Problem:
Problem D: Discrete Speed

Consider car trips in a country where there is no friction.
Cars in this country do not have engines.
Once a car started to move at a speed, it keeps moving at the same speed.
There are acceleration devices on some points on the road,
where a car can increase or decrease its speed by 1.
It can also keep its speed there.
Your job in this problem is to write a program
which determines the route with the shortest time
to travel from a starting city to a goal city.

There are several cities in the country,
and a road network connecting them.
Each city has an acceleration device.
As mentioned above, if a car arrives at a city at a speed v ,
it leaves the city at one of v - 1, v , or v + 1.
The first road leaving the starting city must be run at the speed 1.
Similarly, the last road arriving at the goal city must be run at the speed 1.

The starting city and the goal city are given.
The problem is to find the best route which leads to the goal city
going through several cities on the road network.
When the car arrives at a city, it cannot immediately go back the road
it used to reach the city. No U-turns are allowed.
Except this constraint, one can choose any route on the road network.
It is allowed to visit the same city or use the same road multiple times.
The starting city and the goal city may be visited during the trip.

For each road on the network, its distance and speed limit are given.
A car must run a road at a speed less than or equal to its speed limit.
The time needed to run a road is the distance divided by the speed.
The time needed within cities including that for acceleration or deceleration should be ignored.

Input

The input consists of multiple datasets, each in the following format.

n m

s g

x 1 y 1 d 1 c 1

...

xm ym dm cm

Every input item in a dataset is a non-negative integer.
Input items in the same line are separated by a space.

The first line gives the size of the road network.
n is the number of cities in the network.
You can assume that the number of cities is between 2 and 30, inclusive.
m is the number of roads between cities, which may be zero.

The second line gives the trip.
s is the city index of the starting city.
g is the city index of the goal city.
s is not equal to g .
You can assume that all city indices in a dataset (including the above two)
are between 1 and n , inclusive.

The following m lines give the details of roads between cities.
The i -th road connects two cities with city indices
xi and yi ,
and has a distance di (1 ≤ i ≤ m ).
You can assume that the distance is between 1 and 100, inclusive.
The speed limit of the road is specified by ci .
You can assume that the speed limit is between 1 and 30, inclusive.

No two roads connect the same pair of cities.
A road never connects a city with itself.
Each road can be traveled in both directions.

The last dataset is followed by a line containing two zeros
(separated by a space).

Output

For each dataset in the input, one line should be output as specified below.
An output line should not contain extra characters such as spaces.

If one can travel from the starting city to the goal city,
the time needed for the best route (a route with the shortest time)
should be printed.
The answer should not have an error greater than 0.001.
You may output any number of digits after the decimal point,
provided that the above accuracy condition is satisfied.

If it is impossible to reach the goal city,
the string "unreachable" should be printed.
Note that all the letters of "unreachable" are in lowercase.

Sample Input

2 0
1 2
5 4
1 5
1 2 1 1
2 3 2 2
3 4 2 2
4 5 1 1
6 6
1 6
1 2 2 1
2 3 2 1
3 6 2 1
1 4 2 30
4 5 3 30
5 6 2 30
6 7
1 6
1 2 1 30
2 3 1 30
3 1 1 30
3 4 100 30
4 5 1 30
5 6 1 30
6 4 1 30
0 0

Output for the Sample Input

unreachable
4.00000
5.50000
11.25664

Solution in Ruby (judge verdict: Wrong Answer)

class Heap
	require 'forwardable'
	extend Forwardable

	def initialize
		@array = []
		@index = {}
	end

	def add(obj, key)
		@array << [obj, key]
		@index[obj] = @array.size - 1
		lift_up(@array.size - 1)
	end

	def swap(i, j)
		u = @array[i]
		v = @array[j]
		@index[u[0]], @index[v[0]] = j, i
		@array[i], @array[j] = v, u
	end

	def lift_up(i)
		return if i == 0
		j = (i - 1) / 2
		if @array[i][1] < @array[j][1]
			swap(i, j)
			lift_up(j)
		end
	end

	def del_min
		raise "heap is empty!" if empty?
		min = @array[0]
		if size > 1
			swap(0, @array.size-1)
			@array.pop
			lift_down(0)
		else
			@array.pop
		end
		@index.delete(min[0])
		min
	end

	def inc_key(obj, diff)
		i = @index[obj]
		@array[i][1] += diff
		lift_down(i)
	end

	def dec_key(obj, diff)
		i = @index[obj]
		@array[i][1] -= diff
		lift_up(i)
	end

	def update_key(obj, new_key)
		i = @index[obj]
		return unless i
		return if new_key >= @array[i][1]
		@array[i][1] = new_key
		lift_up(i)
	end

	def min_obj
		@array[0][0]
	end

	def min_key
		@array[0][1]
	end

	def lift_down(i)
		j = 2 * i + 1
		k = j + 1
		if k >= size
			return if j >= size
			if @array[i][1] > @array[j][1]
				swap(i, j)
				lift_down(j)
			end
		else
			l = (@array[j][1] < @array[k][1] ? j : k)
			if @array[i][1] > @array[l][1]
				swap(i, l)
				lift_down(l)
			end
		end
	end

	def_delegators :@array, :empty?, :size
end

module Graphical
	require 'set'

	def reachable_from(s, max_depth = nil)
		reached = Set[s]
		queue = [s]
		depth = 0
		loop {
			break if max_depth && depth >= max_depth
			next_queue = []
			until queue.empty?
				u = queue.shift
				each_out_connection_of(u){|v, _|
					next if reached.include?(v)
					reached << v
					next_queue << v
				}
			end
			break if next_queue.empty?
			queue = next_queue
			depth += 1
		}
		return reached
	end

	def dijkstra(start)
		h = Heap.new
		reached = Set[start]
		h.add(start, 0)

		dist = {}
		until h.empty?
			u, key = h.del_min
			return key if block_given? && yield(u, key)
			dist[u] = key
			each_out_connection_of(u){|v, l|
				d = dist[u] + l
				if reached.include?(v)
					h.update_key(v, d)
				else
					h.add(v, d)
					reached << v
				end
			}
		end
		block_given? ? nil : dist
	end

	def bidir_dijkstra(s, ts)
		f_dist = {}
		b_dist = {}

		f_heap = Heap.new
		b_heap = Heap.new

		f_heap.add(s, 0)
		ts.each{|t|
			b_heap.add(t, 0)
		}

		f_reached = Set[s]
		b_reached = Set[*ts]

		f_current_dist = b_current_dist = 0
		best = Float::INFINITY

		loop {
			if !f_heap.empty?
				u, key = f_heap.del_min
				f_dist[u] = key
				f_current_dist = key
				each_out_connection_of(u){|v, l|
					d = f_dist[u] + l
					if b_dist[v]
						path_len = d + b_dist[v]
						best = path_len if path_len < best
					end
					if f_reached.include?(v)
						f_heap.update_key(v, d)
					else
						f_heap.add(v, d)
						f_reached << v
					end
				}
			end

			if !b_heap.empty?
				u, key = b_heap.del_min
				b_dist[u] = key
				b_current_dist = key
				each_in_connection_of(u){|v, l|
					d = b_dist[u] + l
					if f_dist[v]
						path_len = d + f_dist[v]
						best = path_len if path_len < best
					end
					if b_reached.include?(v)
						b_heap.update_key(v, d)
					else
						b_heap.add(v, d)
						b_reached << v
					end
				}
			end

			return best if f_current_dist + b_current_dist >= best
			return nil if f_heap.empty? && b_heap.empty?
		}
	end

	def prim(start)
		h = Heap.new
		h.add(start, 0)
		reached = Set[start]
		tree_len = 0

		until h.empty?
			u, key = h.del_min
			tree_len += key

			each_connection_of(u){|v, d|
				if reached.include?(v)
					h.update_key(v, d)
				else
					h.add(v, d)
					reached << v
				end
			}
		end
		tree_len
	end
end

class Graph
	include Graphical

	def initialize(*options)
		@options = Set[*options]
	end

	def add_node(property = {})
		Node.new(property)
	end

	def add_edge(u, v, property = {})
		e = Edge.new(u, v, property)
		u.add_edge(e)
		v.add_edge(e)
		e
	end
	
	def del_edge(e)
		u, v = e.from, e.to
		u.del_edge(e)
		v.del_edge(e)
		e
	end

	class Node
		def initialize(property)
			@property = property
			@edges = Set.new
		end

		def add_edge(e)
			@edges << e
		end

		def del_edge(e)
			@edges.delete(e)
		end

		def to_s
			@property[:name] || super
		end

		def each_edge
			@edges.each{|e| 
				yield e
			}
		end

		attr_reader :property
	end

	class Edge
		def initialize(u, v, property)
			@opposite = {u => v, v => u}
			@property = property
		end
		attr_reader :opposite, :property

		def to_s
			@property[:name] || super
		end

		def length
			@property[:length] || 1
		end
	end

	def each_out_connection_of(u)
		u.each_edge{|e|
			yield(e.opposite[u], e.length)
		}
	end

	alias :each_in_connection_of :each_out_connection_of
end

itr = 0
loop {
	n, m = gets.split.map &:to_i
	break if n == 0
	start, goal = gets.split.map &:to_i

	$g = Graph.new
	$nodes = (0..n).map {|i| $g.add_node({:index => i}) }
	$limit = {}
	m.times {
		i, j, d, c = gets.split.map &:to_i
		$g.add_edge($nodes[i], $nodes[j], {:length => d, :limit => c})
	}

	unless $g.reachable_from($nodes[start]).include?($nodes[goal])
		puts 'unreachable'
		next
	end

	extend Graphical
	def each_out_connection_of(a)
		u, used_edge, speed = a
		u.each_edge{|e|
			next if e == used_edge
			v = e.opposite[u]
			[speed-1, speed, speed+1].each{|next_speed|
				next if next_speed <= 0
				yield([v, e, next_speed], e.length / next_speed.to_f) if next_speed <= e.property[:limit]
			}
		}
	end

	def each_in_connection_of(a)
		w, used_edge, speed = a
		v = used_edge.opposite[w]
		v.each_edge{|e|
			next if e == used_edge
			u = e.opposite[v]
			[speed-1, speed, speed+1].each{|prev_speed|
				next if prev_speed <= 0
				yield([v, e, prev_speed], used_edge.length / speed.to_f) if prev_speed <= e.property[:limit]
			}
		}
	end

	goals = []
	$nodes[goal].each_edge{|e|
		goals << [$nodes[goal], e, 1]
	}

	t = bidir_dijkstra([$nodes[start], nil, 0], goals) 
	printf("%.7f\n", t);
}