Fastest way to get a positive modulo in C/C++

Question

Often in my inner loops I need to index an array in a "wrap-around" way, so that (for example) if the array size is 100 and my code asks for element -2, it should be given element 98. In many high level languages such as Python, one can do this simply with my_array[index % array_size], but for some reason C's integer arithmetic (usually) rounds toward zero instead of consistently rounding down, and consequently its modulo operator returns a negative result when given a negative first argument.

Often I know that index will not be less than -array_size, and in these cases I just do my_array[(index + array_size) % array_size]. However, sometimes this can't be guaranteed, and for those cases I would like to know the fastest way to implement an always-positive modulo function. There are several "clever" ways to do it without branching, such as

inline int positive_modulo(int i, int n) {
    return (n + (i % n)) % n;
}

or

inline int positive_modulo(int i, int n) {
    return (i % n) + (n * (i < 0));
}

Of course I can profile these to find out which is the fastest on my system, but I can't help worrying that I might have missed a better one, or that what's fast on my machine might be slow on a different one.

So is there a standard way to do this, or some clever trick that I've missed that's likely to be the fastest possible way?

Also, I know it's probably wishful thinking, but if there's a way of doing this that can be auto-vectorised, that would be amazing.

Answer 1

An old-school way to get the optional addend using twos-complement sign-bit propagation:

int positive_mod(int i, int m)
{
    /* constexpr */ int shift = CHAR_BIT * sizeof i - 1;
    int r = i % m;
    return r + (r >> shift & m);
}

I need to index an array in a "wrap-around" way

as another answer points out, if you're worried about arrays with negative sizes you should use more mathematically-pure, general methods.

Answer 2

If you want to optimize the following function while preserving the behavior for negative operands

inline int positive_modulo(int i, int n) {
    return (n + (i % n)) % n;
}

You can use the following implementation

inline int positive_modulo(int i, int n) {
    int m = i % n;
    if ((m != 0) & ((i ^ n) < 0)) m += n;
    return m;
}

This implementation will give the same result as the first implementation for negative operands such as positive_modulo(-7, 3), positive_modulo(7, -3) or positive_modulo(-7, -3).

This implementation also fixes the undefined behavior when the second operand is the maximum integer value 2147483647 or the minimum integer value -2147483648 (assuming 32-bit integer).

---

Note: (i ^ n) < 0 is true when i and n have opposite signs, false otherwise.

Answer 3

Fastest way to get a positive modulo in C/C++

The following fast? - maybe not as fast as others, yet is simple and functionally correct for all¹ a,b -- unlike others.

int modulo_Euclidean(int a, int b) {
  int m = a % b;
  if (m < 0) {
    // m += (b < 0) ? -b : b; // Avoid this form: -b is UB when b == INT_MIN
    m = (b < 0) ? m - b : m + b;
  }
  return m;
}

[Edit 2022]

From here, added tests to handle INT_MIN mod -1 and detect mod 0.

int modulo_Euclidean2(int a, int b) {
  if (b == 0) TBD_Code(); // Perhaps return -1 to indicate failure?
  if (b == -1) return 0; // This test needed to prevent UB of `INT_MIN % -1`.
  int m = a % b;
  if (m < 0) {
    // m += (b < 0) ? -b : b; // Avoid this form: it is UB when b == INT_MIN
    m = (b < 0) ? m - b : m + b;
  }
  return m;
}

---

Various other answers have mod(a,b) weaknesses especially when b < 0.

See Euclidean division for ideas about b < 0

---

inline int positive_modulo(int i, int n) {
    return (i % n + n) % n;
}

Fails when i % n + n overflows (think large i, n) - Undefined behavior.

---

return i & (n-1);

Relies on n as a power of two. (Fair that the answer does mention this.)

---

int positive_mod(int i, int n)
{
    /* constexpr */ int shift = CHAR_BIT*sizeof i - 1;
    int m = i%n;
    return m+ (m>>shift & n);
}

Often fails when n < 0. e, g, positive_mod(-2,-3) --> -5

---

int32_t positive_modulo(int32_t number, int32_t modulo) {
    return (number + ((int64_t)modulo << 32)) % modulo;
}

Obliges using 2 integer widths. (Fair that the answer does mention this.)
Fails with modulo < 0. positive_modulo(2, -3) --> -1.

---

inline int positive_modulo(int i, int n) {
    int tmp = i % n;
    return tmp ? i >= 0 ? tmp : tmp + n : 0;
}

Often fails when n < 0. e, g, positive_modulo(-2,-3) --> -5

---

¹ Exceptions: In C, a%b is not defined when a/b overflows as in a/0 or INT_MIN/-1.

Answer 4

Rather than try to find a generic way to compute a positive modulus, the simplest solution to your problem is to plainly convert negative indices into a positive value. This will be branchless on a processor with conditional moves. If you don't need extra bounds checks this will produce a minimal amount of code with no divide or multiply ops.

char pos_index(char *my_array, size_t array_size, int index) {
  if(index < 0)
    index = array_size - index;
  return my_array[index];
}

With modular bounds you have the cost of one extra divide:

char pos_index(char *my_array, size_t array_size, int index) {
  index = index % array_size; // Remainder; could be negative
  if(index < 0)
    index = array_size - index;
  return my_array[index];
}

Being less clever is often a better approach with modern hardware and compilers. Conditional moves are a superpower you can leverage to tame otherwise stall inducing code.

Answer 5

If you want to avoid all conditional paths (including the conditional move generated above, (For example if you need this code to vectorize, or to run in constant time), You can use the sign bit as a mask:

unsigned modulo(int value, unsigned m) {
  int shift_width = sizeof(int) * 8 - 1;
  int tweak = (value >> shift_width);
  int mod = ((value - tweak) % (int) m) + tweak;
  mod += (tweak & m);
  return mod;
}

Here are the quickbench results You can see that on gcc it's better in the generic case. For clang it's the same speed in the generic case, because clang generates the branch free code in the generic case. The technique is useful regardless, because the compiler can't always be relied on to produce the particular optimization, and you may have to roll it by hand for vector code.

Answer 6

Most of the time, compilers are very good at optimizing your code, so it is usually best to keep your code readable (for both compilers and other developers to know what you are doing).

Since your array size is always positive, I suggest you to define the quotient as unsigned. The compiler will optimize small if/else blocks into conditional instructions which have no branches:

unsigned modulo( int value, unsigned m) {
    int mod = value % (int)m;
    if (mod < 0) {
        mod += m;
    }
    return mod;
}

This creates a very small function without branches:

modulo(int, unsigned int):
        mov     eax, edi
        cdq
        idiv    esi
        add     esi, edx
        mov     eax, edx
        test    edx, edx
        cmovs   eax, esi
        ret

For example modulo(-5, 7) returns 2.

Unfortunately, since the quotient is not known they must perform an integer division, which is a bit slow compared to other integer operations. If you know the sizes of your array are power of two, I recommend keeping these function definitions in a header, so that the compiler can optimize them into a more efficient function. Here is the function unsigned modulo256(int v) { return modulo(v,256); }:

modulo256(int):                          # @modulo256(int)
        mov     edx, edi
        sar     edx, 31
        shr     edx, 24
        lea     eax, [rdi+rdx]
        movzx   eax, al
        sub     eax, edx
        lea     edx, [rax+256]
        test    eax, eax
        cmovs   eax, edx
        ret

See assembly: https://gcc.godbolt.org/z/DG7jMw

See comparison with most voted answer: http://quick-bench.com/oJbVwLr9G5HJb0oRaYpQOCec4E4

Edit: turns out Clang is able to generate a function without any conditional move instructions (which cost more than regular arithmetic operations). This difference is completely negligible in the general case due to the fact that the integral division takes around 70% of the total time.

Basically, Clang shifts value right to extend its sign bit to the whole width of m (that is 0xffffffff when negative and 0 otherwise) which is used to mask the second operand in mod + m.

unsigned modulo (int value, unsigned m) {
    int mod = value % (int)m;
    m &= mod >> std::numeric_limits<int>::digits;
    return mod + m;
}

Answer 7

If you can afford to promote to a larger type (and do your modulo on the larger type), this code does a single modulo and no if:

int32_t positive_modulo(int32_t number, int32_t modulo) {
    return (number + ((int64_t)modulo << 32)) % modulo;
}

Answer 8

Your second example is better than the first. A multiplication is a more complex operation than an if/else operation, so use this:

inline int positive_modulo(int i, int n) {
    int tmp = i % n;
    return tmp ? i >= 0 ? tmp : tmp + n : 0;
}

Answer 9

You can as well do array[(i+array_size*N) % array_size], where N is large enough integer to guarantee positive argument, but small enough for not to overflow.

When the array_size is constant, there are techniques to calculate the modulus without division. Besides of power of two approach, one can calculate a weighted sum of bitgroups multiplied by the 2^i % n, where i is the least significant bit in each group:

e.g. 32-bit integer 0xaabbccdd % 100 = dd + cc*[2]56 + bb*[655]36 + aa*[167772]16, having the maximum range of (1+56+36+16)*255 = 27795. With repeated applications and different subdivision one can reduce the operation to few conditional subtractions.

Common practises also include approximation of division with reciprocal of 2^32 / n, which usually can handle reasonably large range of arguments.

 i - ((i * 655)>>16)*100; // (gives 100*n % 100 == 100 requiring adjusting...)

Answer 10

The standard way I learned is

inline int positive_modulo(int i, int n) {
    return (i % n + n) % n;
}

This function is essentially your first variant without the abs (which, in fact, makes it return the wrong result). I wouldn't be surprised if an optimizing compiler could recognize this pattern and compile it to machine code that computes an "unsigned modulo".

Edit:

Moving on to your second variant: First of all, it contains a bug, too -- the n < 0 should be i < 0.

This variant may not look as if it branches, but on a lot of architectures, the i < 0 will compile into a conditional jump. In any case, it will be at least as fast to replace (n * (i < 0)) with i < 0? n: 0, which avoids the multiplication; in addition, it's "cleaner" because it avoids reinterpreting the bool as an int.

As to which of these two variants is faster, that probably depends on the compiler and processor architecture -- time the two variants and see. I don't think there's a faster way than either of these two variants, though.

Answer 11

Modulo a power of two, the following works (assuming twos complement representation):

return i & (n-1);