Problem:
﻿Score : 400 points

Problem Statement

In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.

In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.

According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

1 \leq N \leq 100

1 \leq t_i \leq 200

1 \leq v_i \leq 100

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N

Output

Print the maximum possible that a train can cover in the run.

Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Sample Input 1

1
100
30

Sample Output 1

2100.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 30 seconds, it accelerates at a rate of 1m/s^2, covering 450 meters.

In the subsequent 40 seconds, it maintains the velocity of 30m/s, covering 1200 meters.

In the last 30 seconds, it decelerates at the acceleration of -1m/s^2, covering 450 meters.

The total distance covered is 450 + 1200 + 450 = 2100 meters.

Sample Input 2

2
60 50
34 38

Sample Output 2

2632.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 34 seconds, it accelerates at a rate of 1m/s^2, covering 578 meters.

In the subsequent 26 seconds, it maintains the velocity of 34m/s, covering 884 meters.

In the subsequent 4 seconds, it accelerates at a rate of 1m/s^2, covering 144 meters.

In the subsequent 8 seconds, it maintains the velocity of 38m/s, covering 304 meters.

In the last 38 seconds, it decelerates at the acceleration of -1m/s^2, covering 722 meters.

The total distance covered is 578 + 884 + 144 + 304 + 722 = 2632 meters.

Sample Input 3

3
12 14 2
6 2 7

Sample Output 3

76.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 6 seconds, it accelerates at a rate of 1m/s^2, covering 18 meters.

In the subsequent 2 seconds, it maintains the velocity of 6m/s, covering 12 meters.

In the subsequent 4 seconds, it decelerates at the acceleration of -1m/s^2, covering 16 meters.

In the subsequent 14 seconds, it maintains the velocity of 2m/s, covering 28 meters.

In the last 2 seconds, it decelerates at the acceleration of -1m/s^2, covering 2 meters.

The total distance covered is 18 + 12 + 16 + 28 + 2 = 76 meters.

Sample Input 4

1
9
10

Sample Output 4

20.250000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 4.5 seconds, it accelerates at a rate of 1m/s^2, covering 10.125 meters.

In the last 4.5 seconds, it decelerates at the acceleration of -1m/s^2, covering 10.125 meters.

The total distance covered is 10.125 + 10.125 = 20.25 meters.

Sample Input 5

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70

Sample Output 5

20291.000000000000

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

#[allow(unused_macros)]
macro_rules! debug {
  ($($e:expr), *) => {println!(concat!($(stringify!($e), " = {:?}\n"), *), $($e), *)}
}

fn main() {
  let _ = get::val::<usize>();
  let ts = get::list::<f64>();
  let mut vs = get::list::<f64>();
  
  vs.push(0.0);
  let mut ans = 0.0;
  let mut speed = 0.0; 
  for (&t,v) in ts.iter().zip(vs.windows(2)) {
    let vmax = t / 2.0 + (v[1]-speed) / 2.0;
    let d = |x,y| if f64::max(x,y) == y {0.0} else {x-y};
  
    let ac = f64::min(vmax, d(v[0],speed)) * f64::min(vmax, d(v[0],speed)) / 2.0 + speed * f64::min(vmax, d(v[0],speed));
    let kp = v[0] * f64::max(0.0, t - d(v[0],speed) - d(v[0],v[1]));
    let rd = f64::min(d(t,vmax), d(v[0],v[1])) * f64::min(d(t,vmax), d(v[0],v[1])) / 2.0 + v[1] * f64::min(d(t,vmax), d(v[0],v[1]));
    
    ans += ac + kp + rd;
    speed = f64::min(v[0], f64::min(speed + t, v[1]))
  }
  
  println!("{}", ans)
}

#[allow(dead_code)]
mod get {
  use std::io::*;
  use std::str::*;

  pub fn val<T: FromStr>() -> T {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    buf.trim_right().parse::<T>().ok().unwrap()
  }

  pub fn vals<T: FromStr>(n: usize) -> Vec<T> {
    let mut vec: Vec<T> = vec![];
    for _ in 0 .. n {
      vec.push(val());
    }
    vec
  }

  pub fn tuple<T1: FromStr, T2: FromStr>() -> (T1, T2) {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    let mut it = buf.trim_right().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    (x, y)
  }

  pub fn tuples<T1: FromStr, T2: FromStr>(n: usize) -> Vec<(T1, T2)> {
    let mut vec: Vec<(T1, T2)> = vec![];
    for _ in 0 .. n {
      vec.push(tuple());
    }
    vec
  }

  pub fn tuple3<T1: FromStr, T2: FromStr, T3: FromStr>() -> (T1, T2, T3) {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    let mut it = buf.trim_right().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    let z = it.next().unwrap().parse::<T3>().ok().unwrap();
    (x, y, z)
  }

  pub fn tuple3s<T1: FromStr, T2: FromStr, T3: FromStr>(n: usize) -> Vec<(T1, T2, T3)> {
    let mut vec: Vec<(T1, T2, T3)> = vec![];
    for _ in 0 .. n {
      vec.push(tuple3());
    }
    vec
  }

  pub fn list<T: FromStr>() -> Vec<T> {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    buf.trim_right().split_whitespace().map(|t| t.parse::<T>().ok().unwrap()).collect()
  }

  pub fn lists<T: FromStr>(h: usize) -> Vec<Vec<T>> {
    let mut mat: Vec<Vec<T>> = vec![];
    for _ in 0 .. h {
      mat.push(list());
    }
    mat
  }

  pub fn chars() -> Vec<char> {
    let mut buf = String::new();
    let s = stdin();
    s.lock().read_line(&mut buf).ok();
    buf.trim_right().chars().collect()
  }
}