Score : $600$ points

问题描述

我们有一个 $N$ 面骰子，每个面分别显示从 $1$ 到 $N$ 的整数，并且出现每个整数的概率相等。
当骰子顶部朝上的面显示整数 $X$ 时，我们称骰子当前显示的值为 $X$。
最初，骰子显示的整数为 $S$。

你可以对这个骰子进行以下两种操作任意次（包括零次），并且可以按任意顺序执行：

• 支付 $A$ 日元（日本货币）以“增加”骰子上显示的数值，即当骰子当前显示 $X$ 时，重新放置它使其显示 $X+1$。但当骰子显示 $N$ 时，不能进行此操作。
• 支付 $B$ 日元来重掷骰子，之后骰子将以相等的概率显示 $1$ 到 $N$ 之间的某个整数。

从初始状态下骰子显示 $S$ 开始，考虑使其最终显示 $T$。
请计算并输出在采用最优策略以最小化期望成本时，完成这一目标所需的最小期望成本。

以上为通义千问 qwen-max 翻译，仅供参考。

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 $10^{-5}$.

Sample Input 1

5 2 4 10 4

Sample Output 1

15.0000000000000000

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

Sample Input 2

10 6 6 1 2

Sample Output 2

0.0000000000000000

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

Sample Input 3

1000000000 1000000000 1 1000000000 1000000000

Sample Output 3

1000000000000000000.0000000000000000

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

update @ 2024/3/10 09:52:18