Update  November 15, 2013: Full source code of the sample software also made available (besides the already available library code).
Update  April 12, 2013: A much improved and more efficient version of Math Processor calculation engine as well as a complete new software making use of it made available. Primarily the new optimizations have been implemented in 'repeat' and 'while' loops. It also contains bug fixes and provides serialization mechanism primarily for persisting the state on disk. A comparison might help:
{a=2;:b=4;:time=ticks();:repeat(100000){4*(4*atan(1/5.0)atan(1/239.0))+a+b;:} ticks()  time:}
The old engine took about 4 seconds on my machine to execute the above code. However, in the new WPF based executable it takes just around 1 second!
Table of Contents
 Introduction
 Using the Sample Applications
 Introduction to Math Processor
 Using the Math Processor Library
 Understanding the Core Engine
1. Introduction
First, the best thing about this article is that we will try to make everything extremely simple.
We will not use BIG words and complex theories. You are NOT required to
be a master of (or even familiar with) the details that are
taught in colleges when you study computer languages, parsers, compiler
etc. What distinguishes this article from numerous other resources on the Internet, as well as on the Code Project, about this subject is this very simplicity of use.
Also, we are going to try something more interesting and challenging than a mere mathematical expression parser. We will define a simple programming language and we will create an interpreter for it. The engine we create will allow operations on large sets of numeric data, matrices and Boolean values. It will also give us a unified platform supporting function calls, user defined functions, extensibility through addons and a lot more!
Before we get into details, let's have a look at a few examples of the commands that the parser/engine we are going to create will execute. I hope that these examples will help you understand the scope of this article as well as the sample applications and the class library before we start the main discussion.
Commands 
Result 
Comment 
>> 2+6*(12/25) 
4.8800 
Evaluation of a simple arithmetic expression 
>> x = array(32, 30, 36, 34, 33, 35, 40, 39)
>> d = sum(abs(x  median(x))) / count(x) 
2.6250 
Deviation of the given data set using the Median. 
>> primes(1000000) 
2 3 5 ... 
The first one million prime numbers 
>> prime(10) 
29 
The 10th prime number 
>> gcd(24,12, 124, 136) 
4 
GCD of the given numbers 
>> a = array(50, 10, 50)
>> b = array(60, 40, 50)
>> c = array(80, 60, 10)
>> m = matrix(a, b, c)
>> d = det(m) 
76000 
Determinant of a matrix 
In addition to the kind of commands given above, our parser also provides a simple graphing module as demonstrated in the next table.
Note: You can copypaste the command blocks given in the first column (including the curly brackets) into the sample application and execute them to see how graphs are created in Math Processor (MP).
Command 
Graph 
Comment 
{
x= vectorin(0, 0.05, 2*pi);:
y= sin(x);:
plot(x, y, x, y*1.5, x, y*2, x, y*2.5);:
}


Sine curves. 
{
theta= vectorin(0, 0.05, 2*pi);:
x= 4 * cos(theta) ^ 3;:
y= 4 * sin(theta) ^ 3;:
plot(x, y, green);:
}


Astroid (hypocycloid of 4 cups). 
{
t = vectorin(16, 0.05, 16);:
x = sin(8*t/5) * cos(t);:
y = sin(8*t/5) * sin(t);:
plot(x, y);:
}


A polar curve. 
To make life more interesting, we will also implement a loosely typed language called Math Processor Command Language (MPCL). Here is an interesting example of a User Defined Function taken from the MP documentation page (to get more details on it see User Defined Functions).
spirograph = function (v_R, v_r, v_p, v_nRotations, s_color)
{
t = vectorin(0, 0.05, 2 * pi * v_nRotations);:
x = (v_R + v_r) * cos(t) + v_p * cos((v_R + v_r) * t / v_r);:
y = (v_R + v_r)* sin(t) + v_p * sin((v_R + v_r) * t / v_r);:
plot(x, y, s_color):
}
This function will draw some very interesting Spirograph patterns using the MP graphing module.
>> spirograph(53, 29, 40, 30, gray)
>> spirograph(52, 25, 36, 25, green)
>> spirograph(43, 19, 17, 39, red)
You can have a look at some of the artwork I created using a more sophisticated implementation of the above technique as a separate little software.
If you are interested in getting more details on how this user defined function actually created those patterns, have a look at this Wikipedia article.
2. Using the Sample Applications
Update August 28, 2013: This section was written for the old versions. Since a new software with complete documentation became available (top downloads), I deleted the contents of this section so as to make the article easier/smaller for new readers. Anyone interested can still read the old stuff on the previous versions of the article (no one need to, though). The old downloads can be got from these links
3. Introduction to Math Processor
I hope that by this time you have played with the Math Processor application a bit. If not, I recommend that you do so now and try to get familiar with its working. I know it sounds cumbersome; but believe me, you will find it worth the effort as we move forward.
The Math Processor engine provides a lot more functions and commands than discussed in this article. However, mentioning everything the engine does is beyond the scope of this article as well as unnecessary. If you are interested in knowing more about the facilities provided by Math Processor, you can have a look at the Math Processor Documentation.
3.1. The Road Ahead
We will be building an interpreter for math expressions as well as a simple little language MPCL (but still more powerful than you probably think). More on MPCL and what we will be implementing in next section. Before we get into that, here is what we will NOT be doing:
 Support symbolic math
 Perform arbitrary precision calculations
 Implement things the hard way
Despite the above limitations, the engine is still reasonably powerful. In fact, it is a perfect calculator for most people's everyday calculation work (well, those people who do math). Since MP's first working version, I have exclusively used it for 99.9% of my calculation needs!
3.2. Math Processor Command Language (MPCL)
MPCL is a simple loosely typed language the Math Processor parser/calculating engine can interpret at run time to carry out the supported operations. MPCL supports:
 Basic math expressions containing the common operations like +, , *, /, % etc.
 Boolean/logical operations
 Operations on numbers (including lists of numbers)
 Operations on Matrices
 Conditional Processing using the if and ifelse constructs
 Loops (repeat and while)
 Certain directives to control a few aspects of the processing environment
 Inbuilt functions to expose the major facilities of the
System.Math
class of the .Net framework as well as a lot of other operations. For the complete list of supported functions, visit the Math Processor Functions page.
 User defined functions
 An extremely simple mechanism to allow programmers to write more useful functions and plug them into the main engine at run time.
4. Using the Math Processor Library
Math Processor provides a unified application storage and communication mechanism within the internal modules as well as for the programming interface through the use of class Token
. A Token
is capable of storing text, numbers, matrices, Boolean data and a lot more things that MP uses internally. What is contained within a Token
depends upon its type, which is determined by a public property TokenType of type TokenType
. This property is just an enum defined as under:
public enum TokenType
{
Void, Error, Operator, Bool, Vector, Matrix, Text, Function, UserFunction, Directive,
Block, LoopOrCondition, FunctionDefiner, Break
}
To the user of Math Processor, not all members of this enum are of interest. A few are for internal use only. We will learn about the important ones later in this text. The different fields defined in class Token
are declared as under:
TokenType type;
string name = "";
string data = "";
List<double> vector = new List<double>();
double extra;
And that's all that we need to support all the kinds of data we want to process. From numbers to Boolean values, from lists to matrices and all that. Amazing! Isn't it? The trick is to have different interpretations of the fields according to the type of the token. So for a Token of type Void, most fields mean nothing. It is just what it is called: Void. For a Boolean type, a 0 in the list would mean false and a 1 would mean true. For a matrix, we will make use of 'extra' to store the number of rows. We will store the actual matrix values in the List<double>
field. We will compute the rows and columns for the matrix at run time.
Please note that by stuffing everything inside a Token
, we have traded some efficiency (in terms of speed and memory) for simplicity. I believe the tradeoff here is worth the cost, as we have avoided the need to create complex data structures. You can devise a more complex model if you wished. For me, life is much simpler without parse tables, expression trees and a thousand look up fields!
Now that we have understood the role as well as form of the class Token
it is time to discover how to make use of the Math Processor library. There are two possibilities:
 Directly send the raw expressions to the parsing engine, obtain the result and do whatever is needed with it.
 Create instances of class
Token
in code and then use different public methods of the class library to manipulate them.
Both the approaches are quite simple. However, the first one is the preferred way to do things in most situations. In the sample applications we have primarily used the first approach. However, in the Console based sample we have also shown how to use the second method.
The second method is needed only when you want to extend the functionality of the application by providing more functions or when you want to directly work with the already defined functions inside the main class library. We shall see a use of this approach a bit later.
The use of the first method is just as easy as calling the following static method of MathProcessor.Calculator
class:
public static Token ProcessCommand (string expression) { ... }
The parameter expression is any text that is a valid MPCL command. For a broad list of what constitutes a valid MPCL command please refer to Section 3.2. The result returned from this method would be a Token
of any of the following types:
TokenType.Void
The operation completed successfully but nothing remarkable was returned to the caller. Note that the operation itself may have produced variables or functions etc. depending on the nature of the operation performed.
TokenType.Error
The operation resulted in an error.
TokenType.Text
The resulting Token
is just plain text.
TokenType.Vector
The returned Token
contains one or more numeric values.
TokenType.Matrix
The returned Token
is a matrix. 
TokenType.Bool
The returned Token
contains Boolean data (true/false).
Another notable fact is that some operations produce intermediate results before the whole block of code is parsed and executed. For example, the following block of MPCL code produces output multiple times by use of the repeat loop (paste it in the demo application to see what it does!):
{
format(G);:
a = 1;:
repeat(10)
{
echo("4 x ", a, "= ", 4*a):
a = a + 1;:
}:
}
In order to inform the frontend about the intermediate output, the Calculator
class raises events using the following delegate:
public delegate void IntermediateResult(Token result);
Now that we have understood the communication mechanism, let's build a very simple front end for the MP engine. Perform the following steps:
 Create a new Console application in Visual Studio.
 Add a reference to "MathProcessor.dll" (available with downloads of this article)
 Modify your Program class as under:
class Program
{
static void Main(string[] args)
{
Calculator.IntermediateResultProduced += new IntermediateResult(IntermediateResultProduced);
string input = "";
Console.ForegroundColor = ConsoleColor.White;
Console.WriteLine("Enter MP expressions to process. Enter 'exit' to quit:");
Console.ForegroundColor = ConsoleColor.Gray;
while (true)
{
input = Console.ReadLine();
if (input.Trim().ToLower() == "exit")
{
break;
}
Token result = Calculator.ProcessCommand(input);
DisplayResult(result);
}
}
static void IntermediateResultProduced(Token result)
{
DisplayResult(result);
}
static private void DisplayResult(Token result)
{
if (result.TokenType == TokenType.Error)
{
Console.ForegroundColor = ConsoleColor.Red;
Console.WriteLine(">> " + result.GetString());
Console.ForegroundColor = ConsoleColor.Gray;
}
else if (result.TokenType == TokenType.Matrix)
{
int rows = (int)result.Extra;
if (rows > 0)
{
List<double> data = result.Vector.ToList();
Token temp;
temp = new Token(TokenType.Vector, data.GetRange(0, result.Count / rows).ToArray());
Console.WriteLine(">> " + temp.GetString());
for (int i = result.Count / rows; i < result.Count; i += result.Count / rows)
{
temp = new Token(TokenType.Vector, data.GetRange(i, result.Count / rows).ToArray());
Console.WriteLine(" " + temp.GetString());
}
}
}
else if (result.TokenType != TokenType.Void)
{
Console.WriteLine(">> " + result.GetString());
}
}
}
 That's it!
After writing this little program, almost all the MP functions are now available to us on the command line. However the function plot will not be available, as it creates a Windows Form and calls its Show() method, which requires a message loop not available to us in this example. The sample application uses a workaround. Please have a look at the downloadable code to see how the workaround is implemented.
4.1. Extending the Core by Adding More Command Line Functions
Another problem we have is the lack of our ability to paste multiline code blocks in the Console application. Such an operation is not directly available. Let's write a new MP command line function to allow us to 'paste' text content directly from the Clipboard (the sample application contains this functionality). Add a reference to Windows.Forms
assembly to the project and put the following method in your Program
class:
public static Token Paste(string operation, List<Token> arguments)
{
try
{
string text = System.Windows.Forms.Clipboard.GetText();
return Calculator.ProcessCommand(text);
}
catch (Exception)
{
return new Token(TokenType.Error, "Could not paste text from Clipboard.");
}
}
Now modify your Main()
method as under:
[STAThread]
static void Main(string[] args)
{
Function.AddDirective("paste", Paste);
}
Please note the use of [STAThread]
Attribute with the Main()
method. This is needed for working with the Clipboard. After making the above changes, you will have a new directive called "paste" available to you. You can also make it a function, if you so desired, like this:
Function.AddFunction("paste", Paste);
In MPCL, the difference between a directive and a function is that a function is always followed by a pair of parentheses containing zero or more parameters. Also the parameters to functions are processed before the function is called. For a directive, the parameters are not perprocessed. For more detail about directives, please refer to Math Processor Directives page.
Most of the times you will be working with functions. The directives are just a few in number and mostly deal with the configuration of the engine and are not meant for real processing.
5. Understanding the Core Engine
Let's begin with a simple diagram showing the workflow of Math Processor.
The expression from the client is first converted into tokens by the parser/tokenizer. The list of tokens is then converted from Infix to Postfix (Reverse Polish) notation. After that, the tokens are processed by the calculation engine and the result is sent back to the client.
The conversion from Infix to Postfix form is done to facilitate the evaluation phase. Due to the way expressions are represented in Postfix notation, changing the priority of operations does not require use of parentheses, which simplifies the evaluation process. You can see the conversion from Infix to Postfix notation using the postfix directive of Math Processor. Here are a couple of examples:
>> postfix 2*9+3
>> 2 9 * 3 +
>> postfix 2*(9+3)
>> 2 9 3 + *
In the above examples, we can see there is not need to use parentheses in the postfix form, as the priority of operations is inbuilt in the very expressions. In the first example the multiplication takes precedence according to general rules of mathematics. However, in the second example, the Infix form changes the priority by use of parentheses. This changed priority is reflected in the Postfix form through reordering of the tokens, which enables us to get rid of the parentheses.
So far so good. But how does the process start and continue and end? Actually it is done in different stages. First, we need to parse the ordinary expression to create the tokens. Then we need to arrange these tokens in a suitable way for processing. After that comes the actual calculation. Finally the result is returned back to the user.
The whole process takes place inside 7 core classes consisting of just a few hundred lines of code. The details of these classes are as under:
5.1. Token
This is the ubiquitous data holder of Math Processor. We already discussed Token
in detail earlier in this text in Part4.
5.2. Tokenizer
This class is responsible for parsing the input and converting it into Token
objects. The entire process takes place inside a static function:
public static Token TokenizeString(String expression, out List<token> tokens)
The first parameter 'expression' is the expression to be tokenized. The expression is any valid MPCL command (consisting of math expressions, functions, directives and so on). The second parameter out List<Token> tokens
would contain the result of what tokens were found in the input expression. Lastly, the return value will tell about the result of the parsing and can be either of type TokenType.Void
in case of success or TokenType.Error
in case of erroneous input expression.
5.3. Variables
This class is responsible for storing reserved words and named constants and variables. It defines a small inner class NamedToken
to store the named Token
objects. A variable can be created using the assignment operator. Some MP functions can also create variables. The name of a variable must start with a letter or underscore and can consist of letters and numbers. Here is how a variable can be created and used:
>> a = 10
>> 10
>> b = a + 100
>> 110
5.4. Function
This class stores all the functions that can be used using MPCL function calls. Every MP commandline function should be of the form defined by the following delegate:
public delegate Token FunctionDelegate(string operation, List<token> arguments);
Let's see the whole process in action. MP provides a function named contains, which is used to check if a number is present in a matrix or array of numbers. This is how this function is implemented:
public static Token Contains(string operation, List<Token> arguments)
{
if (arguments.Count != 2)
return Token.Error("Exactly two arguments expected");
if ((arguments[0].TokenType != TokenType.Matrix && arguments[0].TokenType != TokenType.Vector) 
(arguments[1].TokenType != TokenType.Matrix && arguments[1].TokenType != TokenType.Vector) 
arguments[1].Count != 1)
return Token.Error("First argument should be an array or matrix and second a single numeric value");
if (arguments[0].Vector.Contains(arguments[1].FirstValue))
return new Token(TokenType.Bool, 1, 1);
else
return new Token(TokenType.Bool, 1, 0);
}
Let's now see how we can use this function:
>> a = array(1,2,3,4);
>> contains(a, 1)
>> true
>> contains(a, 5)
>> false
>> if (contains(a, 2)) { echo("Number found in the array."): }
>> Number found in the array.
5.5. FunctionDefiner
This class is responsible for creating, managing and executing User Defined Functions, which are a very important means to allow the users to create small amounts of code to extend the functionality of the engine at run time. User defined functions are not really efficient due to their interpreted nature. However, they are still useful for doing small chunks of work you need to perform repeatedly.
We already saw a very interesting example of a User Defined Function earlier in this text when we created some very interesting curves (Spirograph curves). To get more details and see other examples of User Defined functions, you can refer to the User Defined Functions page of the online MP documentation.
5.6. BlockCommands
This class is responsible for managing and executing the if/ifelse, while and repeat constructs. The reason that these constructs are put together is their common syntactical form:
key_word(expression) { one or more statements }
For an explanation of each of these see the Conditional Statements and Loops pages.
5.7. Calculator
We already saw the Calculator.ProcessCommand()
method in action, which was used to send all the expressions to the engine for processing. Besides providing that facility, this class also contains all the code to carry out basic arithmetic calculations involving the supported operators like +, , * etc. It also handles the conversion from Infix to Postfix notation, determines the priority of operators and applies these operators to operands to obtain the results.
Conclusion
Finally, we have reached the end. A lot more remains to be said as the subject is quite vast. I hope you found the article, the main engine and the sample applications helpful. In case there remains anything you found unclear or lacking in this article or you think any specific topic needs more coverage, please let me know. I will try to meet your expectations in a revised version or another article.
I also hope I was able to deliver my promise of creating a simple parsing library and calculation engine for you to use in your own projects. Please do provide your feedback; it will help improve this article as well as the Math Processor engine.