Introduction to Parsing - 5

In the last part we have improved our parser so that it can now handle basic arithmetics and floating number. This time, we’ll teach it how to call functions!

Identifiers in the lexer

Our lexer is currently working with this grammar:

token : number | OPERATOR;

number : DIGIT [DIGIT]* ['.' DIGIT*];

OPERATOR: '+' | '-' | '*' | '/' | '(' | ')';

DIGIT: '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9';

To add support for functions, we are going to add a new rule:





With these rules we have allowed identifiers to contain digits, but not in the first position: they must start with a letter.

Our token rule changes as follows:

token : number | identifier | OPERATOR;

Let’s see the code. First we have to define a new token type:

enum class TokenType
    IDENTIFIER,			// This is the new one

We can now add some simple tests:

    CASE("lexing 'sin x'") {
        std::istringstream input{"sin x"};
        Lexer lexer(input);

        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "sin"));

        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "x"));

    CASE("lexing 'cos (3 * x ) + sin x'") {
        std::istringstream input{"cos (3 * x ) + sin x"};
        Lexer lexer(input);

        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "cos"));
        EXPECT(lexer.nextToken() == Token(TokenType::OPERATOR, "("));
        EXPECT(lexer.nextToken() == Token(TokenType::NUMBER, "3"));
        EXPECT(lexer.nextToken() == Token(TokenType::OPERATOR, "*"));
        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "x"));
        EXPECT(lexer.nextToken() == Token(TokenType::OPERATOR, ")"));
        EXPECT(lexer.nextToken() == Token(TokenType::OPERATOR, "+"));
        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "sin"));
        EXPECT(lexer.nextToken() == Token(TokenType::IDENTIFIER, "x"));


Modifying the lexer so that it passes the tests is quite simple. As we can see, the rule for identifier is quite similar to the one for number. Similarly, the new function parseIdentifier is quite similar to the old parseNumber:

Token Lexer::parseIdentifier()
    // Add the first character to the identifier
    std::string identifier{next_};

    // Match more identifier parts
    while (!atEof_ && isIdentifierPart(next_)) {
        identifier += next_;

    return Token(TokenType::IDENTIFIER, identifier);

bool Lexer::isIdentifierStart(char candidate) const
    return std::isalpha(candidate);

bool Lexer::isIdentifierPart(char candidate) const
    return isIdentifierStart(candidate) || std::isdigit(candidate);

We also have to change nextToken to handle the new rule:

Token Lexer::nextToken()
    if (atEof_) {
        return Token(TokenType::END_OF_INPUT, "");

    if (std::isdigit(next_)) {
        return parseNumber();
    } else if (isIdentifierStart(next_)) {
        return parseIdentifier();
    } else {
        return parseOperator();

With this changes, we can pass the tests!

Functions in the parser

Now that we have support for identifiers in our lexer, we can move to the parser. For the moment we are going to have a static catalog of functions; eventually we’ll let the user extend it. For us, a “function” in our language will be a C++ function taking one double and returning a double:

    using doubleToDoubleFunction = double(*)(double);

This is a relatively new syntax introduced in C++11. It means that doubleToDoubleFunction is a pointer to a function taking a double and returning a double. Using C and C++98’s syntax, we would have written this:

    typedef double(*doubleToDoubleFunction)(double);

Personally I find the new syntax quite nice and readable. Moving on, we are going to define the catalog of functions supported as this:

    std::map<std::string, doubleToDoubleFunction> functions_ {
        {"exp", std::exp},
        {"log", std::log},
        {"sin", std::sin},
        {"cos", std::cos},
        {"tan", std::tan}

Here we are using two new C++11 constructs at once. By using an initializer list we can immediately fill our map, which is a great improvement over the C++98 syntax. Furthermore, we are initializing a member variable directly at the member’s declaration, and not in the constructor. This is only a cosmetic change, but a very welcome one, especially in classes with many constructors.

Let’s now focus on the grammar. How are we going to extend it to allow function calls? Let’s first think about some examples: we definitely want sin 3 to be a valid function call. But what about sin 3 * x? It’s quite reasonable to interpret it either as (sin 3) * x or as sin( 3 * x ). We are going to settle for the first alternative, mostly because we’d like to interpret the (similar) case sin 3 + x + y as (sin 3) + x + y rather than any other alternatives. So, the new grammar is going to be as follows:

expression: term [ ('+'|'-') term ]*;

term : factor [ ('*'|'/') factor]*;

factor :   NUMBER
         | '(' expression ')'
         | function-call;

function-call: identifier factor;

We have defined function-call to be an identifier, i.e. the function name, followed by a factor. With this definition, we’ve implemented the rules we defined above. Furthermore, this syntax allows quite natural expressions such as sin(42), since (42) is a valid factor. If we had decided that we wanted sin 3 * x to mean sin( 3 * x ), we could have done it quite easily by changing the function-call definition to be an identifier followed by a term.

Anyway, time for the code. The new tests are as follows:

    CASE("parsing 'exp 1") {
        std::istringstream input{"exp 1"};
        Parser parser(input);

        EXPECT(approx(M_E) == parser.evalNextExpression());

    CASE("parsing 'foo 1") {
        std::istringstream input{"foo 1"};
        Parser parser(input);

        EXPECT_THROWS_AS(parser.evalNextExpression(), UnknownFunctionName);

    CASE("parsing 'sin(2 * 3.141592653)") {
        std::istringstream input{"sin(2 * 3.141592653)"};
        Parser parser(input);

        EXPECT(approx(0.) == parser.evalNextExpression());

    CASE("parsing 'exp") {
        std::istringstream input{"exp"};
        Parser parser(input);

        EXPECT_THROWS_AS(parser.evalNextExpression(), InvalidInputException);

We have also defined a trivial new exception class UnknownFunctionName, to be thrown when the user attempts to call an invalid function.

In the actual parser code, we first have changed evalNextFactor to match the updated grammar:

double Parser::evalNextFactor()
    if (nextToken_.getTokenType() == TokenType::NUMBER) {
        double value = atof(nextToken_.getContent().c_str());
        return value;
    } else if (nextToken_.getTokenType()  == TokenType::OPERATOR
        && nextToken_.getContent() == "(") {
        return evalNextParenthesisFactor();
    } else if (nextToken_.getTokenType() == TokenType::IDENTIFIER) {
        return evalNextFunctionCall();
    } else {
        throw InvalidInputException("Found an unexpected token: " + nextToken_.getContent());

Then we have written the new evalNextFunctionCall and a simple helper function:

double Parser::evalNextFunctionCall() {
    // Match the function name
    std::string functionName = nextToken_.getContent();

    // Is it a valid function?
    doubleToDoubleFunction f = lookupFunctionByName(functionName);

    // Eval its argument
    double argumentValue = evalNextFactor();

    // Call the function!
    return f(argumentValue);

Parser::doubleToDoubleFunction Parser::lookupFunctionByName(const std::string &name)
    auto it = functions_.find(name);
    if (it == functions_.end()) {
        throw UnknownFunctionName(name);
    return it->second;

I hope the evalNextFunctionCall code is clear. The code in lookupFunctionByName is a bit more C++-esque than usual, so it might require a bit of explanation.

Most of the C++ standard library, including the standard containers such as std::vector or std::map, work with the concept of iterators. Iterators are usually specified in pairs, with the first pointing to the first item of a range and the second pointing to one past the last item. Most containers have two member functions, begin, and end, which return iterators to the first and the (invalid) one-past-the-last items respectively. To see an example:

std::vector<char> v { 'a', 'b', 'c'};
f(v.begin(), v.end());       // the whole vector
f(v.begin(), v.begin() + 1); // a range containing [a]
f(v.begin(), v.begin() + 2); // a range containing [a, b]
f(v.begin(), v.begin());     // an empty range

To return to our code, std:map has a function find, which returns an iterator to the searched item or map::end if the item could not be found. The last piece of the puzzle is that it->second: map’s iterators are instances of std::pair, which is a simple class in the C++ standard library representing a pair of object, with two members: first and second. For map’s iterators, first points to the key and second to the value. I hope this clarifies lookupFunctionByName.


Our parser has learnt how to call functions. Next time, we’ll add to it the ability to handle variables!

Update: part 6 is available.