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function isAnagram(word1, word2) {
  if (!word1 || !word2 || !word1.length || !word2.length) {
    throw new Error('isAnagram requires two strings to be passed.')
  }

  var length1 = word1.length;
  var length2 = word2.length;

  if (length1 !== length2) {
    return false;
  }

  var word1hash = 1;
  var word2hash = 1;

  var tab = {'q': 0.6931471805599453, 'w': 1.0986122886681098, 'e': 1.6094379124341003, 'r': 1.9459101490553132, 't': 2.3978952727983707, 'y': 2.5649493574615367, 'u': 2.833213344056216, 'i': 2.9444389791664403, 'o': 3.1354942159291497, 'p': 3.367295829986474, 'a': 3.4339872044851463, 's': 3.6109179126442243, 'd': 3.713572066704308, 'f': 3.7612001156935624, 'g': 3.8501476017100584, 'h': 3.970291913552122, 'j': 4.07753744390572, 'k': 4.110873864173311, 'l': 4.204692619390966, 'z': 4.2626798770413155, 'x': 4.290459441148391, 'c': 4.3694478524670215, 'v': 4.418840607796598, 'b': 4.48863636973214, 'n': 4.574710978503383, 'm': 4.61512051684126, 'Q': 0.6931471805599453, 'W': 1.0986122886681098, 'E': 1.6094379124341003, 'R': 1.9459101490553132, 'T': 2.3978952727983707, 'Y': 2.5649493574615367, 'U': 2.833213344056216, 'I': 2.9444389791664403, 'O': 3.1354942159291497, 'P': 3.367295829986474, 'A': 3.4339872044851463, 'S': 3.6109179126442243, 'D': 3.713572066704308, 'F': 3.7612001156935624, 'G': 3.8501476017100584, 'H': 3.970291913552122, 'J': 4.07753744390572, 'K': 4.110873864173311, 'L': 4.204692619390966, 'Z': 4.2626798770413155, 'X': 4.290459441148391, 'C': 4.3694478524670215, 'V': 4.418840607796598, 'B': 4.48863636973214, 'N': 4.574710978503383, 'M': 4.61512051684126, ' ': 0.0
}

  for (var i = 0; i < length1; i++) {
    word1hash += tab[word1[i]]
  }

  for (var i = 0; i < length2; i++) {
    word2hash += tab[word2[i]]
  }

  return word1hash == word2hash;
}

function isAnagram(word1, word2) {
  if (!word1 || !word2 || !word1.length || !word2.length) {
    throw new Error('isAnagram requires two strings to be passed.')
  }

  var length1 = word1.length;
  var length2 = word2.length;

  var word1hash = 1;
  var word2hash = 1;

  var tab = {'q': 0.6931471805599453, 'w': 1.0986122886681098, 'e': 1.6094379124341003, 'r': 1.9459101490553132, 't': 2.3978952727983707, 'y': 2.5649493574615367, 'u': 2.833213344056216, 'i': 2.9444389791664403, 'o': 3.1354942159291497, 'p': 3.367295829986474, 'a': 3.4339872044851463, 's': 3.6109179126442243, 'd': 3.713572066704308, 'f': 3.7612001156935624, 'g': 3.8501476017100584, 'h': 3.970291913552122, 'j': 4.07753744390572, 'k': 4.110873864173311, 'l': 4.204692619390966, 'z': 4.2626798770413155, 'x': 4.290459441148391, 'c': 4.3694478524670215, 'v': 4.418840607796598, 'b': 4.48863636973214, 'n': 4.574710978503383, 'm': 4.61512051684126, 'Q': 0.6931471805599453, 'W': 1.0986122886681098, 'E': 1.6094379124341003, 'R': 1.9459101490553132, 'T': 2.3978952727983707, 'Y': 2.5649493574615367, 'U': 2.833213344056216, 'I': 2.9444389791664403, 'O': 3.1354942159291497, 'P': 3.367295829986474, 'A': 3.4339872044851463, 'S': 3.6109179126442243, 'D': 3.713572066704308, 'F': 3.7612001156935624, 'G': 3.8501476017100584, 'H': 3.970291913552122, 'J': 4.07753744390572, 'K': 4.110873864173311, 'L': 4.204692619390966, 'Z': 4.2626798770413155, 'X': 4.290459441148391, 'C': 4.3694478524670215, 'V': 4.418840607796598, 'B': 4.48863636973214, 'N': 4.574710978503383, 'M': 4.61512051684126, ' ': 0.0
}

  for (var i = 0; i < length1; i++) {
    word1hash += tab[word1[i]]
  }

  for (var i = 0; i < length2; i++) {
    word2hash += tab[word2[i]]
  }

  return word1hash == word2hash;
}

Edit 3: removed equal length check.

Edit 2

I know this is silly, but adding logarithms of prime numbers is actually a tiny bit faster. (benchmark)

function isAnagram(word1, word2) {
  if (!word1 || !word2 || !word1.length || !word2.length) {
    throw new Error('isAnagram requires two strings to be passed.')
  }

  var length1 = word1.length;
  var length2 = word2.length;

  if (length1 !== length2) {
    return false;
  }

  var word1hash = 1;
  var word2hash = 1;

  var tab = {'q': 0.6931471805599453, 'w': 1.0986122886681098, 'e': 1.6094379124341003, 'r': 1.9459101490553132, 't': 2.3978952727983707, 'y': 2.5649493574615367, 'u': 2.833213344056216, 'i': 2.9444389791664403, 'o': 3.1354942159291497, 'p': 3.367295829986474, 'a': 3.4339872044851463, 's': 3.6109179126442243, 'd': 3.713572066704308, 'f': 3.7612001156935624, 'g': 3.8501476017100584, 'h': 3.970291913552122, 'j': 4.07753744390572, 'k': 4.110873864173311, 'l': 4.204692619390966, 'z': 4.2626798770413155, 'x': 4.290459441148391, 'c': 4.3694478524670215, 'v': 4.418840607796598, 'b': 4.48863636973214, 'n': 4.574710978503383, 'm': 4.61512051684126, 'Q': 0.6931471805599453, 'W': 1.0986122886681098, 'E': 1.6094379124341003, 'R': 1.9459101490553132, 'T': 2.3978952727983707, 'Y': 2.5649493574615367, 'U': 2.833213344056216, 'I': 2.9444389791664403, 'O': 3.1354942159291497, 'P': 3.367295829986474, 'A': 3.4339872044851463, 'S': 3.6109179126442243, 'D': 3.713572066704308, 'F': 3.7612001156935624, 'G': 3.8501476017100584, 'H': 3.970291913552122, 'J': 4.07753744390572, 'K': 4.110873864173311, 'L': 4.204692619390966, 'Z': 4.2626798770413155, 'X': 4.290459441148391, 'C': 4.3694478524670215, 'V': 4.418840607796598, 'B': 4.48863636973214, 'N': 4.574710978503383, 'M': 4.61512051684126, ' ': 0.0
}

  for (var i = 0; i < length1; i++) {
    word1hash += tab[word1[i]]
  }

  for (var i = 0; i < length2; i++) {
    word2hash += tab[word2[i]]
  }

  return word1hash == word2hash;
}

Edit 1

I guess we have a speed contest now, and this version is a great deal faster than all its precedents (benchmark).

function isAnagram(word1, word2) {
  if (!word1 || !word2 || !word1.length || !word2.length) {
    throw new Error('isAnagram requires two strings to be passed.')
  }

  var nword1 = word1;
  var nword2 = word2;

  var length1 = nword1.length;
  var length2 = nword2.length;

  if (length1 !== length2) {
    return false;
  }

  var word1hash = 1;
  var word2hash = 1;

  var tab = {'q': 2, 'w': 3, 'e': 5, 'r': 7, 't': 11, 'y': 13, 'u': 17, 'i': 19, 'o': 23, 'p': 29, 'a': 31, 's': 37, 'd': 41, 'f': 43, 'g': 47, 'h': 53, 'j': 59, 'k': 61, 'l': 67, 'z': 71, 'x': 73, 'c': 79, 'v': 83, 'b': 89, 'n': 97, 'm': 101, 'Q': 2, 'W': 3, 'E': 5, 'R': 7, 'T': 11, 'Y': 13, 'U': 17, 'I': 19, 'O': 23, 'P': 29, 'A': 31, 'S': 37, 'D': 41, 'F': 43, 'G': 47, 'H': 53, 'J': 59, 'K': 61, 'L': 67, 'Z': 71, 'X': 73, 'C': 79, 'V': 83, 'B': 89, 'N': 97, 'M': 101, ' ': 1}

  for (var i = 0; i < length1; i++) {
    word1hash *= tab[word1[i]]
  }

  for (var i = 0; i < length2; i++) {
    word2hash *= tab[word2[i]]
  }

  return word1hash == word2hash;
}