OMP usage should be reviewed

[dasergatskov]

I think OMP usage shoud be optional. Apple really does not like omp and compiling with OMP on apple is somewhat involved.

Did you try to benchmark your actual OMP code? I briefly looked at it and it looks suspicious to me. In fact I think it most likely hurt than help.
E.g. with default OMP setting on Apple MacOS (M4 Pro) (14 threads):

 octave:9> A = mpfr_t (randn (30), 113);
octave:10> tic; B = A + 1e-20; toc
warning: mpfr_t.m (plus at line 381): Inexact operation.

Suppress MPFR_T inexactness warning messages with:

	warning ('off', 'mpfr_t:inexactOperation')
Elapsed time is 0.287056 seconds.

With % OMP_NUM_THREADS=1 octave

octave:2> A = mpfr_t (randn (30), 113);
octave:3> tic; B = A + 1e-20; toc
warning: mpfr_t.m (plus at line 381): Inexact operation.

Suppress MPFR_T inexactness warning messages with:

	warning ('off', 'mpfr_t:inexactOperation')
Elapsed time is 0.00146723 seconds.

[dasergatskov]

I think attached diff fixes some of the problems: it scales the number of threads with problem complexity (I ballpark the cost of the extra thread and the numbers can be adjusted further); removes (flatten / collaps) nested OMP pragmas; moved OMP pragma from the inner loop to the outside.
It improved my (casual) benchmarks, please check your. E.g.:

octave:1> pkg load apa
APA is ready to use.
    Detected MPFR version: 4.2.2
    Detected  GMP version: 6.3.0
octave:2> A = mpfr_t (randn (30), 113);
octave:3> tic; B = A + 1e-20; toc
warning: mpfr_t.m (plus at line 381): Inexact operation.

Suppress MPFR_T inexactness warning messages with:

	warning ('off', 'mpfr_t:inexactOperation')
Elapsed time is 0.0013659 seconds.
octave:4> A = mpfr_t (randn (300), 113);
octave:5> tic; B = A + 1e-20; toc
warning: mpfr_t.m (plus at line 381): Inexact operation.

Suppress MPFR_T inexactness warning messages with:

	warning ('off', 'mpfr_t:inexactOperation')
Elapsed time is 0.00598788 seconds.
octave:6> A = mpfr_t (randn (3000), 113);
octave:7> tic; B = A + 1e-20; toc
warning: mpfr_t.m (plus at line 381): Inexact operation.

Suppress MPFR_T inexactness warning messages with:

	warning ('off', 'mpfr_t:inexactOperation')
Elapsed time is 0.27712 seconds.

The added function get_optimal_threads in mex_mpfr_algorithms.h shows how one can conditionally include OMP code.

UPDATE:
I was getting intermittent error

octave:6> A = mpfr_t (randn (3000), 113);

error: mex_apa_interface: mex_mpfr_interface.c:150:mex_mpfr_interface():cmd[1003]:idx Invalid MPFR variable indices.

(the error was kind of "sticky" in particular Octave session -- oce I got it it would reproduce 100% until I restart Octave). It usually (but not always) happens if I run the code directly after I installed APA package.
So I tried a few different things which did not really fix the problem but improved APA speed by about 2x.

optimize_das.diff.gz

At this moment I pretty much run out of ideas what could it be. I suspect the MEX (at least Octave's MEX) does nor play well with OpenMP. So one thing to try is to change the code so or any OpenMP region inside the MEX file, do not let worker threads touch MEX/Octave-managed arrays directly. But that a lot of work and no guarantee for a fix.

[Schamschula]

Speaking of libomp on macOS:

While attempting to create a Portfile of apa 1.1.1 for MacPorts I get

Executing:  cd "/opt/local/var/macports/build/octave-apa-1fa3225d/work/apa" && /usr/bin/arch -arch $OCTAVE_ARCH /opt/local/bin/octave-cli -q -f -H --eval 'try; pkg prefix /opt/local/var/macports/build/octave-apa-1fa3225d/work/destroot/opt/local/share/octave/packages /opt/local/var/macports/build/octave-apa-1fa3225d/work/destroot/opt/local/lib/octave/packages; pkg install -verbose -nodeps -local /opt/local/var/macports/build/octave-apa-1fa3225d/work/tmp-build/apa-1.1.1-any-none.tar.gz; catch; disp(lasterror.message); exit(1); end_try_catch;' 
mkdir (/opt/local/var/macports/build/octave-apa-1fa3225d/work/.tmp/oct-6mKOdS)
untar (/opt/local/var/macports/build/octave-apa-1fa3225d/work/tmp-build/apa-1.1.1-any-none.tar.gz, /opt/local/var/macports/build/octave-apa-1fa3225d/work/.tmp/oct-6mKOdS)
ld: library 'omp' not found
clang++: error: linker command failed with exit code 1 (use -v to see invocation)
warning: mkoctfile: building exited with failure status
MEX interface creation failed: mex: building exited with failure status

MPFR cannot be used.
Command failed:  cd "/opt/local/var/macports/build/octave-apa-1fa3225d/work/apa" && /usr/bin/arch -arch $OCTAVE_ARCH /opt/local/bin/octave-cli -q -f -H --eval 'try; pkg prefix /opt/local/var/macports/build/octave-apa-1fa3225d/work/destroot/opt/local/share/octave/packages /opt/local/var/macports/build/octave-apa-1fa3225d/work/destroot/opt/local/lib/octave/packages; pkg install -verbose -nodeps -local /opt/local/var/macports/build/octave-apa-1fa3225d/work/tmp-build/apa-1.1.1-any-none.tar.gz; catch; disp(lasterror.message); exit(1); end_try_catch;' 
Exit code: 1
Error: Failed to destroot octave-apa: command execution failed

libomp.dylib is installed under /opt/local/lib/libomp/, but I had to help install_apa.m to find it. I needed to patch
that file as well as inst/mex/mex_apa_interface.h to look for libomp/omp.h rather than omp.h.

See: macports/macports-ports@745ff1b

[siko1056]

Thank you for the benchmark and porting effort @dasergatskov and @Schamschula ,
I agree, that the OpenMP idea was pre-mature "optimization", which might not work well in the context of MPFR and deeper nested datatypes than PODs. Let me soon make OpenMP an optional feature, until I can show it really brings a performance benefit for simplicity.