Add 'fixnum' library to utils/

utils/fixnum/int64.dart

+// Copyright (c) 2012, the Dart project authors.  Please see the AUTHORS file
+// for details. All rights reserved. Use of this source code is governed by a
+// BSD-style license that can be found in the LICENSE file.
+
+/**
+ * An immutable 64-bit signed integer, in the range [-2^63, 2^63 - 1].
+ * Arithmetic operations may overflow in order to maintain this range.
+ */
+class int64 implements intx {
+  // Note: instances of int64 are immutable outside of this library,
+  // therefore we may return a reference to an existing instance.
+  // We take care to perform mutation only on internally-generated
+  // instances before they are exposed to external code.
+
+  // Note: several functions require _BITS == 22 -- do not change this value

[Mads Ager (google), 2012/05/15 07:37:39]

End comment with a period.

Also, I think it would be helpful if you added an overall comment about the
representation of a 64-bit integer here. I can deduce it from the below
constants but spelling it out might not be a bad idea.

I was looking for the fields actually representing the value. I expected to find
them either at the top or at the bottom of the file. Maybe we can move things
around to make it a bit easier to find fields, constructors, public methods,
internal methods?

+  static final int _BITS = 22;
+  static final int _BITS01 = 44; // 2 * BITS
+  static final int _BITS2 = 20; // 64 - BITS01
+  static final int _MASK = 4194303; // (1 << BITS) - 1
+  static final int _MASK_2 = 1048575; // (1 << BITS2) - 1
+  static final int _SIGN_BIT = 19; // BITS2 - 1
+  static final int _SIGN_BIT_VALUE = 524288; // 1 << SIGN_BIT
+
+  static int64 _remainder;
+
+  static int64 _MAX_VALUE;
+  static int64 _MIN_VALUE;
+  static int64 _ZERO;
+  static int64 _ONE;
+  static int64 _TWO;
+
+  /**
+   * The maximum positive value attainable by an [int64], namely
+   * 9,223,372,036,854,775,807.
+   */
+  static int64 get MAX_VALUE() {
+    if (_MAX_VALUE == null) {
+      _MAX_VALUE = new int64._bits(_MASK, _MASK, _MASK_2 >> 1);
+    }
+    return _MAX_VALUE;
+  }
+
+  /**
+   * The minimum positive value attainable by an [int64], namely
+   * -9,223,372,036,854,775,808.
+   */
+  static int64 get MIN_VALUE() {
+    if (_MIN_VALUE == null) {
+      _MIN_VALUE = new int64._bits(0, 0, _SIGN_BIT_VALUE);
+    }
+    return _MIN_VALUE;
+  }
+
+  /**
+   * An [int64] constant equal to 0.
+   */
+  static int64 get ZERO() {
+    if (_ZERO == null) {
+      _ZERO = new int64();
+    }
+    return _ZERO;
+  }
+
+  /**
+   * An [int64] constant equal to 1.
+   */
+  static int64 get ONE() {
+    if (_ONE == null) {
+      _ONE = new int64._bits(1, 0, 0);
+    }
+    return _ONE;
+  }
+
+  /**
+   * An [int64] constant equal to 2.
+   */
+  static int64 get TWO() {
+    if (_TWO == null) {
+      _TWO = new int64._bits(2, 0, 0);
+    }
+    return _TWO;
+  }
+
+  /**
+   * Parses a [String] in a given [radix] between 2 and 16 and returns an
+   * [int64].
+   */
+  // TODO(rice) - make this faster by converting several digits at once
+  static int64 parseRadix(String s, int radix) {
+    if ((radix <= 1) || (radix > 16)) {
+      throw "Bad radix: $radix";
+    }
+    int64 x = ZERO;
+    int i = 0;
+    bool negative = false;
+    if (s[0] == '-') {
+      negative = true;
+      i++;
+    }
+    for (; i < s.length; i++) {
+      int c = s.charCodeAt(i);
+      int digit = int32._decodeHex(c);
+      if (digit < 0 || digit >= radix) {
+        throw new Exception("Non-radix char code: $c");
+      }
+      x = (x * radix) + digit;
+    }
+    return negative ? -x : x;
+  }
+
+  /**
+   * Parses a decimal [String] and returns an [int64].
+   */
+  static int64 parseInt(String s) => parseRadix(s, 10);
+
+  /**
+   * Parses a hexadecimal [String] and returns an [int64].
+   */
+  static int64 parseHex(String s) => parseRadix(s, 16);
+
+  // Low, middle, and high bits.  _l and _m are in the range
+  // [0, 2^22 - 1] and _h is in the range [0, 2^20 - 1].
+  int _l, _m, _h;
+
+  /**
+   * Constructs an [int64] equal to 0.
+   */
+  int64() : _l = 0, _m = 0, _h = 0;
+
+  /**
+   * Constructs an [int64] with a given bitwise representation.  No validation
+   * is performed.
+   */
+  int64._bits(int this._l, int this._m, int this._h);
+
+  /**
+   * Constructs an [int64] with the same value as an existing [int64].
+   */
+  int64._copy(int64 other) {
+    _l = other._l;
+    _m = other._m;
+    _h = other._h;
+  }
+
+  // Determine whether the platform supports ints greater than 2^53
+  // without loss of precision.
+  static bool _haveBigIntsCached = null;
+
+  static bool get _haveBigInts() {
+    if (_haveBigIntsCached == null) {
+      var x = 9007199254740992;
+      var y = x + 1;
+      var same = y == x;
+      _haveBigIntsCached = !same;
+    }
+    return _haveBigIntsCached;
+  }
+
+  /**
+   * Constructs an [int64] with a given [int] value.
+   */
+  int64.fromInt(int value) {
+    bool negative = false;
+    if (value < 0) {
+      negative = true;
+      value = -value - 1;
+    }
+    if (_haveBigInts) {
+      _l = value & _MASK;
+      _m = (value >> _BITS) & _MASK;
+      _h = (value >> _BITS01) & _MASK_2;
+    } else {
+      // Avoid using bitwise operations that coerce their input to 32 bits
+      _h = value ~/ 17592186044416; // 2^44
+      value -= _h * 17592186044416;
+      _m = value ~/ 4194304; // 2^22
+      value -= _m * 4194304;
+      _l = value;
+    }
+
+    if (negative) {
+      _l = ~_l & _MASK;
+      _m = ~_m & _MASK;
+      _h = ~_h & _MASK_2;
+    }
+  }
+
+  factory int64.fromBytes(List<int> bytes) {
+    int top = bytes[7] & 0xff; 
+    top <<= 8;
+    top |= bytes[6] & 0xff;
+    top <<= 8;
+    top |= bytes[5] & 0xff;
+    top <<= 8;
+    top |= bytes[4] & 0xff;
+
+    int bottom = bytes[3] & 0xff; 
+    bottom <<= 8;
+    bottom |= bytes[2] & 0xff;
+    bottom <<= 8;
+    bottom |= bytes[1] & 0xff;
+    bottom <<= 8;
+    bottom |= bytes[0] & 0xff;
+
+    return new int64.fromInts(top, bottom);
+  }
+
+  factory int64.fromBytesBigEndian(List<int> bytes) {
+    int top = bytes[0] & 0xff; 
+    top <<= 8;
+    top |= bytes[1] & 0xff;
+    top <<= 8;
+    top |= bytes[2] & 0xff;
+    top <<= 8;
+    top |= bytes[3] & 0xff;
+
+    int bottom = bytes[4] & 0xff; 
+    bottom <<= 8;
+    bottom |= bytes[5] & 0xff;
+    bottom <<= 8;
+    bottom |= bytes[6] & 0xff;
+    bottom <<= 8;
+    bottom |= bytes[7] & 0xff;
+
+    return new int64.fromInts(top, bottom);
+ }
+
+  /**
+   * Constructs an [int64] from a pair of 32-bit integers having the value
+   * [:((top & 0xffffffff) << 32) | (bottom & 0xffffffff):].
+   */
+  int64.fromInts(int top, int bottom) {
+    top &= 0xffffffff;
+    bottom &= 0xffffffff;
+    _l = bottom & _MASK;
+    _m = ((top & 0xfff) << 10) | ((bottom >> _BITS) & 0x3ff);
+    _h = (top >> 12) & _MASK_2;
+  }
+
+  int64 _promote(other) {
+    if (other == null) {
+      throw new NullPointerException();
+    } else if (other is intx) {
+      other = other.toInt64();
+    } else if (other is int) {
+      other = new int64.fromInt(other);
+    }
+    if (other is !int64) {
+      throw new Exception("Can't promote $other to int64");
+    }
+    return other;
+  }
+
+  int64 operator +(other) {
+    int64 o = _promote(other);
+    int sum0 = _l + o._l;
+    int sum1 = _m + o._m + (sum0 >> _BITS);
+    int sum2 = _h + o._h + (sum1 >> _BITS);
+
+    int64 result = new int64._bits(sum0 & _MASK, sum1 & _MASK, sum2 & _MASK_2);
+    return result;
+  }
+
+  int64 operator -(other) {
+    int64 o = _promote(other);
+    int sum0 = _l - o._l;
+    int sum1 = _m - o._m + (sum0 >> _BITS);
+    int sum2 = _h - o._h + (sum1 >> _BITS);
+
+    return new int64._bits(sum0 & _MASK, sum1 & _MASK, sum2 & _MASK_2);
+  }
+
+  int64 operator negate() {
+    // Like 0 - this
+    int sum0 = -_l;
+    int sum1 = -_m + (sum0 >> _BITS);
+    int sum2 = -_h + (sum1 >> _BITS);
+
+    return new int64._bits(sum0 & _MASK, sum1 & _MASK, sum2 & _MASK_2);
+  }
+
+  int64 operator *(other) {
+    int64 o = _promote(other);
+    // Grab 13-bit chunks
+    int a0 = _l & 0x1fff;
+    int a1 = (_l >> 13) | ((_m & 0xf) << 9);
+    int a2 = (_m >> 4) & 0x1fff;
+    int a3 = (_m >> 17) | ((_h & 0xff) << 5);
+    int a4 = (_h & 0xfff00) >> 8;
+
+    int b0 = o._l & 0x1fff;
+    int b1 = (o._l >> 13) | ((o._m & 0xf) << 9);
+    int b2 = (o._m >> 4) & 0x1fff;
+    int b3 = (o._m >> 17) | ((o._h & 0xff) << 5);
+    int b4 = (o._h & 0xfff00) >> 8;
+
+    // Compute partial products
+    // Optimization: if b is small, avoid multiplying by parts that are 0
+    int p0 = a0 * b0; // << 0
+    int p1 = a1 * b0; // << 13
+    int p2 = a2 * b0; // << 26
+    int p3 = a3 * b0; // << 39
+    int p4 = a4 * b0; // << 52
+
+    if (b1 != 0) {
+      p1 += a0 * b1;
+      p2 += a1 * b1;
+      p3 += a2 * b1;
+      p4 += a3 * b1;
+    }
+    if (b2 != 0) {
+      p2 += a0 * b2;
+      p3 += a1 * b2;
+      p4 += a2 * b2;
+    }
+    if (b3 != 0) {
+      p3 += a0 * b3;
+      p4 += a1 * b3;
+    }
+    if (b4 != 0) {
+      p4 += a0 * b4;
+    }
+
+    // Accumulate into 22-bit chunks:
+    // .........................................c10|...................c00|
+    // |....................|..................xxxx|xxxxxxxxxxxxxxxxxxxxxx| p0
+    // |....................|......................|......................|
+    // |....................|...................c11|......c01.............|
+    // |....................|....xxxxxxxxxxxxxxxxxx|xxxxxxxxx.............| p1
+    // |....................|......................|......................|
+    // |.................c22|...............c12....|......................|
+    // |..........xxxxxxxxxx|xxxxxxxxxxxxxxxxxx....|......................| p2
+    // |....................|......................|......................|
+    // |.................c23|..c13.................|......................|
+    // |xxxxxxxxxxxxxxxxxxxx|xxxxx.................|......................| p3
+    // |....................|......................|......................|
+    // |.........c24........|......................|......................|
+    // |xxxxxxxxxxxx........|......................|......................| p4
+
+    int c00 = p0 & 0x3fffff;
+    int c01 = (p1 & 0x1ff) << 13;
+    int c0 = c00 + c01;
+
+    int c10 = p0 >> 22;
+    int c11 = p1 >> 9;
+    int c12 = (p2 & 0x3ffff) << 4;
+    int c13 = (p3 & 0x1f) << 17;
+    int c1 = c10 + c11 + c12 + c13;
+
+    int c22 = p2 >> 18;
+    int c23 = p3 >> 5;
+    int c24 = (p4 & 0xfff) << 8;
+    int c2 = c22 + c23 + c24;
+
+    // Propagate high bits from c0 -> c1, c1 -> c2
+    c1 += c0 >> _BITS;
+    c0 &= _MASK;
+    c2 += c1 >> _BITS;
+    c1 &= _MASK;
+    c2 &= _MASK_2;
+
+    return new int64._bits(c0, c1, c2);
+  }
+
+  int64 operator %(other) {
+    if (other.isZero()) {
+      throw new IntegerDivisionByZeroException();
+    }
+    if (this.isZero()) {
+      return ZERO;
+    }
+    int64 o = _promote(other).abs();
+    _divMod(this, o, true);
+    return _remainder < 0 ? (_remainder + o) : _remainder;
+  }
+
+  int64 operator ~/(other) => _divMod(this, _promote(other), false);
+
+  // int64 remainder(other) => this - (this ~/ other) * other;
+  int64 remainder(other) {
+    if (other.isZero()) {
+      throw new IntegerDivisionByZeroException();
+    }
+    int64 o = _promote(other).abs();
+    _divMod(this, o, true);
+    return _remainder;
+  }
+
+  int64 operator &(other) {
+    int64 o = _promote(other);
+    int a0 = _l & o._l;
+    int a1 = _m & o._m;
+    int a2 = _h & o._h;
+    return new int64._bits(a0, a1, a2);
+  }
+
+  int64 operator |(other) {
+    int64 o = _promote(other);
+    int a0 = _l | o._l;
+    int a1 = _m | o._m;
+    int a2 = _h | o._h;
+    return new int64._bits(a0, a1, a2);
+  }
+
+  int64 operator ^(other) {
+    int64 o = _promote(other);
+    int a0 = _l ^ o._l;
+    int a1 = _m ^ o._m;
+    int a2 = _h ^ o._h;
+    return new int64._bits(a0, a1, a2);
+  }
+
+  int64 operator ~() {
+    var result = new int64._bits((~_l) & _MASK, (~_m) & _MASK, (~_h) & _MASK_2);
+    return result;
+  }
+
+  int64 operator <<(int n) {
+    if (n < 0) {
+      throw new IllegalArgumentException("$n");
+    }
+    n &= 63;
+
+    int res0, res1, res2;
+    if (n < _BITS) {
+      res0 = _l << n;
+      res1 = (_m << n) | (_l >> (_BITS - n));
+      res2 = (_h << n) | (_m >> (_BITS - n));
+    } else if (n < _BITS01) {
+      res0 = 0;
+      res1 = _l << (n - _BITS);
+      res2 = (_m << (n - _BITS)) | (_l >> (_BITS01 - n));
+    } else {
+      res0 = 0;
+      res1 = 0;
+      res2 = _l << (n - _BITS01);
+    }
+
+    return new int64._bits(res0 & _MASK, res1 & _MASK, res2 & _MASK_2);
+  }
+
+  int64 operator >>(int n) {
+    if (n < 0) {
+      throw new IllegalArgumentException("$n");
+    }
+    n &= 63;
+
+    int res0, res1, res2;
+
+    // Sign extend h(a)
+    int a2 = _h;
+    bool negative = (a2 & _SIGN_BIT_VALUE) != 0;
+    if (negative) {
+      a2 |= ~_MASK_2;
+    }
+
+    if (n < _BITS) {
+      res2 = a2 >> n;
+      res1 = (_m >> n) | (a2 << (_BITS - n));
+      res0 = (_l >> n) | (_m << (_BITS - n));
+    } else if (n < _BITS01) {
+      res2 = negative ? _MASK_2 : 0;
+      res1 = a2 >> (n - _BITS);
+      res0 = (_m >> (n - _BITS)) | (a2 << (_BITS01 - n));
+    } else {
+      res2 = negative ? _MASK_2 : 0;
+      res1 = negative ? _MASK : 0;
+      res0 = a2 >> (n - _BITS01);
+    }
+
+    return new int64._bits(res0 & _MASK, res1 & _MASK, res2 & _MASK_2);
+  }
+
+  int64 shiftRightUnsigned(int n) {
+    if (n < 0) {
+      throw new IllegalArgumentException("$n");
+    }
+    n &= 63;
+
+    int res0, res1, res2;
+    int a2 = _h & _MASK_2;
+    if (n < _BITS) {
+      res2 = a2 >> n; // was >>>
+      res1 = (_m >> n) | (a2 << (_BITS - n));
+      res0 = (_l >> n) | (_m << (_BITS - n));
+    } else if (n < _BITS01) {
+      res2 = 0;
+      res1 = a2 >> (n - _BITS); // was >>>
+      res0 = (_m >> (n - _BITS)) | (_h << (_BITS01 - n));
+    } else {
+      res2 = 0;
+      res1 = 0;
+      res0 = a2 >> (n - _BITS01); // was >>>
+    }
+
+    return new int64._bits(res0 & _MASK, res1 & _MASK, res2 & _MASK_2);
+  }
+
+  /**
+   * Returns [true] if this [int64] has the same numeric value as the
+   * given object.  The argument may be an [int] or an [intx].
+   */
+  bool operator ==(other) {
+    if (other == null) {
+      return false;
+    }
+    int64 o = _promote(other);
+    return _l == o._l && _m == o._m && _h == o._h;
+  }
+
+  int compareTo(Comparable other) {
+    int64 o = _promote(other);
+    int signa = _h >> (_BITS2 - 1);
+    int signb = o._h >> (_BITS2 - 1);
+    if (signa != signb) {
+      return signa == 0 ? 1 : -1;
+    }
+    if (_h > o._h) {
+      return 1;
+    } else if (_h < o._h) {
+      return -1;
+    }
+    if (_m > o._m) {
+      return 1;
+    } else if (_m < o._m) {
+      return -1;
+    }
+    if (_l > o._l) {
+      return 1;
+    } else if (_l < o._l) {
+      return -1;
+    }
+    return 0;
+  }
+
+  bool operator <(other) {
+    return this.compareTo(other) < 0;
+  }
+
+  bool operator <=(other) {
+    return this.compareTo(other) <= 0;
+  }
+
+  bool operator >(other) {
+    return this.compareTo(other) > 0;
+  }
+
+  bool operator >=(other) {
+    return this.compareTo(other) >= 0;
+  }
+
+  bool isEven() => (_l & 0x1) == 0;
+  bool isMaxValue() => (_h == _MASK_2 >> 1) && _m == _MASK && _l == _MASK;
+  bool isMinValue() => _h == _SIGN_BIT_VALUE && _m == 0 && _l == 0;
+  bool isNegative() => (_h >> (_BITS2 - 1)) != 0;
+  bool isOdd() => (_l & 0x1) == 1;
+  bool isZero() => _h == 0 && _m == 0 && _l == 0;
+
+  /**
+   * Returns a hash code based on all the bits of this [int64].
+   */
+  int hashCode() {
+    int bottom = ((_m & 0x3ff) << 22) | _l;
+    int top = (_h << 12) | ((_m >> 10) & 0xfff);
+    return bottom ^ top;
+  }
+
+  int64 abs() {
+    return this < 0 ? -this : this;
+  }
+
+  /**
+   * Returns the number of leading zeros in this [int64] as an [int]
+   * between 0 and 64.
+   */
+  int numberOfLeadingZeros() {
+    int b2 = int32._numberOfLeadingZeros(_h);
+    if (b2 == 32) {
+      int b1 = int32._numberOfLeadingZeros(_m);
+      if (b1 == 32) {
+        return int32._numberOfLeadingZeros(_l) + 32;

[Mads Ager (google), 2012/05/15 07:37:39]

Should we either make the computation explicit here as below of just use
constants below as well?

+      } else {
+        return b1 + _BITS2 - (32 - _BITS);
+      }
+    } else {
+      return b2 - (32 - _BITS2);
+    }
+  }
+
+  /**
+   * Returns the number of trailing zeros in this [int64] as an [int]
+   * between 0 and 64.
+   */
+  int numberOfTrailingZeros() {
+    int zeros = int32._numberOfTrailingZeros(_l);
+    if (zeros < 32) {
+      return zeros;
+    }
+
+    zeros = int32._numberOfTrailingZeros(_m);
+    if (zeros < 32) {
+      return 22 + zeros;
+    }
+
+    zeros = int32._numberOfTrailingZeros(_h);
+    if (zeros < 32) {
+      return 44 + zeros;
+    }
+    // All zeros
+    return 64;
+  }
+
+  List<int> toBytes() {
+    List<int> result = new List<int>(8);
+    result[0] = _l & 0xff;
+    result[1] = (_l >> 8) & 0xff;
+    result[2] = ((_m << 6) & 0xfc) | ((_l >> 16) & 0x3f);
+    result[3] = (_m >> 2) & 0xff;
+    result[4] = (_m >> 10) & 0xff;
+    result[5] = ((_h << 4) & 0xf0) | ((_m >> 18) & 0xf);
+    result[6] = (_h >> 4) & 0xff;
+    result[7] = (_h >> 12) & 0xff;
+    return result;
+  }
+
+  int toInt() {
+    int l = _l;
+    int m = _m;
+    int h = _h;
+    bool negative = false;
+    if ((_h & _SIGN_BIT_VALUE) != 0) {
+      l = ~_l & _MASK;
+      m = ~_m & _MASK;
+      h = ~_h & _MASK_2;
+      negative = true;
+    }
+
+    int result;
+    if (_haveBigInts) {
+      result = (h << 44) | (m << 22) | l;
+    } else {
+      result = (h * 17592186044416) + (m * 4194304) + l;
+    }
+    return negative ? -result - 1 : result;
+  }
+
+  /**
+   * Returns an [int32] containing the low 32 bits of this [int64].
+   */
+  int32 toInt32() {
+    return new int32.fromInt(((_m & 0x3ff) << 22) | _l);
+  }
+
+  /**
+   * Returns [this].
+   */
+  int64 toInt64() => this;
+
+  /**
+   * Returns the value of this [int64] as a decimal [String].
+   */
+  // TODO(rice) - make this faster by converting several digits at once
+  String toString() {
+    int64 a = this;
+    if (a.isZero()) {
+      return "0";
+    }
+    if (a.isMinValue()) {
+      return "-9223372036854775808";
+    }
+
+    String result = "";
+    bool negative = false;
+    if (a.isNegative()) {
+      negative = true;
+      a = -a;
+    }
+
+    int64 ten = new int64._bits(10, 0, 0);
+    while (!a.isZero()) {
+      a = _divMod(a, ten, true);
+      result = "${_remainder._l}$result";
+    }
+    return negative ? "-$result" : result;
+  }
+
+  String _hexDigit(int digit) => "0123456789ABCDEF"[digit];
+
+  // TODO(rice) - make this faster by avoiding arithmetic
+  String toHexString() {
+    int64 x = new int64._copy(this);
+    if (isZero()) {
+      return "0";
+    }
+    String hexStr = "";
+    int64 digit_f = new int64.fromInt(0xf);
+    while (!x.isZero()) {
+      int digit = x._l & 0xf;
+      hexStr = "${_hexDigit(digit)}$hexStr";
+      x = x.shiftRightUnsigned(4);
+    }
+    return hexStr;
+  }
+
+  // Precompute the radix strings for MIN_VALUE to avoid the problem
+  // of overflow of -MIN_VALUE.
+  List<String> _minValues = const <String>[
+      null, null,
+      "-1000000000000000000000000000000000000000000000000000000000000000", // 2
+      "-2021110011022210012102010021220101220222", // base 3
+      "-20000000000000000000000000000000", // base 4
+      "-1104332401304422434310311213", // base 5
+      "-1540241003031030222122212", // base 6
+      "-22341010611245052052301", // base 7
+      "-1000000000000000000000", // base 8
+      "-67404283172107811828", // base 9
+      "-9223372036854775808", // base 10
+      "-1728002635214590698", // base 11
+      "-41A792678515120368", // base 12
+      "-10B269549075433C38", // base 13
+      "-4340724C6C71DC7A8", // base 14
+      "-160E2AD3246366808", // base 15
+      "-8000000000000000" // base 16
+  ];
+
+  String toRadixString(int radix) {
+    if ((radix <= 1) || (radix > 16)) {
+      throw "Bad radix: $radix";
+    }
+    int64 a = this;
+    if (a.isZero()) {
+      return "0";
+    }
+    if (a.isMinValue()) {
+      return _minValues[radix];
+    }
+
+    String result = "";
+    bool negative = false;
+    if (a.isNegative()) {
+      negative = true;
+      a = -a;
+    }
+
+    int64 r = new int64._bits(radix, 0, 0);
+    while (!a.isZero()) {
+      a = _divMod(a, r, true);
+      result = "${_hexDigit(_remainder._l)}$result";
+    }
+    return negative ? "-$result" : result;
+  }
+
+  // Implementation of ~/ and %
+
+  // Note: mutates [this]
+  void _negate() {
+    int neg0 = (~_l + 1) & _MASK;
+    int neg1 = (~_m + (neg0 == 0 ? 1 : 0)) & _MASK;
+    int neg2 = (~_h + ((neg0 == 0 && neg1 == 0) ? 1 : 0)) & _MASK_2;
+
+    _l = neg0;
+    _m = neg1;
+    _h = neg2;
+  }
+
+  // Note: mutates [this]
+  void _setBit(int bit) {
+    if (bit < _BITS) {
+      _l |= 0x1 << bit;
+    } else if (bit < _BITS01) {
+      _m |= 0x1 << (bit - _BITS);
+    } else {
+      _h |= 0x1 << (bit - _BITS01);
+    }
+  }
+
+  // Note: mutates [this]
+  void _toShru1() {
+    int a2 = _h;
+    int a1 = _m;
+    int a0 = _l;
+
+    _h = a2 >> 1;
+    _m = (a1 >> 1) | ((a2 & 0x1) << (_BITS - 1));
+    _l = (a0 >> 1) | ((a1 & 0x1) << (_BITS - 1));
+  }
+
+  /**
+   * Attempt to subtract b from a if a >= b:
+   *
+   * if (a >= b) {
+   *   a -= b;
+   *   return true;
+   * } else {
+   *   return false;
+   * }
+   */
+  // Note: mutates [a]
+  static bool _trialSubtract(int64 a, int64 b) {
+    // Early exit
+    int sum2 = a._h - b._h;
+    if (sum2 < 0) {
+      return false;
+    }
+
+    int sum0 = a._l - b._l;
+    int sum1 = a._m - b._m + (sum0 >> _BITS);
+    sum2 += (sum1 >> _BITS);
+
+    if (sum2 < 0) {
+      return false;
+    }
+
+    a._l = sum0 & _MASK;
+    a._m = sum1 & _MASK;
+    a._h = sum2 & _MASK_2;
+
+    return true;
+  }
+
+  // Note: mutates [a] via _trialSubtract
+  static int64 _divModHelper(int64 a, int64 b,
+    bool negative, bool aIsNegative, bool aIsMinValue,
+    bool computeRemainder) {
+
+    // Align the leading one bits of a and b by shifting b left
+    int shift = b.numberOfLeadingZeros() - a.numberOfLeadingZeros();
+    int64 bshift = b << shift;
+
+    // Quotient must be a new instance since we mutate it
+    int64 quotient = new int64();
+    while (shift >= 0) {
+      bool gte = _trialSubtract(a, bshift);
+      if (gte) {
+        quotient._setBit(shift);
+        if (a.isZero()) {
+          break;
+        }
+      }
+
+      bshift._toShru1();
+      shift--;
+    }
+
+    if (negative) {
+      quotient._negate();
+    }
+
+    if (computeRemainder) {
+      if (aIsNegative) {
+        _remainder = -a;
+        if (aIsMinValue) {
+          _remainder = _remainder - ONE;
+        }
+      } else {
+        _remainder = a;
+      }
+    }
+
+    return quotient;
+  }
+
+  int64 _divModByMinValue(bool computeRemainder) {
+    // MIN_VALUE / MIN_VALUE == 1, remainder = 0
+    // (x != MIN_VALUE) / MIN_VALUE == 0, remainder == x
+    if (isMinValue()) {
+      if (computeRemainder) {
+        _remainder = ZERO;
+      }
+      return ONE;
+    }
+    if (computeRemainder) {
+      _remainder = this;
+    }
+    return ZERO;
+  }
+
+  /**
+   * this &= ((1L << bits) - 1)
+   */
+  // Note: mutates [this]
+  int64 _maskRight(int bits) {
+    int b0, b1, b2;
+    if (bits <= _BITS) {
+      b0 = _l & ((1 << bits) - 1);
+      b1 = b2 = 0;
+    } else if (bits <= _BITS01) {
+      b0 = _l;
+      b1 = _m & ((1 << (bits - _BITS)) - 1);
+      b2 = 0;
+    } else {
+      b0 = _l;
+      b1 = _m;
+      b2 = _h & ((1 << (bits - _BITS01)) - 1);
+    }
+
+    _l = b0;
+    _m = b1;
+    _h = b2;
+  }
+
+  int64 _divModByShift(int64 a, int bpower, bool negative, bool aIsCopy,
+      bool aIsNegative, bool computeRemainder) {
+    int64 c = a >> bpower;
+    if (negative) {
+      c._negate();
+    }
+
+    if (computeRemainder) {
+      if (!aIsCopy) {
+        a = new int64._copy(a);
+      }
+      a._maskRight(bpower);
+      if (aIsNegative) {
+        a._negate();
+      }
+      _remainder = a;
+    }
+    return c;
+  }
+
+  /**
+   * Return the exact log base 2 of this, or -1 if this is not a power of two.
+   */
+  int _powerOfTwo() {
+    // Power of two or 0
+    int l = _l;
+    if ((l & (l - 1)) != 0) {
+      return -1;
+    }
+    int m = _m;
+    if ((m & (m - 1)) != 0) {
+      return -1;
+    }
+    int h = _h;
+    if ((h & (h - 1)) != 0) {
+      return -1;
+    }
+    if (h == 0 && m == 0 && l == 0) {
+      return -1;
+    }
+    if (h == 0 && m == 0 && l != 0) {
+      return int32._numberOfTrailingZeros(l);
+    }
+    if (h == 0 && m != 0 && l == 0) {
+      return int32._numberOfTrailingZeros(m) + _BITS;
+    }
+    if (h != 0 && m == 0 && l == 0) {
+      return int32._numberOfTrailingZeros(h) + _BITS01;
+    }
+
+    return -1;
+  }
+
+  int64 _divMod(int64 a, int64 b, bool computeRemainder) {
+    if (b.isZero()) {
+      throw new IntegerDivisionByZeroException();
+    }
+    if (a.isZero()) {
+      if (computeRemainder) {
+        _remainder = ZERO;
+      }
+      return ZERO;
+    }
+    // MIN_VALUE / MIN_VALUE = 1, anything other a / MIN_VALUE is 0
+    if (b.isMinValue()) {
+      return a._divModByMinValue(computeRemainder);
+    }
+    // Normalize b to abs(b), keeping track of the parity in 'negative'.
+    // We can do this because we have already ensured that b != MIN_VALUE.
+    bool negative = false;
+    if (b.isNegative()) {
+      b = -b;
+      negative = !negative;
+    }
+    // If b == 2^n, bpower will be n, otherwise it will be -1
+    int bpower = b._powerOfTwo();
+
+    // True if the original value of a is negative
+    bool aIsNegative = false;
+    // True if the original value of a is Long.MIN_VALUE
+    bool aIsMinValue = false;
+
+    /*
+     * Normalize a to a positive value, keeping track of the sign change in
+     * 'negative' (which tracks the sign of both a and b and is used to
+     * determine the sign of the quotient) and 'aIsNegative' (which is used to
+     * determine the sign of the remainder).
+     *
+     * For all values of a except MIN_VALUE, we can just negate a and modify
+     * negative and aIsNegative appropriately. When a == MIN_VALUE, negation is
+     * not possible without overflowing 64 bits, so instead of computing
+     * abs(MIN_VALUE) / abs(b) we compute (abs(MIN_VALUE) - 1) / abs(b). The
+     * only circumstance under which these quotients differ is when b is a power
+     * of two, which will divide abs(MIN_VALUE) == 2^64 exactly. In this case,
+     * we can get the proper result by shifting MIN_VALUE in unsigned fashion.
+     *
+     * We make a single copy of a before the first operation that needs to
+     * modify its value.
+     */
+    bool aIsCopy = false;
+    if (a.isMinValue()) {
+      aIsMinValue = true;
+      aIsNegative = true;
+      // If b is not a power of two, treat -a as MAX_VALUE (instead of the
+      // actual value (MAX_VALUE + 1)).
+      if (bpower == -1) {
+        a = new int64._copy(MAX_VALUE);
+        aIsCopy = true;
+        negative = !negative;
+      } else {
+        // Signed shift of MIN_VALUE produces the right answer
+        int64 c = a >> bpower;
+        if (negative) {
+          c._negate();
+        }
+        if (computeRemainder) {
+          _remainder = ZERO;
+        }
+        return c;
+      }
+    } else if (a.isNegative()) {
+      aIsNegative = true;
+      a = -a;
+      aIsCopy = true;
+      negative = !negative;
+    }
+
+    // Now both a and b are non-negative
+    // If b is a power of two, just shift
+    if (bpower != -1) {
+      return _divModByShift(a, bpower, negative, aIsCopy, aIsNegative,
+        computeRemainder);
+    }
+
+    // if a < b, the quotient is 0 and the remainder is a
+    if (a < b) {
+      if (computeRemainder) {
+        if (aIsNegative) {
+          _remainder = -a;
+        } else {
+          _remainder = aIsCopy ? a : new int64._copy(a);
+        }
+      }
+      return ZERO;
+    }
+
+    // Generate the quotient using bit-at-a-time long division
+    return _divModHelper(aIsCopy ? a : new int64._copy(a), b, negative,
+        aIsNegative, aIsMinValue, computeRemainder);
+  }
+}