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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Numerics;
using static System.Math;
using static System.Array;
using static AtCoder.IO_ShortCut;
using static AtCoder.Tool;
namespace AtCoder
{
    class AC
    {
        const int MOD = 1000000007;
        //const int MOD = 998244353;
        const int INF = int.MaxValue / 2;
        const long SINF = long.MaxValue / 2;
        const double EPS = 1e-8;
        static readonly int[] dI = { 0, 1, 0, -1, 1, -1, -1, 1 };
        static readonly int[] dJ = { 1, 0, -1, 0, 1, 1, -1, -1 };
        static List<List<int>> G = new List<List<int>>();
        //static List<List<Edge>> G = new List<List<Edge>>();
        //static List<Edge> E = new List<Edge>();
        static void Main(string[] args)
        {
            //var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            //Console.SetOut(sw);

            long L, R;
            Input(out L, out R);
            long ans = 0;
            int pL = 0;
            int pR = 0;
            for(var i = 0; i < 64; i++)
            {
                if (bit(L, i)) pL = i;
                if (bit(R, i)) pR = i;
            }
            /*OutL(Convert.ToString(R, 2));
            OutL(Convert.ToString(L, 2));
            Out_Sep(pL, pR);
            int[] cnt = { 0, 0, 0, 0, 0, 0, 0 };
            for(var y = L; y <= R; y++)
            {
                for(var x = L; x <= R; x++)
                {
                    if ((x ^ y) != y % x) continue;
                    for(var i = 7; i >= 0; i--)
                    {
                        if (bit(x, i) && bit(y, i))
                        {
                            Out_Sep(Convert.ToString(y, 2), " ", Convert.ToString(x, 2));
                            cnt[i]++;
                            break;
                        }
                    }
                }
            }
            Out_Sep(cnt);*/

            for(var d = 0; d < 63; d++)
            {
                var dp = new long[d + 1, 2, 2];
                if (d < pL || pR < d) continue;
                int sl = d == pL ? 0 : 1;
                int sr = d == pR ? 0 : 1;
                dp[0, sr, sl] = 1;
                for (var i = 1; i <= d; i++)
                {
                    int s = d - i;
                    //0,0
                    if (!bit(L, s))
                    {
                        dp[i, (bit(R, s) ? 1 : 0), 0] += dp[i - 1, 0, 0];
                        dp[i, (bit(R, s) ? 1 : 0), 0] %= MOD;
                    }
                    if (bit(R, s))
                    {
                        if (bit(L, s))
                        {
                            dp[i, 0, 0] += dp[i - 1, 0, 0];
                            dp[i, 0, 0] %= MOD;
                        }
                        else
                        {
                            dp[i, 0, 1] += dp[i - 1, 0, 0];
                            dp[i, 0, 1] %= MOD;
                            dp[i, 0, 0] += dp[i - 1, 0, 0];
                            dp[i, 0, 0] %= MOD;
                        }
                    }

                    //1,0
                    if (!bit(L, s))
                    {
                        dp[i, 1, 0] += dp[i - 1, 1, 0];
                        dp[i, 1, 0] %= MOD;
                    }
                    if (bit(L, s))
                    {
                        dp[i, 1, 0] += dp[i - 1, 1, 0];
                        dp[i, 1, 0] %= MOD;
                    }
                    else
                    {
                        dp[i, 1, 1] += dp[i - 1, 1, 0];
                        dp[i, 1, 1] %= MOD;

                        dp[i, 1, 0] += dp[i - 1, 1, 0];
                        dp[i, 1, 0] %= MOD;
                    }

                    //0,1
                    if (bit(R, s))
                    {
                        dp[i, 1, 1] += dp[i - 1, 0, 1];
                        dp[i, 1, 1] %= MOD;
                    }
                    else
                    {
                        dp[i, 0, 1] += dp[i - 1, 0, 1];
                        dp[i, 0, 1] %= MOD;
                    }

                    if (bit(R, s))
                    {
                        dp[i, 0, 1] += 2 * dp[i - 1, 0, 1];
                        dp[i, 0, 1] %= MOD;
                    }

                    //1,1
                    dp[i, 1, 1] += dp[i - 1, 1, 1];
                    dp[i, 1, 1] %= MOD;

                    dp[i, 1, 1] += 2 * dp[i - 1, 1, 1];
                    dp[i, 1, 1] %= MOD;
                }
                for (var i = 0; i < 2; i++) for (var j = 0; j < 2; j++) ans = (ans + dp[d, i, j]) % MOD;
                //Out_Sep(dp[d, 0, 0], dp[d, 0, 1], dp[d, 1, 0], dp[d, 1, 1]);
            }
            
            OutL(ans);
            //Console.Out.Flush();
        }
        static bool bit(long x,int dig)
        {
            return ((1L << dig) & x) != 0;
        }
        struct Edge
        {
            public int from;

            public int to;
            public long dist;
            public Edge(int t, long c)
            {
                from = -1;
                to = t;
                dist = c;
            }
            public Edge(int f, int t, long c)
            {
                from = f;
                to = t;
                dist = c;
            }
        }
    }
    static class IO_ShortCut
    {
        public static string[] ReadSplit => Console.ReadLine().Split();
        public static int[] ReadSplitInt => ConvertAll(Console.ReadLine().Split(), int.Parse);
        public static long[] ReadSplitLong => ConvertAll(Console.ReadLine().Split(), long.Parse);
        public static double[] ReadSplit_Double => ConvertAll(Console.ReadLine().Split(), double.Parse);
        public static string Str => Console.ReadLine();
        public static int Int => int.Parse(Console.ReadLine());
        public static long Long => long.Parse(Console.ReadLine());
        public static double Double => double.Parse(Console.ReadLine());
        public static T Conv<T>(string input) => (T)Convert.ChangeType(input, typeof(T));
        public static void Input<T>(out T a) => a = Conv<T>(Console.ReadLine());
        public static void Input<T, U>(out T a, out U b)
        { var q = ReadSplit;a = Conv<T>(q[0]);b = Conv<U>(q[1]); }
        public static void Input<T, U, V>(out T a, out U b, out V c)
        { var q = ReadSplit;a = Conv<T>(q[0]);b = Conv<U>(q[1]);c = Conv<V>(q[2]); }
        public static void Input<T, U, V, W>(out T a, out U b, out V c, out W d)
        { var q = ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]);d = Conv<W>(q[3]); }

        public static void OutL(object s) => Console.WriteLine(s);
        public static void Out_Sep<T>(IEnumerable<T> s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_Sep<T>(IEnumerable<T> s, string sep) => Console.WriteLine(string.Join($"{sep}", s));
        public static void Out_Sep(params object[] s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_One(object s) => Console.Write($"{s} ");
        public static void Out_One(object s, string sep) => Console.Write($"{s}{sep}");
    }
    public static class Tool
    {
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = ConvertAll(array, x => initialvalue);
        }
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Display<T>(T[,] array2d, int n, int m)
        {
            for (var i = 0; i < n; i++)
            {
                for (var j = 0; j < m; j++)
                {
                    Console.Write($"{array2d[i, j]} ");
                }
                Console.WriteLine();
            }
        }
        static public long LPow(int a, int b) => (long)Pow(a, b);
    }
    class Modulo
    {
        private int M;
        private readonly long[] m_facs;
        public long Mul(long a, long b)
        {
            return ((a * b) % M);
        }
        public Modulo(long n, int m)
        {
            M = m;
            m_facs = new long[n + 1];
            m_facs[0] = 1;
            for (long i = 1; i <= n; ++i)
            {
                m_facs[i] = Mul(m_facs[i - 1], i);
            }
        }
        public long Fac(long n)
        {
            return m_facs[n];
        }
        public long Pow(long a, long m)
        {
            switch (m)
            {
                case 0:
                    return 1L;
                case 1:
                    return a;
                default:
                    long p1 = Pow(a, m / 2);
                    long p2 = Mul(p1, p1);
                    return ((m % 2) == 0) ? p2 : Mul(p2, a);
            }
        }
        public long Div(long a, long b)
        {
            return Mul(a, Pow(b, M - 2));
        }
        public long Ncr(long n, long r)
        {
            if (n < r) { return 0; }
            if (n == r) { return 1; }
            long res = Fac(n);
            res = Div(res, Fac(r));
            res = Div(res, Fac(n - r));
            return res;
        }
    }
}