Digital Speedometer

A digital speedometer shows a vehicle'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.

Figure 1: Sensor reading from the Sample Input, with $t\_f=0.25$ and $t\_r=0.75$.

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 non-zero speed, no matter how small, must display as non-zero to indicate that the vehicle is in motion.

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 end-of-file 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 is the list of speeds, one speed per line, smoothed to integer values appropriate to $t\_f$ and $t\_r$.