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Problem D
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
\[ ^ na=\underbrace{a^{a^{\cdot ^{\cdot ^{\cdot ^{a}}}}}}_ n. \]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 $10^{-5}$.
| 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 |
