Slower performance for 32-bit integers

[tdulcet]

Thank you for creating this wonderful library!

I compared the performance to other similar libraries/programs and it is significantly faster for larger 128-bit integers. For example, compared to GNU factor, it is over 330× faster:

$ ./time.sh -i -r 5 "seq $(bc <<<'(2^127) - (10^2) - 1') $(bc <<<'(2^127) - 1') | "{factor,./gnu/factor,./factoring/gcc_example,./factoring/clang_example}
Benchmark #1: seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | factor
  Time (mean ± σ std dev):     143.1562s ±  0.5082s          [User: 143.1556s, System: 0.0000s]
  Range (min … median … max):  142.371s … 143.422s … 143.732s   CPU: 100.0%, 5 runs

Benchmark #2: seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./gnu/factor
  Time (mean ± σ std dev):     142.3252s ±  0.7053s          [User: 141.9832s, System: 0.0060s]
  Range (min … median … max):  141.651s … 141.996s … 143.588s   CPU:  99.8%, 5 runs

Benchmark #3: seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./factoring/gcc_example
  Time (mean ± σ std dev):      0.9602s ±  0.7591s          [User: 0.5456s, System: 0.0080s]
  Range (min … median … max):   0.507s …  0.564s …  2.473s   CPU:  57.7%, 5 runs

Benchmark #4: seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./factoring/clang_example
  Time (mean ± σ std dev):      0.4254s ±  0.1519s          [User: 0.3314s, System: 0.0080s]
  Range (min … median … max):   0.312s …  0.356s …  0.719s   CPU:  79.8%, 5 runs

Summary
  #4 ‘seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./factoring/clang_example’ ran
  336.521 ± 120.198 times (33,552.1%) faster than #1 ‘seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | factor’
  334.568 ± 119.506 times (33,356.8%) faster than #2 ‘seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./gnu/factor’
    2.257 ± 1.958 times (125.7%) faster than #3 ‘seq 170141183460469231731687303715884105627 170141183460469231731687303715884105727 | ./factoring/gcc_example’

For a couple of 128-bit integers with large 64-bit factors (from the GNU test suite), it is almost 2,000× faster:

$ ./time.sh -i -m 2 "echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | "{factor,./gnu/factor,./factoring/gcc_example,./factoring/clang_example}
Benchmark #1: echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | factor
  Time (mean ± σ std dev):     319.4225s ±  0.3625s          [User: 319.4225s, System: 0.0000s]
  Range (min … median … max):  319.060s … 319.422s … 319.785s   CPU: 100.0%, 2 runs

Benchmark #2: echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./gnu/factor
  Time (mean ± σ std dev):     318.5655s ±  0.4765s          [User: 317.7500s, System: 0.0005s]
  Range (min … median … max):  318.089s … 318.566s … 319.042s   CPU:  99.7%, 2 runs

Benchmark #3: echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./factoring/gcc_example
  Time (mean ± σ std dev):      1.1580s ±  0.8340s          [User: 0.2590s, System: 0.0340s]
  Range (min … median … max):   0.324s …  1.158s …  1.992s   CPU:  25.3%, 2 runs

Benchmark #4: echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./factoring/clang_example
  Time (mean ± σ std dev):      0.1634s ±  0.0169s          [User: 0.1365s, System: 0.0059s]
  Range (min … median … max):   0.152s …  0.158s …  0.219s   CPU:  87.2%, 13 runs

Summary
  #4 ‘echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./factoring/clang_example’ ran
1,955.034 ± 202.348 times (195,403.4%) faster than #1 ‘echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | factor’
1,949.789 ± 201.814 times (194,878.9%) faster than #2 ‘echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./gnu/factor’
    7.088 ± 5.157 times (608.8%) faster than #3 ‘echo 170141183460469225450570946617781744489 170141183460469229545748130981302223887 | ./factoring/gcc_example’

However, for smaller integers under 32-bits, it seems to be up to 11× slower than GNU factor:

$ ./time.sh -i -r 5 'seq 2 10000000 | '{factor,./gnu/factor,./uu/factor,./factoring/gcc_example,./factoring/clang_example,'./numbers -p'}
Benchmark #1: seq 2 10000000 | factor
  Time (mean ± σ std dev):      2.2944s ±  0.0112s          [User: 2.0142s, System: 0.4204s]
  Range (min … median … max):   2.281s …  2.291s …  2.313s   CPU: 106.1%, 5 runs

Benchmark #2: seq 2 10000000 | ./gnu/factor
  Time (mean ± σ std dev):      2.2020s ±  0.0087s          [User: 1.8820s, System: 0.4534s]
  Range (min … median … max):   2.188s …  2.201s …  2.213s   CPU: 106.1%, 5 runs

Benchmark #3: seq 2 10000000 | ./uu/factor
  Time (mean ± σ std dev):     12.6390s ±  0.0296s          [User: 10.2822s, System: 2.4632s]
  Range (min … median … max):  12.591s … 12.637s … 12.684s   CPU: 100.8%, 5 runs

Benchmark #4: seq 2 10000000 | ./factoring/gcc_example
  Time (mean ± σ std dev):     15.2668s ±  0.1457s          [User: 12.6782s, System: 2.7610s]
  Range (min … median … max):  15.093s … 15.291s … 15.512s   CPU: 101.1%, 5 runs

Benchmark #5: seq 2 10000000 | ./factoring/clang_example
  Time (mean ± σ std dev):     15.2658s ±  0.0899s          [User: 12.7338s, System: 2.7064s]
  Range (min … median … max):  15.172s … 15.212s … 15.419s   CPU: 101.1%, 5 runs

Benchmark #6: seq 2 10000000 | ./numbers -p
  Time (mean ± σ std dev):     19.5844s ±  0.2708s          [User: 17.0100s, System: 2.7530s]
  Range (min … median … max):  19.241s … 19.561s … 20.072s   CPU: 100.9%, 5 runs

Summary
  #2 ‘seq 2 10000000 | ./gnu/factor’ ran
    1.042 ± 0.007 times (4.2%) faster than #1 ‘seq 2 10000000 | factor’
    5.740 ± 0.026 times (474.0%) faster than #3 ‘seq 2 10000000 | ./uu/factor’
    6.933 ± 0.072 times (593.3%) faster than #4 ‘seq 2 10000000 | ./factoring/gcc_example’
    6.933 ± 0.049 times (593.3%) faster than #5 ‘seq 2 10000000 | ./factoring/clang_example’
    8.894 ± 0.128 times (789.4%) faster than #6 ‘seq 2 10000000 | ./numbers -p’

It seems to get progressively faster as the integers get larger. For integers less than 10⁴, it is about 10.9× slower, for 10⁵ about 8.1× slower, for 10⁶ about 7.2× slower, for 10⁷ about 6.9× slower and for 10⁸ about 4.6× slower.

The above benchmarks were created with my Benchmarking Tool and include the system GNU factor (factor), a locally compiled version of GNU factor with -O3 and LTO enabled (./gnu/factor), uutils' Rust port of GNU factor (./uu/factor), a simple wrapper program for this library adapted from your example without CMake compiled with both GCC (./factoring/gcc_example) and Clang (./factoring/clang_example) and with inline ASM enabled, and lastly my simple Numbers Tool program which includes a partial C++ port of GNU factor (./numbers -p).

Hopefully I am using your library correctly to get maximum performance. For reference, the wrapper program I wrote for your library is available here: https://gist.github.com/tdulcet/6393865769644aaf9244561f3eb4e9c2. The commands I used to compile and run it are on top of the file. I used GCC 11.4 and Clang 14.0. Note that Clang did generate 42 compiler warnings, but in my benchmarking it was always faster than GCC with your library.