Description

The player starts with a prize of $1, and is asked a sequence of n questions. For each question, he may

• quit and keep his prize.

<li>answer the question. If wrong, he quits with nothing. If correct, the prize is doubled, and he continues with the next question. </li>

After the last question, he quits with his prize. The player wants to maximize his expected prize.

Once each question is asked, the player is able to assess the probability p that he will be able to answer it. For each question, we assume that p is a random variable uniformly distributed over the range t .. 1.

Input

Input is a number of lines, each with two numbers: an integer 1 <= n <= 30, and a real 0 <= t <= 1. Input is terminated by a line containing 0 0. This line should not be processed.

Output

For each input n and t, print the player's expected prize, if he plays the best strategy. Output should be rounded to three fractional digits.

1 0.5
1 0.3
2 0.6
24 0.25
0 0

1.500
1.357
2.560
230.138

Source

Waterloo local 2005.09.17