Part $1$: Postfix Expression Evaluation

Create a Java program that reads and evaluates postfix expressions containing single$-$digit non$-$negative integers $(0$ to $9)$ and five arithmetic operators $(+,-,*,/,^).$ You$$ll also allow users to define single$-$letter variables, like a $=3,$ before evaluating expressions that incorporate these variables. For example, in the input a b $+$ c $*,$ if a $=3,$ b $=4,$ and c $=5,$ the program should evaluate the expression correctly, treating it as $34+5*.$

To make this more challenging, you need to:

$-$ Implement error handling to detect and attempt to correct common errors in the postfix expression $($e$.$g$.,$ too many operators or missing operands$).$ For instance, if the input is $53+*,$ the program should suggest removing $*$ to make it a valid expression.

$-$ Optimize the expression: look for parts of the expression that could be precomputed to speed up evaluation. For instance, $23+4*$ can be simplified to $20$ before evaluating.

$-$ Support floating$-$point variables and conversions between integer and floating$-$point values.

Next,

$$ implement a timer to measure how long it takes for the program to evaluate expressions, and analyze the impact of any optimizations.

$$ Provide at least five test examples in your report, showing the infix and postfix versions of each expression, the final result, and the evaluation time.

Draw a diagram outlining the steps taken to evaluate an expression in your program. This diagram should include each step, from reading