G - Roll or Increment

Score : $600$ points

Problem Statement

We have a $N$-face die (singular of dice) that shows integers from $1$ through $N$ with equal probability.
Below, the die is said to be showing an integer $X$ when it is placed so that the top face is the face with the integer $X$.
Initially, the die shows the integer $S$.

You can do the following two operations on this die any number (possibly zero) of times in any order.

• Pay $A$ yen (the Japanese currency) to "increase" the value shown by the die by $1$, that is, reposition it to show $X+1$ when it currently shows $X$. This operation cannot be done when the die shows $N$.
• Pay $B$ yen to recast the die, after which it will show an integer between $1$ and $N$ with equal probability.

Consider making the die show $T$ from the initial state where it shows $S$.
Print the minimum expected value of the cost required to do so when the optimal strategy is used to minimize this expected value.

Constraints

• $1 \leq N \leq 10^9$
• $1 \leq S, T \leq N$
• $1 \leq A, B \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $S$ $T$ $A$ $B$

Output

Print the answer. Your output will be considered correct when its absolute or relative error is at most $10^{-5}$.

5 2 4 10 4

15.0000000000000000

When the optimal strategy is used to minimize the expected cost, it will be $15$ yen.

10 6 6 1 2

0.0000000000000000

The die already shows $T$ in the initial state, which means no operation is needed.

1000000000 1000000000 1 1000000000 1000000000

1000000000000000000.0000000000000000

Your output will be considered correct when its absolute or relative error is at most $10^{-5}$.