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This is the algorithm by GiNaC. In theory one
could extend this to add a caching mechanism such as

    complex(kind=prec) :: cache(size(x),MPLMaxQ)

    do q=1,j
      cache(:,q) = x**q/q**m
    enddo
    do q=1,MPLMaxQ
      res = t(1)

      ! Fortran uses Column-major order, hence cache(:,q) is
      ! faster than cache(q,:).
      cache(:,q+j-1) = x**(q+j-1)/(q+j-1)**m
      t(j) = t(j) + cache(j,q)
      do k=1,j-1
        t(j-k) = t(j-k) + t(j-k+1) * cache(j-k,k+q)
      enddo

      if (mod(q,2) .eq. 1) then
        if (abs(t(1)-res).lt.MPLdel) exit
      endif
    enddo

In practice this doesn't really help because any
time saved with the cache is paid back through the
allocation and clearing of cache(:,:). Both
variations work similarly well now. If at some
point we might need MPLs with many more arguments
(size(x)), this might change.

I've studied the timing a bit and the message is
quite clear:

 1) Use -O3
 2) Use -march=native and -mtune=native (in some
    cases it might be better to use actually work
    out what the architecture is as Kalby Lake
    (7th gen i5) is misdetected as Broadwell (5th
    gen))
 3) even though -ffast-math speeds the code up
    tremendously it also produces very wrong
    results.

Below a list of G/s

Nothing           :  5907.27,  5852.97,  4255.59,  5627.56,  5886.03
O3                :  9780.68, 11269.85, 11464.97, 10475.08, 10966.49
O3+unroll         : 11385.20, 10785.49, 11361.03, 10225.69, 11134.86
O3+tree vec       : 11028.18, 11232.13, 11349.96, 11257.04, 11410.13
O3+native         :  9124.84,  8609.82,  9330.82,  9912.89,  9503.70
O3+skylake        : 11894.75, 11966.64, 11882.61, 12000.73, 11666.96
O3+march+mtune    : 11818.07, 11943.54, 11963.75, 11780.81, 11560.69
O3+native+nati    : 11390.44, 11873.35, 11827.96, 11781.51, 11725.69
O3+ffast-math     : 19014.69, 19016.52, 18849.96, 19213.97, 19067.80
O3+un+vec+nat+nat : 11521.01, 11666.27, 11341.51, 11290.41, 11488.92
O3+vec+nat+nat    : 10712.55, 11211.78, 11328.53, 11140.49, 11298.79
O3+un+nat+nat     : 11702.13, 11442.44, 11680.73, 11498.26, 11677.28

unroll: -funroll-loops
tree vec: -ftree-vectorize
skylake: -march=skylake
native: -march=native
march+mtune: -march=skylake -mtune=skylake