Order of Operation: Operator Precedence and Associativity

Can you trust C++ to do complicated arithmetic? Yes, but you must know the rules C++ uses. For example, many expressions involve more than one operator. That can raise questions about which operator gets applied first. For example, consider this statement:

int flyingpigs = 3 + 4 * 5;  // 35 or 23?

The 4 appears to be an operand for both the + and * operators. When more than one operator can be applied to the same operand, C++ uses precedence rules to decide which operator is used first. The arithmetic operators follow the usual algebraic precedence, with multiplication, division, and the taking of the modulus done before addition and subtraction. Thus 3 + 4 * 5 means 3 + (4 * 5), not (3 + 4) * 5. So the answer is 23, not 35. Of course, you can use parentheses to enforce your own priorities. Appendix D, “Operator Precedence,” shows precedence for all the C++ operators. Note that *, /, and % are all in the same row in Appendix D. That means they have equal precedence. Similarly, addition and subtraction share a lower precedence.

Sometimes the precedence list is not enough. Consider the following statement:

float logs = 120 / 4 * 5;    // 150 or 6?

Once again, 4 is an operand for two operators. But the / and * operators have the same precedence, so precedence alone doesn’t tell the program whether to first divide 120 by 4 or multiply 4 by 5. Because the first choice leads to a result of 150 and the second to a result of 6, the choice is an important one. When two operators have the same precedence, C++ looks at whether the operators have a left-to-right associativity or a right-to-left associativity. Left-to-right associativity means that if two operators acting on the same operand have the same precedence, you apply the left-hand operator first. For right-to-left associativity, you apply the right-hand operator first. The associativity information, too, is in Appendix D. Appendix D shows that multiplication and division associate left-to-right. That means you use 4 with the leftmost operator first. That is, you divide 120 by 4, get 30 as a result, and then multiply the result by 5 to get 150.

Note that the precedence and associativity rules come into play only when two operators share the same operand. Consider the following expression:

int dues = 20 * 5 + 24 * 6;

Operator precedence tells you two things: The program must evaluate 20 * 5 before doing addition, and the program must evaluate 24 * 6 before doing addition. But neither precedence nor associativity says which multiplication takes place first. You might think that associativity says to do the leftmost multiplication first, but in this case, the two * operators do not share a common operand, so the rules don’t apply. In fact, C++ leaves it to the implementation to decide which order works best on a system. For this example, either order gives the same result, but there are circumstances in which the order can make a difference. You’ll see one in Chapter 5, which discusses the increment operator.