15.16.3 Remainder Operator %

And on the pedestal these words appear:
"My name is Ozymandias, king of kings:
Look on my works, ye Mighty, and despair!"
Nothing beside remains.
—Percy Bysshe Shelley, Ozymandias (1817)

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 (§5.6.2) 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 (§20.11.14).

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 where q is an integer that is negative only if is negative and positive only if 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