CF1734F.Zeros and Ones

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.

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 ( $1 \leq n,m \leq 10^{18}$ ).

For each testcase, output a non-negative integer — the Hamming distance between the two required strings.

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 $6$ .