Faster modulus in C/C#?

Question

Is there a trick for creating a faster integer modulus than the standard % operator for particular bases?

For my program, I'd be looking for around 1000-4000 (e.g. n%2048). Is there a quicker way to perform n modulus 2048 than simply: n%2048?

Answer 1

If the denominator is known at compile time to be a power of 2, like your example of 2048, you could subtract 1 and do a bitwise-and.

That is:

n % m == n & (m - 1) 

...where m is a power of 2.

For example:

22 % 8 == 6

         Dec       Bin
       -----     -----
22 & (8 - 1) =   10110 
               & 00111 
               -------
           6 =   00110

Bear in mind that a good compiler will have its own optimizations for %, maybe even enough to be as fast as the above technique. Arithmetic operators tend to be pretty heavily optimized.

Answer 2

For powers of two 2^n, all you have to do is zero out all bits except the last n bits.

For example (assuming 32 bit integers):

x%2 is equivalent to x & 0x00000001

x%4 is equivalent to x & 0x00000003

In general x % (2^n) is equal to x & (2^n-1). Written out in C, this would be x & ((1<<n)-1).

This is because 2^n gives you a 1 in the n+1th bit (from the right). So 2^n-1 will give you n ones on the right, and zeros on the left.

Answer 3

Branchless non-power-of-two modulus is possible by precomputing magic constants at run-time, to implement division using a multiply-add-shift.

This is roughly 2x faster than the built-in modulo operator % on my Intel Core i5.

I'm surprised it's not more dramatic, as x86 CPU div instructions can have latencies as high as 80-90 cycles for 64-bit division on some CPUs, compared to mul at 3 cycles and bitwise ops at 1 cycle each.

Proof of concept and timings shown below. series_len refers to the number of modulus ops performed in series on a single var. That's to prevent the CPU from hiding latencies through parallelization.

---

#include <stdint.h>
#include <stdlib.h>
#include <stdio.h>
#include <sys/time.h>

typedef int32_t s32;
typedef uint32_t u32;
typedef uint64_t u64;

#define NUM_NUMS 1024
#define NUM_RUNS 500
#define MAX_NUM UINT32_MAX
#define MAX_DEN 1024

struct fastdiv {
    u32 mul;
    u32 add;
    s32 shift;
    u32 _odiv;  /* save original divisor for modulo calc */
};

static u32 num[NUM_NUMS];
static u32 den[NUM_NUMS];
static struct fastdiv fd[NUM_NUMS];

/* hash of results to prevent gcc from optimizing out our ops */
static u32 cookie = 0;

/* required for magic constant generation */
u32 ulog2(u32 v) {
    u32 r, shift;
    r =     (v > 0xFFFF) << 4; v >>= r;
    shift = (v > 0xFF  ) << 3; v >>= shift; r |= shift;
    shift = (v > 0xF   ) << 2; v >>= shift; r |= shift;
    shift = (v > 0x3   ) << 1; v >>= shift; r |= shift;
                                            r |= (v >> 1);
    return r;
}

/* generate constants for implementing a division with multiply-add-shift */
void fastdiv_make(struct fastdiv *d, u32 divisor) {
    u32 l, r, e;
    u64 m;

    d->_odiv = divisor;
    l = ulog2(divisor);
    if (divisor & (divisor - 1)) {
        m = 1ULL << (l + 32);
        d->mul = (u32)(m / divisor);
        r = (u32)m - d->mul * divisor;
        e = divisor - r;
        if (e < (1UL << l)) {
            ++d->mul;
            d->add = 0;
        } else {
            d->add = d->mul;
        }
        d->shift = l;
    } else {
        if (divisor == 1) {
            d->mul = 0xffffffff;
            d->add = 0xffffffff;
            d->shift = 0;
        } else {
            d->mul = 0x80000000;
            d->add = 0;
            d->shift = l-1;
        }
    }
}

/* 0: use function that checks for a power-of-2 modulus (speedup for POTs)
 * 1: use inline macro */
#define FASTMOD_BRANCHLESS 0

#define fastdiv(v,d) ((u32)(((u64)(v)*(d)->mul + (d)->add) >> 32) >> (d)->shift)
#define _fastmod(v,d) ((v) - fastdiv((v),(d)) * (d)->_odiv)

#if FASTMOD_BRANCHLESS
#define fastmod(v,d) _fastmod((v),(d))
#else
u32 fastmod(u32 v, struct fastdiv *d) {
    if (d->mul == 0x80000000) {
        return (v & ((1 << d->shift) - 1));
    }
    return _fastmod(v,d);
}
#endif

u32 random32(u32 upper_bound) {
    return arc4random_uniform(upper_bound);
}

u32 random32_range(u32 lower_bound, u32 upper_bound) {
    return random32(upper_bound - lower_bound) + lower_bound;
}

void fill_arrays() {
    int i;
    for (i = 0; i < NUM_NUMS; ++i) {
        num[i] = random32_range(MAX_DEN, MAX_NUM);
        den[i] = random32_range(1, MAX_DEN);
        fastdiv_make(&fd[i], den[i]);
    }
}

void fill_arrays_pot() {
    u32 log_bound, rand_log;
    int i;

    log_bound = ulog2(MAX_DEN);
    for (i = 0; i < NUM_NUMS; ++i) {
        num[i] = random32_range(MAX_DEN, MAX_NUM);
        rand_log = random32(log_bound) + 1;
        den[i] = 1 << rand_log;
        fastdiv_make(&fd[i], den[i]);
    }
}

u64 clock_ns() {
    struct timeval tv;
    gettimeofday(&tv, NULL);
    return tv.tv_sec*1000000000 + tv.tv_usec*1000;
}

void use_value(u32 v) {
    cookie += v;
}

int main(int argc, char **arg) {
    u64 builtin_npot_ns;
    u64 builtin_pot_ns;
    u64 branching_npot_ns;
    u64 branching_pot_ns;
    u64 branchless_npot_ns;
    u64 branchless_pot_ns;
    u64 t0, t1;
    u32 v;
    int s, r, i, j;
    int series_len;

    builtin_npot_ns = builtin_pot_ns = 0;
    branching_npot_ns = branching_pot_ns = 0;
    branchless_npot_ns = branchless_pot_ns = 0;

    for (s = 5; s >= 0; --s) {
        series_len = 1 << s;
        for (r = 0; r < NUM_RUNS; ++r) {
            /* built-in NPOT */
            fill_arrays();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v /= den[i];
                }
                use_value(v);
            }
            t1 = clock_ns();
            builtin_npot_ns += (t1 - t0) / NUM_NUMS;

            /* built-in POT */
            fill_arrays_pot();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v /= den[i];
                }
                use_value(v);
            }
            t1 = clock_ns();
            builtin_pot_ns += (t1 - t0) / NUM_NUMS;

            /* branching NPOT */
            fill_arrays();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v = fastmod(v, fd+i);
                }
                use_value(v);
            }
            t1 = clock_ns();
            branching_npot_ns += (t1 - t0) / NUM_NUMS;

            /* branching POT */
            fill_arrays_pot();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v = fastmod(v, fd+i);
                }
                use_value(v);
            }
            t1 = clock_ns();
            branching_pot_ns += (t1 - t0) / NUM_NUMS;

            /* branchless NPOT */
            fill_arrays();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v = _fastmod(v, fd+i);
                }
                use_value(v);
            }
            t1 = clock_ns();
            branchless_npot_ns += (t1 - t0) / NUM_NUMS;

            /* branchless POT */
            fill_arrays_pot();
            t0 = clock_ns();
            for (i = 0; i < NUM_NUMS; ++i) {
                v = num[i];
                for (j = 0; j < series_len; ++j) {
                    v = _fastmod(v, fd+i);
                }
                use_value(v);
            }
            t1 = clock_ns();
            branchless_pot_ns += (t1 - t0) / NUM_NUMS;
        }

        builtin_npot_ns /= NUM_RUNS;
        builtin_pot_ns /= NUM_RUNS;
        branching_npot_ns /= NUM_RUNS;
        branching_pot_ns /= NUM_RUNS;
        branchless_npot_ns /= NUM_RUNS;
        branchless_pot_ns /= NUM_RUNS;

        printf("series_len = %d\n", series_len);
        printf("----------------------------\n");
        printf("builtin_npot_ns    : %llu ns\n", builtin_npot_ns);
        printf("builtin_pot_ns     : %llu ns\n", builtin_pot_ns);
        printf("branching_npot_ns  : %llu ns\n", branching_npot_ns);
        printf("branching_pot_ns   : %llu ns\n", branching_pot_ns);
        printf("branchless_npot_ns : %llu ns\n", branchless_npot_ns);
        printf("branchless_pot_ns  : %llu ns\n\n", branchless_pot_ns);
    }
    printf("cookie=%u\n", cookie);
}

---

Results

Intel Core i5 (MacBookAir7,2), macOS 10.11.6, clang 8.0.0

series_len = 32
----------------------------
builtin_npot_ns    : 218 ns
builtin_pot_ns     : 225 ns
branching_npot_ns  : 115 ns
branching_pot_ns   : 42 ns
branchless_npot_ns : 110 ns
branchless_pot_ns  : 110 ns

series_len = 16
----------------------------
builtin_npot_ns    : 87 ns
builtin_pot_ns     : 89 ns
branching_npot_ns  : 47 ns
branching_pot_ns   : 19 ns
branchless_npot_ns : 45 ns
branchless_pot_ns  : 45 ns

series_len = 8
----------------------------
builtin_npot_ns    : 32 ns
builtin_pot_ns     : 34 ns
branching_npot_ns  : 18 ns
branching_pot_ns   : 10 ns
branchless_npot_ns : 17 ns
branchless_pot_ns  : 17 ns

series_len = 4
----------------------------
builtin_npot_ns    : 15 ns
builtin_pot_ns     : 16 ns
branching_npot_ns  : 8 ns
branching_pot_ns   : 3 ns
branchless_npot_ns : 7 ns
branchless_pot_ns  : 7 ns

series_len = 2
----------------------------
builtin_npot_ns    : 8 ns
builtin_pot_ns     : 7 ns
branching_npot_ns  : 4 ns
branching_pot_ns   : 2 ns
branchless_npot_ns : 2 ns
branchless_pot_ns  : 2 ns

Answer 4

You could zero out the high order bits i.e.

x = 11 = 1011
x % 4 = 3 = 0011

so for x % 4 you could just take the last 2 bits - I'm not sure what would happen if negative numbers were used though

Answer 5

Here's a few techniques that replicate the modulus operation.

Of those benchmarked, this was the fastest (modified to fit your 2048 scenario). As long as your "max" isn't millions and in the 1000-4000 range you mentioned, it may work faster for you too:

int threshold = 2048; //the number to mod by
int max = 1000; //the number on the left. Ex: 1000 % 2048
int total = 0;
int y = 0;
for (int x = 0; x < max; x++)
{
    if (y > (threshold - 1))
    {
        y = 0;
        total += x;
    }
    y += 1;
}
return total;

Give it a go. It performed faster on the author's machine at various settings, so should perform admirably well for you too.

Answer 6

If you are dividing by literals that are powers of two, then the answer is probably No: Any decent compiler will automatically turn such expressions into a variation of an AND operation, which is pretty close to optimal.

Answer 7

The fastest way to multiply/divide unsigned integers numbers is by bit shifting them left or right. Shift operations match directly to CPU commands. For example, 3 << 2 =6, while 4>>1 = 2.

You can use the same trick to calculate the module: Shift an integer far enough to the left so that only the remainder bits are left, then shift it back right so you can check the remainder value.

On the other hand, integer modulo also exists as a CPU command. If the integer modulo operator maps to this command in optimized builds, you will not see any improvement by using the bit shift trick.

The following code caclulates 7%4 by shifting far enough that only the 2 last bits are left (since 4=2^2). This means that we need to shift 30 bits:

uint i=7;
var modulo=((i<<30)>>30);

The result is 3

EDIT:

I just read all the solutions proposing simply erasing the higher order bits. It has the same effect, but a lot simpler and direct.