third_party/bigint/BigInteger.cc - pdfium - Git at Google

 // Copyright 2014 PDFium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.

 // Original code by Matt McCutchen, see the LICENSE file.

 #include "BigInteger.hh"

 BigInteger& BigInteger::operator =(const BigInteger &x) {
 	// Calls like a = a have no effect
 	if (this == &x)
 		return *this;
 	// Copy sign
 	sign = x.sign;
 	// Copy the rest
 	mag = x.mag;
 	return *this;
 }

 BigInteger::BigInteger(const Blk *b, Index blen, Sign s) : mag(b, blen) {
 	switch (s) {
 	case zero:
 		if (!mag.isZero())
             abort();
 		sign = zero;
 		break;
 	case positive:
 	case negative:
 		// If the magnitude is zero, force the sign to zero.
 		sign = mag.isZero() ? zero : s;
 		break;
 	default:
 		/* g++ seems to be optimizing out this case on the assumption
 		 * that the sign is a valid member of the enumeration.  Oh well. */
         abort();
 	}
 }

 BigInteger::BigInteger(const BigUnsigned &x, Sign s) : mag(x) {
 	switch (s) {
 	case zero:
 		if (!mag.isZero())
             abort();
 		sign = zero;
 		break;
 	case positive:
 	case negative:
 		// If the magnitude is zero, force the sign to zero.
 		sign = mag.isZero() ? zero : s;
 		break;
 	default:
 		/* g++ seems to be optimizing out this case on the assumption
 		 * that the sign is a valid member of the enumeration.  Oh well. */
         abort();
 	}
 }

 /* CONSTRUCTION FROM PRIMITIVE INTEGERS
  * Same idea as in BigUnsigned.cc, except that negative input results in a
  * negative BigInteger instead of an exception. */

 // Done longhand to let us use initialization.
 BigInteger::BigInteger(unsigned long  x) : mag(x) { sign = mag.isZero() ? zero : positive; }
 BigInteger::BigInteger(unsigned int   x) : mag(x) { sign = mag.isZero() ? zero : positive; }
 BigInteger::BigInteger(unsigned short x) : mag(x) { sign = mag.isZero() ? zero : positive; }

 // For signed input, determine the desired magnitude and sign separately.

 namespace {
 	template <class X, class UX>
 	BigInteger::Blk magOf(X x) {
 		/* UX(...) cast needed to stop short(-2^15), which negates to
 		 * itself, from sign-extending in the conversion to Blk. */
 		return BigInteger::Blk(x < 0 ? UX(-x) : x);
 	}
 	template <class X>
 	BigInteger::Sign signOf(X x) {
 		return (x == 0) ? BigInteger::zero
 			: (x > 0) ? BigInteger::positive
 			: BigInteger::negative;
 	}
 }

 BigInteger::BigInteger(long  x) : sign(signOf(x)), mag(magOf<long , unsigned long >(x)) {}
 BigInteger::BigInteger(int   x) : sign(signOf(x)), mag(magOf<int  , unsigned int  >(x)) {}
 BigInteger::BigInteger(short x) : sign(signOf(x)), mag(magOf<short, unsigned short>(x)) {}

 // CONVERSION TO PRIMITIVE INTEGERS

 /* Reuse BigUnsigned's conversion to an unsigned primitive integer.
  * The friend is a separate function rather than
  * BigInteger::convertToUnsignedPrimitive to avoid requiring BigUnsigned to
  * declare BigInteger. */
 template <class X>
 inline X convertBigUnsignedToPrimitiveAccess(const BigUnsigned &a) {
 	return a.convertToPrimitive<X>();
 }

 template <class X>
 X BigInteger::convertToUnsignedPrimitive() const {
 	if (sign == negative)
         abort();
 	else
 		return convertBigUnsignedToPrimitiveAccess<X>(mag);
 }

 /* Similar to BigUnsigned::convertToPrimitive, but split into two cases for
  * nonnegative and negative numbers. */
 template <class X, class UX>
 X BigInteger::convertToSignedPrimitive() const {
 	if (sign == zero)
 		return 0;
 	else if (mag.getLength() == 1) {
 		// The single block might fit in an X.  Try the conversion.
 		Blk b = mag.getBlock(0);
 		if (sign == positive) {
 			X x = X(b);
 			if (x >= 0 && Blk(x) == b)
 				return x;
 		} else {
 			X x = -X(b);
 			/* UX(...) needed to avoid rejecting conversion of
 			 * -2^15 to a short. */
 			if (x < 0 && Blk(UX(-x)) == b)
 				return x;
 		}
 		// Otherwise fall through.
 	}
     abort();
 }

 unsigned long  BigInteger::toUnsignedLong () const { return convertToUnsignedPrimitive<unsigned long >       (); }
 unsigned int   BigInteger::toUnsignedInt  () const { return convertToUnsignedPrimitive<unsigned int  >       (); }
 unsigned short BigInteger::toUnsignedShort() const { return convertToUnsignedPrimitive<unsigned short>       (); }
 long           BigInteger::toLong         () const { return convertToSignedPrimitive  <long , unsigned long> (); }
 int            BigInteger::toInt          () const { return convertToSignedPrimitive  <int  , unsigned int>  (); }
 short          BigInteger::toShort        () const { return convertToSignedPrimitive  <short, unsigned short>(); }

 // COMPARISON
 BigInteger::CmpRes BigInteger::compareTo(const BigInteger &x) const {
 	// A greater sign implies a greater number
 	if (sign < x.sign)
 		return less;
 	else if (sign > x.sign)
 		return greater;
 	else switch (sign) {
 		// If the signs are the same...
 	case zero:
 		return equal; // Two zeros are equal
 	case positive:
 		// Compare the magnitudes
 		return mag.compareTo(x.mag);
 	case negative:
 		// Compare the magnitudes, but return the opposite result
 		return CmpRes(-mag.compareTo(x.mag));
 	default:
         abort();
 	}
 }

 /* COPY-LESS OPERATIONS
  * These do some messing around to determine the sign of the result,
  * then call one of BigUnsigned's copy-less operations. */

 // See remarks about aliased calls in BigUnsigned.cc .
 #define DTRT_ALIASED(cond, op) \
 	if (cond) { \
 		BigInteger tmpThis; \
 		tmpThis.op; \
 		*this = tmpThis; \
 		return; \
 	}

 void BigInteger::add(const BigInteger &a, const BigInteger &b) {
 	DTRT_ALIASED(this == &a || this == &b, add(a, b));
 	// If one argument is zero, copy the other.
 	if (a.sign == zero)
 		operator =(b);
 	else if (b.sign == zero)
 		operator =(a);
 	// If the arguments have the same sign, take the
 	// common sign and add their magnitudes.
 	else if (a.sign == b.sign) {
 		sign = a.sign;
 		mag.add(a.mag, b.mag);
 	} else {
 		// Otherwise, their magnitudes must be compared.
 		switch (a.mag.compareTo(b.mag)) {
 		case equal:
 			// If their magnitudes are the same, copy zero.
 			mag = 0;
 			sign = zero;
 			break;
 			// Otherwise, take the sign of the greater, and subtract
 			// the lesser magnitude from the greater magnitude.
 		case greater:
 			sign = a.sign;
 			mag.subtract(a.mag, b.mag);
 			break;
 		case less:
 			sign = b.sign;
 			mag.subtract(b.mag, a.mag);
 			break;
 		}
 	}
 }

 void BigInteger::subtract(const BigInteger &a, const BigInteger &b) {
 	// Notice that this routine is identical to BigInteger::add,
 	// if one replaces b.sign by its opposite.
 	DTRT_ALIASED(this == &a || this == &b, subtract(a, b));
 	// If a is zero, copy b and flip its sign.  If b is zero, copy a.
 	if (a.sign == zero) {
 		mag = b.mag;
 		// Take the negative of _b_'s, sign, not ours.
 		// Bug pointed out by Sam Larkin on 2005.03.30.
 		sign = Sign(-b.sign);
 	} else if (b.sign == zero)
 		operator =(a);
 	// If their signs differ, take a.sign and add the magnitudes.
 	else if (a.sign != b.sign) {
 		sign = a.sign;
 		mag.add(a.mag, b.mag);
 	} else {
 		// Otherwise, their magnitudes must be compared.
 		switch (a.mag.compareTo(b.mag)) {
 			// If their magnitudes are the same, copy zero.
 		case equal:
 			mag = 0;
 			sign = zero;
 			break;
 			// If a's magnitude is greater, take a.sign and
 			// subtract a from b.
 		case greater:
 			sign = a.sign;
 			mag.subtract(a.mag, b.mag);
 			break;
 			// If b's magnitude is greater, take the opposite
 			// of b.sign and subtract b from a.
 		case less:
 			sign = Sign(-b.sign);
 			mag.subtract(b.mag, a.mag);
 			break;
 		}
 	}
 }

 void BigInteger::multiply(const BigInteger &a, const BigInteger &b) {
 	DTRT_ALIASED(this == &a || this == &b, multiply(a, b));
 	// If one object is zero, copy zero and return.
 	if (a.sign == zero || b.sign == zero) {
 		sign = zero;
 		mag = 0;
 		return;
 	}
 	// If the signs of the arguments are the same, the result
 	// is positive, otherwise it is negative.
 	sign = (a.sign == b.sign) ? positive : negative;
 	// Multiply the magnitudes.
 	mag.multiply(a.mag, b.mag);
 }

 /*
  * DIVISION WITH REMAINDER
  * Please read the comments before the definition of
  * `BigUnsigned::divideWithRemainder' in `BigUnsigned.cc' for lots of
  * information you should know before reading this function.
  *
  * Following Knuth, I decree that x / y is to be
  * 0 if y==0 and floor(real-number x / y) if y!=0.
  * Then x % y shall be x - y*(integer x / y).
  *
  * Note that x = y * (x / y) + (x % y) always holds.
  * In addition, (x % y) is from 0 to y - 1 if y > 0,
  * and from -(|y| - 1) to 0 if y < 0.  (x % y) = x if y = 0.
  *
  * Examples: (q = a / b, r = a % b)
  *	a	b	q	r
  *	===	===	===	===
  *	4	3	1	1
  *	-4	3	-2	2
  *	4	-3	-2	-2
  *	-4	-3	1	-1
  */
 void BigInteger::divideWithRemainder(const BigInteger &b, BigInteger &q) {
 	// Defend against aliased calls;
 	// same idea as in BigUnsigned::divideWithRemainder .
 	if (this == &q)
         abort();
 	if (this == &b || &q == &b) {
 		BigInteger tmpB(b);
 		divideWithRemainder(tmpB, q);
 		return;
 	}

 	// Division by zero gives quotient 0 and remainder *this
 	if (b.sign == zero) {
 		q.mag = 0;
 		q.sign = zero;
 		return;
 	}
 	// 0 / b gives quotient 0 and remainder 0
 	if (sign == zero) {
 		q.mag = 0;
 		q.sign = zero;
 		return;
 	}

 	// Here *this != 0, b != 0.

 	// Do the operands have the same sign?
 	if (sign == b.sign) {
 		// Yes: easy case.  Quotient is zero or positive.
 		q.sign = positive;
 	} else {
 		// No: harder case.  Quotient is negative.
 		q.sign = negative;
 		// Decrease the magnitude of the dividend by one.
 		mag--;
 		/*
 		 * We tinker with the dividend before and with the
 		 * quotient and remainder after so that the result
 		 * comes out right.  To see why it works, consider the following
 		 * list of examples, where A is the magnitude-decreased
 		 * a, Q and R are the results of BigUnsigned division
 		 * with remainder on A and |b|, and q and r are the
 		 * final results we want:
 		 *
 		 *	a	A	b	Q	R	q	r
 		 *	-3	-2	3	0	2	-1	0
 		 *	-4	-3	3	1	0	-2	2
 		 *	-5	-4	3	1	1	-2	1
 		 *	-6	-5	3	1	2	-2	0
 		 *
 		 * It appears that we need a total of 3 corrections:
 		 * Decrease the magnitude of a to get A.  Increase the
 		 * magnitude of Q to get q (and make it negative).
 		 * Find r = (b - 1) - R and give it the desired sign.
 		 */
 	}

 	// Divide the magnitudes.
 	mag.divideWithRemainder(b.mag, q.mag);

 	if (sign != b.sign) {
 		// More for the harder case (as described):
 		// Increase the magnitude of the quotient by one.
 		q.mag++;
 		// Modify the remainder.
 		mag.subtract(b.mag, mag);
 		mag--;
 	}

 	// Sign of the remainder is always the sign of the divisor b.
 	sign = b.sign;

 	// Set signs to zero as necessary.  (Thanks David Allen!)
 	if (mag.isZero())
 		sign = zero;
 	if (q.mag.isZero())
 		q.sign = zero;

 	// WHEW!!!
 }

 // Negation
 void BigInteger::negate(const BigInteger &a) {
 	DTRT_ALIASED(this == &a, negate(a));
 	// Copy a's magnitude
 	mag = a.mag;
 	// Copy the opposite of a.sign
 	sign = Sign(-a.sign);
 }

 // INCREMENT/DECREMENT OPERATORS

 // Prefix increment
 BigInteger& BigInteger::operator ++() {
 	if (sign == negative) {
 		mag--;
 		if (mag == 0)
 			sign = zero;
 	} else {
 		mag++;
 		sign = positive; // if not already
 	}
 	return *this;
 }

 // Postfix increment
 BigInteger BigInteger::operator ++(int) {
 	BigInteger temp(*this);
 	operator ++();
 	return temp;
 }

 // Prefix decrement
 BigInteger& BigInteger::operator --() {
 	if (sign == positive) {
 		mag--;
 		if (mag == 0)
 			sign = zero;
 	} else {
 		mag++;
 		sign = negative;
 	}
 	return *this;
 }

 // Postfix decrement
 BigInteger BigInteger::operator --(int) {
 	BigInteger temp(*this);
 	operator --();
 	return temp;
 }
	// Copyright 2014 PDFium Authors. All rights reserved.
	// Use of this source code is governed by a BSD-style license that can be
	// found in the LICENSE file.

	// Original code by Matt McCutchen, see the LICENSE file.

	#include "BigInteger.hh"

	BigInteger& BigInteger::operator =(const BigInteger &x) {
	// Calls like a = a have no effect
	if (this == &x)
	return *this;
	// Copy sign
	sign = x.sign;
	// Copy the rest
	mag = x.mag;
	return *this;
	}

	BigInteger::BigInteger(const Blk *b, Index blen, Sign s) : mag(b, blen) {
	switch (s) {
	case zero:
	if (!mag.isZero())
	abort();
	sign = zero;
	break;
	case positive:
	case negative:
	// If the magnitude is zero, force the sign to zero.
	sign = mag.isZero() ? zero : s;
	break;
	default:
	/* g++ seems to be optimizing out this case on the assumption
	* that the sign is a valid member of the enumeration. Oh well. */
	abort();
	}
	}

	BigInteger::BigInteger(const BigUnsigned &x, Sign s) : mag(x) {
	switch (s) {
	case zero:
	if (!mag.isZero())
	abort();
	sign = zero;
	break;
	case positive:
	case negative:
	// If the magnitude is zero, force the sign to zero.
	sign = mag.isZero() ? zero : s;
	break;
	default:
	/* g++ seems to be optimizing out this case on the assumption
	* that the sign is a valid member of the enumeration. Oh well. */
	abort();
	}
	}

	/* CONSTRUCTION FROM PRIMITIVE INTEGERS
	* Same idea as in BigUnsigned.cc, except that negative input results in a
	* negative BigInteger instead of an exception. */

	// Done longhand to let us use initialization.
	BigInteger::BigInteger(unsigned long x) : mag(x) { sign = mag.isZero() ? zero : positive; }
	BigInteger::BigInteger(unsigned int x) : mag(x) { sign = mag.isZero() ? zero : positive; }
	BigInteger::BigInteger(unsigned short x) : mag(x) { sign = mag.isZero() ? zero : positive; }

	// For signed input, determine the desired magnitude and sign separately.

	namespace {
	template <class X, class UX>
	BigInteger::Blk magOf(X x) {
	/* UX(...) cast needed to stop short(-2^15), which negates to
	* itself, from sign-extending in the conversion to Blk. */
	return BigInteger::Blk(x < 0 ? UX(-x) : x);
	}
	template <class X>
	BigInteger::Sign signOf(X x) {
	return (x == 0) ? BigInteger::zero
	: (x > 0) ? BigInteger::positive
	: BigInteger::negative;
	}
	}

	BigInteger::BigInteger(long x) : sign(signOf(x)), mag(magOf<long , unsigned long >(x)) {}
	BigInteger::BigInteger(int x) : sign(signOf(x)), mag(magOf<int , unsigned int >(x)) {}
	BigInteger::BigInteger(short x) : sign(signOf(x)), mag(magOf<short, unsigned short>(x)) {}

	// CONVERSION TO PRIMITIVE INTEGERS

	/* Reuse BigUnsigned's conversion to an unsigned primitive integer.
	* The friend is a separate function rather than
	* BigInteger::convertToUnsignedPrimitive to avoid requiring BigUnsigned to
	* declare BigInteger. */
	template <class X>
	inline X convertBigUnsignedToPrimitiveAccess(const BigUnsigned &a) {
	return a.convertToPrimitive<X>();
	}

	template <class X>
	X BigInteger::convertToUnsignedPrimitive() const {
	if (sign == negative)
	abort();
	else
	return convertBigUnsignedToPrimitiveAccess<X>(mag);
	}

	/* Similar to BigUnsigned::convertToPrimitive, but split into two cases for
	* nonnegative and negative numbers. */
	template <class X, class UX>
	X BigInteger::convertToSignedPrimitive() const {
	if (sign == zero)
	return 0;
	else if (mag.getLength() == 1) {
	// The single block might fit in an X. Try the conversion.
	Blk b = mag.getBlock(0);
	if (sign == positive) {
	X x = X(b);
	if (x >= 0 && Blk(x) == b)
	return x;
	} else {
	X x = -X(b);
	/* UX(...) needed to avoid rejecting conversion of
	* -2^15 to a short. */
	if (x < 0 && Blk(UX(-x)) == b)
	return x;
	}
	// Otherwise fall through.
	}
	abort();
	}

	unsigned long BigInteger::toUnsignedLong () const { return convertToUnsignedPrimitive<unsigned long > (); }
	unsigned int BigInteger::toUnsignedInt () const { return convertToUnsignedPrimitive<unsigned int > (); }
	unsigned short BigInteger::toUnsignedShort() const { return convertToUnsignedPrimitive<unsigned short> (); }
	long BigInteger::toLong () const { return convertToSignedPrimitive <long , unsigned long> (); }
	int BigInteger::toInt () const { return convertToSignedPrimitive <int , unsigned int> (); }
	short BigInteger::toShort () const { return convertToSignedPrimitive <short, unsigned short>(); }

	// COMPARISON
	BigInteger::CmpRes BigInteger::compareTo(const BigInteger &x) const {
	// A greater sign implies a greater number
	if (sign < x.sign)
	return less;
	else if (sign > x.sign)
	return greater;
	else switch (sign) {
	// If the signs are the same...
	case zero:
	return equal; // Two zeros are equal
	case positive:
	// Compare the magnitudes
	return mag.compareTo(x.mag);
	case negative:
	// Compare the magnitudes, but return the opposite result
	return CmpRes(-mag.compareTo(x.mag));
	default:
	abort();
	}
	}

	/* COPY-LESS OPERATIONS
	* These do some messing around to determine the sign of the result,
	* then call one of BigUnsigned's copy-less operations. */

	// See remarks about aliased calls in BigUnsigned.cc .
	#define DTRT_ALIASED(cond, op) \
	if (cond) { \
	BigInteger tmpThis; \
	tmpThis.op; \
	*this = tmpThis; \
	return; \
	}

	void BigInteger::add(const BigInteger &a, const BigInteger &b) {
	DTRT_ALIASED(this == &a \|\| this == &b, add(a, b));
	// If one argument is zero, copy the other.
	if (a.sign == zero)
	operator =(b);
	else if (b.sign == zero)
	operator =(a);
	// If the arguments have the same sign, take the
	// common sign and add their magnitudes.
	else if (a.sign == b.sign) {
	sign = a.sign;
	mag.add(a.mag, b.mag);
	} else {
	// Otherwise, their magnitudes must be compared.
	switch (a.mag.compareTo(b.mag)) {
	case equal:
	// If their magnitudes are the same, copy zero.
	mag = 0;
	sign = zero;
	break;
	// Otherwise, take the sign of the greater, and subtract
	// the lesser magnitude from the greater magnitude.
	case greater:
	sign = a.sign;
	mag.subtract(a.mag, b.mag);
	break;
	case less:
	sign = b.sign;
	mag.subtract(b.mag, a.mag);
	break;
	}
	}
	}

	void BigInteger::subtract(const BigInteger &a, const BigInteger &b) {
	// Notice that this routine is identical to BigInteger::add,
	// if one replaces b.sign by its opposite.
	DTRT_ALIASED(this == &a \|\| this == &b, subtract(a, b));
	// If a is zero, copy b and flip its sign. If b is zero, copy a.
	if (a.sign == zero) {
	mag = b.mag;
	// Take the negative of _b_'s, sign, not ours.
	// Bug pointed out by Sam Larkin on 2005.03.30.
	sign = Sign(-b.sign);
	} else if (b.sign == zero)
	operator =(a);
	// If their signs differ, take a.sign and add the magnitudes.
	else if (a.sign != b.sign) {
	sign = a.sign;
	mag.add(a.mag, b.mag);
	} else {
	// Otherwise, their magnitudes must be compared.
	switch (a.mag.compareTo(b.mag)) {
	// If their magnitudes are the same, copy zero.
	case equal:
	mag = 0;
	sign = zero;
	break;
	// If a's magnitude is greater, take a.sign and
	// subtract a from b.
	case greater:
	sign = a.sign;
	mag.subtract(a.mag, b.mag);
	break;
	// If b's magnitude is greater, take the opposite
	// of b.sign and subtract b from a.
	case less:
	sign = Sign(-b.sign);
	mag.subtract(b.mag, a.mag);
	break;
	}
	}
	}

	void BigInteger::multiply(const BigInteger &a, const BigInteger &b) {
	DTRT_ALIASED(this == &a \|\| this == &b, multiply(a, b));
	// If one object is zero, copy zero and return.
	if (a.sign == zero \|\| b.sign == zero) {
	sign = zero;
	mag = 0;
	return;
	}
	// If the signs of the arguments are the same, the result
	// is positive, otherwise it is negative.
	sign = (a.sign == b.sign) ? positive : negative;
	// Multiply the magnitudes.
	mag.multiply(a.mag, b.mag);
	}

	/*
	* DIVISION WITH REMAINDER
	* Please read the comments before the definition of
	* `BigUnsigned::divideWithRemainder' in `BigUnsigned.cc' for lots of
	* information you should know before reading this function.
	*
	* Following Knuth, I decree that x / y is to be
	* 0 if y==0 and floor(real-number x / y) if y!=0.
	* Then x % y shall be x - y*(integer x / y).
	*
	* Note that x = y * (x / y) + (x % y) always holds.
	* In addition, (x % y) is from 0 to y - 1 if y > 0,
	* and from -(\|y\| - 1) to 0 if y < 0. (x % y) = x if y = 0.
	*
	* Examples: (q = a / b, r = a % b)
	* a b q r
	* === === === ===
	* 4 3 1 1
	* -4 3 -2 2
	* 4 -3 -2 -2
	* -4 -3 1 -1
	*/
	void BigInteger::divideWithRemainder(const BigInteger &b, BigInteger &q) {
	// Defend against aliased calls;
	// same idea as in BigUnsigned::divideWithRemainder .
	if (this == &q)
	abort();
	if (this == &b \|\| &q == &b) {
	BigInteger tmpB(b);
	divideWithRemainder(tmpB, q);
	return;
	}

	// Division by zero gives quotient 0 and remainder *this
	if (b.sign == zero) {
	q.mag = 0;
	q.sign = zero;
	return;
	}
	// 0 / b gives quotient 0 and remainder 0
	if (sign == zero) {
	q.mag = 0;
	q.sign = zero;
	return;
	}

	// Here *this != 0, b != 0.

	// Do the operands have the same sign?
	if (sign == b.sign) {
	// Yes: easy case. Quotient is zero or positive.
	q.sign = positive;
	} else {
	// No: harder case. Quotient is negative.
	q.sign = negative;
	// Decrease the magnitude of the dividend by one.
	mag--;
	/*
	* We tinker with the dividend before and with the
	* quotient and remainder after so that the result
	* comes out right. To see why it works, consider the following
	* list of examples, where A is the magnitude-decreased
	* a, Q and R are the results of BigUnsigned division
	* with remainder on A and \|b\|, and q and r are the
	* final results we want:
	*
	* a A b Q R q r
	* -3 -2 3 0 2 -1 0
	* -4 -3 3 1 0 -2 2
	* -5 -4 3 1 1 -2 1
	* -6 -5 3 1 2 -2 0
	*
	* It appears that we need a total of 3 corrections:
	* Decrease the magnitude of a to get A. Increase the
	* magnitude of Q to get q (and make it negative).
	* Find r = (b - 1) - R and give it the desired sign.
	*/
	}

	// Divide the magnitudes.
	mag.divideWithRemainder(b.mag, q.mag);

	if (sign != b.sign) {
	// More for the harder case (as described):
	// Increase the magnitude of the quotient by one.
	q.mag++;
	// Modify the remainder.
	mag.subtract(b.mag, mag);
	mag--;
	}

	// Sign of the remainder is always the sign of the divisor b.
	sign = b.sign;

	// Set signs to zero as necessary. (Thanks David Allen!)
	if (mag.isZero())
	sign = zero;
	if (q.mag.isZero())
	q.sign = zero;

	// WHEW!!!
	}

	// Negation
	void BigInteger::negate(const BigInteger &a) {
	DTRT_ALIASED(this == &a, negate(a));
	// Copy a's magnitude
	mag = a.mag;
	// Copy the opposite of a.sign
	sign = Sign(-a.sign);
	}

	// INCREMENT/DECREMENT OPERATORS

	// Prefix increment
	BigInteger& BigInteger::operator ++() {
	if (sign == negative) {
	mag--;
	if (mag == 0)
	sign = zero;
	} else {
	mag++;
	sign = positive; // if not already
	}
	return *this;
	}

	// Postfix increment
	BigInteger BigInteger::operator ++(int) {
	BigInteger temp(*this);
	operator ++();
	return temp;
	}

	// Prefix decrement
	BigInteger& BigInteger::operator --() {
	if (sign == positive) {
	mag--;
	if (mag == 0)
	sign = zero;
	} else {
	mag++;
	sign = negative;
	}
	return *this;
	}

	// Postfix decrement
	BigInteger BigInteger::operator --(int) {
	BigInteger temp(*this);
	operator --();
	return temp;
	}