Description:
Let $$$S$$$ be the Thue-Morse sequence. In other words, $$$S$$$ is the $$$0$$$-indexed binary string with infinite length that can be constructed as follows:
- Initially, let $$$S$$$ be "0".
- Then, we perform the following operation infinitely many times: concatenate $$$S$$$ with a copy of itself with flipped bits.For example, here are the first four iterations: Iteration$$$S$$$ before iteration$$$S$$$ before iteration with flipped bitsConcatenated $$$S$$$1010120110011030110100101101001401101001100101100110100110010110$$$\ldots$$$$$$\ldots$$$$$$\ldots$$$$$$\ldots$$$
You are given two positive integers $$$n$$$ and $$$m$$$. Find the number of positions where the strings $$$S_0 S_1 \ldots S_{m-1}$$$ and $$$S_n S_{n + 1} \ldots S_{n + m - 1}$$$ are different.
Input Format:
Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of the test cases follows.
The first and only line of each test case contains two positive integers, $$$n$$$ and $$$m$$$ respectively ($$$1 \leq n,m \leq 10^{18}$$$).
Output Format:
For each testcase, output a non-negative integer — the Hamming distance between the two required strings.
Note:
The string $$$S$$$ is equal to 0110100110010110....
In the first test case, $$$S_0$$$ is "0", and $$$S_1$$$ is "1". The Hamming distance between the two strings is $$$1$$$.
In the second test case, $$$S_0 S_1 \ldots S_9$$$ is "0110100110", and $$$S_5 S_6 \ldots S_{14}$$$ is "0011001011". The Hamming distance between the two strings is $$$6$$$.