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 Ruby (2.3.3) (judge verdict: Runtime Error)

#!/usr/bin/ruby
#require 'yaml'

input = STDIN.readlines
#input = DATA.readlines
n_, t, v = input.map{|l| l.chomp.split(/\s+/).map{|m| m.to_f}.flatten}
n = n_.first.to_i + 1
t <<  + 1
v << 0

accel_point = [[0, 0]]
current_speed = 0
current_time  = 0

n.times do |i|
#  puts "#{i}: t=#{t[i]}, v=#{v[i]}"
  t[i] += current_time # 制限区間の長さ
#  puts "#{i}: t=#{t[i]}, v=#{v[i]}"
  if v[i] < current_speed
    # 手前から減速
    # 減速しきれるか判定
    loop do
      (t1, v1), (t2, v2) = accel_point.last(2)
#      puts " -> brake: #{t1}, #{v1}, #{t2}, #{v2}"
      accel_point.pop
      if (v[i] + t2 - t1) > v1
        # 減速可能
        if (v1 == v2)
          # 一定速度中
#          puts "teisoku"
          accel_point << [current_time - (v2 - v[i]), v2]
        else
          # 加速中
#          puts "kasoku"
          _t = (current_speed - v[i]).to_f / 2
          accel_point << [current_time - _t, current_speed - _t]
        end
        break
      elsif (v[i] + t2 - t1) == v1
        # 丁度減速
#        puts " just"
        break
      else
        # 減速不能
#        puts " impossible..."
        accel_point.pop
      end
    end
    #lim_t = current_time - (current_speed- v[i])
    #accel_point << [lim_t, current_speed]
    accel_point << [current_time, v[i]]
    accel_point << [t[i], v[i]]
  elsif v[i] > current_speed
    # すぐに加速
    #accel_point << [current_time, current_speed]
    if (t[i] - current_time) > (v[i] - current_speed)
      # 途中で制限にかかる
      accel_point << [current_time + v[i] - current_speed, v[i]]
      accel_point << [t[i], v[i]]
#      puts accel_point.to_yaml
    else
      # 最後まで加速できる
      accel_point << [t[i], current_speed + t[i] - current_time]
    end
  else
    # 一定速度
    #accel_point << [current_time, current_speed]
    accel_point << [t[i], current_speed]
  end
  current_time, current_speed = accel_point.last
#  puts "current_time=#{current_time}, current_speed=#{current_speed}"
#  puts accel_point.to_yaml
end
#puts accel_point.to_yaml

# 距離を計算
dist = 0.0
accel_point.each_cons(2) do |a1, a2|
  dist += (a1[1]+a2[1]).to_f/2*(a2[0]-a1[0]).to_f
#  puts dist
end
puts dist.to_f