 16) {
     unsigned MaskAmt = Radix - 1;
 
     while (Tmp.getBoolValue()) {
       unsigned Digit = unsigned(Tmp.getRawData()[0]) & MaskAmt;
       Str.push_back(Digits[Digit]);
       Tmp.lshrInPlace(ShiftAmt);
     }
   } else {
-    APInt divisor(Tmp.getBitWidth(), Radix);
-    APInt APdigit;
     while (Tmp.getBoolValue()) {
-      udivrem(Tmp, divisor, Tmp, APdigit);
-      unsigned Digit = (unsigned)APdigit.getZExtValue();
+      uint64_t Digit;
+      udivrem(Tmp, Radix, Tmp, Digit);
       assert(Digit < Radix && "divide failed");
       Str.push_back(Digits[Digit]);
     }
   }
 
   // Reverse the digits before returning.
   std::reverse(Str.begin()+StartDig, Str.end());
 }

unittests/ADT/APIntTest.cpp

 
 TEST(APIntTest, divrem_big7) {
   // Yet another test for KnuthDiv rare step D6.
   testDiv({224, "800000008000000200000005", 16},
           {224, "fffffffd", 16},
           {224, "80000000800000010000000f", 16});
 }
 
+void testDiv(APInt a, uint64_t b, APInt c) {
+  auto p = a * b + c;
+
+  APInt q;
+  uint64_t r;
+  // Unsigned division will only work if our original number wasn't negative.
+  if (!a.isNegative()) {
+    q = p.udiv(b);
+    r = p.urem(b);
+    EXPECT_EQ(a, q);
+    EXPECT_EQ(c, r);
+    APInt::udivrem(p, b, q, r);
+    EXPECT_EQ(a, q);
+    EXPECT_EQ(c, r);
+  }
+  q = p.sdiv(b);
+  r = p.srem(b);
+  EXPECT_EQ(a, q);
+  if (c.isNegative())
+    EXPECT_EQ(-c, -r); // Need to negate so the uint64_t compare will work.
+  else
+    EXPECT_EQ(c, r);
+  int64_t sr;
+  APInt::sdivrem(p, b, q, sr);
+  EXPECT_EQ(a, q);
+  if (c.isNegative())
+    EXPECT_EQ(-c, -sr); // Need to negate so the uint64_t compare will work.
+  else
+    EXPECT_EQ(c, sr);
+}
+
+TEST(APIntTest, divremuint) {
+  // Single word APInt
+  testDiv(APInt{64, 9},
+          2,
+          APInt{64, 1});
+
+  // Single word negative APInt
+  testDiv(-APInt{64, 9},
+          2,
+          -APInt{64, 1});
+
+  // Multiword dividend with only one significant word.
+  testDiv(APInt{256, 9},
+          2,
+          APInt{256, 1});
+
+  // Negative dividend.
+  testDiv(-APInt{256, 9},
+          2,
+          -APInt{256, 1});
+
+  // Multiword dividend
+  testDiv(APInt{1024, 19}.shl(811),
+          4356013, // one word
+          APInt{1024, 1});
+}
+
 TEST(APIntTest, fromString) {
   EXPECT_EQ(APInt(32, 0), APInt(32,   "0", 2));
   EXPECT_EQ(APInt(32, 1), APInt(32,   "1", 2));
   EXPECT_EQ(APInt(32, 2), APInt(32,  "10", 2));
   EXPECT_EQ(APInt(32, 3), APInt(32,  "11", 2));
   EXPECT_EQ(APInt(32, 4), APInt(32, "100", 2));
 
   EXPECT_EQ(APInt(32, 0), APInt(32,   "+0", 2));