libdivide

libdivide.h is a header-only C/C++ library for optimizing integer division. Integer division is one of the slowest instructions on most CPUs e.g. on current x64 CPUs a 64-bit integer division has a latency of up to 90 clock cycles whereas a multiplication has a latency of only 3 clock cycles. libdivide allows you to replace expensive integer division instructions by a sequence of shift, add and multiply instructions that will calculate the integer division much faster.

On current CPUs you can get a speedup of up to 10x for 64-bit integer division and a speedup of up to to 5x for 32-bit integer division when using libdivide. libdivide also supports SSE2, AVX2 and AVX512 vector division which provides an even larger speedup. You can test how much speedup you can achieve on your CPU using the benchmark program.

libdivide is compatible with 8-bit microcontrollers, such as the AVR series: the CI build includes a AtMega2560 target. Since low end hardware such as this often do not include a hardware divider, libdivide is particularly useful. In addition to the runtime C & C++ APIs, a set of predefined macros and templates is included to speed up division by 16-bit constants: division by a 16-bit constant is not optimized by avr-gcc on 8-bit systems.

See https://libdivide.com for more information on libdivide.

C++ example

The first code snippet divides all integers in a vector using integer division. This is slow as integer division is at least one order of magnitude slower than any other integer arithmetic operation on current CPUs.

void divide(std::vector<int64_t>& vect, int64_t divisor)
{
    // Slow, uses integer division
    for (auto& n : vect)
        n /= divisor;
}

The second code snippet runs much faster, it uses libdivide to compute the integer division using a sequence of shift, add and multiply instructions hence avoiding the slow integer division operation.

#include "libdivide.h"

void divide(std::vector<int64_t>& vect, int64_t divisor)
{
    libdivide::divider<int64_t> fast_d(divisor);

    // Fast, computes division using libdivide
    for (auto& n : vect)
        n /= fast_d;
}

Generally libdivide will give a significant speedup if:

The divisor is only known at runtime
The divisor is reused multiple times e.g. in a loop

C example

You first need to generate a libdivide divider using one of the libdivide_*_gen functions (*: s32, u32, s64, u64) which can then be used to compute the actual integer division using the corresponding libdivide_*_do function.

#include "libdivide.h"

void divide(int64_t *array, size_t size, int64_t divisor)
{
    struct libdivide_s64_t fast_d = libdivide_s64_gen(divisor);

    // Fast, computes division using libdivide
    for (size_t i = 0; i < size; i++)
        array[i] = libdivide_s64_do(array[i], &fast_d);
}

API reference

Branchfull vs branchfree

The default libdivide divider makes use of branches to compute the integer division. When the same divider is used inside a hot loop as in the C++ example section the CPU will accurately predict the branches and there will be no performance slowdown. Often the compiler is even able to move the branches outside the body of the loop hence completely eliminating the branches, this is called loop-invariant code motion.

libdivide also has a branchfree divider type which computes the integer division without using any branch instructions. The branchfree divider generally uses a few more instructions than the default branchfull divider. The main use case for the branchfree divider is when you have an array of different divisors and you need to iterate over the divisors. In this case the default branchfull divider would exhibit poor performance as the CPU won't be able to correctly predict the branches.

#include "libdivide.h"

// 64-bit branchfree divider type
using branchfree_t = libdivide::branchfree_divider<uint64_t>;

uint64_t divide(uint64_t x, std::vector<branchfree_t>& vect)
{
    uint64_t sum = 0;

    for (auto& fast_d : vect)
        sum += x / fast_d;

    return sum;
}

Caveats of branchfree divider:

Unsigned branchfree divider cannot be 1
Faster for unsigned types than for signed types

Vector division

libdivide supports SSE2, AVX2 and AVX512 vector division on x86 and x64 CPUs. In the example below we divide the packed 32-bit integers inside an AVX512 vector using libdivide. libdivide supports 32-bit and 64-bit vector division for both signed and unsigned integers.

#include "libdivide.h"

void divide(std::vector<__m512i>& vect, uint32_t divisor)
{
    libdivide::divider<uint32_t> fast_d(divisor);

    // AVX512 vector division
    for (auto& n : vect)
        n /= fast_d;
}

Note that you need to define one of macros below to enable vector division:

LIBDIVIDE_SSE2
LIBDIVIDE_AVX2
LIBDIVIDE_AVX512
LIBDIVIDE_NEON

Performance Tips

If possible use unsigned integer types because libdivide's unsigned division is measurably faster than its signed division. This is especially true for the branchfree divider.
Try both the default branchfull divider and the branchfree divider in your program and choose the one that performs best. The branchfree divider is more likely to get auto vectorized by the compiler (if you compile with e.g. -march=native). But don't forget that the unsigned branchfree divider cannot be 1.
Vector division is much faster for 32-bit than for 64-bit. This is because there are currently no vector multiplication instructions on x86 to efficiently calculate 64-bit * 64-bit to 128-bit.

Build instructions

libdivide has one test program and two benchmark programs which can be built using cmake and a recent C++ compiler that supports C++11 or later. Optionally libdivide.h can also be installed to /usr/local/include.

cmake .
make -j
sudo make install

Tester program

You can pass the tester program one or more of the following arguments: u32, s32, u64, s64 to test the four cases (signed, unsigned, 32-bit, or 64-bit), or run it with no arguments to test all four. The tester will verify the correctness of libdivide via a set of randomly chosen numerators and denominators, by comparing the result of libdivide's division to hardware division. It will stop with an error message as soon as it finds a discrepancy.

Benchmark program

You can pass the benchmark program one or more of the following arguments: u16, s16, u32, s32, u64, s64 to compare libdivide's speed against hardware division. benchmark tests a simple function that inputs an array of random numerators and a single divisor, and returns the sum of their quotients. It tests this using both hardware division, and the various division approaches supported by libdivide, including vector division.

It will output data like this:

 #   system  scalar  scl_bf  vector  vec_bf   gener   algo
 1   9.684   0.792   0.783   0.426   0.426    1.346   0
 2   9.078   0.781   1.609   0.426   1.529    1.346   0
 3   9.078   1.355   1.609   1.334   1.531   29.045   1
 4   9.076   0.787   1.609   0.426   1.529    1.346   0
 5   9.074   1.349   1.609   1.334   1.531   29.045   1
 6   9.078   1.349   1.609   1.334   1.531   29.045   1
...

It will keep going as long as you let it, so it's best to stop it when you are happy with the denominators tested. These columns have the following significance. All times are in nanoseconds, lower is better.

     #:  The divisor that is tested
system:  Hardware divide time
scalar:  libdivide time, using scalar division
scl_bf:  libdivide time, using scalar branchfree division
vector:  libdivide time, using vector division
vec_bf:  libdivide time, using vector branchfree division
 gener:  Time taken to generate the divider struct
  algo:  The algorithm used.

The benchmark program will also verify that each function returns the same value, so benchmark is valuable for its verification as well.

Contributing

Although there are no individual unit tests, the supplied cmake builds do include several safety nets:

They compile with:
- All warnings on and;
- Warnings as errors
The CI build will build and run with sanitizers on ; these are available as part of the cmake build: -DCMAKE_BUILD_TYPE=Sanitize
The cmake and CI builds will compile and run both C and C++ test programs.

Before sending in patches, build and run at least the tester and benchmark using the supplied cmake scripts on at least MSVC and GCC (or Clang).

ridiculousfish / libdivide

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