Equation (expression) parser with precedence?

Question

I\'ve developed an equation parser using a simple stack algorithm that will handle binary (+, -, |, &, *, /, etc) operators, unary (!) operators, and parenthesis.

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Answer 1

Algorithm could be easily encoded in C as recursive descent parser.

#include 
#include 

/*
 *  expression -> sum
 *  sum -> product | product "+" sum
 *  product -> term | term "*" product
 *  term -> number | expression
 *  number -> [0..9]+
 */

typedef struct {
    int value;
    const char* context;
} expression_t;

expression_t expression(int value, const char* context) {
    return (expression_t) { value, context };
}

/* begin: parsers */

expression_t eval_expression(const char* symbols);

expression_t eval_number(const char* symbols) {
    // number -> [0..9]+
    double number = 0;        
    while (isdigit(*symbols)) {
        number = 10 * number + (*symbols - '0');
        symbols++;
    }
    return expression(number, symbols);
}

expression_t eval_term(const char* symbols) {
    // term -> number | expression
    expression_t number = eval_number(symbols);
    return number.context != symbols ? number : eval_expression(symbols);
}

expression_t eval_product(const char* symbols) {
    // product -> term | term "*" product
    expression_t term = eval_term(symbols);
    if (*term.context != '*')
        return term;

    expression_t product = eval_product(term.context + 1);
    return expression(term.value * product.value, product.context);
}

expression_t eval_sum(const char* symbols) {
    // sum -> product | product "+" sum
    expression_t product = eval_product(symbols);
    if (*product.context != '+')
        return product;

    expression_t sum = eval_sum(product.context + 1);
    return expression(product.value + sum.value, sum.context);
}

expression_t eval_expression(const char* symbols) {
    // expression -> sum
    return eval_sum(symbols);
}

/* end: parsers */

int main() {
    const char* expression = "1+11*5";
    printf("eval(\"%s\") == %d\n", expression, eval_expression(expression).value);

    return 0;
}

_{next libs might be useful: yupana - strictly arithmetic operations; tinyexpr - arithmetic operations + C math functions + one provided by user; mpc - parser combinators}

Explanation

Let's capture sequence of symbols that represent algebraic expression. First one is a number, that is a decimal digit repeated one or more times. _{We will refer such notation as production rule.}

number -> [0..9]+

Addition operator with its operands is another rule. It is either number or any symbols that represents sum "*" sum sequence.

sum -> number | sum "+" sum

Try substitute number into sum "+" sum that will be number "+" number which in turn could be expanded into [0..9]+ "+" [0..9]+ that finally could be reduced to 1+8 which is correct addition expression.

Other substitutions will also produce correct expression: sum "+" sum -> number "+" sum -> number "+" sum "+" sum -> number "+" sum "+" number -> number "+" number "+" number -> 12+3+5

Bit by bit we could resemble set of production rules _{aka grammar} that express all possible algebraic expression.

expression -> sum
sum -> difference | difference "+" sum
difference -> product | difference "-" product
product -> fraction | fraction "*" product
fraction -> term | fraction "/" term
term -> "(" expression ")" | number
number -> digit+                                                                    

To control operator precedence alter position of its production rule against others. Look at grammar above and note that production rule for * is placed below + this will force product evaluate before sum. Implementation just combines pattern recognition with evaluation and thus closely mirrors production rules.

expression_t eval_product(const char* symbols) {
    // product -> term | term "*" product
    expression_t term = eval_term(symbols);
    if (*term.context != '*')
        return term;

    expression_t product = eval_product(term.context + 1);
    return expression(term.value * product.value, product.context);
}

Here we eval term first and return it if there is no * character after it _{this is left choise in our production rule} otherwise - evaluate symbols after and return term.value * product.value _{this is right choise in our production rule i.e. term "*" product}