This repository contains experimental functions for computing the novel "positional population count" (pospopcnt) statistics using fast SIMD instructions. Given a stream of k-bit words, we seek to count the number of set bits in positions 0, 1, 2, ..., k-1. This problem is a generalization of the population-count problem where we count the sum total of set bits in a k-bit word.
These functions can be applied to any packed 1-hot 8-, 16-, 32-, or 64-bit primitive, for example in machine learning/deep learning. Using large registers (AVX-512), we can achieve ~50 GB/s (~0.120 CPU cycles / int) throughput (25 billion 16-bit integers / second or 200 billion one-hot vectors / second).
For production applications please use libalgebra.h.
The first set of subroutines are described in the paper:
- Efficient Computation of Positional Population Counts Using SIMD Instructions by Marcus D. R. Klarqvist and Wojciech Muła and Daniel Lemire
This benchmark shows the speedup of the 3 pospopcnt algorithms used on x86 CPUs compared to the efficient auto-vectorization of pospopcnt_u16_scalar_naive for different array sizes (in number of 2-byte values). See Results for additional information.
| Algorithm | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 65536 |
|---|---|---|---|---|---|---|---|---|
| pospopcnt_u16_sse_blend_popcnt_unroll8 | 2.09 | 3.16 | 2.35 | 1.88 | 1.67 | 1.56 | 1.5 | 1.44 |
| pospopcnt_u16_avx512_blend_popcnt_unroll8 | 1.78 | 3.61 | 3.61 | 3.59 | 3.68 | 3.65 | 3.67 | 3.7 |
| pospopcnt_u16_avx512_adder_forest | 0.77 | 0.9 | 3.24 | 3.96 | 4.96 | 5.87 | 6.52 | 7.24 |
| pospopcnt_u16_avx512_harley_seal | 0.52 | 0.74 | 1.83 | 2.64 | 4.06 | 6.43 | 9.41 | 16.28 |
Compared to a naive unvectorized solution (pospopcnt_u16_scalar_naive_nosimd):
| Algorithm | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 65536 |
|---|---|---|---|---|---|---|---|---|
| pospopcnt_u16_sse_blend_popcnt_unroll8 | 8.28 | 9.84 | 10.55 | 11 | 11.58 | 11.93 | 12.13 | 12.28 |
| pospopcnt_u16_avx512_blend_popcnt_unroll8 | 7.07 | 11.25 | 16.21 | 21 | 25.49 | 27.91 | 29.73 | 31.55 |
| pospopcnt_u16_avx512_adder_forest | 3.05 | 2.82 | 14.53 | 23.13 | 34.37 | 44.91 | 52.78 | 61.68 |
| pospopcnt_u16_avx512_harley_seal | 2.07 | 2.3 | 8.21 | 15.41 | 28.17 | 49.14 | 76.11 | 138.71 |
The host architecture used is a 10 nm Cannon Lake Core i3-8121U with gcc (GCC) 7.3.1 20180303 (Red Hat 7.3.1-5).
For Linux/Mac: Compile the test suite with: make and run ./bench. For Windows: run cmake . then make. The test suite require c++11 whereas the example and functions require only c99. For more detailed test, see Instrumented tests (Linux specific).
Include pospopcnt.h and pospopcnt.c in your project. Then use the wrapper function for pospopcnt:
pospopcnt_u16(datain, length, target_counters);See example.c for a complete example. Compile with make example.
This is a collaborative effort between Marcus D. R. Klarqvist (@klarqvist), Wojciech Muła (@WojciechMula), and Daniel Lemire (@lemire). We acknowledge @aqrit for contributing with the algorithms pospopcnt_u16_sse2_sad and pospopcnt_u16_scalar_umul128. M.D.R.K. acknowledge James Bonfield (@jkbonfield) for informal discussions that resulted in this work.
We simulated 1 million FLAG fields using a uniform distrubtion U(min,max) with the arguments {1,65536} for 1,000 repetitions using a single core. Numbers represent the average throughput in MB/s (1 MB = 1024b) or average number of CPU cycles per integer. The host architecture used is a 10 nm Cannon Lake Core i3-8121U, a 14 nm Sky Lake Xeon W-2104, and a 22 nm Haswell Xeon E5-2697 v3.
Throughput in CPU cycles / 16-bit integer (lower is better):
| Algorithm | Cannon Lake | Sky Lake | Haswell |
|---|---|---|---|
| pospopcnt_u16_scalar_naive | 2.049 | 3.016 | 3.778 |
| pospopcnt_u16_scalar_naive_nosimd | 17.521 | 17.847 | 18.031 |
| pospopcnt_u16_scalar_partition | 3.079 | 3.042 | 3.358 |
| pospopcnt_u16_scalar_hist1x4 | 2.844 | 2.953 | 3.119 |
| pospopcnt_u16_sse_single | 3.853 | 4.023 | 4.305 |
| pospopcnt_u16_sse_mula | 2.074 | 1.620 | 2.133 |
| pospopcnt_u16_sse_mula_unroll4 | 1.569 | 1.396 | 1.709 |
| pospopcnt_u16_sse_mula_unroll8 | 1.427 | 1.348 | 1.500 |
| pospopcnt_u16_sse_mula_unroll16 | 1.379 | 1.407 | 1.534 |
| pospopcnt_u16_sse2_sad | 1.002 | 1.004 | 1.365 |
| pospopcnt_u16_sse2_csa | 0.342 | 0.356 | 0.428 |
| pospopcnt_u16_avx2_popcnt | 2.378 | 2.334 | 3.013 |
| pospopcnt_u16_avx2 | 2.023 | 3.025 | 4.012 |
| pospopcnt_u16_avx2_naive_counter | 2.022 | 3.023 | 3.905 |
| pospopcnt_u16_avx2_single | 2.698 | 2.937 | 3.589 |
| pospopcnt_u16_avx2_lemire | 2.862 | 1.919 | 2.187 |
| pospopcnt_u16_avx2_lemire2 | 1.708 | 1.139 | 1.503 |
| pospopcnt_u16_avx2_mula | 1.100 | 1.035 | 1.305 |
| pospopcnt_u16_avx2_mula_unroll4 | 0.833 | 0.775 | 0.879 |
| pospopcnt_u16_avx2_mula_unroll8 | 0.745 | 0.709 | 0.768 |
| pospopcnt_u16_avx2_mula_unroll16 | 0.725 | 0.741 | 0.805 |
| pospopcnt_u16_avx2_mula3 | 0.414 | 0.416 | 0.479 |
| pospopcnt_u16_avx2_csa | 0.199 | 0.270 | 0.308 |
| pospopcnt_u16_avx512 | 1.501 | 1.616 | - |
| pospopcnt_u16_avx512_popcnt32_mask | 0.910 | 1.375 | - |
| pospopcnt_u16_avx512_popcnt64_mask | 0.879 | 0.995 | - |
| pospopcnt_u16_avx512_masked_ops | 1.835 | 2.000 | - |
| pospopcnt_u16_avx512_popcnt | 1.663 | 1.741 | - |
| pospopcnt_u16_avx512_mula | 0.830 | 0.803 | - |
| pospopcnt_u16_avx512_mula_unroll4 | 0.642 | 0.582 | - |
| pospopcnt_u16_avx512_mula_unroll8 | 0.552 | 0.588 | - |
| pospopcnt_u16_avx512_mula2 | 0.509 | 0.510 | - |
| pospopcnt_u16_avx512_mula3 | 0.280 | 0.353 | - |
| pospopcnt_u16_avx512_csa | 0.121 | 0.264 | - |
Throughput in MB/s (higher is better):
| Algorithm | Cannon Lake | Sky Lake | Haswell |
|---|---|---|---|
| pospopcnt_u16_scalar_naive | 2952.55 | 1816.52 | 1319.05 |
| pospopcnt_u16_scalar_naive_nosimd | 345.284 | 331.54 | 263.482 |
| pospopcnt_u16_scalar_partition | 1962.29 | 1794.31 | 1428.73 |
| pospopcnt_u16_scalar_hist1x4 | 2126.36 | 1842.85 | 1575.02 |
| pospopcnt_u16_sse_single | 1569.83 | 1343.2 | 1149.01 |
| pospopcnt_u16_sse_mula | 2916.44 | 3238.28 | 2251.89 |
| pospopcnt_u16_sse_mula_unroll4 | 3845.46 | 3754.62 | 2464.27 |
| pospopcnt_u16_sse_mula_unroll8 | 4238.55 | 3884.62 | 3254.86 |
| pospopcnt_u16_sse_mula_unroll16 | 4374.65 | 3682.14 | 3137.09 |
| pospopcnt_u16_sse2_sad | 6035.91 | 5183.01 | 3732.58 |
| pospopcnt_u16_sse2_csa | 17498.6 | 12466.3 | 10422.7 |
| pospopcnt_u16_avx2_popcnt | 2539.75 | 2223.02 | 1647.11 |
| pospopcnt_u16_avx2 | 2994.27 | 1796 | 1134.65 |
| pospopcnt_u16_avx2_naive_counter | 2994.27 | 1809.63 | 1112.81 |
| pospopcnt_u16_avx2_single | 2230.82 | 1835.75 | 1340.37 |
| pospopcnt_u16_avx2_lemire | 2114.58 | 2619.98 | 2228.21 |
| pospopcnt_u16_avx2_lemire2 | 3519.09 | 4618.28 | 3266.01 |
| pospopcnt_u16_avx2_mula | 5480.89 | 3653.92 | 2435.95 |
| pospopcnt_u16_avx2_mula_unroll4 | 7224.81 | 6055.08 | 3981.94 |
| pospopcnt_u16_avx2_mula_unroll8 | 8081.99 | 7143.63 | 5211.34 |
| pospopcnt_u16_avx2_mula_unroll16 | 8329.03 | 6811.96 | 5512.57 |
| pospopcnt_u16_avx2_mula3 | 14671.9 | 11089.2 | 7817 |
| pospopcnt_u16_avx2_csa | 28899.2 | 15381.8 | 14559.9 |
| pospopcnt_u16_avx512 | 4023.94 | 3249.32 | - |
| pospopcnt_u16_avx512_popcnt32_mask | 4721.16 | 3776.93 | - |
| pospopcnt_u16_avx512_popcnt64_mask | 6860.97 | 5155 | - |
| pospopcnt_u16_avx512_masked_ops | 3299.91 | 4119.54 | - |
| pospopcnt_u16_avx512_popcnt | 3571.81 | 2989.57 | - |
| pospopcnt_u16_avx512_mula | 7252.28 | 6294.88 | - |
| pospopcnt_u16_avx512_mula_unroll4 | 9395.81 | 8365.56 | - |
| pospopcnt_u16_avx512_mula_unroll8 | 10899.1 | 8292.82 | - |
| pospopcnt_u16_avx512_mula2 | 11846.9 | 9633.07 | - |
| pospopcnt_u16_avx512_mula3 | 21430.9 | 11995.9 | - |
| pospopcnt_u16_avx512_csa | 48906.4 | 17339.5 | - |
If you are running Linux, you can run tests that take advantage of Performance Counters for Linux (PCL). This allows for programmatic discovery and enumeration of all counters and events. Compile this test suite with: make instrumented_benchmark. Running the output executable requires root (sudo) access to the host machine. Pass the -v (verbose) flag to get a detailed report of performance counters (example output from the Cannon Lake machine):
$ make instrumented_benchmark
$ sudo ./instrumented_benchmark -v
n = 10000000
pospopcnt_u16_scalar_naive_nosimd instructions per cycle 3.82, cycles per 16-bit word: 17.524, instructions per 16-bit word 67.002
min: 675172 cycles, 2581464 instructions, 1 branch mis., 4 cache ref., 0 cache mis.
avg: 681460.1 cycles, 2581464.2 instructions, 2.2 branch mis., 41.2 cache ref., 0.6 cache mis.
pospopcnt_u16_scalar_naive instructions per cycle 1.99, cycles per 16-bit word: 2.059, instructions per 16-bit word 4.104
min: 79312 cycles, 158101 instructions, 1 branch mis., 83 cache ref., 0 cache mis.
avg: 79454.0 cycles, 158101.0 instructions, 2.0 branch mis., 133.4 cache ref., 0.0 cache mis.
pospopcnt_u16_scalar_partition instructions per cycle 2.60, cycles per 16-bit word: 3.090, instructions per 16-bit word 8.038
min: 119056 cycles, 309690 instructions, 1 branch mis., 4 cache ref., 0 cache mis.
avg: 120536.1 cycles, 309690.0 instructions, 1.2 branch mis., 31.8 cache ref., 0.3 cache mis.
pospopcnt_u16_hist1x4 instructions per cycle 2.03, cycles per 16-bit word: 2.851, instructions per 16-bit word 5.788
min: 109856 cycles, 223008 instructions, 2 branch mis., 10 cache ref., 0 cache mis.
avg: 111365.9 cycles, 223008.0 instructions, 2.4 branch mis., 61.1 cache ref., 0.3 cache mis.
pospopcnt_u16_sse_single instructions per cycle 2.73, cycles per 16-bit word: 3.616, instructions per 16-bit word 9.874
min: 139336 cycles, 380410 instructions, 3 branch mis., 129 cache ref., 0 cache mis.
avg: 139815.0 cycles, 380410.0 instructions, 5.8 branch mis., 206.7 cache ref., 0.8 cache mis.
pospopcnt_u16_sse_mula instructions per cycle 3.00, cycles per 16-bit word: 2.084, instructions per 16-bit word 6.252
min: 80282 cycles, 240888 instructions, 1 branch mis., 7 cache ref., 0 cache mis.
avg: 80492.4 cycles, 240888.0 instructions, 2.1 branch mis., 29.6 cache ref., 0.4 cache mis.
pospopcnt_u16_sse_mula_unroll4 instructions per cycle 3.35, cycles per 16-bit word: 1.577, instructions per 16-bit word 5.284
min: 60757 cycles, 203573 instructions, 2 branch mis., 29 cache ref., 0 cache mis.
avg: 61098.4 cycles, 203573.0 instructions, 2.3 branch mis., 64.2 cache ref., 0.2 cache mis.
pospopcnt_u16_sse_mula_unroll8 instructions per cycle 4.08, cycles per 16-bit word: 1.433, instructions per 16-bit word 5.846
min: 55194 cycles, 225221 instructions, 2 branch mis., 12 cache ref., 0 cache mis.
avg: 55304.1 cycles, 225221.0 instructions, 2.2 branch mis., 35.0 cache ref., 0.4 cache mis.
pospopcnt_u16_sse_mula_unroll16 instructions per cycle 3.62, cycles per 16-bit word: 1.385, instructions per 16-bit word 5.018
min: 53377 cycles, 193332 instructions, 2 branch mis., 45 cache ref., 0 cache mis.
avg: 53743.4 cycles, 193332.0 instructions, 2.4 branch mis., 93.8 cache ref., 0.5 cache mis.
pospopcnt_u16_sse_sad instructions per cycle 3.08, cycles per 16-bit word: 0.985, instructions per 16-bit word 3.033
min: 37943 cycles, 116844 instructions, 1 branch mis., 65 cache ref., 0 cache mis.
avg: 38069.8 cycles, 116844.0 instructions, 2.1 branch mis., 100.1 cache ref., 0.1 cache mis.
pospopcnt_u16_avx2_popcnt instructions per cycle 2.56, cycles per 16-bit word: 2.379, instructions per 16-bit word 6.100
min: 91645 cycles, 235038 instructions, 1 branch mis., 154 cache ref., 0 cache mis.
avg: 92870.1 cycles, 235038.0 instructions, 4.2 branch mis., 243.9 cache ref., 2.5 cache mis.
pospopcnt_u16_avx2 instructions per cycle 2.01, cycles per 16-bit word: 2.038, instructions per 16-bit word 4.105
min: 78506 cycles, 158140 instructions, 2 branch mis., 131 cache ref., 0 cache mis.
avg: 78592.1 cycles, 158140.0 instructions, 2.3 branch mis., 197.1 cache ref., 0.8 cache mis.
pospopcnt_u16_avx2_naive_counter instructions per cycle 2.02, cycles per 16-bit word: 2.034, instructions per 16-bit word 4.104
min: 78360 cycles, 158124 instructions, 1 branch mis., 102 cache ref., 0 cache mis.
avg: 78479.2 cycles, 158124.0 instructions, 2.3 branch mis., 162.3 cache ref., 0.8 cache mis.
pospopcnt_u16_avx2_single instructions per cycle 2.02, cycles per 16-bit word: 2.029, instructions per 16-bit word 4.104
min: 78191 cycles, 158113 instructions, 1 branch mis., 125 cache ref., 0 cache mis.
avg: 78324.6 cycles, 158113.0 instructions, 2.3 branch mis., 179.0 cache ref., 0.6 cache mis.
pospopcnt_u16_avx2_lemire instructions per cycle 2.80, cycles per 16-bit word: 2.861, instructions per 16-bit word 8.006
min: 110245 cycles, 308436 instructions, 1 branch mis., 44 cache ref., 0 cache mis.
avg: 110818.9 cycles, 308436.0 instructions, 1.1 branch mis., 103.3 cache ref., 0.4 cache mis.
pospopcnt_u16_avx2_lemire2 instructions per cycle 3.17, cycles per 16-bit word: 1.697, instructions per 16-bit word 5.382
min: 65395 cycles, 207368 instructions, 1 branch mis., 8 cache ref., 0 cache mis.
avg: 66261.5 cycles, 207368.0 instructions, 1.1 branch mis., 42.6 cache ref., 0.6 cache mis.
pospopcnt_u16_avx2_mula instructions per cycle 2.89, cycles per 16-bit word: 1.104, instructions per 16-bit word 3.190
min: 42544 cycles, 122897 instructions, 2 branch mis., 10 cache ref., 0 cache mis.
avg: 42804.7 cycles, 122897.0 instructions, 3.1 branch mis., 40.5 cache ref., 0.4 cache mis.
pospopcnt_u16_avx2_mula2 instructions per cycle 2.94, cycles per 16-bit word: 1.454, instructions per 16-bit word 4.269
min: 56002 cycles, 164472 instructions, 3 branch mis., 8 cache ref., 0 cache mis.
avg: 56256.0 cycles, 164472.0 instructions, 3.3 branch mis., 43.8 cache ref., 0.4 cache mis.
pospopcnt_u16_avx2_mula_unroll4 instructions per cycle 3.19, cycles per 16-bit word: 0.847, instructions per 16-bit word 2.706
min: 32631 cycles, 104244 instructions, 1 branch mis., 10 cache ref., 0 cache mis.
avg: 32779.7 cycles, 104244.0 instructions, 1.9 branch mis., 38.2 cache ref., 0.3 cache mis.
pospopcnt_u16_avx2_mula_unroll8 instructions per cycle 3.95, cycles per 16-bit word: 0.756, instructions per 16-bit word 2.986
min: 29123 cycles, 115056 instructions, 2 branch mis., 10 cache ref., 0 cache mis.
avg: 29201.8 cycles, 115056.0 instructions, 2.2 branch mis., 41.4 cache ref., 0.5 cache mis.
pospopcnt_u16_avx2_mula_unroll16 instructions per cycle 3.49, cycles per 16-bit word: 0.738, instructions per 16-bit word 2.574
min: 28416 cycles, 99177 instructions, 3 branch mis., 31 cache ref., 0 cache mis.
avg: 28535.7 cycles, 99177.0 instructions, 3.6 branch mis., 84.7 cache ref., 0.5 cache mis.
pospopcnt_u16_avx512 instructions per cycle 2.07, cycles per 16-bit word: 1.511, instructions per 16-bit word 3.133
min: 58218 cycles, 120696 instructions, 1 branch mis., 30 cache ref., 0 cache mis.
avg: 58299.8 cycles, 120696.0 instructions, 2.1 branch mis., 80.8 cache ref., 0.6 cache mis.
pospopcnt_u16_avx512_popcnt32_mask instructions per cycle 3.21, cycles per 16-bit word: 0.818, instructions per 16-bit word 2.629
min: 31534 cycles, 101304 instructions, 1 branch mis., 9 cache ref., 0 cache mis.
avg: 31807.5 cycles, 101304.0 instructions, 2.1 branch mis., 47.0 cache ref., 0.3 cache mis.
pospopcnt_u16_avx512_popcnt64_mask instructions per cycle 3.10, cycles per 16-bit word: 0.835, instructions per 16-bit word 2.593
min: 32189 cycles, 99912 instructions, 1 branch mis., 60 cache ref., 0 cache mis.
avg: 32391.7 cycles, 99912.0 instructions, 2.6 branch mis., 93.8 cache ref., 0.7 cache mis.
pospopcnt_u16_avx512_popcnt instructions per cycle 1.96, cycles per 16-bit word: 1.676, instructions per 16-bit word 3.287
min: 64590 cycles, 126635 instructions, 1 branch mis., 90 cache ref., 0 cache mis.
avg: 64755.2 cycles, 126635.0 instructions, 2.3 branch mis., 169.1 cache ref., 1.3 cache mis.
pospopcnt_u16_avx512_mula instructions per cycle 2.41, cycles per 16-bit word: 0.752, instructions per 16-bit word 1.815
min: 28975 cycles, 69920 instructions, 1 branch mis., 17 cache ref., 0 cache mis.
avg: 29182.1 cycles, 69920.0 instructions, 1.7 branch mis., 50.0 cache ref., 0.2 cache mis.
pospopcnt_u16_avx512_mula_unroll4 instructions per cycle 2.52, cycles per 16-bit word: 0.624, instructions per 16-bit word 1.573
min: 24054 cycles, 60599 instructions, 1 branch mis., 28 cache ref., 0 cache mis.
avg: 24189.0 cycles, 60599.0 instructions, 1.3 branch mis., 69.2 cache ref., 0.4 cache mis.
pospopcnt_u16_avx512_mula_unroll8 instructions per cycle 3.08, cycles per 16-bit word: 0.556, instructions per 16-bit word 1.713
min: 21410 cycles, 66003 instructions, 1 branch mis., 16 cache ref., 0 cache mis.
avg: 21472.3 cycles, 66003.0 instructions, 1.9 branch mis., 47.8 cache ref., 0.5 cache mis.
pospopcnt_u16_avx2_mula3 instructions per cycle 3.02, cycles per 16-bit word: 0.436, instructions per 16-bit word 1.318
min: 16804 cycles, 50796 instructions, 1 branch mis., 49 cache ref., 0 cache mis.
avg: 17025.3 cycles, 50796.0 instructions, 3.0 branch mis., 135.1 cache ref., 1.1 cache mis.
pospopcnt_u16_avx512_mula3 instructions per cycle 2.20, cycles per 16-bit word: 0.291, instructions per 16-bit word 0.640
min: 11194 cycles, 24644 instructions, 0 branch mis., 76 cache ref., 0 cache mis.
avg: 11295.2 cycles, 24644.0 instructions, 0.2 branch mis., 147.0 cache ref., 1.5 cache mis.
pospopcnt_u16_avx2_csa instructions per cycle 2.99, cycles per 16-bit word: 0.256, instructions per 16-bit word 0.765
min: 9854 cycles, 29492 instructions, 1 branch mis., 193 cache ref., 0 cache mis.
avg: 10114.0 cycles, 29492.0 instructions, 1.3 branch mis., 288.8 cache ref., 1.9 cache mis.
pospopcnt_u16_avx512_csa instructions per cycle 2.09, cycles per 16-bit word: 0.168, instructions per 16-bit word 0.350
min: 6463 cycles, 13496 instructions, 0 branch mis., 342 cache ref., 0 cache mis.
avg: 6656.0 cycles, 13496.0 instructions, 1.0 branch mis., 434.4 cache ref., 2.3 cache mis.
pospopcnt_u16_avx512_mula2 instructions per cycle 1.99, cycles per 16-bit word: 0.517, instructions per 16-bit word 1.029
min: 19917 cycles, 39640 instructions, 1 branch mis., 94 cache ref., 0 cache mis.
avg: 19988.3 cycles, 39640.0 instructions, 1.4 branch mis., 166.3 cache ref., 0.2 cache mis.