Problem:
Score : 600 points

Problem Statement

Snuke is standing on a two-dimensional plane. In one operation, he can move by 1 in the positive x-direction, or move by 1 in the positive y-direction.

Let us define a function f(r, c) as follows:

f(r,c) := (The number of paths from the point (0, 0) to the point (r, c) that Snuke can trace by repeating the operation above)

Given are integers r_1, r_2, c_1, and c_2.
Find the sum of f(i, j) over all pair of integers (i, j) such that r_1 ≤ i ≤ r_2 and c_1 ≤ j ≤ c_2, and compute this value modulo (10^9+7).

Constraints

1 ≤ r_1 ≤ r_2 ≤ 10^6

1 ≤ c_1 ≤ c_2 ≤ 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

r_1 c_1 r_2 c_2

Output

Print the sum of f(i, j) modulo (10^9+7).

Sample Input 1

1 1 2 2

Sample Output 1

14

For example, there are two paths from the point (0, 0) to the point (1, 1): (0,0) → (0,1) → (1,1) and (0,0) → (1,0) → (1,1), so f(1,1)=2.

Similarly, f(1,2)=3, f(2,1)=3, and f(2,2)=6. Thus, the sum is 14.

Sample Input 2

314 159 2653 589

Sample Output 2

602215194

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::io::{stdin, Read};
const MOD: usize = 1000000007;

fn gcd(a: usize, b: usize) -> usize {
    match b {
        0 => a,
        _ => gcd(b, a % b),
    }
}

fn powmod(base: usize, exp: usize, m: usize) -> usize {
    let b: Vec<bool> = format!("{:b}", exp)
        .chars()
        .map(|x| x.to_digit(10).unwrap() != 0)
        .collect();
    let mut v = 1;
    for bi in b {
        v *= v;
        v %= m;
        if bi {
            v *= base;
            v %= m
        }
    }
    return v % m;
}

fn invmod(x: usize, m: usize) -> usize {
    powmod(x, m - 2, m)
}

fn prime_factors(x: usize) -> Vec<usize> {
    let mut res = vec![];
    let mut y = x;
    let mut p = 2;
    let upper = (y as f64).sqrt() as usize;
    while y > 1 && p <= upper {
        match y % p {
            0 => {
                res.push(p);
                y /= p;
            }
            _ => {
                p += 1;
            }
        }
    }
    if y > 1 {
        res.push(y)
    }
    return res;
}

fn get_factorials(n: usize, m: usize) -> Vec<usize> {
    let mut res = vec![0; n as usize];
    res[0] = 1;
    for i in 1..(n as usize) {
        res[i] = (res[i - 1] * (i as usize)) % m;
    }
    res
}

fn get_facinvs(n: usize, m: usize, facts: &Vec<usize>) -> Vec<usize> {
    let mut res = vec![0; n];
    res[n - 1] = invmod(facts[n - 1], m);
    for i in (0..(n - 1)).rev() {
        res[i] = (res[i + 1] * (i + 1)) % m;
    }
    res
}

fn combmod(a: usize, b: usize, m: usize, facts: &Vec<usize>, invs: &Vec<usize>) -> usize {
    if a < b {
        0
    } else {
        (((facts[a as usize] * invs[b as usize]) % m) * invs[(a - b) as usize]) % m
    }
}

macro_rules! parse {
    ($iter: ident) => {
        $iter.next().unwrap().parse().unwrap()
    };
    ($iter: ident,2) => {
        (parse!($iter), parse!($iter))
    };
    ($iter: ident,3) => {
        (parse!($iter), parse!($iter), parse!($iter))
    };
    ($iter: ident,4) => {
        (parse!($iter), parse!($iter), parse!($iter), parse!($iter))
    };
    ($iter: ident, $n: ident) => {
        (&mut $iter).take($n).map(|e| e.parse().unwrap()).collect();
    };
}

fn main() {
    let mut s = String::new();
    stdin().read_to_string(&mut s).unwrap();
    let mut it = s.split_whitespace();
    let (r1, c1, r2, c2): (usize, usize, usize, usize) = parse!(it, 4);
    let facts = get_factorials(2000010, MOD);
    let invs = get_facinvs(2000010, MOD, &facts);
    let f = |i, j| combmod(i + j, i, MOD, &facts, &invs);
    let g = |i, j| -> usize { (0..(i + 1)).map(|x| f(x + 1, j)).sum() };
    let ret = ((g(r2, c2) + g(r1 - 1, c1 - 1)) as i64) - ((g(r2, c1 - 1) + g(r1 - 1, c2)) as i64);
    let ret = ret % (MOD as i64);
    let ret = if ret < 0 { ret + (MOD as i64) } else { ret };
    println!("{}", ret);
}