Problem: 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.)

Submission attempts (in order):
import bisect
import heapq
import itertools
import math
import operator
import os
import re
import string
import sys
from collections import Counter, deque, defaultdict
from copy import deepcopy
from decimal import Decimal
from fractions import gcd
from functools import lru_cache, reduce
from operator import itemgetter, mul, add, xor

import numpy as np

if os.getenv("LOCAL"):
    sys.stdin = open("_in.txt", "r")

sys.setrecursionlimit(2147483647)
INF = float("inf")
IINF = 10 ** 18
MOD = 10 ** 9 + 7

L, R = list(map(int, sys.stdin.readline().split()))


# ans = 0

# for x in range(1, R + 1):
#     for y in range(x, R + 1):



#             ans += 1
# print(ans)
# print(cnt)

def count(R):
    if R == 1:
        return 1
    if R == 0:
        return 0
    
    dp = np.zeros((R.bit_length(), R.bit_length() + 1, 2), dtype=int)
    dp[-1][0][0] = 1
    for d in reversed(range(R.bit_length() - 1)):
        if R >> d & 1:
            for n in range(R.bit_length()):
                dp[d][n][1] += dp[d + 1][n][0]
                dp[d][n + 1][0] += dp[d + 1][n][0]
                dp[d][n][1] += dp[d + 1][n][1]
                dp[d][n + 1][1] += dp[d + 1][n][1]
        else:
            for n in range(R.bit_length()):
                dp[d][n][0] += dp[d + 1][n][0]
                dp[d][n][1] += dp[d + 1][n][1]
                dp[d][n + 1][1] += dp[d + 1][n][1]
        # print(dp[d], R >> d & 1)
        # if R >> d & 1:
        #     for n in range(R.bit_length()):
        #         dp[d][n + 1][1] += dp[d + 1][n][1]
        #         dp[d][n][1] += dp[d + 1][n][1]
        #         dp[d][n][1] += dp[d + 1][n][0]
        #         dp[d][n + 1][0] += dp[d + 1][n][0]
        # else:
        #     for n in range(R.bit_length()):
        #         dp[d][n][1] += dp[d + 1][n][1]
        #         dp[d][n + 1][1] += dp[d + 1][n][1]
        #         dp[d][n][0] += dp[d + 1][n][0]

    ans = (3 ** (R.bit_length() - 1) - 1) // 2 % MOD
    for i, r in enumerate(dp[0]):
        if r.sum():
            ans += pow(2, i, MOD) * r.sum()
            ans %= MOD
    return int(ans)


# print(count(R))
print((count(R) - count(L - 1)) % MOD)
# print(ans)


import os
import os
import sys
from collections import defaultdict

import numpy as np

if os.getenv("LOCAL"):
    sys.stdin = open("_in.txt", "r")

sys.setrecursionlimit(2147483647)
INF = float("inf")
IINF = 10 ** 18
MOD = 10 ** 9 + 7

L, R = list(map(int, sys.stdin.readline().split()))


def test(L, R):
    ans = 0
    cnt = defaultdict(int)
    for x in range(L, R + 1):
        for y in range(x, R + 1):
            if y % x == y ^ x:
                cnt[x + y % x] += 1
                ans += 1
    return ans


def solve(L, R):
    ret = L == 1
    for msb in range(R.bit_length()):
        r = min(R, (1 << msb + 1) - 1)
        l = max(L, (1 << msb))
        if l > r:
            continue
        # print(bin(r), bin(l))
        
        dp = np.zeros((r.bit_length(), 2, 2), dtype=int)
        for d in reversed(range(r.bit_length() - 1)):
            dp[-1][0][0] = 1
            if r >> d & 1:
                if l >> d & 1:
                    # r == 1, l == 1
                    # y1, x1
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    dp[d][0][1] += dp[d + 1][0][1]
                    dp[d][0][0] += dp[d + 1][0][0]
                    # y1, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][0][1] += dp[d + 1][0][1]
                    # y0, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][1] += dp[d + 1][0][1]
                else:
                    # r == 1, l == 0
                    # y1, x1
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][1] += dp[d + 1][1][0]
                    dp[d][0][1] += dp[d + 1][0][1]
                    dp[d][0][1] += dp[d + 1][0][0]
                    # y1, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    dp[d][0][1] += dp[d + 1][0][1]
                    dp[d][0][0] += dp[d + 1][0][0]
                    # y0, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    dp[d][1][1] += dp[d + 1][0][1]
                    dp[d][1][0] += dp[d + 1][0][0]
            else:
                if l >> d & 1:
                    # r == 0, l == 1
                    # y1, x1
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    # y1, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    # y0, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][0][1] += dp[d + 1][0][1]
                else:
                    # r == 0, l == 0
                    # y1, x1
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][1] += dp[d + 1][1][0]
                    # y1, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    # y0, x0
                    dp[d][1][1] += dp[d + 1][1][1]
                    dp[d][1][0] += dp[d + 1][1][0]
                    dp[d][0][1] += dp[d + 1][0][1]
                    dp[d][0][0] += dp[d + 1][0][0]
            dp[d] %= MOD
        ret += dp[0].sum()
        ret %= MOD
    return int(ret)


print(solve(L, R))
# for l in range(1, R + 1):
#     for r in range(l, R + 1):
#         t1, t2 = solve(l, r), test(l, r)
#         if t1 != t2:
#             print(bin(l), bin(r), l, r, t1, t2)


Verdicts: Wrong Answer

Accepted