Problem H
Digital Speedometer
A digital speedometer shows a vehicleâ€™s speed as integer miles per hour. There are occasions when the sensed speed varies between two integer values, such as during cruise control. Using a single threshold to round between adjacent integers often makes the display toggle rapidly between the two integers, which is distracting to the driver.
Your team must implement a smoothing technique for the display using separate rising and falling thresholds ($t_ r$ and $t_ f$, $t_ f < t_ r$, respectively). See Figure 1 for a graphical depiction of the Sample Input for use with the following rules.
Each sensed speed, $s$, falls between two adjacent integers $i$ and $j$, $i \le s < j$, where $j = i + 1$. When displaying the sensed speed $s$ as an integer:

When $s$ falls between $i$ and $i+t_ f$, $s$ is displayed as $i$.

When $s$ falls between $i+t_ r$ and $j$, $s$ is displayed as $j$.

When $s$ falls between $i+t_ f$ and $i+t_ r$, $s$ is displayed as $i$ if the most recent preceding value for $s$ outside of range $[i+t_ f, i+t_ r]$ is less than $i+t_ r$, and $s$ is displayed as $j$ if the most recent preceding value for $s$ outside of range $[i+t_ f, i+t_ r]$ is greater than $i+t_ r$.

Any sensed speed, $0 < s < 1$, must display as $1$ because any nonzero speed, no matter how small, must display as nonzero to indicate that the vehicle is in motion.
Input
The first line of input contains $t_ f$, the falling threshold. The second line of input contains $t_ r$, the rising threshold. The speed sensor reports $s$ in increments of $0.1$ mph. The thresholds are always set halfway between speed increments. All remaining lines until endoffile are successive decimal speeds, $s$, in miles per hour, one speed per line. The third line of input, which is the first measured speed, will always be $0$. There are at most $1000$ observed speeds $s$ in input.
\[ 0 < t_ f,t_ r < 1; \ \ \ \ t_ f < t_ r; \ \ \ \ 0 \le s \le 120 \]Output
Output is the list of speeds, one speed per line, smoothed to integer values appropriate to $t_ f$ and $t_ r$.
Sample Explanation
Input 
Output 
Explanation 
0.25 
Value of $t_ f$. 

0.75 
Value of $t_ r$. 

0 
0 
Initial input. 
2.0 
2 
Input greater than $0$, below threshold of $2.25$. 
5.7 
5 
Input greater than $2.0$, in threshold range. 
5.8 
6 
Input greater than $2.0$, exceeds upper threshold of $5.75$. 
5.7 
6 
Input less than $5.8$, in threshold range. 
5.2 
5 
Input less than $5.8$, below threshold of $5.25$. 
5.7 
5 
Input greater than $5.2$, in threshold range. 
0.8 
1 
Input greater than $0$ and less than $1$. 
0.2 
1 
Input greater than $0$ and less than $1$. 
Sample Input 1  Sample Output 1 

0.25 0.75 0 2.0 5.7 5.8 5.7 5.2 5.7 0.8 0.2 
0 2 5 6 6 5 5 1 1 