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 Rust (1.15.1) (judge verdict: Accepted)

fn read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

const MODULO: usize = 1_000_000_007;

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let l: usize = scan(tokens);
    let r: usize = scan(tokens);
    println!("{}", f(l, r));
}

fn f(l: usize, r: usize) -> usize {
    // #{ (x, y) | l <= x <= y <= r, p(x, y) }
    if l == r {
        1
    } else if l % 2 == 1 {
        (g(l, r) + f(l + 1, r)) % MODULO
    } else if r % 2 == 0 {
        (f(l, r - 1) + h(l, r)) % MODULO
    } else {
        3 * f(l / 2, r / 2) % MODULO
    }
}

fn g(l: usize, r: usize) -> usize {
    // #{ y | l <= y <= r, p(l, y) }
    if l == r {
        1
    } else if r % 2 == 0 {
        (g(l, r - 1) + p(l, r)) % MODULO
    } else if l == 1 {
        1
    } else if l % 2 == 1 {
        g(l / 2, r / 2)
    } else {
        2 * g(l / 2, r / 2) % MODULO
    }
}

fn h(l: usize, r: usize) -> usize {
    // #{ x | l <= x <= r, p(x, r) }
    if l == r {
        1
    } else if l % 2 == 1 {
        (p(l, r) + h(l + 1, r)) % MODULO
    } else if r % 2 == 0 {
        h(l / 2, r / 2)
    } else {
        2 * h(l / 2, r / 2) % MODULO
    }
}

fn p(x: usize, y: usize) -> usize {
    if y % x == y ^ x {
        1
    } else {
        0
    }
}