Category: math library
Release Date: 05/2012
Technologies: C++, Bison, Flex, GoogleTest
Platforms: Linux, Mac OS X
Download from: Github

C++ library which parses human-like arithmetic expressions like the following:

-3xy^2 - 3(xy + 3)(-5x + y)

To understand better, let's make some examples comparing them to the more traditional computer-like notation:

MExpr notationTraditional computer-like notation
3xy3 * x * y
-3(x + y)-3 * (x + y)
(x + y)(x + 4)(x + y) * (x + 4)


Implicit multiplications

The parser interprets the text to detect implicit multiplication. For example the expression xy is interpreted as x * y, and the expression -3(x + y)(x + 4) as -3 * (x + y) * (x + 4).

One character variables

Like the arithmetic expression which we are used to write in a sheet of paper, the variables are single-character words like: x, y, etc.

That has opened the possibility to recognize implicit multiplications in a expression like: xy^2.

That is also an assumption which we have in our minds, and that allow us to interprets xy as a multiplication between the variables x and y, rather than the single variable xy.

Bytecode compilation

The library parses the input string and then it builds an abstract syntax tree. For example, with the expression -3xy^2, the parser builds the following abstract syntax tree:

[ * ]─[ -1 ]
  └───[ * ]─[ 3 ]
        └───[ * ]─[ x ]
              └───[ ^ ]─[ y ]
                    └───[ 2 ]

At this point, the library can directly evaluate the expression using the tree (browsing it recursively).

In some cases, for example when you want to draw a plot, you need to evaluate the same expression changing only the value of a variable. For this reason, the library can "compile" the AST to have a more efficient representation of the expression. The generated code is a simple bytecode, that uses a stack to compute operations (similarly to the Java bytecode).

This is the representation of the bytecode generated using the previous expression:

VAL: -1
VAL: 3
VAR: x
VAR: y
VAL: 2


MExpr has the possibility to use functions inside the expression.

The library comes with all the functions of math.h. Furthermore, one can also insert new custom functions that can do almost anything!

All the function names starts with the underscore character, for example _f(x). This notation is important because it allows to disambiguate expressions like f(x). In fact, that could be interprets as a multiplication between the variables f and x, or as the call of the function f passing x as argument. On the contrary _f(x) has only one interpretation.

Function overloading

The functions can be overloaded. For example, one can define the function _sum(a,b) that adds two numbers. Furthermore, one can also define _sum(a,b,c). The parser can manage overloaded functions distinguishing the functions by the number of parameters.

Dynamic environment

The parser has a dynamic environment, that maintains all the value associated to a variable symbol. Moreover, for each function symbol, it maintains a pointer to the associated function.

This approach allows one to define a variable or a function even after the expression parsing (as well as the expression compilation). For the same reason one can redefine a previously defined function, or change the value of a variable (useful if one is drawing a plot).

How to use it

The following examples introduces how to use MExpr in your own project. Those examples are also available in the test folder and compiled in the build/test/ folder.

Example 1

This example shows how to create an expression, assign some values to its variables and print its value.

Expression* e = new Expression("-3(4xy^2x-2x)(8x^-(3x)+2y^-2)");
e->setVariable('x', 4);
e->setVariable('y', -5);
cout << e->evaluate() << endl;

Example 2

This example shows how MExpr could be used to repeatedly evaluate the same expression, only changing the value of a variable. The expression is previously compiled, to allow a faster evaluation.

Expression* e = new Expression("x^2");
e->compile(); //To allow a faster evaluations (e.g. to draw a plot)
for (int i = 0; i < 10; i++) {
    e->setVariable('x', i);
    cout << e->evaluate() << ' ';
cout << endl;

Example 3

This example shows how to print in the console the tree representation and the bytecode representation of the expression.

Expression* e = new Expression("-3xy^2");
cout << "Expression AST: " << endl;
string* s = e->getExprTreeString();
cout << *s;
delete s;
cout << endl << "Compiled expression: " << endl;
s = e->getExprCodeString();
cout << *s;
delete s;

Example 4

This example shows how to create and use a custom function inside the expression.

 // myDiv(a, b) = a / b
void myDiv(MExpr::StackType* s) {
    MExpr::ValueType arg1, arg2;
    arg1 = s->stack[s->stp - 2];
    arg2 = s->stack[s->stp - 1];
    s->stack[s->stp - 2] = arg1 / arg2;
int main(void) {
    Expression* e = new Expression("_div(16,4)");
    e->setFunction("_div", &myDiv, 2);
    cout << e->evaluate() << endl;
    return 0;

Compile your code that uses MExpr

The Makefile is configured to create a shared library, you can use it with your C++ programs dynamically linking this library.

g++ -o myExample -lmexpr myExample.cpp

How to compile MExpr

  • Ensure that you satisfy the requirements.
  • Open the Makefile and set your OS changing the OperatingSystem variable.

Run the command you need

  • make all to compile the library
  • make run-tests to run the tests
  • make clean-gtest to remove the GoogleTest library folder
  • make install to install the library in /usr/local/lib

Common issues

  • Mac OS X, by default, doesn't have the /usr/local/lib and /usr/local/include folders, check that you have these folders.
  • You can have some problems compiling your examples if you don't install it before. If you don't want to install it, ensure that the DYLD_LIBRARY_PATH (or the LD_LIBRARY_PATH) was proudly configured.
  • Ensure that you have installed wget, that is used to download the GoogleTest library


Tested operating systems: OSX, Linux

To compile the library you need: bison, flex and g++.

GoogleTests library: but is automatically downloaded in the gtest folder during the building process (i.e. make all).


This project is licensed under the MIT license