Problem:
Score : 600 points

Problem Statement

Given are integers L and R. Find the number, modulo 10^9 + 7, of pairs of integers (x, y) (L \leq x \leq y \leq R) such that the remainder when y is divided by x is equal to y \mbox{ XOR } x.

What is \mbox{ XOR }?

The XOR of integers A and B, A \mbox{ XOR } B, is defined as follows:

When A \mbox{ XOR } B is written in base two, the digit in the 2^k's place (k \geq 0) is 1 if either A or B, but not both, has 1 in the 2^k's place, and 0 otherwise.

For example, 3 \mbox{ XOR } 5 = 6. (In base two: 011 \mbox{ XOR } 101 = 110.)

Constraints

1 \leq L \leq R \leq 10^{18}

Input

Input is given from Standard Input in the following format:

L R

Output

Print the number of pairs of integers (x, y) (L \leq x \leq y \leq R) satisfying the condition, modulo 10^9 + 7.

Sample Input 1

2 3

Sample Output 1

3

Three pairs satisfy the condition: (2, 2), (2, 3), and (3, 3).

Sample Input 2

10 100

Sample Output 2

604

Sample Input 3

1 1000000000000000000

Sample Output 3

68038601

Be sure to compute the number modulo 10^9 + 7.

Solution in Ruby (2.3.3) (judge verdict: Accepted)

L, R = gets.split.map(&:to_i)
M = 10**9 + 7

#  y % x == y ^ x  ==>  y.to_s(2).length == x.to_s(2).length && y - x == y ^ x
#  1-1 == 1^1 = 0, 1-0 == 1^0 = 1, 0-0 == 0^0 = 0, but 0-1 = -1 != 0^1 = 1
m = (10**18).to_s(2).length
#  dp[digit][L<=x?][y<=R?][msb?]
dp = Array.new(m + 1){Array.new(2){Array.new(2){Array.new(2, 0)}}}
dp[m][0][0][0] = 1  # x, y = 1, 1
c3 = [0, 1].repeated_permutation(3)
c2 = [[0, 0], [0, 1], [1, 1]]  # x, y != 1, 0
(m - 1).downto(0).each do |i|
  l, r = L >> i & 1, R >> i & 1
  c3.each do |j, k, s|
    d0 = dp[i + 1][j][k][s] % M
    c2.each do |x, y|
      next if s == 0 && x != y
      next if j == 0 && x == 0 && l == 1
      next if k == 0 && y == 1 && r == 0
      nj, nk, ns = j, k, s
      nj = 1 if x == 1 && l == 0
      nk = 1 if y == 0 && r == 1
      ns = 1 if x == 1 && y == 1
      dp[i][nj][nk][ns] += d0
    end
  end
end
puts dp[0].flatten.inject(:+) % M