Unfortunately, Bessie is way too tired to think about the problem, but since the answer to each test case is either "yes" or "no," she has a plan! Precisely, she decides to repeatedly submit the following nondeterministic solution:

if input == sample_input:
  print sample_output
else:
  print "yes" or "no" each with probability 1/2, independently for each test case

Note that for all test cases besides the sample, this program may produce a different output when resubmitted, so the number of test cases that it passes will vary.

Bessie knows that she cannot submit more than $K$ ($1\le K\le 10^9$) times in total because then she will certainly be disqualified. What is the maximum possible expected value of Bessie's final score, assuming that she follows the optimal strategy?

SAMPLE INPUT:

2 3

SAMPLE OUTPUT:

1.875

In this example, Bessie should keep resubmitting until she has reached $3$ submissions or she receives full credit. Bessie will receive full credit with probability $\frac{7}{8}$ and half credit with probability $\frac{1}{8}$, so the expected value of Bessie's final score under this strategy is $\frac{7}{8}\cdot 2+\frac{1}{8}\cdot 1=\frac{15}{8}=1.875$. As we see from this formula, the expected value of Bessie's score can be calculated by taking the sum over $x$ of $p(x) \cdot x$, where $p(x)$ is the probability of receiving a score of $x$.

SAMPLE INPUT:

4 2

SAMPLE OUTPUT:

2.8750000000000000000

Here, Bessie should only submit twice if she passes fewer than $3$ test cases on her first try.

Problem credits: Benjamin Qi