Tetration

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*

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 $\frac{1}{e}\leq N\leq e$.

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

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 $10^{-5}$.

Sample Input 1 | Sample Output 1 |
---|---|

2.000000 |
1.414214 |

Sample Input 2 | Sample Output 2 |
---|---|

1.000000 |
1.000000 |

Sample Input 3 | Sample Output 3 |
---|---|

1.500000 |
1.310371 |