Let's make an expression evaluator that takes strings like "2 * 3 + 1.5", evaluates them, and returns the result (7.5). The four arithmetic operators and parentheses are allowed in the expression. You can use this simple program instead of your desktop calculator, or you can embed it in a larger application (for example, it can be useful in accounting software).

You may think about reading an expression sequentially and do the calculations (for the example above: 2 * 3 = 6, then 6 + 1.5 = 7.5). But, if you do so, you will get wrong results for expressions like "3 – 1 / 2" or "1 + 3 * (25 – 4)". The operators have different priorities; that's why you need a more complex algorithm.

## Recursive descent parsing

A popular approach (called recursive descent parsing) views an expression as a hierarchical structure. On the top level, an expression consists of several **summands**:

Summands are separated by "+" and "–" signs. They include single numbers (1.5, 2, 8) and more complex expressions [3.4 * (25 – 4) / 2]. (Strictly speaking, 8 is not a summand here, but a subtrahend. For the purposes of this article, we will refer to the parts of expression separated by "–" as summands, though a mathematician would say it's a misnomer.)

Each summand, in turn, consist of several **factors** (we will call "factors" the things separated by "*" and "/"):

The summand "1.5" can be viewed as a degenerated case: it constists of one factor, 1.5. The summand "3.4 * (25 – 4) / 2" consists of three factors.

There are 2 types of factors: **atoms** and **subexpressions in parenthesis**. In our simple expression evaluator, an atom is just a number (in more complex translators, atoms can also include variable names). Subexpressions can be parsed in the same way as the whole expression (you again found the summands inside the brackets, then factors, and then atoms).

So, we have **an hierarchy**, where atoms constitute the lowest level, then they are combined into factors, and the factors are combined into summands. A natural way to parse such hierarchical expression is to use recursion:

```
double ParseAtom(char*& expr) {
// Read the number from string
char* end_ptr;
double res = strtod(expr, &end_ptr);
// Advance the pointer and return the result
expr = end_ptr;
return res;
}
// Parse multiplication and division
double ParseFactors(char*& expr) {
double num1 = ParseAtom(expr);
for(;;) {
// Save the operation
char op = *expr;
if(op != '/' && op != '*')
return num1;
expr++;
double num2 = ParseAtom(expr);
// Perform the saved operation
if(op == '/')
num1 /= num2;
else
num1 *= num2;
}
}
// Parse addition and subtraction
double ParseSummands(char*& expr) {
double num1 = ParseFactors(expr);
for(;;) {
char op = *expr;
if(op != '-' && op != '+')
return num1;
expr++;
double num2 = ParseFactors(expr);
if(op == '-')
num1 -= num2;
else
num1 += num2;
}
}
double EvaluateTheExpression(char* expr) {
return ParseSummands(expr);
};
```

The hierarchical structure is directly reflected in this code.

## Comments to the code

We need to save the operation, because we cannot calculate the result when the second operand is still unknown (e.g., you cannot evaluate "2*3" when you have read only "2*"). The *for(;;)* loop allows us to parse several summands, e.g., in "1+2+3".

The program uses **double data type**, which is okay for most applications. If you want to use this code for accounting, you should change to fixed-point data type. Don't use floating-point arithmetic when dealing with money. (If you need an explanation, read about Currency type in Visual Basic 6.0. Also peruse this example in C# and this article from IBM.)

Error handling, parentheses and unary minus parsing are also added to the program below. Only three error messages are used: "parentheses don't match", "invalid character", and "division by zero". When the interpreter sees an unexpected character anywhere, it returns the "invalid character" error code. The position in string where the error occurred is also shown. If there are several errors, only one of them will be reported.

**Download the expression evaluator**, 40 KB

## Recommended reading

- Recursive decent parsing at Wikipedia. A formal definition of the term.
- Jack Crenshaw's tutorial
- "The C++ Programming Language" by Stroustrup has a chapter on expression evaluators.

The next part:

**Code generator**

## 13 comments

Ten recent comments are shown below. Show all comments

Nicholas Wirth's "Compiler Construction"

http://www.oberon.ethz.ch/WirthPubl/CBEAll.pdf

The code and the target are Oberon. Oberon is the language designed to be small and explainable so it's convenient to demonstrate the concepts, and it's not hard to understand.

Pretty nice, although I like Reverse Polish Notation better. Just combine the RPN expression evaluation algorithm with Djikstra's infix to postfix(RPN) algorithm, and you have expression evaluator that is highly configurable - you can implement whatever operators you want (very easy, just add a new operator with its symbol, precedence, left or right association, number of arguments and function that will process the data), have functions as operators, have variables (very easy to add), strings and string variables (a bit more work, but still fairly easy) and at a bit higher level you can put language constructs like loops.

Voila, you've got yourself a very basic scripting language without ever knowing about flex or bison. ;)

good efforts!i appriacet all ur works

Of all the examples Ive perused, this is the easiest to get your head around recursive descent parsing. Also, I have to mention Game Scripting Mastery to anyone without a deep knowledge of this subject (grammars etc). Look for a torrent on google if you're interested.

You're a star! Thank you!

Superb, clear code, and written as a Class too.

Thank you so much for this extremely clear, concise, simple and VALUABLE article! Thanks to this - and the code you provided - I was able to reverse-engineer a perfectly operational version for BlitzMax in a single day... And I hardly even know any C++! This here is a treasure, and makes it so easy to grasp that I was even able to add exponents into the heirarchy with no bother at all!

Thank you!

I know this post is ancient, but I really appreciate finding your equation solver for C++. I wrote a Python solver for the Mathler Hard Mode puzzle on the internet here in Feb 2022, and it ran 13 minutes. I ported it to C++ but there's no solver utility so I looked around at several and like yours the best. Now the solve runs in 39 seconds:-)

If you happen to see this, thank you for the handy tool I can use for some other puzzle solving perhaps. I'll also study it a bit and learn from it I hope. I modified it very slightly to work as a function inside my main program. Converting my character array to a string usable by your solver took a little learning on my part.

I'm just north of San Diego, California, by the way.