在许多 64 位位掩码上分别计算每个位位置，使用 AVX 而不是 AVX2

Question

(Related: How to quickly count bits into separate bins in a series of ints on Sandy Bridge? is an earlier duplicate of this, with some different answers. Editor's note: the answers here are probably better.

Also, an AVX2 version of a similar problem, with many bins for a whole row of bits much wider than one uint64_t : Improve column population count algorithm )

---

I am working on a project in C where I need to go through tens of millions of masks (of type ulong (64-bit)) and update an array (called target ) of 64 short integers (uint16) based on a simple rule:

// for any given mask, do the following loop
for (i = 0; i < 64; i++) {
    if (mask & (1ull << i)) {
        target[i]++
    }
}

The problem is that I need do the above loops on tens of millions of masks and I need to finish in less than a second. Wonder if there are any way to speed it up, like using some sort special assembly instruction that represents the above loop.

Currently I use gcc 4.8.4 on ubuntu 14.04 (i7-2670QM, supporting AVX, not AVX2) to compile and run the following code and took about 2 seconds. Would love to make it run under 200ms.

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <sys/time.h>
#include <sys/stat.h>

double getTS() {
    struct timeval tv;
    gettimeofday(&tv, NULL);
    return tv.tv_sec + tv.tv_usec / 1000000.0;
}
unsigned int target[64];

int main(int argc, char *argv[]) {
    int i, j;
    unsigned long x = 123;
    unsigned long m = 1;
    char *p = malloc(8 * 10000000);
    if (!p) {
        printf("failed to allocate\n");
        exit(0);
    }
    memset(p, 0xff, 80000000);
    printf("p=%p\n", p);
    unsigned long *pLong = (unsigned long*)p;
    double start = getTS();
    for (j = 0; j < 10000000; j++) {
        m = 1;
        for (i = 0; i < 64; i++) {
            if ((pLong[j] & m) == m) {
                target[i]++;
            }
            m = (m << 1);
        }
    }
    printf("took %f secs\n", getTS() - start);
    return 0;
}

Thanks in advance!

Answer 1

On my system, a 4 year old MacBook (2.7 GHz intel core i5) with clang-900.0.39.2 -O3 , your code runs in 500ms.

Just changing the inner test to if ((pLong[j] & m) != 0) saves 30%, running in 350ms.

Further simplifying the inner part to target[i] += (pLong[j] >> i) & 1; without a test brings it down to 280ms.

Further improvements seem to require more advanced techniques such as unpacking the bits into blocks of 8 ulongs and adding those in parallel, handling 255 ulongs at a time.

Here is an improved version using this method. it runs in 45ms on my system.

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

double getTS() {
    struct timeval tv;
    gettimeofday(&tv, NULL);
    return tv.tv_sec + tv.tv_usec / 1000000.0;
}

int main(int argc, char *argv[]) {
    unsigned int target[64] = { 0 };
    unsigned long *pLong = malloc(sizeof(*pLong) * 10000000);
    int i, j;

    if (!pLong) {
        printf("failed to allocate\n");
        exit(1);
    }
    memset(pLong, 0xff, sizeof(*pLong) * 10000000);
    printf("p=%p\n", (void*)pLong);
    double start = getTS();
    uint64_t inflate[256];
    for (i = 0; i < 256; i++) {
        uint64_t x = i;
        x = (x | (x << 28));
        x = (x | (x << 14));
        inflate[i] = (x | (x <<  7)) & 0x0101010101010101ULL;
    }
    for (j = 0; j < 10000000 / 255 * 255; j += 255) {
        uint64_t b[8] = { 0 };
        for (int k = 0; k < 255; k++) {
            uint64_t u = pLong[j + k];
            for (int kk = 0; kk < 8; kk++, u >>= 8)
                b[kk] += inflate[u & 255];
        }
        for (i = 0; i < 64; i++)
            target[i] += (b[i / 8] >> ((i % 8) * 8)) & 255;
    }
    for (; j < 10000000; j++) {
        uint64_t m = 1;
        for (i = 0; i < 64; i++) {
            target[i] += (pLong[j] >> i) & 1;
            m <<= 1;
        }
    }
    printf("target = {");
    for (i = 0; i < 64; i++)
        printf(" %d", target[i]);
    printf(" }\n");
    printf("took %f secs\n", getTS() - start);
    return 0;
}

The technique for inflating a byte to a 64-bit long are investigated and explained in the answer: https://stackoverflow.com/a/55059914/4593267 . I made the target array a local variable, as well as the inflate array, and I print the results to ensure the compiler will not optimize the computations away. In a production version you would compute the inflate array separately.

Using SIMD directly might provide further improvements at the expense of portability and readability. This kind of optimisation is often better left to the compiler as it can generate specific code for the target architecture. Unless performance is critical and benchmarking proves this to be a bottleneck, I would always favor a generic solution.

A different solution by njuffa provides similar performance without the need for a precomputed array. Depending on your compiler and hardware specifics, it might be faster.

Answer 2

Related:

• an earlier duplicate has some alternate ideas: How to quickly count bits into separate bins in a series of ints on Sandy Bridge? .
• Harold's answer on AVX2 column population count algorithm over each bit-column separately .
• Matrix transpose and population count has a couple useful answers with AVX2, including benchmarks. It uses 32-bit chunks instead of 64-bit.

Also: https://github.com/mklarqvist/positional-popcount has SSE blend, various AVX2, various AVX512 including Harley-Seal which is great for large arrays, and various other algorithms for positional popcount. Possibly only for uint16_t , but most could be adapted for other word widths. I think the algorithm I propose below is what they call adder_forest .

---

Your best bet is SIMD, using AVX1 on your Sandybridge CPU. Compilers aren't smart enough to auto-vectorize your loop-over-bits for you, even if you write it branchlessly to give them a better chance.

And unfortunately not smart enough to auto-vectorize the fast version that gradually widens and adds.

---

See is there an inverse instruction to the movemask instruction in intel avx2? for a summary of bitmap -> vector unpack methods for different sizes. Ext3h's suggestion in another answer is good: Unpack bits to something narrower than the final count array gives you more elements per instruction. Bytes is efficient with SIMD, and then you can do up to 255 vertical paddb without overflow, before unpacking to accumulate into the 32-bit counter array.

It only takes 4x 16-byte __m128i vectors to hold all 64 uint8_t elements, so those accumulators can stay in registers, only adding to memory when widening out to 32-bit counters in an outer loop.

The unpack doesn't have to be in-order : you can always shuffle target[] once at the very end, after accumulating all the results.

The inner loop could be unrolled to start with a 64 or 128-bit vector load, and unpack 4 or 8 different ways using pshufb ( _mm_shuffle_epi8 ).

---

An even better strategy is to widen gradually

Starting with 2-bit accumulators, then mask/shift to widen those to 4-bit. So in the inner-most loop most of the operations are working with "dense" data, not "diluting" it too much right away. Higher information / entropy density means that each instruction does more useful work.

Using SWAR techniques for 32x 2-bit add inside scalar or SIMD registers is easy / cheap because we need to avoid the possibility of carry out the top of an element anyway. With proper SIMD, we'd lose those counts, with SWAR we'd corrupt the next element.

uint64_t x = *(input++);        // load a new bitmask
const uint64_t even_1bits = 0x5555555555555555;  // 0b...01010101;

uint64_t lo = x & even_1bits;
uint64_t hi = (x>>1) & even_1bits;            // or use ANDN before shifting to avoid a MOV copy

accum2_lo += lo;   // can do up to 3 iterations of this without overflow
accum2_hi += hi;   // because a 2-bit integer overflows at 4

Then you repeat up to 4 vectors of 4-bit elements, then 8 vectors of 8-bit elements, then you should widen all the way to 32 and accumulate into the array in memory because you'll run out of registers anyway, and this outer outer loop work is infrequent enough that we don't need to bother with going to 16-bit. (Especially if we manually vectorize).

Biggest downside: this doesn't auto-vectorize, unlike @njuffa's version. But with gcc -O3 -march=sandybridge for AVX1 (then running the code on Skylake), this running scalar 64-bit is actually still slightly faster than 128-bit AVX auto-vectorized asm from @njuffa's code.

But that's timing on Skylake, which has 4 scalar ALU ports (and mov-elimination), while Sandybridge lacks mov-elimination and only has 3 ALU ports, so the scalar code will probably hit back-end execution-port bottlenecks. (But SIMD code may be nearly as fast, because there's plenty of AND / ADD mixed with the shifts, and SnB does have SIMD execution units on all 3 of its ports that have any ALUs on them. Haswell just added port 6, for scalar-only including shifts and branches.)

With good manual vectorization, this should be a factor of almost 2 or 4 faster.

But if you have to choose between this scalar or @njuffa's with AVX2 autovectorization, @njuffa's is faster on Skylake with -march=native

If building on a 32-bit target is possible/required, this suffers a lot (without vectorization because of using uint64_t in 32-bit registers), while vectorized code barely suffers at all (because all the work happens in vector regs of the same width).

// TODO: put the target[] re-ordering somewhere
// TODO: cleanup for N not a multiple of 3*4*21 = 252
// TODO: manual vectorize with __m128i, __m256i, and/or __m512i

void sum_gradual_widen (const uint64_t *restrict input, unsigned int *restrict target, size_t length)
{
    const uint64_t *endp = input + length - 3*4*21;     // 252 masks per outer iteration
    while(input <= endp) {
        uint64_t accum8[8] = {0};     // 8-bit accumulators
        for (int k=0 ; k<21 ; k++) {
            uint64_t accum4[4] = {0};  // 4-bit accumulators can hold counts up to 15.  We use 4*3=12
            for(int j=0 ; j<4 ; j++){
                uint64_t accum2_lo=0, accum2_hi=0;
                for(int i=0 ; i<3 ; i++) {  // the compiler should fully unroll this
                    uint64_t x = *input++;    // load a new bitmask
                    const uint64_t even_1bits = 0x5555555555555555;
                    uint64_t lo = x & even_1bits; // 0b...01010101;
                    uint64_t hi = (x>>1) & even_1bits;  // or use ANDN before shifting to avoid a MOV copy
                    accum2_lo += lo;
                    accum2_hi += hi;   // can do up to 3 iterations of this without overflow
                }

                const uint64_t even_2bits = 0x3333333333333333;
                accum4[0] +=  accum2_lo       & even_2bits;  // 0b...001100110011;   // same constant 4 times, because we shift *first*
                accum4[1] += (accum2_lo >> 2) & even_2bits;
                accum4[2] +=  accum2_hi       & even_2bits;
                accum4[3] += (accum2_hi >> 2) & even_2bits;
            }
            for (int i = 0 ; i<4 ; i++) {
                accum8[i*2 + 0] +=   accum4[i] & 0x0f0f0f0f0f0f0f0f;
                accum8[i*2 + 1] +=  (accum4[i] >> 4) & 0x0f0f0f0f0f0f0f0f;
            }
        }

        // char* can safely alias anything.
        unsigned char *narrow = (uint8_t*) accum8;
        for (int i=0 ; i<64 ; i++){
            target[i] += narrow[i];
        }
    }
    /* target[0] = bit 0
     * target[1] = bit 8
     * ...
     * target[8] = bit 1
     * target[9] = bit 9
     * ...
     */
    // TODO: 8x8 transpose
}

We don't care about order, so accum4[0] has 4-bit accumulators for every 4th bit, for example. The final fixup needed (but not yet implemented) at the very end is an 8x8 transpose of the uint32_t target[64] array, which can be done efficiently using unpck and vshufps with only AVX1. ( Transpose an 8x8 float using AVX/AVX2 ). And also a cleanup loop for the last up to 251 masks.

We can use any SIMD element width to implement these shifts; we have to mask anyway for widths lower than 16-bit (SSE/AVX doesn't have byte-granularity shifts, only 16-bit minimum.)

Benchmark results on Arch Linux i7-6700k from @njuffa's test harness, with this added. ( Godbolt ) N = (10000000 / (3*4*21) * 3*4*21) = 9999864 (ie 10000000 rounded down to a multiple of the 252 iteration "unroll" factor, so my simplistic implementation is doing the same amount of work, not counting re-ordering target[] which it doesn't do, so it does print mismatch results. But the printed counts match another position of the reference array.)

I ran the program 4x in a row (to make sure the CPU was warmed up to max turbo) and took one of the runs that looked good (none of the 3 times abnormally high).

ref: the best bit-loop (next section)
fast: @njuffa's code. (auto-vectorized with 128-bit AVX integer instructions).
gradual: my version (not auto-vectorized by gcc or clang, at least not in the inner loop.) gcc and clang fully unroll the inner 12 iterations.

• gcc8.2 -O3 -march=sandybridge -fpie -no-pie
  ref: 0.331373 secs, fast: 0.011387 secs, gradual: 0.009966 secs
• gcc8.2 -O3 -march=sandybridge -fno-pie -no-pie
  ref: 0.397175 secs, fast: 0.011255 secs, gradual: 0.010018 secs
• clang7.0 -O3 -march=sandybridge -fpie -no-pie
  ref: 0.352381 secs, fast: 0.011926 secs, gradual: 0.009269 secs (very low counts for port 7 uops, clang used indexed addressing for stores)
• clang7.0 -O3 -march=sandybridge -fno-pie -no-pie
  ref: 0.293014 secs , fast: 0.011777 secs, gradual: 0.009235 secs

-march=skylake (allowing AVX2 for 256-bit integer vectors) helps both, but @njuffa's most because more of it vectorizes (including its inner-most loop):

• gcc8.2 -O3 -march=skylake -fpie -no-pie
  ref: 0.328725 secs, fast: 0.007621 secs, gradual: 0.010054 secs (gcc shows no gain for "gradual", only "fast")

• gcc8.2 -O3 -march=skylake -fno-pie -no-pie
  ref: 0.333922 secs, fast: 0.007620 secs, gradual: 0.009866 secs

• clang7.0 -O3 -march=skylake -fpie -no-pie
  ref: 0.260616 secs, fast: 0.007521 secs, gradual: 0.008535 secs (IDK why gradual is faster than -march=sandybridge; it's not using BMI1 andn . I guess because it's using 256-bit AVX2 for the k=0..20 outer loop with vpaddq )

• clang7.0 -O3 -march=skylake -fno-pie -no-pie
  ref: 0.259159 secs , fast: 0.007496 secs , gradual: 0.008671 secs

Without AVX, just SSE4.2: ( -march=nehalem ), bizarrely clang's gradual is faster than with AVX / tune=sandybridge. "fast" is only barely slower than with AVX.

• gcc8.2 -O3 -march=skylake -fno-pie -no-pie
  ref: 0.337178 secs, fast: 0.011983 secs, gradual: 0.010587 secs
• clang7.0 -O3 -march=skylake -fno-pie -no-pie
  ref: 0.293555 secs , fast: 0.012549 secs, gradual: 0.008697 secs

-fprofile-generate / -fprofile-use help some for GCC, especially for the "ref" version where it doesn't unroll at all by default.

I highlighted the best, but often they're within measurement noise margin of each other. It's unsurprising the -fno-pie -no-pie was sometimes faster: indexing static arrays with [disp32 + reg] is not an indexed addressing mode, just base + disp32, so it doesn't ever unlaminate on Sandybridge-family CPUs.

But with gcc sometimes -fpie was faster; I didn't check but I assume gcc just shot itself in the foot somehow when 32-bit absolute addressing was possible. Or just innocent-looking differences in code-gen happened to cause alignment or uop-cache problems; I didn't check in detail.

---

For SIMD, we can simply do 2 or 4x uint64_t in parallel, only accumulating horizontally in the final step where we widen bytes to 32-bit elements. (Perhaps by shuffling in-lane and then using pmaddubsw with a multiplier of _mm256_set1_epi8(1) to add horizontal byte pairs into 16-bit elements.)

TODO: manually-vectorized __m128i and __m256i (and __m512i ) versions of this. Should be close to 2x, 4x, or even 8x faster than the "gradual" times above. Probably HW prefetch can still keep up with it, except maybe an AVX512 version with data coming from DRAM, especially if there's contention from other threads. We do a significant amount of work per qword we read.

---

Obsolete code: improvements to the bit-loop

Your portable scalar version can be improved, too, speeding it up from ~1.92 seconds ( with a 34% branch mispredict rate overall , with the fast loops commented out!) to ~0.35sec ( clang7.0 -O3 -march=sandybridge ) with a properly random input on 3.9GHz Skylake. Or 1.83 sec for the branchy version with != 0 instead of == m , because compilers fail to prove that m always has exactly 1 bit set and/or optimize accordingly.

(vs. 0.01 sec for @njuffa's or my fast version above, so this is pretty useless in an absolute sense, but it's worth mentioning as a general optimization example of when to use branchless code.)

If you expect a random mix of zeros and ones, you want something branchless that won't mispredict. Doing += 0 for elements that were zero avoids that, and also means that the C abstract machine definitely touches that memory regardless of the data.

Compilers aren't allowed to invent writes, so if they wanted to auto-vectorize your if() target[i]++ version, they'd have to use a masked store like x86 vmaskmovps to avoid a non-atomic read / rewrite of unmodified elements of target . So some hypothetical future compiler that can auto-vectorize the plain scalar code would have an easier time with this.

Anyway, one way to write this is target[i] += (pLong[j] & m != 0); , using bool->int conversion to get a 0 / 1 integer.

But we get better asm for x86 (and probably for most other architectures) if we just shift the data and isolate the low bit with &1 . Compilers are kinda dumb and don't seem to spot this optimization. They do nicely optimize away the extra loop counter, and turn m <<= 1 into add same,same to efficiently left shift, but they still use xor-zero / test / setne to create a 0 / 1 integer.

An inner loop like this compiles slightly more efficiently (but still much much worse than we can do with SSE2 or AVX, or even scalar using @chrqlie's lookup table which will stay hot in L1d when used repeatedly like this, allowing SWAR in uint64_t ):

    for (int j = 0; j < 10000000; j++) {
#if 1  // extract low bit directly
        unsigned long long tmp = pLong[j];
        for (int i=0 ; i<64 ; i++) {   // while(tmp) could mispredict, but good for sparse data
            target[i] += tmp&1;
            tmp >>= 1;
        }
#else // bool -> int shifting a mask
        unsigned long m = 1;
        for (i = 0; i < 64; i++) {
            target[i]+= (pLong[j] & m) != 0;
            m = (m << 1);
        }
#endif

Note that unsigned long is not guaranteed to be a 64-bit type, and isn't in x86-64 System V x32 (ILP32 in 64-bit mode), and Windows x64. Or in 32-bit ABIs like i386 System V.

Compiled on the Godbolt compiler explorer by gcc, clang, and ICC , it's 1 fewer uops in the loop with gcc. But all of them are just plain scalar, with clang and ICC unrolling by 2.

# clang7.0 -O3 -march=sandybridge
.LBB1_2:                            # =>This Loop Header: Depth=1
   # outer loop loads a uint64 from the src
    mov     rdx, qword ptr [r14 + 8*rbx]
    mov     rsi, -256
.LBB1_3:                            #   Parent Loop BB1_2 Depth=1
                                    # do {
    mov     edi, edx
    and     edi, 1                              # isolate the low bit
    add     dword ptr [rsi + target+256], edi   # and += into target

    mov     edi, edx
    shr     edi
    and     edi, 1                              # isolate the 2nd bit
    add     dword ptr [rsi + target+260], edi

    shr     rdx, 2                              # tmp >>= 2;

    add     rsi, 8
    jne     .LBB1_3                       # } while(offset += 8 != 0);

This is slightly better than we get from test / setnz . Without unrolling, bt / setc might have been equal, but compilers are bad at using bt to implement bool (x & (1ULL << n)) , or bts to implement x |= 1ULL << n .

If many words have their highest set bit far below bit 63, looping on while(tmp) could be a win . Branch mispredicts make it not worth it if it only saves ~0 to 4 iterations most of the time, but if it often saves 32 iterations, that could really be worth it. Maybe unroll in the source so the loop only tests tmp every 2 iterations (because compilers won't do that transformation for you), but then the loop branch can be shr rdx, 2 / jnz .

On Sandybridge-family, this is 11 fused-domain uops for the front end per 2 bits of input. ( add [mem], reg with a non-indexed addressing mode micro-fuses the load+ALU, and the store-address+store-data, everything else is single-uop. add/jcc macro-fuses. See Agner Fog's guide, and https://stackoverflow.com/tags/x86/info ). So it should run at something like 3 cycles per 2 bits = one uint64_t per 96 cycles. (Sandybridge doesn't "unroll" internally in its loop buffer, so non-multiple-of-4 uop counts basically round up, unlike on Haswell and later).

vs. gcc's not-unrolled version being 7 uops per 1 bit = 2 cycles per bit. If you compiled with gcc -O3 -march=native -fprofile-generate / test-run / gcc -O3 -march=native -fprofile-use , profile-guided optimization would enable loop unrolling.

This is probably slower than a branchy version on perfectly predictable data like you get from memset with any repeating byte pattern . I'd suggest filling your array with randomly-generated data from a fast PRNG like an SSE2 xorshift+, or if you're just timing the count loop then use anything you want, like rand() .

Answer 3

One way of speeding this up significantly, even without AVX, is to split the data into blocks of up to 255 elements, and accumulate the bit counts byte-wise in ordinary uint64_t variables. Since the source data has 64 bits, we need an array of 8 byte-wise accumulators. The first accumulator counts bits in positions 0, 8, 16, ... 56, second accumulator counts bits in positions 1, 9, 17, ... 57; and so on. After we are finished processing a block of data, we transfers the counts form the byte-wise accumulator into the target counts. A function to update the target counts for a block of up to 255 numbers can be coded in a straightforward fashion according to the description above, where BITS is the number of bits in the source data:

/* update the counts of 1-bits in each bit position for up to 255 numbers */
void sum_block (const uint64_t *pLong, unsigned int *target, int lo, int hi)
{
    int jj, k, kk;
    uint64_t byte_wise_sum [BITS/8] = {0};
    for (jj = lo; jj < hi; jj++) {
        uint64_t t = pLong[jj];
        for (k = 0; k < BITS/8; k++) {
            byte_wise_sum[k] += t & 0x0101010101010101;
            t >>= 1;
        }
    }
    /* accumulate byte sums into target */
    for (k = 0; k < BITS/8; k++) {
        for (kk = 0; kk < BITS; kk += 8) {
            target[kk + k] += (byte_wise_sum[k] >> kk) & 0xff;
        }
    }
}

The entire ISO-C99 program, which should be able to run on at least Windows and Linux platforms is shown below. It initializes the source data with a PRNG, performs a correctness check against the asker's reference implementation, and benchmarks both the reference code and the accelerated version. On my machine (Intel Xeon E3-1270 v2 @ 3.50 GHz), when compiled with MSVS 2010 at full optimization ( /Ox ), the output of the program is:

p=0000000000550040
ref took 2.020282 secs, fast took 0.027099 secs

where ref refers to the asker's original solution. The speed-up here is about a factor 74x. Different speed-ups will be observed with other (and especially newer) compilers.

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>

#if defined(_WIN32)
#if !defined(WIN32_LEAN_AND_MEAN)
#define WIN32_LEAN_AND_MEAN
#endif
#include <windows.h>
double second (void)
{
    LARGE_INTEGER t;
    static double oofreq;
    static int checkedForHighResTimer;
    static BOOL hasHighResTimer;

    if (!checkedForHighResTimer) {
        hasHighResTimer = QueryPerformanceFrequency (&t);
        oofreq = 1.0 / (double)t.QuadPart;
        checkedForHighResTimer = 1;
    }
    if (hasHighResTimer) {
        QueryPerformanceCounter (&t);
        return (double)t.QuadPart * oofreq;
    } else {
        return (double)GetTickCount() * 1.0e-3;
    }
}
#elif defined(__linux__) || defined(__APPLE__)
#include <stddef.h>
#include <sys/time.h>
double second (void)
{
    struct timeval tv;
    gettimeofday(&tv, NULL);
    return (double)tv.tv_sec + (double)tv.tv_usec * 1.0e-6;
}
#else
#error unsupported platform
#endif

/*
  From: geo <gmars...@gmail.com>
  Newsgroups: sci.math,comp.lang.c,comp.lang.fortran
  Subject: 64-bit KISS RNGs
  Date: Sat, 28 Feb 2009 04:30:48 -0800 (PST)

  This 64-bit KISS RNG has three components, each nearly
  good enough to serve alone.    The components are:
  Multiply-With-Carry (MWC), period (2^121+2^63-1)
  Xorshift (XSH), period 2^64-1
  Congruential (CNG), period 2^64
*/
static uint64_t kiss64_x = 1234567890987654321ULL;
static uint64_t kiss64_c = 123456123456123456ULL;
static uint64_t kiss64_y = 362436362436362436ULL;
static uint64_t kiss64_z = 1066149217761810ULL;
static uint64_t kiss64_t;
#define MWC64  (kiss64_t = (kiss64_x << 58) + kiss64_c, \
                kiss64_c = (kiss64_x >> 6), kiss64_x += kiss64_t, \
                kiss64_c += (kiss64_x < kiss64_t), kiss64_x)
#define XSH64  (kiss64_y ^= (kiss64_y << 13), kiss64_y ^= (kiss64_y >> 17), \
                kiss64_y ^= (kiss64_y << 43))
#define CNG64  (kiss64_z = 6906969069ULL * kiss64_z + 1234567ULL)
#define KISS64 (MWC64 + XSH64 + CNG64)

#define N          (10000000)
#define BITS       (64)
#define BLOCK_SIZE (255)

/* cupdate the count of 1-bits in each bit position for up to 255 numbers */
void sum_block (const uint64_t *pLong, unsigned int *target, int lo, int hi)
{
    int jj, k, kk;
    uint64_t byte_wise_sum [BITS/8] = {0};
    for (jj = lo; jj < hi; jj++) {
        uint64_t t = pLong[jj];
        for (k = 0; k < BITS/8; k++) {
            byte_wise_sum[k] += t & 0x0101010101010101;
            t >>= 1;
        }
    }
    /* accumulate byte sums into target */
    for (k = 0; k < BITS/8; k++) {
        for (kk = 0; kk < BITS; kk += 8) {
            target[kk + k] += (byte_wise_sum[k] >> kk) & 0xff;
        }
    }
}

int main (void) 
{
    double start_ref, stop_ref, start, stop;
    uint64_t *pLong;
    unsigned int target_ref [BITS] = {0};
    unsigned int target [BITS] = {0};
    int i, j;

    pLong = malloc (sizeof(pLong[0]) * N);
    if (!pLong) {
        printf("failed to allocate\n");
        return EXIT_FAILURE;
    }
    printf("p=%p\n", pLong);

    /* init data */
    for (j = 0; j < N; j++) {
        pLong[j] = KISS64;
    }

    /* count bits slowly */
    start_ref = second();
    for (j = 0; j < N; j++) {
        uint64_t m = 1;
        for (i = 0; i < BITS; i++) {
            if ((pLong[j] & m) == m) {
                target_ref[i]++;
            }
            m = (m << 1);
        }
    }
    stop_ref = second();

    /* count bits fast */
    start = second();
    for (j = 0; j < N / BLOCK_SIZE; j++) {
        sum_block (pLong, target, j * BLOCK_SIZE, (j+1) * BLOCK_SIZE);
    }
    sum_block (pLong, target, j * BLOCK_SIZE, N);
    stop = second();

    /* check whether result is correct */
    for (i = 0; i < BITS; i++) {
        if (target[i] != target_ref[i]) {
            printf ("error @ %d: res=%u ref=%u\n", i, target[i], target_ref[i]);
        }
    }

    /* print benchmark results */
    printf("ref took %f secs, fast took %f secs\n", stop_ref - start_ref, stop - start);
    return EXIT_SUCCESS;
}

Answer 4

For starters, the problem of unpacking the bits, because seriously, you do not want to test each bit individually.

So just follow the following strategy for unpacking the bits into bytes of a vector: https://stackoverflow.com/a/24242696/2879325

Now that you have padded each bit to 8 bits, you can just do this for blocks of up to 255 bitmasks at a time, and accumulate them all into a single vector register. After that, you would have to expect potential overflows, so you need to transfer.

After each block of 255, unpack again to 32bit, and add into the array. (You don't have to do exactly 255, just some convenient number less than 256 to avoid overflow of byte accumulators).

At 8 instructions per bitmask (4 per each lower and higher 32-bit with AVX2) - or half that if you have AVX512 available - you should be able to achieve a throughput of about half a billion bitmasks per second and core on an recent CPU.

---

typedef uint64_t T;
const size_t bytes = 8;
const size_t bits = bytes * 8;
const size_t block_size = 128;

static inline __m256i expand_bits_to_bytes(uint32_t x)
{
    __m256i xbcast = _mm256_set1_epi32(x);    // we only use the low 32bits of each lane, but this is fine with AVX2

    // Each byte gets the source byte containing the corresponding bit
    const __m256i shufmask = _mm256_set_epi64x(
        0x0303030303030303, 0x0202020202020202,
        0x0101010101010101, 0x0000000000000000);
    __m256i shuf = _mm256_shuffle_epi8(xbcast, shufmask);

    const __m256i andmask = _mm256_set1_epi64x(0x8040201008040201);  // every 8 bits -> 8 bytes, pattern repeats.
    __m256i isolated_inverted = _mm256_andnot_si256(shuf, andmask);

    // this is the extra step: byte == 0 ? 0 : -1
    return _mm256_cmpeq_epi8(isolated_inverted, _mm256_setzero_si256());
}

void bitcount_vectorized(const T *data, uint32_t accumulator[bits], const size_t count)
{
    for (size_t outer = 0; outer < count - (count % block_size); outer += block_size)
    {
        __m256i temp_accumulator[bits / 32] = { _mm256_setzero_si256() };
        for (size_t inner = 0; inner < block_size; ++inner) {
            for (size_t j = 0; j < bits / 32; j++)
            {
                const auto unpacked = expand_bits_to_bytes(static_cast<uint32_t>(data[outer + inner] >> (j * 32)));
                temp_accumulator[j] = _mm256_sub_epi8(temp_accumulator[j], unpacked);
            }
        }
        for (size_t j = 0; j < bits; j++)
        {
            accumulator[j] += ((uint8_t*)(&temp_accumulator))[j];
        }
    }
    for (size_t outer = count - (count % block_size); outer < count; outer++)
    {
        for (size_t j = 0; j < bits; j++)
        {
            if (data[outer] & (T(1) << j))
            {
                accumulator[j]++;
            }
        }
    }
}

void bitcount_naive(const T *data, uint32_t accumulator[bits], const size_t count)
{
    for (size_t outer = 0; outer < count; outer++)
    {
        for (size_t j = 0; j < bits; j++)
        {
            if (data[outer] & (T(1) << j))
            {
                accumulator[j]++;
            }
        }
    }
}

Depending on the chose compiler, the vectorized form achieved roughly a factor 25 speedup over the naive one.

On a Ryzen 5 1600X, the vectorized form roughly achieved the predicted throughput of ~600,000,000 elements per second.

Surprisingly, this is actually still 50% slower than the solution proposed by @njuffa.

Answer 5

See

Efficient Computation of Positional Population Counts Using SIMD Instructions by Marcus DR Klarqvist, Wojciech Muła, Daniel Lemire (7 Nov 2019)

Faster Population Counts using AVX2 Instructions by Wojciech Muła, Nathan Kurz, Daniel Lemire (23 Nov 2016).

Basically, each full adder compresses 3 inputs to 2 outputs. So one can eliminate an entire 256-bit word for the price of 5 logic instructions. The full adder operation could be repeated until registers become exhausted. Then results in the registers are accumulated (as seen in most of the other answers).

Positional popcnt for 16-bit subwords is implemented here: https://github.com/mklarqvist/positional-popcount

// Carry-Save Full Adder (3:2 compressor)
b ^= a;
a ^= c;
c ^= b; // xor sum
b |= a;
b ^= c; // carry

Note: the accumulate step for positional-popcnt is more expensive than for normal simd popcnt . Which I believe makes it feasible to add a couple of half-adders to the end of the CSU, it might pay to go all the way up to 256 words before accumulating.