In computer programming, a pure function is a function that has the following properties:^[1]^[2]

the function return values are identical for identical arguments (no variation with local static variables, non-local variables, mutable reference arguments or input streams), and
the function application has no side effects (no mutation of local static variables, non-local variables, mutable reference arguments or input/output streams).

Thus a pure function is a computational analogue of a mathematical function. Some authors, particularly from the imperative language community, use the term "pure" for all functions that just have the above property 2^[3]^[4] (discussed below).

Examples

Pure functions

The following examples of C++ functions are pure:

floor, returning the floor of a number;
max, returning the maximum of two values.
the function f, defined as
```
void f() {
  static std::atomic<unsigned int> x = 0;
  ++x;
}
```
Although this code sample looks like it is not pure, it actually is. The value of x can be only observed inside other invocations of f(), and as f() does not communicate the value of x to its environment, it is indistinguishable from function void f() {} that does nothing. Note that x is std::atomic so that modifications from multiple threads executing f() concurrently do not result in a data race, which has undefined behavior in C and C++.

Impure functions

The following C++ functions are impure as they lack the above property 1:

because of return value variation with a static variable

int f() {
  static int x = 0;
  ++x;
  return x;
}

because of return value variation with a non-local variable
```
int f() {
  return x;
}
```
For the same reason, e.g. the C++ library function sin() is not pure, since its result depends on the IEEE rounding mode which can be changed at runtime.
because of return value variation with a mutable reference argument
```
int f(int* x) {
  return *x;
}
```

because of return value variation with an input stream

int f() {
  int x = 0;
  std::cin >> x;
  return x;
}

The following C++ functions are impure as they lack the above property 2:

because of mutation of a local static variable

void f() {
  static int x = 0;
  ++x;
}

because of mutation of a non-local variable
```
void f() {
  ++x;
}
```
because of mutation of a mutable reference argument
```
void f(int* x) {
  ++*x;
}
```

because of mutation of an output stream

void f() {
  std::cout << "Hello, world!" << std::endl;
}

The following C++ functions are impure as they lack both the above properties 1 and 2:

because of return value variation with a local static variable and mutation of a local static variable
```
int f() {
  static int x = 0;
  ++x;
  return x;
}
```
because of return value variation with an input stream and mutation of an input stream
```
int f() {
  int x = 0;
  std::cin >> x;
  return x;
}
```

I/O in pure functions

I/O is inherently impure: input operations undermine referential transparency, and output operations create side effects. Nevertheless, there is a sense in which function can perform input or output and still be pure, if the sequence of operations on the relevant I/O devices is modeled explicitly as both an argument and a result, and I/O operations are taken to fail when the input sequence does not describe the operations actually taken since the program began execution.

The second point ensures that the only sequence usable as an argument must change with each I/O action; the first allows different calls to an I/O-performing function to return different results on account of the sequence arguments having changed.^[5]^[6]

The I/O monad is a programming idiom typically used to perform I/O in pure functional languages.

Compiler optimizations

Functions that have just the above property 2 allow for compiler optimization techniques such as common subexpression elimination and loop optimization similar to arithmetic operators.^[3] A C++ example is the length method, returning the size of a string, which depends on the memory contents where the string points to, therefore lacking the above property 1. Nevertheless, in a single-threaded environment, the following C++ code

std::string s = "Hello, world!";
int a[10] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
int l = 0;

for (int i = 0; i < 10; ++i) {
  l += s.length() + a[i];
}

can be optimized such that the value of s.length() is computed only once, before the loop.

Some programming languages allow for declaring a pure property to a function:

In Fortran and D, the pure keyword can be used to declare a function to be just side-effect free (i.e. have just the above property 2).^[7] The compiler may be able to deduce property 1 on top of the declaration.^[8]
In the GCC, the pure attribute specifies property 2, while the const attribute specifies a truly pure function with both properties.^[9]
Languages offering compile-time function execution may require functions to be pure, sometimes with the addition of some other constraints. Examples include constexpr of C++ (both properties).^[10]

Unit testing

Since pure functions have identical return values for identical arguments, they are well suited to unit testing.