JavaScriptでbig-integerでできること

目次

1. 概要
2. 定数
3. メソッド
4. 静的メソッド
5. モジュールの利用

概要

JavScriptのモジュールbig-integerを使ってできることのメモ

https://www.npmjs.com/package/big-integer

定数

bigInt.one; // bigInt(1)に相当
bigInt.zero; // bigInt(0)に相当
bigInt.minusOne; // bigInt(-1)に相当

メソッド

abs

絶対値

bigInt(-45).abs(); // 45
bigInt(45).abs(); // 45

add, plus

加算

bigInt(5).add(7); // 12
bigInt(5).plus(7); // 12

and

ビット単位のand演算を行う。

bigInt(6).and(3); // 2
bigInt(6).and(-3); // 4

bitLength

数値をバイナリで表すのに必要な桁数

bigInt(5).bitLength(); // 3（5 101は3桁のバイナリであるため）

compare compareTo

数値の比較。compareToはcompareのエイリアス

bigInt(5).compare(5); // 5 == 5 => 0
bigInt(5).compare(4); // 5 > 4 => 1
bigInt(4).compare(5); // 4 < 5 => -1

compareAmb

絶対値としての比較

bigInt(5).compareAbs(-5); // 0
bigInt(5).compareAbs(4); // 1
bigInt(4).compareAbs(-5); // -1

divide, over

整数の除算

bigInt(59).divide(5); // 11
bigInt(59).over(5); // 11

divmod

除算を実行し、「除算の結果」と「あまり」を返す。

bigInt(59).divmod(5); // {quotient: bigInt(11), remainder: bigInt(4) }
bigInt(-5).divmod(2); // {quotient: bigInt(-2),

equals, eq

2つの数値が等しいか。eqはequalsのエイリアス

bigInt(5).equals(5); // true
bigInt(4).equals(7); // false

greater, gt

最初の数値が2番目の数値より大きいかどうか計算。同じ値の場合はfalse。gtはエイリアス

bigInt(5).greater(6); // false
bigInt(5).greater(5); // false
bigInt(5).greater(4); // true

greaterOrEquals, geq

最初の数値が2番目の数値より大きいかどうか判定。同じ値の場合はtrue。geqはエイリアス

bigInt(5).greaterOrEquals(6); // false
bigInt(5).greaterOrEquals(5); // true
bigInt(5).greaterOrEquals(4); // true

isDivisibleBy

割り切れるかどうか

bigInt(999).isDivisibleBy(333); // true
bigInt(99).isDivisibleBy(5); // false

isEven

偶数判定

bigInt(6).isEven(); // true
bigInt(3).isEven(); // false

isNegative

負の数の場合trueを返す

bigInt(-23).isNegative(); // true
bigInt(50).isNegative(); // false

isOdd

奇数判定

bigInt(13).isOdd(); // true
bigInt(40).isOdd(); // false

isPositive

正の整数か

bigInt(54).isPositive(); // true
bigInt(-1).isPositive(); // false

isPrime

素数判定

bigInt(5).isPrime(); // true
bigInt(6).isPrime(); // false

isProbablePrime

数値が素数である可能性が高い時にtrueを返す。出なければfalse。ミラーラビンの素数判定によって計算される

bigInt(5).isProbablePrime(); // true
bigInt(49).isProbablePrime(); // false
bigInt(1729).isProbablePrime(); // false
bigInt(1729).isProbablePrime(1, () => 0.1); // false
bigInt(1729).isProbablePrime(1, () => 0.2); // true

isUnit

数値が1 もしくは -1 のときにtrueを返す

bigInt.one.isUnit(); // true
bigInt.minusOne.isUnit(); // true
bigInt(5).isUnit(); // false

isZero

数値が0のときにtrueを返す

bigInt.zero.isZero(); // true
bigInt('-0').isZero(); // true
bigInt(50).isZero(); // false

lesser, lt

最初の数値が2番目の数値より小さいかどうか判定。ltはエイリアス

bigInt(5).lesser(6); // true
bigInt(5).lesser(5); // false
bigInt(5).lesser(4); // false

lesserOrEquals, leq

最初の数値が2番目の数値以下かどうか判定。leqはエイリアス

bigInt(5).lesserOrEquals(6); // true
bigInt(5).lesserOrEquals(5); // true
bigInt(5).lesserOrEquals(4); // false

minus, subtract

減算をする

bigInt(3).minus(5); // -2
bigInt(3).subtract(5); // -2

mod, remaider

除算を実行し、商を無視して剰余​​を返します。remaiderはmodのエイリアス。

bigInt(59).mod(5); // 4
bigInt(-5).mod(2); // -1

modInv

モジュラ逆数(乗法)を求める

bigInt(3).modInv(11); // 4
bigInt(42).modInv(2017); // 1969

modPow

冪剰余を求める

bigInt(10).modPow(3, 30); // 10

multiply, times

乗算

bigInt(111).multiply(111); // 12321
bigInt(111).times(111); // 12321

next

1つ加算する

bigInt(6).next(); // 7

not

ビット単位のnot演算を実行する

bigInt(10).not(); // -11
bigInt(0).not(); // -1

notEquals, neq

2つの数値が等しくないかどうかを判定。neqはエイリアス

bigInt(5).notEquals(5); // false
bigInt(4).notEquals(7); // true

or

ビット単位のOR演算を実行します

bigInt(13).or(10); // 15
bigInt(13).or(-8); // -3

pow

べき乗を実行します。指数が0より小さい場合、0を返します。

bigInt(16).pow(16); // 18446744073709551616

prev

数値から1を引く

bigInt(6).prev(); // 5

shiftLeft

左シフト

bigInt(8).shiftLeft(2); // 32
bigInt(8).shiftLeft(-2); // 2

shiftRight

右シフト

bigInt(8).shiftRight(2); // 2
bigInt(8).shiftRight(-2); // 32

square

2乗する

bigInt(3).square(); // 9

toArray

プロパティ「value」および「isNegative」を持つbigIntオブジェクトに変換します。「value」は、指定された基数を法とする整数の配列です。「isNegative」は、結果の符号を表すbool値です。

bigInt('1e9').toArray(10); // { value: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], isNegative: false }
bigInt('1e9').toArray(16); // { value: [3, 11, 9, 10, 12, 10, 0, 0], isNegative: false }
bigInt(567890).toArray(100); // { value: [56, 78, 90], isNegative: false }
bigInt(12345).toArray(-10); // { value: [2, 8, 4, 6, 5], isNegative: false }
bigInt(-15).toArray(1); // { value: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], isNegative: true }
bigInt(-15).toArray(-1); // { value: [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], isNegative: false }
bigInt(0).toArray(0); // { value: [0], isNegative: false }
bigInt(1).toArray(0); // Error: Cannot convert nonzero numbers to base 0.

toJsNumber

bigIntオブジェクトをJSの数値型に変換します。ただし、〜-9007199254740992、9007199254740992〜は精度を失います。

bigInt('18446744073709551616').toJSNumber(); // 18446744073709552000

xor

ビット単位のxorを計算します。

bigInt(12).xor(5); // 9
bigInt(12).xor(-5); // -9

静的メソッド

fromArray

配列からbigIntオブジェクトを生成する。第二引数で基数を指定できる。第三引数で正負を選択する。

bigInt.fromArray([1, 2, 3, 4, 5], 10); // 12345
bigInt.fromArray([1, 0, 0], 2, true); // -4

gcd

最大公約数の計算

bigInt.gcd(42, 56); // 14

isInstance

bigIntmオブジェクトかの判定

bigInt.isInstance(bigInt(14)); // true
bigInt.isInstance(14); // false

lcm

最小公倍数の計算

bigInt.lcm(21, 6); // 42

max

最大値の計算

bigInt.max(77, 432); // 432

min

最小値の計算

bigInt.min(77, 432); // 77

randBetween

乱数の生成

モジュールの利用

タグ

<script src="https://peterolson.github.io/BigInteger.js/BigInteger.min.js"></script>

npm

npm install big-integer

ワンライナー

var bigInt = function(undefined) {
  'use strict';
  var BASE = 1e7,
    LOG_BASE = 7,
    MAX_INT = 9007199254740992,
    MAX_INT_ARR = smallToArray(MAX_INT),
    LOG_MAX_INT = Math.log(MAX_INT);
  function Integer(v, radix) {
    if (typeof v === 'undefined') return Integer[0];
    if (typeof radix !== 'undefined') {
      return +radix === 10
        ? parseValue(v)
        : parseBase(v, radix);
    }
    return parseValue(v);
  }
  function BigInteger(value, sign) {
    this.value = value;
    this.sign = sign;
    this.isSmall = false;
  }
  BigInteger.prototype = Object.create(Integer.prototype);
  function SmallInteger(value) {
    this.value = value;
    this.sign = value < 0;
    this.isSmall = true;
  }
  SmallInteger.prototype = Object.create(Integer.prototype);
  function isPrecise(n) {
    return -MAX_INT < n && n < MAX_INT;
  }
  function smallToArray(n) {
    if (n < 1e7) return [n];
    if (n < 1e14) return [n % 1e7, Math.floor(n / 1e7)];
    return [n % 1e7, Math.floor(n / 1e7) % 1e7, Math.floor(n / 1e14)];
  }
  function arrayToSmall(arr) {
    trim(arr);
    var length = arr.length;
    if (length < 4 && compareAbs(arr, MAX_INT_ARR) < 0) {
      switch (length) {
        case 0:
          return 0;
        case 1:
          return arr[0];
        case 2:
          return arr[0] + arr[1] * BASE;
        default:
          return arr[0] + (arr[1] + arr[2] * BASE) * BASE;
      }
    }
    return arr;
  }
  function trim(v) {
    var i = v.length;
    while (v[--i] === 0);
    v.length = i + 1;
  }
  function createArray(length) {
    var x = new Array(length);
    var i = -1;
    while (++i < length) x[i] = 0;
    return x;
  }
  function truncate(n) {
    if (n > 0) return Math.floor(n);
    return Math.ceil(n);
  }
  function add(a, b) {
    var l_a = a.length,
      l_b = b.length,
      r = new Array(l_a),
      carry = 0,
      base = BASE,
      sum,
      i;
    for (i = 0; i < l_b; i++) {
      sum = a[i] + b[i] + carry;
      carry = sum >= base ? 1 : 0;
      r[i] = sum - carry * base;
    }
    while (i < l_a) {
      sum = a[i] + carry;
      carry = sum === base ? 1 : 0;
      r[i++] = sum - carry * base;
    }
    if (carry > 0) r.push(carry);
    return r;
  }
  function addAny(a, b) {
    if (a.length >= b.length) return add(a, b);
    return add(b, a);
  }
  function addSmall(a, carry) {
    var l = a.length, r = new Array(l), base = BASE, sum, i;
    for (i = 0; i < l; i++) {
      sum = a[i] - base + carry;
      carry = Math.floor(sum / base);
      r[i] = sum - carry * base;
      carry += 1;
    }
    while (carry > 0) {
      r[i++] = carry % base;
      carry = Math.floor(carry / base);
    }
    return r;
  }
  BigInteger.prototype.add = function(v) {
    var n = parseValue(v);
    if (this.sign !== n.sign) return this.subtract(n.negate());
    var a = this.value, b = n.value;
    if (n.isSmall) return new BigInteger(addSmall(a, Math.abs(b)), this.sign);
    return new BigInteger(addAny(a, b), this.sign);
  };
  BigInteger.prototype.plus = BigInteger.prototype.add;
  SmallInteger.prototype.add = function(v) {
    var n = parseValue(v);
    var a = this.value;
    if (a < 0 !== n.sign) return this.subtract(n.negate());
    var b = n.value;
    if (n.isSmall) {
      if (isPrecise(a + b)) return new SmallInteger(a + b);
      b = smallToArray(Math.abs(b));
    }
    return new BigInteger(addSmall(b, Math.abs(a)), a < 0);
  };
  SmallInteger.prototype.plus = SmallInteger.prototype.add;
  function subtract(a, b) {
    var a_l = a.length,
      b_l = b.length,
      r = new Array(a_l),
      borrow = 0,
      base = BASE,
      i,
      difference;
    for (i = 0; i < b_l; i++) {
      difference = a[i] - borrow - b[i];
      if (difference < 0) {
        difference += base;
        borrow = 1;
      } else borrow = 0;
      r[i] = difference;
    }
    for (i = b_l; i < a_l; i++) {
      difference = a[i] - borrow;
      if (difference < 0) difference += base;
      else {
        r[i++] = difference;
        break;
      }
      r[i] = difference;
    }
    for (; i < a_l; i++) r[i] = a[i];
    trim(r);
    return r;
  }
  function subtractAny(a, b, sign) {
    var value;
    if (compareAbs(a, b) >= 0) value = subtract(a, b);
    else {
      value = subtract(b, a);
      sign = !sign;
    }
    value = arrayToSmall(value);
    if (typeof value === 'number') {
      if (sign) value = -value;
      return new SmallInteger(value);
    }
    return new BigInteger(value, sign);
  }
  function subtractSmall(a, b, sign) {
    var l = a.length, r = new Array(l), carry = -b, base = BASE, i, difference;
    for (i = 0; i < l; i++) {
      difference = a[i] + carry;
      carry = Math.floor(difference / base);
      difference %= base;
      r[i] = difference < 0 ? difference + base : difference;
    }
    r = arrayToSmall(r);
    if (typeof r === 'number') {
      if (sign) r = -r;
      return new SmallInteger(r);
    }
    return new BigInteger(r, sign);
  }
  BigInteger.prototype.subtract = function(v) {
    var n = parseValue(v);
    if (this.sign !== n.sign) return this.add(n.negate());
    var a = this.value, b = n.value;
    if (n.isSmall) return subtractSmall(a, Math.abs(b), this.sign);
    return subtractAny(a, b, this.sign);
  };
  BigInteger.prototype.minus = BigInteger.prototype.subtract;
  SmallInteger.prototype.subtract = function(v) {
    var n = parseValue(v);
    var a = this.value;
    if (a < 0 !== n.sign) return this.add(n.negate());
    var b = n.value;
    if (n.isSmall) return new SmallInteger(a - b);
    return subtractSmall(b, Math.abs(a), a >= 0);
  };
  SmallInteger.prototype.minus = SmallInteger.prototype.subtract;
  BigInteger.prototype.negate = function() {
    return new BigInteger(this.value, !this.sign);
  };
  SmallInteger.prototype.negate = function() {
    var sign = this.sign;
    var small = new SmallInteger(-this.value);
    small.sign = !sign;
    return small;
  };
  BigInteger.prototype.abs = function() {
    return new BigInteger(this.value, false);
  };
  SmallInteger.prototype.abs = function() {
    return new SmallInteger(Math.abs(this.value));
  };
  function multiplyLong(a, b) {
    var a_l = a.length,
      b_l = b.length,
      l = a_l + b_l,
      r = createArray(l),
      base = BASE,
      product,
      carry,
      i,
      a_i,
      b_j;
    for (i = 0; i < a_l; ++i) {
      a_i = a[i];
      for (var j = 0; j < b_l; ++j) {
        b_j = b[j];
        product = a_i * b_j + r[i + j];
        carry = Math.floor(product / base);
        r[i + j] = product - carry * base;
        r[i + j + 1] += carry;
      }
    }
    trim(r);
    return r;
  }
  function multiplySmall(a, b) {
    var l = a.length, r = new Array(l), base = BASE, carry = 0, product, i;
    for (i = 0; i < l; i++) {
      product = a[i] * b + carry;
      carry = Math.floor(product / base);
      r[i] = product - carry * base;
    }
    while (carry > 0) {
      r[i++] = carry % base;
      carry = Math.floor(carry / base);
    }
    return r;
  }
  function shiftLeft(x, n) {
    var r = [];
    while (n-- > 0) r.push(0);
    return r.concat(x);
  }
  function multiplyKaratsuba(x, y) {
    var n = Math.max(x.length, y.length);
    if (n <= 30) return multiplyLong(x, y);
    n = Math.ceil(n / 2);
    var b = x.slice(n), a = x.slice(0, n), d = y.slice(n), c = y.slice(0, n);
    var ac = multiplyKaratsuba(a, c),
      bd = multiplyKaratsuba(b, d),
      abcd = multiplyKaratsuba(addAny(a, b), addAny(c, d));
    var product = addAny(
      addAny(ac, shiftLeft(subtract(subtract(abcd, ac), bd), n)),
      shiftLeft(bd, 2 * n),
    );
    trim(product);
    return product;
  }
  function useKaratsuba(l1, l2) {
    return -.012 * l1 - .012 * l2 + 15e-6 * l1 * l2 > 0;
  }
  BigInteger.prototype.multiply = function(v) {
    var n = parseValue(v),
      a = this.value,
      b = n.value,
      sign = this.sign !== n.sign,
      abs;
    if (n.isSmall) {
      if (b === 0) return Integer[0];
      if (b === 1) return this;
      if (b === -1) return this.negate();
      abs = Math.abs(b);
      if (abs < BASE) return new BigInteger(multiplySmall(a, abs), sign);
      b = smallToArray(abs);
    }
    if (useKaratsuba(a.length, b.length)) {
      return new BigInteger(multiplyKaratsuba(a, b), sign);
    }
    return new BigInteger(multiplyLong(a, b), sign);
  };
  BigInteger.prototype.times = BigInteger.prototype.multiply;
  function multiplySmallAndArray(a, b, sign) {
    if (a < BASE) return new BigInteger(multiplySmall(b, a), sign);
    return new BigInteger(multiplyLong(b, smallToArray(a)), sign);
  }
  SmallInteger.prototype._multiplyBySmall = function(a) {
    if (isPrecise(a.value * this.value)) {
      return new SmallInteger(a.value * this.value);
    }
    return multiplySmallAndArray(
      Math.abs(a.value),
      smallToArray(Math.abs(this.value)),
      this.sign !== a.sign,
    );
  };
  BigInteger.prototype._multiplyBySmall = function(a) {
    if (a.value === 0) return Integer[0];
    if (a.value === 1) return this;
    if (a.value === -1) return this.negate();
    return multiplySmallAndArray(
      Math.abs(a.value),
      this.value,
      this.sign !== a.sign,
    );
  };
  SmallInteger.prototype.multiply = function(v) {
    return parseValue(v)._multiplyBySmall(this);
  };
  SmallInteger.prototype.times = SmallInteger.prototype.multiply;
  function square(a) {
    var l = a.length,
      r = createArray(l + l),
      base = BASE,
      product,
      carry,
      i,
      a_i,
      a_j;
    for (i = 0; i < l; i++) {
      a_i = a[i];
      carry = 0 - a_i * a_i;
      for (var j = i; j < l; j++) {
        a_j = a[j];
        product = 2 * (a_i * a_j) + r[i + j] + carry;
        carry = Math.floor(product / base);
        r[i + j] = product - carry * base;
      }
      r[i + l] = carry;
    }
    trim(r);
    return r;
  }
  BigInteger.prototype.square = function() {
    return new BigInteger(square(this.value), false);
  };
  SmallInteger.prototype.square = function() {
    var value = this.value * this.value;
    if (isPrecise(value)) return new SmallInteger(value);
    return new BigInteger(square(smallToArray(Math.abs(this.value))), false);
  };
  function divMod1(a, b) {
    var a_l = a.length,
      b_l = b.length,
      base = BASE,
      result = createArray(b.length),
      divisorMostSignificantDigit = b[b_l - 1],
      lambda = Math.ceil(base / (2 * divisorMostSignificantDigit)),
      remainder = multiplySmall(a, lambda),
      divisor = multiplySmall(b, lambda),
      quotientDigit,
      shift,
      carry,
      borrow,
      i,
      l,
      q;
    if (remainder.length <= a_l) remainder.push(0);
    divisor.push(0);
    divisorMostSignificantDigit = divisor[b_l - 1];
    for (shift = a_l - b_l; shift >= 0; shift--) {
      quotientDigit = base - 1;
      if (remainder[shift + b_l] !== divisorMostSignificantDigit) {
        quotientDigit = Math.floor(
          (remainder[shift + b_l] * base + remainder[shift + b_l - 1])
            / divisorMostSignificantDigit,
        );
      }
      carry = 0;
      borrow = 0;
      l = divisor.length;
      for (i = 0; i < l; i++) {
        carry += quotientDigit * divisor[i];
        q = Math.floor(carry / base);
        borrow += remainder[shift + i] - (carry - q * base);
        carry = q;
        if (borrow < 0) {
          remainder[shift + i] = borrow + base;
          borrow = -1;
        } else {
          remainder[shift + i] = borrow;
          borrow = 0;
        }
      }
      while (borrow !== 0) {
        quotientDigit -= 1;
        carry = 0;
        for (i = 0; i < l; i++) {
          carry += remainder[shift + i] - base + divisor[i];
          if (carry < 0) {
            remainder[shift + i] = carry + base;
            carry = 0;
          } else {
            remainder[shift + i] = carry;
            carry = 1;
          }
        }
        borrow += carry;
      }
      result[shift] = quotientDigit;
    }
    remainder = divModSmall(remainder, lambda)[0];
    return [arrayToSmall(result), arrayToSmall(remainder)];
  }
  function divMod2(a, b) {
    var a_l = a.length,
      b_l = b.length,
      result = [],
      part = [],
      base = BASE,
      guess,
      xlen,
      highx,
      highy,
      check;
    while (a_l) {
      part.unshift(a[--a_l]);
      trim(part);
      if (compareAbs(part, b) < 0) {
        result.push(0);
        continue;
      }
      xlen = part.length;
      highx = part[xlen - 1] * base + part[xlen - 2];
      highy = b[b_l - 1] * base + b[b_l - 2];
      if (xlen > b_l) highx = (highx + 1) * base;
      guess = Math.ceil(highx / highy);
      do {
        check = multiplySmall(b, guess);
        if (compareAbs(check, part) <= 0) break;
        guess--;
      } while (guess);
      result.push(guess);
      part = subtract(part, check);
    }
    result.reverse();
    return [arrayToSmall(result), arrayToSmall(part)];
  }
  function divModSmall(value, lambda) {
    var length = value.length,
      quotient = createArray(length),
      base = BASE,
      i,
      q,
      remainder,
      divisor;
    remainder = 0;
    for (i = length - 1; i >= 0; --i) {
      divisor = remainder * base + value[i];
      q = truncate(divisor / lambda);
      remainder = divisor - q * lambda;
      quotient[i] = q | 0;
    }
    return [quotient, remainder | 0];
  }
  function divModAny(self, v) {
    var value, n = parseValue(v);
    var a = self.value, b = n.value;
    var quotient;
    if (b === 0) throw new Error('Cannot divide by zero');
    if (self.isSmall) {
      if (n.isSmall) {
        return [new SmallInteger(truncate(a / b)), new SmallInteger(a % b)];
      }
      return [Integer[0], self];
    }
    if (n.isSmall) {
      if (b === 1) return [self, Integer[0]];
      if (b == -1) return [self.negate(), Integer[0]];
      var abs = Math.abs(b);
      if (abs < BASE) {
        value = divModSmall(a, abs);
        quotient = arrayToSmall(value[0]);
        var remainder = value[1];
        if (self.sign) remainder = -remainder;
        if (typeof quotient === 'number') {
          if (self.sign !== n.sign) quotient = -quotient;
          return [new SmallInteger(quotient), new SmallInteger(remainder)];
        }
        return [
          new BigInteger(quotient, self.sign !== n.sign),
          new SmallInteger(remainder),
        ];
      }
      b = smallToArray(abs);
    }
    var comparison = compareAbs(a, b);
    if (comparison === -1) return [Integer[0], self];
    if (comparison === 0) {
      return [Integer[self.sign === n.sign ? 1 : -1], Integer[0]];
    }
    if (a.length + b.length <= 200) value = divMod1(a, b);
    else value = divMod2(a, b);
    quotient = value[0];
    var qSign = self.sign !== n.sign, mod = value[1], mSign = self.sign;
    if (typeof quotient === 'number') {
      if (qSign) quotient = -quotient;
      quotient = new SmallInteger(quotient);
    } else quotient = new BigInteger(quotient, qSign);
    if (typeof mod === 'number') {
      if (mSign) mod = -mod;
      mod = new SmallInteger(mod);
    } else mod = new BigInteger(mod, mSign);
    return [quotient, mod];
  }
  BigInteger.prototype.divmod = function(v) {
    var result = divModAny(this, v);
    return { quotient: result[0], remainder: result[1] };
  };
  SmallInteger.prototype.divmod = BigInteger.prototype.divmod;
  BigInteger.prototype.divide = function(v) {
    return divModAny(this, v)[0];
  };
  SmallInteger.prototype.over =
    SmallInteger.prototype.divide =
    BigInteger
      .prototype.over =
      BigInteger.prototype.divide;
  BigInteger.prototype.mod = function(v) {
    return divModAny(this, v)[1];
  };
  SmallInteger.prototype.remainder =
    SmallInteger.prototype.mod =
    BigInteger
      .prototype.remainder =
      BigInteger.prototype.mod;
  BigInteger.prototype.pow = function(v) {
    var n = parseValue(v), a = this.value, b = n.value, value, x, y;
    if (b === 0) return Integer[1];
    if (a === 0) return Integer[0];
    if (a === 1) return Integer[1];
    if (a === -1) return n.isEven() ? Integer[1] : Integer[-1];
    if (n.sign) return Integer[0];
    if (!n.isSmall) {
      throw new Error('The exponent ' + n.toString() + ' is too large.');
    }
    if (this.isSmall) {
      if (isPrecise(value = Math.pow(a, b))) {
        return new SmallInteger(truncate(value));
      }
    }
    x = this;
    y = Integer[1];
    while (true) {
      if (b & 1 === 1) {
        y = y.times(x);
        --b;
      }
      if (b === 0) break;
      b /= 2;
      x = x.square();
    }
    return y;
  };
  SmallInteger.prototype.pow = BigInteger.prototype.pow;
  BigInteger.prototype.modPow = function(exp, mod) {
    exp = parseValue(exp);
    mod = parseValue(mod);
    if (mod.isZero()) throw new Error('Cannot take modPow with modulus 0');
    var r = Integer[1], base = this.mod(mod);
    while (exp.isPositive()) {
      if (base.isZero()) return Integer[0];
      if (exp.isOdd()) r = r.multiply(base).mod(mod);
      exp = exp.divide(2);
      base = base.square().mod(mod);
    }
    return r;
  };
  SmallInteger.prototype.modPow = BigInteger.prototype.modPow;
  function compareAbs(a, b) {
    if (a.length !== b.length) return a.length > b.length ? 1 : -1;
    for (var i = a.length - 1; i >= 0; i--) {
      if (a[i] !== b[i]) return a[i] > b[i] ? 1 : -1;
    }
    return 0;
  }
  BigInteger.prototype.compareAbs = function(v) {
    var n = parseValue(v), a = this.value, b = n.value;
    if (n.isSmall) return 1;
    return compareAbs(a, b);
  };
  SmallInteger.prototype.compareAbs = function(v) {
    var n = parseValue(v), a = Math.abs(this.value), b = n.value;
    if (n.isSmall) {
      b = Math.abs(b);
      return a === b ? 0 : a > b ? 1 : -1;
    }
    return -1;
  };
  BigInteger.prototype.compare = function(v) {
    if (v === Infinity) return -1;
    if (v === -Infinity) return 1;
    var n = parseValue(v), a = this.value, b = n.value;
    if (this.sign !== n.sign) return n.sign ? 1 : -1;
    if (n.isSmall) return this.sign ? -1 : 1;
    return compareAbs(a, b) * (this.sign ? -1 : 1);
  };
  BigInteger.prototype.compareTo = BigInteger.prototype.compare;
  SmallInteger.prototype.compare = function(v) {
    if (v === Infinity) return -1;
    if (v === -Infinity) return 1;
    var n = parseValue(v), a = this.value, b = n.value;
    if (n.isSmall) return a == b ? 0 : a > b ? 1 : -1;
    if (a < 0 !== n.sign) return a < 0 ? -1 : 1;
    return a < 0 ? 1 : -1;
  };
  SmallInteger.prototype.compareTo = SmallInteger.prototype.compare;
  BigInteger.prototype.equals = function(v) {
    return this.compare(v) === 0;
  };
  SmallInteger.prototype.eq =
    SmallInteger.prototype.equals =
    BigInteger
      .prototype.eq =
      BigInteger.prototype.equals;
  BigInteger.prototype.notEquals = function(v) {
    return this.compare(v) !== 0;
  };
  SmallInteger.prototype.neq =
    SmallInteger.prototype.notEquals =
    BigInteger
      .prototype.neq =
      BigInteger.prototype.notEquals;
  BigInteger.prototype.greater = function(v) {
    return this.compare(v) > 0;
  };
  SmallInteger.prototype.gt =
    SmallInteger.prototype.greater =
    BigInteger
      .prototype.gt =
      BigInteger.prototype.greater;
  BigInteger.prototype.lesser = function(v) {
    return this.compare(v) < 0;
  };
  SmallInteger.prototype.lt =
    SmallInteger.prototype.lesser =
    BigInteger
      .prototype.lt =
      BigInteger.prototype.lesser;
  BigInteger.prototype.greaterOrEquals = function(v) {
    return this.compare(v) >= 0;
  };
  SmallInteger.prototype.geq =
    SmallInteger.prototype.greaterOrEquals =
    BigInteger.prototype.geq =
      BigInteger.prototype.greaterOrEquals;
  BigInteger.prototype.lesserOrEquals = function(v) {
    return this.compare(v) <= 0;
  };
  SmallInteger.prototype.leq =
    SmallInteger.prototype.lesserOrEquals =
    BigInteger.prototype.leq =
      BigInteger.prototype.lesserOrEquals;
  BigInteger.prototype.isEven = function() {
    return (this.value[0] & 1) === 0;
  };
  SmallInteger.prototype.isEven = function() {
    return (this.value & 1) === 0;
  };
  BigInteger.prototype.isOdd = function() {
    return (this.value[0] & 1) === 1;
  };
  SmallInteger.prototype.isOdd = function() {
    return (this.value & 1) === 1;
  };
  BigInteger.prototype.isPositive = function() {
    return !this.sign;
  };
  SmallInteger.prototype.isPositive = function() {
    return this.value > 0;
  };
  BigInteger.prototype.isNegative = function() {
    return this.sign;
  };
  SmallInteger.prototype.isNegative = function() {
    return this.value < 0;
  };
  BigInteger.prototype.isUnit = function() {
    return false;
  };
  SmallInteger.prototype.isUnit = function() {
    return Math.abs(this.value) === 1;
  };
  BigInteger.prototype.isZero = function() {
    return false;
  };
  SmallInteger.prototype.isZero = function() {
    return this.value === 0;
  };
  BigInteger.prototype.isDivisibleBy = function(v) {
    var n = parseValue(v);
    var value = n.value;
    if (value === 0) return false;
    if (value === 1) return true;
    if (value === 2) return this.isEven();
    return this.mod(n).equals(Integer[0]);
  };
  SmallInteger.prototype.isDivisibleBy = BigInteger.prototype.isDivisibleBy;
  function isBasicPrime(v) {
    var n = v.abs();
    if (n.isUnit()) return false;
    if (n.equals(2) || n.equals(3) || n.equals(5)) return true;
    if (n.isEven() || n.isDivisibleBy(3) || n.isDivisibleBy(5)) return false;
    if (n.lesser(25)) return true;
  }
  BigInteger.prototype.isPrime = function() {
    var isPrime = isBasicPrime(this);
    if (isPrime !== undefined) return isPrime;
    var n = this.abs(), nPrev = n.prev();
    var a = [2, 3, 5, 7, 11, 13, 17, 19], b = nPrev, d, t, i, x;
    while (b.isEven()) b = b.divide(2);
    for (i = 0; i < a.length; i++) {
      x = bigInt(a[i]).modPow(b, n);
      if (x.equals(Integer[1]) || x.equals(nPrev)) continue;
      for (t = true, d = b; t && d.lesser(nPrev); d = d.multiply(2)) {
        x = x.square().mod(n);
        if (x.equals(nPrev)) t = false;
      }
      if (t) return false;
    }
    return true;
  };
  SmallInteger.prototype.isPrime = BigInteger.prototype.isPrime;
  BigInteger.prototype.isProbablePrime = function(iterations) {
    var isPrime = isBasicPrime(this);
    if (isPrime !== undefined) return isPrime;
    var n = this.abs();
    var t = iterations === undefined ? 5 : iterations;
    for (var i = 0; i < t; i++) {
      var a = bigInt.randBetween(2, n.minus(2));
      if (!a.modPow(n.prev(), n).isUnit()) return false;
    }
    return true;
  };
  SmallInteger.prototype.isProbablePrime = BigInteger.prototype.isProbablePrime;
  BigInteger.prototype.modInv = function(n) {
    var t = bigInt.zero,
      newT = bigInt.one,
      r = parseValue(n),
      newR = this.abs(),
      q,
      lastT,
      lastR;
    while (!newR.equals(bigInt.zero)) {
      q = r.divide(newR);
      lastT = t;
      lastR = r;
      t = newT;
      r = newR;
      newT = lastT.subtract(q.multiply(newT));
      newR = lastR.subtract(q.multiply(newR));
    }
    if (!r.equals(1)) {
      throw new Error(
        this.toString() + ' and ' + n.toString() + ' are not co-prime',
      );
    }
    if (t.compare(0) === -1) t = t.add(n);
    if (this.isNegative()) return t.negate();
    return t;
  };
  SmallInteger.prototype.modInv = BigInteger.prototype.modInv;
  BigInteger.prototype.next = function() {
    var value = this.value;
    if (this.sign) return subtractSmall(value, 1, this.sign);
    return new BigInteger(addSmall(value, 1), this.sign);
  };
  SmallInteger.prototype.next = function() {
    var value = this.value;
    if (value + 1 < MAX_INT) return new SmallInteger(value + 1);
    return new BigInteger(MAX_INT_ARR, false);
  };
  BigInteger.prototype.prev = function() {
    var value = this.value;
    if (this.sign) return new BigInteger(addSmall(value, 1), true);
    return subtractSmall(value, 1, this.sign);
  };
  SmallInteger.prototype.prev = function() {
    var value = this.value;
    if (value - 1 > -MAX_INT) return new SmallInteger(value - 1);
    return new BigInteger(MAX_INT_ARR, true);
  };
  var powersOfTwo = [1];
  while (2 * powersOfTwo[powersOfTwo.length - 1] <= BASE) {
    powersOfTwo.push(2 * powersOfTwo[powersOfTwo.length - 1]);
  }
  var powers2Length = powersOfTwo.length,
    highestPower2 = powersOfTwo[powers2Length - 1];
  function shift_isSmall(n) {
    return (typeof n === 'number' || typeof n === 'string')
        && +Math.abs(n) <= BASE
      || n instanceof BigInteger && n.value.length <= 1;
  }
  BigInteger.prototype.shiftLeft = function(n) {
    if (!shift_isSmall(n)) {
      throw new Error(String(n) + ' is too large for shifting.');
    }
    n = +n;
    if (n < 0) return this.shiftRight(-n);
    var result = this;
    if (result.isZero()) return result;
    while (n >= powers2Length) {
      result = result.multiply(highestPower2);
      n -= powers2Length - 1;
    }
    return result.multiply(powersOfTwo[n]);
  };
  SmallInteger.prototype.shiftLeft = BigInteger.prototype.shiftLeft;
  BigInteger.prototype.shiftRight = function(n) {
    var remQuo;
    if (!shift_isSmall(n)) {
      throw new Error(String(n) + ' is too large for shifting.');
    }
    n = +n;
    if (n < 0) return this.shiftLeft(-n);
    var result = this;
    while (n >= powers2Length) {
      if (result.isZero() || result.isNegative() && result.isUnit()) {
        return result;
      }
      remQuo = divModAny(result, highestPower2);
      result = remQuo[1].isNegative() ? remQuo[0].prev() : remQuo[0];
      n -= powers2Length - 1;
    }
    remQuo = divModAny(result, powersOfTwo[n]);
    return remQuo[1].isNegative() ? remQuo[0].prev() : remQuo[0];
  };
  SmallInteger.prototype.shiftRight = BigInteger.prototype.shiftRight;
  function bitwise(x, y, fn) {
    y = parseValue(y);
    var xSign = x.isNegative(), ySign = y.isNegative();
    var xRem = xSign ? x.not() : x, yRem = ySign ? y.not() : y;
    var xDigit = 0, yDigit = 0;
    var xDivMod = null, yDivMod = null;
    var result = [];
    while (!xRem.isZero() || !yRem.isZero()) {
      xDivMod = divModAny(xRem, highestPower2);
      xDigit = xDivMod[1].toJSNumber();
      if (xSign) xDigit = highestPower2 - 1 - xDigit;
      yDivMod = divModAny(yRem, highestPower2);
      yDigit = yDivMod[1].toJSNumber();
      if (ySign) yDigit = highestPower2 - 1 - yDigit;
      xRem = xDivMod[0];
      yRem = yDivMod[0];
      result.push(fn(xDigit, yDigit));
    }
    var sum = fn(xSign ? 1 : 0, ySign ? 1 : 0) !== 0 ? bigInt(-1) : bigInt(0);
    for (var i = result.length - 1; i >= 0; i -= 1) {
      sum = sum.multiply(highestPower2).add(bigInt(result[i]));
    }
    return sum;
  }
  BigInteger.prototype.not = function() {
    return this.negate().prev();
  };
  SmallInteger.prototype.not = BigInteger.prototype.not;
  BigInteger.prototype.and = function(n) {
    return bitwise(this, n, function(a, b) {
      return a & b;
    });
  };
  SmallInteger.prototype.and = BigInteger.prototype.and;
  BigInteger.prototype.or = function(n) {
    return bitwise(this, n, function(a, b) {
      return a | b;
    });
  };
  SmallInteger.prototype.or = BigInteger.prototype.or;
  BigInteger.prototype.xor = function(n) {
    return bitwise(this, n, function(a, b) {
      return a ^ b;
    });
  };
  SmallInteger.prototype.xor = BigInteger.prototype.xor;
  var LOBMASK_I = 1 << 30,
    LOBMASK_BI = (BASE & -BASE) * (BASE & -BASE) | LOBMASK_I;
  function roughLOB(n) {
    var v = n.value,
      x = typeof v === 'number'
        ? v | LOBMASK_I
        : v[0] + v[1] * BASE | LOBMASK_BI;
    return x & -x;
  }
  function integerLogarithm(value, base) {
    if (base.compareTo(value) <= 0) {
      var tmp = integerLogarithm(value, base.square(base));
      var p = tmp.p;
      var e = tmp.e;
      var t = p.multiply(base);
      return t.compareTo(value) <= 0
        ? { p: t, e: e * 2 + 1 }
        : { p: p, e: e * 2 };
    }
    return { p: bigInt(1), e: 0 };
  }
  BigInteger.prototype.bitLength = function() {
    var n = this;
    if (n.compareTo(bigInt(0)) < 0) n = n.negate().subtract(bigInt(1));
    if (n.compareTo(bigInt(0)) === 0) return bigInt(0);
    return bigInt(integerLogarithm(n, bigInt(2)).e).add(bigInt(1));
  };
  SmallInteger.prototype.bitLength = BigInteger.prototype.bitLength;
  function max(a, b) {
    a = parseValue(a);
    b = parseValue(b);
    return a.greater(b) ? a : b;
  }
  function min(a, b) {
    a = parseValue(a);
    b = parseValue(b);
    return a.lesser(b) ? a : b;
  }
  function gcd(a, b) {
    a = parseValue(a).abs();
    b = parseValue(b).abs();
    if (a.equals(b)) return a;
    if (a.isZero()) return b;
    if (b.isZero()) return a;
    var c = Integer[1], d, t;
    while (a.isEven() && b.isEven()) {
      d = Math.min(roughLOB(a), roughLOB(b));
      a = a.divide(d);
      b = b.divide(d);
      c = c.multiply(d);
    }
    while (a.isEven()) a = a.divide(roughLOB(a));
    do {
      while (b.isEven()) b = b.divide(roughLOB(b));
      if (a.greater(b)) {
        t = b;
        b = a;
        a = t;
      }
      b = b.subtract(a);
    } while (!b.isZero());
    return c.isUnit() ? a : a.multiply(c);
  }
  function lcm(a, b) {
    a = parseValue(a).abs();
    b = parseValue(b).abs();
    return a.divide(gcd(a, b)).multiply(b);
  }
  function randBetween(a, b) {
    a = parseValue(a);
    b = parseValue(b);
    var low = min(a, b), high = max(a, b);
    var range = high.subtract(low).add(1);
    if (range.isSmall) return low.add(Math.floor(Math.random() * range));
    var length = range.value.length - 1;
    var result = [], restricted = true;
    for (var i = length; i >= 0; i--) {
      var top = restricted ? range.value[i] : BASE;
      var digit = truncate(Math.random() * top);
      result.unshift(digit);
      if (digit < top) restricted = false;
    }
    result = arrayToSmall(result);
    return low.add(
      typeof result === 'number'
        ? new SmallInteger(result)
        : new BigInteger(result, false),
    );
  }
  var parseBase = function(text, base) {
    var length = text.length;
    var i;
    var absBase = Math.abs(base);
    for (var i = 0; i < length; i++) {
      var c = text[i].toLowerCase();
      if (c === '-') continue;
      if (/[a-z0-9]/.test(c)) {
        if (/[0-9]/.test(c) && +c >= absBase) {
          if (c === '1' && absBase === 1) continue;
          throw new Error(c + ' is not a valid digit in base ' + base + '.');
        } else if (c.charCodeAt(0) - 87 >= absBase) {
          throw new Error(c + ' is not a valid digit in base ' + base + '.');
        }
      }
    }
    if (2 <= base && base <= 36) {
      if (length <= LOG_MAX_INT / Math.log(base)) {
        var result = parseInt(text, base);
        if (isNaN(result)) {
          throw new Error(c + ' is not a valid digit in base ' + base + '.');
        }
        return new SmallInteger(parseInt(text, base));
      }
    }
    base = parseValue(base);
    var digits = [];
    var isNegative = text[0] === '-';
    for (i = isNegative ? 1 : 0; i < text.length; i++) {
      var c = text[i].toLowerCase(), charCode = c.charCodeAt(0);
      if (48 <= charCode && charCode <= 57) digits.push(parseValue(c));
      else if (97 <= charCode && charCode <= 122) {
        digits.push(parseValue(c.charCodeAt(0) - 87));
      } else if (c === '<') {
        var start = i;
        do {
          i++;
        } while (text[i] !== '>');
        digits.push(parseValue(text.slice(start + 1, i)));
      } else throw new Error(c + ' is not a valid character');
    }
    return parseBaseFromArray(digits, base, isNegative);
  };
  function parseBaseFromArray(digits, base, isNegative) {
    var val = Integer[0], pow = Integer[1], i;
    for (i = digits.length - 1; i >= 0; i--) {
      val = val.add(digits[i].times(pow));
      pow = pow.times(base);
    }
    return isNegative ? val.negate() : val;
  }
  function stringify(digit) {
    if (digit <= 35) {
      return '0123456789abcdefghijklmnopqrstuvwxyz'.charAt(digit);
    }
    return '<' + digit + '>';
  }
  function toBase(n, base) {
    base = bigInt(base);
    if (base.isZero()) {
      if (n.isZero()) return { value: [0], isNegative: false };
      throw new Error('Cannot convert nonzero numbers to base 0.');
    }
    if (base.equals(-1)) {
      if (n.isZero()) return { value: [0], isNegative: false };
      if (n.isNegative()) {
        return {
          value: [].concat.apply(
            [],
            Array.apply(null, Array(-n)).map(Array.prototype.valueOf, [1, 0]),
          ),
          isNegative: false,
        };
      }
      var arr = Array.apply(null, Array(+n - 1)).map(Array.prototype.valueOf, [
        0,
        1,
      ]);
      arr.unshift([1]);
      return { value: [].concat.apply([], arr), isNegative: false };
    }
    var neg = false;
    if (n.isNegative() && base.isPositive()) {
      neg = true;
      n = n.abs();
    }
    if (base.equals(1)) {
      if (n.isZero()) return { value: [0], isNegative: false };
      return {
        value: Array.apply(null, Array(+n)).map(Number.prototype.valueOf, 1),
        isNegative: neg,
      };
    }
    var out = [];
    var left = n, divmod;
    while (left.isNegative() || left.compareAbs(base) >= 0) {
      divmod = left.divmod(base);
      left = divmod.quotient;
      var digit = divmod.remainder;
      if (digit.isNegative()) {
        digit = base.minus(digit).abs();
        left = left.next();
      }
      out.push(digit.toJSNumber());
    }
    out.push(left.toJSNumber());
    return { value: out.reverse(), isNegative: neg };
  }
  function toBaseString(n, base) {
    var arr = toBase(n, base);
    return (arr.isNegative ? '-' : '') + arr.value.map(stringify).join('');
  }
  BigInteger.prototype.toArray = function(radix) {
    return toBase(this, radix);
  };
  SmallInteger.prototype.toArray = function(radix) {
    return toBase(this, radix);
  };
  BigInteger.prototype.toString = function(radix) {
    if (radix === undefined) radix = 10;
    if (radix !== 10) return toBaseString(this, radix);
    var v = this.value,
      l = v.length,
      str = String(v[--l]),
      zeros = '0000000',
      digit;
    while (--l >= 0) {
      digit = String(v[l]);
      str += zeros.slice(digit.length) + digit;
    }
    var sign = this.sign ? '-' : '';
    return sign + str;
  };
  SmallInteger.prototype.toString = function(radix) {
    if (radix === undefined) radix = 10;
    if (radix != 10) return toBaseString(this, radix);
    return String(this.value);
  };
  BigInteger.prototype.toJSON = SmallInteger.prototype.toJSON = function() {
    return this.toString();
  };
  BigInteger.prototype.valueOf = function() {
    return parseInt(this.toString(), 10);
  };
  BigInteger.prototype.toJSNumber = BigInteger.prototype.valueOf;
  SmallInteger.prototype.valueOf = function() {
    return this.value;
  };
  SmallInteger.prototype.toJSNumber = SmallInteger.prototype.valueOf;
  function parseStringValue(v) {
    if (isPrecise(+v)) {
      var x = +v;
      if (x === truncate(x)) return new SmallInteger(x);
      throw new Error('Invalid integer: ' + v);
    }
    var sign = v[0] === '-';
    if (sign) v = v.slice(1);
    var split = v.split(/e/i);
    if (split.length > 2) {
      throw new Error('Invalid integer: ' + split.join('e'));
    }
    if (split.length === 2) {
      var exp = split[1];
      if (exp[0] === '+') exp = exp.slice(1);
      exp = +exp;
      if (exp !== truncate(exp) || !isPrecise(exp)) {
        throw new Error(
          'Invalid integer: ' + exp + ' is not a valid exponent.',
        );
      }
      var text = split[0];
      var decimalPlace = text.indexOf('.');
      if (decimalPlace >= 0) {
        exp -= text.length - decimalPlace - 1;
        text = text.slice(0, decimalPlace) + text.slice(decimalPlace + 1);
      }
      if (exp < 0) {
        throw new Error('Cannot include negative exponent part for integers');
      }
      text += new Array(exp + 1).join('0');
      v = text;
    }
    var isValid = /^([0-9][0-9]*)$/.test(v);
    if (!isValid) throw new Error('Invalid integer: ' + v);
    var r = [], max = v.length, l = LOG_BASE, min = max - l;
    while (max > 0) {
      r.push(+v.slice(min, max));
      min -= l;
      if (min < 0) min = 0;
      max -= l;
    }
    trim(r);
    return new BigInteger(r, sign);
  }
  function parseNumberValue(v) {
    if (isPrecise(v)) {
      if (v !== truncate(v)) throw new Error(v + ' is not an integer.');
      return new SmallInteger(v);
    }
    return parseStringValue(v.toString());
  }
  function parseValue(v) {
    if (typeof v === 'number') return parseNumberValue(v);
    if (typeof v === 'string') return parseStringValue(v);
    return v;
  }
  for (var i = 0; i < 1e3; i++) {
    Integer[i] = new SmallInteger(i);
    if (i > 0) Integer[-i] = new SmallInteger(-i);
  }
  Integer.one = Integer[1];
  Integer.zero = Integer[0];
  Integer.minusOne = Integer[-1];
  Integer.max = max;
  Integer.min = min;
  Integer.gcd = gcd;
  Integer.lcm = lcm;
  Integer.isInstance = function(x) {
    return x instanceof BigInteger || x instanceof SmallInteger;
  };
  Integer.randBetween = randBetween;
  Integer.fromArray = function(digits, base, isNegative) {
    return parseBaseFromArray(
      digits.map(parseValue),
      parseValue(base || 10),
      isNegative,
    );
  };
  return Integer;
}();