Anthony is just now learning basic math, how exciting! He first learns about addition\[ a+n=a+\underbrace{1+1+\cdots +1}_ n, \]

then multiplication\[ a\times n=\underbrace{a+a+\cdots +a}_ n, \]

exponentiation\[ a^n=\underbrace{a\times a\times \cdots \times a}_ n. \]

and finally, tetration\[ ^na=\underbrace{a^{a^{\cdot ^{\cdot ^{\cdot ^{a}}}}}}_ n. \]

Very quickly, Anthony becomes interested in infinite tetrations, namely\[ ^\infty a={a^{a^{\cdot ^{\cdot ^{\cdot }}}}}. \]

Anthony wonders, given an arbitrary real number $N$, what is the solution to $^\infty a=N$? Unable to figure it out, Anthony has asked you to write a program to help him!

Here’s a fun fact: A solution only exists for $\frac{1}{e}\leq N\leq e$.

-----Input-----
The first line of input contains one real number $N$, $0.36788\leq N\leq 2.718281$.

-----Output-----
Output a single line containing a real number $a$, such that $^\infty a=N$. Your answer will be considered correct if its absolute or relative error doesn’t exceed $10^{-5}$.

-----Examples-----
Sample Input 1:
2.000000
Sample Output 1:
1.414214

Sample Input 2:
1.000000
Sample Output 2:
1.000000

Source: kattis | Difficulty: EASY
Tags: Mathematics


Solutions: