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### Problem Description

A program to illustrate the  use of the "Postfix"  notation  to simplify the evaluation of arithmetic expressions.  Postfix notation is a way to write arithmetic expressions that simplifies their evaluation by computers.  All parentheses are removed and each operation or function is preceded by the values that it needs.

### Background & Techniques

I had used the Postfix data structure previously in programs (Brute Force for example), but was surprised to find that I had not created an object to simplify its use in a more general sense.  I recently posted a program to analyze "age type" story problems and produce the equations which solve the problem.  To actually solve the equations will require an expression evaluator.  This program provides one.

Postfix is also referred to as Reverse Polish Notation (RPN).  RPN was probably the original name, because a Polish mathematician (whose name escapes me) developed it.  Wikipedia has a good Postfix article under "Reverse Polish Notation".

Evaluating an expression using the Postfix structure is a two phase process:

 Convert the input expression in "infix" notation, (the way we normally write expressions), to a list of items in Postfix notation.  Infix separates variables and partial results defined by parentheses by operations such as plus (+), minus (-), multiply (*), and divide (/) operations.  In Postfix form, each binary operation applies to the the two operands which precede it. While building the Postfix list, the implied priority of operations must be honored (multiplications and divisions are performed before additions and subtractions) as well as honoring the use of parentheses to specify phrases to be evaluated first.  So, for example, a+b*c is evaluated by multiplying b*c first and then adding a and the b*c product but (a+b)*c is evaluated by adding a and b and then multiplying that sum by c.  In postfix form, a+b*c   ==> {a, b, c, *, + } and (a+b)*c ==>  {a, b, +, c, *}. Once the Postfix list is built, evaluation proceeds by moving from left to right, replacing each operation and the two numbers preceding it with the result of performing the computation implied by the  operation (add, subtract, multiply, divide).

The program allows the user to define a set of variables and their values in one box and the expressions to be evaluated in another.  The "Evaluate" button displays the results in a third box.   A "Show Solving Steps" checkbox displays the calculation process step by step.

#### Notes for programmers

The program uses a TEval object to implement the evaluation.  The two mandatory methods to call are:

 AddVariable which receives a variable name and the value to substitute when the variable is referenced in an expression. Evaluate which is passed a text string and returns the value of the expression.  If the Evaluate function returns false, a call may be made to GetLastError to retrieve the  text message describing problem found in the expression.

TOpObj is a simple object containing the information about each item in the PostFixList.  A Tokentype field specifies whether the items is a variable, a constant or an operation.  For constants, the value is contained here, for variables, the object contains the index of the variable in a separate Variables list.   For an operation, the operator type is included.

Both phases of the evaluation make use of a temporary stack area.  A "stack" is a last-in, first-out (LIFO) list of items and provides an easy way to reverse the order by using "push" and "pop" operations.  "Push" adds an item to the list, "Pop" retrieves the last item added it and removes it from list.    TEval simulates push and pop using an array (AStack) of TOpObj objects with a counter, SCount.  An item from the Postfix list is added by incrementing SCount and  AStack[SCount] to the object being added.  Pop sets the retrieved item to AStack[SCount] and decrements SCount.

One potentially confusing "trick" used is to store the variable value in the Objects property field of the Variables  Tstringlist.   The Objects property normally holds a pointer to an object to be associated with each string in the list.   Floating point type Single is  4 bytes in length and by typecasting can be stored and retrieved with each in the  Objects property for each  variable defined.    The better solution would be to define a TVariable object containing the values as type Extended  and using Objects entries to hold pointers to these value objects.

There is a lot more that could be said about TEval, but grandkids (and kids) are arriving this afternoon for Thanksgiving celebrations, so I'll let those interested study the code and write with specific questions/suggestions.