CHAPTER 15: Expressions

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15.16 Multiplicative Operators

The operators * , / , and % are called the multiplicative operators. They have the same precedence and are syntactically left-associative (they group left-to-right).

MultiplicativeExpression:

	UnaryExpression

	MultiplicativeExpression * UnaryExpression

	MultiplicativeExpression / UnaryExpression

	MultiplicativeExpression % UnaryExpression

The type of each of the operands of a multiplicative operator must be a primitive numeric type, or a compile-time error occurs. Binary numeric promotion is performed on the operands (S5.6.2). The type of a multiplicative expression is the promoted type of its operands. If this promoted type is int or long , then integer arithmetic is performed; if this promoted type is float or double , then floating-point arithmetic is performed.

15.16.1 Multiplication Operator *

The binary * operator performs multiplication, producing the product of its operands. Multiplication is a commutative operation if the operand expressions have no side effects. While integer multiplication is associative when the operands are all of the same type, floating-point multiplication is not associative.

If an integer multiplication overflows, then the result is the low-order bits of the mathematical product as represented in some sufficiently large two's-complement format. As a result, if overflow occurs, then the sign of the result may not be the same as the sign of the mathematical product of the two operand values.

The result of a floating-point multiplication is governed by the rules of IEEE 754 arithmetic:

• If either operand is NaN, the result is NaN.
• If the result is not NaN, the sign of the result is positive if both operands have the same sign, and negative if the operands have different signs.
• Multiplication of an infinity by a zero results in NaN.
• Multiplication of an infinity by a finite value results in a signed infinity. The sign is determined by the rule stated above.
• In the remaining cases, where neither an infinity or NaN is involved, the product is computed. If the magnitude of the product is too large to represent, we say the operation overflows. The result is then an infinity of appropriate sign. If the magnitude is too small to represent, we say the operation underflows; the result is then a zero of appropriate sign. Otherwise, the product is rounded to the nearest representable value using IEEE 754 round-to-nearest mode. The Java language requires support of gradual underflow as defined by IEEE 754 (S4.2.4).

Despite the fact that overflow, underflow, or loss of information may occur, evaluation of a multiplication operator * never throws a run-time exception.

15.16.2 Division Operator /

The binary / operator performs division, producing the quotient of its operands. The left-hand operand is the dividend and the right-hand operand is the divisor.

Integer division rounds toward 0 . That is, the quotient produced for operands n and d that are integers after binary numeric promotion (S5.6.2) is an integer value q whose magnitude is as large as possible while satisfying ; moreover, q is positive when and n and d have the same sign, but q is negative when and n and d have opposite signs. There is one special case that does not satisfy this rule: if the dividend is the negative integer of largest possible magnitude for its type, and the divisor is -1 , then integer overflow occurs and the result is equal to the dividend. Despite the overflow, no exception is thrown in this case. On the other hand, if the value of the divisor in an integer division is 0 , then an ArithmeticException is thrown.

The result of a floating-point division is determined by the specification of IEEE arithmetic:

• If either operand is NaN, the result is NaN.
• If the result is not NaN, the sign of the result is positive if both operands have the same sign, negative if the operands have different signs.
• Division of an infinity by an infinity results in NaN.
• Division of an infinity by a finite value results in a signed infinity. The sign is determined by the rule stated above.
• Division of a finite value by an infinity results in a signed zero. The sign is determined by the rule stated above.
• Division of a zero by a zero results in NaN; division of zero by any other finite value results in a signed zero. The sign is determined by the rule stated above.
• Division of a nonzero finite value by a zero results in a signed infinity. The sign is determined by the rule stated above.
• In the remaining cases, where neither an infinity, nor a zero, nor NaN is involved, the quotient is computed. If the magnitude of the quotient is too large to represent, we say the operation overflows; the result is then an infinity of appropriate sign. If the magnitude is too small to represent, we say the operation underflows and the result is then a zero of appropriate sign. Otherwise, the quotient is rounded to the nearest representable value using IEEE 754 round-to-nearest mode. The Java language requires support of gradual underflow as defined by IEEE 754 (S4.2.4).

Despite the fact that overflow, underflow, division by zero, or loss of information may occur, evaluation of a floating-point division operator / never throws a run-time exception.

15.16.3 Remainder Operator %

The binary % operator is said to yield the remainder of its operands from an implied division; the left-hand operand is the dividend and the right-hand operand is the divisor.

In C and C++, the remainder operator accepts only integral operands, but in Java, it also accepts floating-point operands.

The remainder operation for operands that are integers after binary numeric promotion (S5.6.2) produces a result value such that (a/b)*b+(a%b) is equal to a . This identity holds even in the special case that the dividend is the negative integer of largest possible magnitude for its type and the divisor is -1 (the remainder is 0 ). It follows from this rule that the result of the remainder operation can be negative only if the dividend is negative, and can be positive only if the dividend is positive; moreover, the magnitude of the result is always less than the magnitude of the divisor. If the value of the divisor for an integer remainder operator is 0 , then an ArithmeticException is thrown.

Examples:

5%3 produces 2							(note that 5/3 produces 1)
5%(-3) produces 2							(note that 5/(-3) produces -1)
(-5)%3 produces -2							(note that (-5)/3 produces -1)
(-5)%(-3) produces -2							(note that (-5)/(-3) produces 1)

The result of a floating-point remainder operation as computed by the % operator is not the same as that produced by the remainder operation defined by IEEE 754. The IEEE 754 remainder operation computes the remainder from a rounding division, not a truncating division, and so its behavior is not analogous to that of the usual integer remainder operator. Instead, the Java language defines % on floating-point operations to behave in a manner analogous to that of the Java integer remainder operator; this may be compared with the C library function fmod . The IEEE 754 remainder operation may be computed by the Java library routine Math.IEEEremainder (S20.11.14).

The result of a Java floating-point remainder operation is determined by the rules of IEEE arithmetic:

• If either operand is NaN, the result is NaN.
• If the result is not NaN, the sign of the result equals the sign of the dividend.
• If the dividend is an infinity, or the divisor is a zero, or both, the result is NaN.
• If the dividend is finite and the divisor is an infinity, the result equals the dividend.
• If the dividend is a zero and the divisor is finite, the result equals the dividend.
• In the remaining cases, where neither an infinity, nor a zero, nor NaN is involved, the floating-point remainder r from the division of a dividend n by a divisor d is defined by the mathematical relation

  r = n - (d · q) where q is an integer that is negative only if n/d is negative and positive only if n/d is positive, and whose magnitude is as large as possible without exceeding the magnitude of the true mathematical quotient of n and d.

Evaluation of a floating-point remainder operator % never throws a run-time exception, even if the right-hand operand is zero. Overflow, underflow, or loss of precision cannot occur.

Examples:

5.0%3.0 produces 2.0
5.0%(-3.0) produces 2.0
(-5.0)%3.0 produces -2.0
(-5.0)%(-3.0) produces -2.0