[WIP] allow for an alternate EOS table for the helmholtz EOS

[zingale]

This will allow for a table generated by our new C++ EOS, which uses better quadrature and precision and should be more accurate overall for the electron-positron component.

[zingale]

Here's a comparison of test_eos with the old and new tables:

            variable name            absolute error            relative error
                                        (||A - B||)         (||A - B||/||A||)
 ----------------------------------------------------------------------------
 level = 0
 density                                          0                         0
 temperature                                      0                         0
 specific_enthalpy                  8.079280964e+19           2.982940776e-06
 specific_energy                    6.095835231e+19           2.980379594e-06
 pressure                           6.068397046e+21           4.547890027e-07
 specific_entropy                        8079280958           2.982939321e-06
 X_hydrogen-1                                     0                         0
 X_helium-3                                       0                         0
 X_helium-4                                       0                         0
 X_carbon-12                                      0                         0
 X_nitrogen-14                                    0                         0
 X_oxygen-16                                      0                         0
 X_neon-20                                        0                         0
 X_magnesium-24                                   0                         0
 X_silicon-28                                     0                         0
 X_sulfur-32                                      0                         0
 X_argon-36                                       0                         0
 X_calcium-40                                     0                         0
 X_titanium-44                                    0                         0
 X_chromium-48                                    0                         0
 X_iron-52                                        0                         0
 X_iron-54                                        0                         0
 X_nickel-56                                      0                         0
 X_neutron                                        0                         0
 X_proton                                         0                         0
 err_T_eos_rh                       4.463145742e-10               1.042990214
 err_rho_eos_tp                         6.667819099               1.051192241
 err_T_eos_rp                       1.097922213e-10              0.9788131459
 err_T_eos_re                       6.344377762e-10              0.9932798702
 err_rho_eos_ps                      2.00036147e-13              0.8982993348
 err_T_eos_ps                       2.727631179e-11              0.8926206267
 err_rho_eos_ph                     9.138613768e-13               3.825521075
 err_T_eos_ph                       6.301744143e-10               1.098603558
 err_rho_eos_th                     2.760014189e-06           1.529514792e-05
 c_v                                2.454254217e+10           2.973657129e-06
 c_p                                1.569158872e+18           4.762952204e-06
 n_e                                4.653342571e+26           1.715461361e-07
 n_p                                              0                         0
 eta                                   0.1070367233           1.150363746e-06
 p_ele                              1.856725216e+21           1.889507739e-07
 p_pos                                            0                         0
 mu                                               0                         0
 mu_e                                             0                         0
 Y_e                                              0                         0
 dp_dT                              6.857777685e+11           1.071302253e-06
 dp_drho                            3.563847897e+14           0.0001098986724
 de_dT                              2.454254217e+10           2.973657129e-06
 de_drho                            6.095841843e+18           2.980380935e-06
 ds_dT                                  2.454254217           2.973657129e-06
 ds_drho                                807928760.9           2.982941798e-06
 dh_dT                              3.262182892e+10           2.975951162e-06
 dh_drho                             8.07928756e+18           2.982941837e-06
 dp_dX_hydrogen-1                   4.477957259e+21           5.292565765e-07
 dp_dX_helium-3                     2.955480122e+21           5.449484792e-07
 dp_dX_helium-4                     3.666564805e+21           4.514851085e-07
 dp_dX_carbon-12                    4.499587302e+21           5.103719659e-07
 dp_dX_nitrogen-14                  4.559088908e+21           5.142248082e-07
 dp_dX_oxygen-16                     4.60371511e+21           5.170862364e-07
 dp_dX_neon-20                      4.666191796e+21           5.210522543e-07
 dp_dX_magnesium-24                 4.707842925e+21           5.236707102e-07
 dp_dX_silicon-28                   4.737593728e+21           5.255286733e-07
 dp_dX_sulfur-32                     4.75990683e+21           5.269154434e-07
 dp_dX_argon-36                     4.777261463e+21            5.27990096e-07
 dp_dX_calcium-40                   4.791145172e+21           5.288473444e-07
 dp_dX_titanium-44                   4.80250457e+21           5.295471004e-07
 dp_dX_chromium-48                  4.811970735e+21           5.301291139e-07
 dp_dX_iron-52                      4.819980567e+21            5.30620796e-07
 dp_dX_iron-54                      4.856270873e+21           5.195117893e-07
 dp_dX_nickel-56                    4.826846138e+21           5.310416625e-07
 dp_dX_neutron                      2.625333235e+21           2.148602067e-07
 dp_dX_proton                       4.477957259e+21           5.292565765e-07
 de_dX_hydrogen-1                   7.163573926e+13           7.963816484e-06
 de_dX_helium-3                     2.967237308e+13           1.078331614e-05
 de_dX_helium-4                     6.260349865e+13           1.157983329e-05
 de_dX_carbon-12                    6.262494428e+13            1.11549118e-05
 de_dX_nitrogen-14                  6.262647611e+13           1.112576112e-05
 de_dX_oxygen-16                    6.262762499e+13           1.110399883e-05
 de_dX_neon-20                      6.262923341e+13           1.107367558e-05
 de_dX_magnesium-24                 6.263030569e+13           1.105355279e-05
 de_dX_silicon-28                   6.263107161e+13            1.10392245e-05
 de_dX_sulfur-32                    6.263164604e+13           1.102850287e-05
 de_dX_argon-36                     6.263209283e+13           1.102017835e-05
 de_dX_calcium-40                   6.263245026e+13           1.101352786e-05
 de_dX_titanium-44                   6.26327427e+13           1.100809257e-05
 de_dX_chromium-48                   6.26329864e+13            1.10035673e-05
 de_dX_iron-52                       6.26331926e+13           1.099974116e-05
 de_dX_iron-54                      6.629110678e+13           1.108852509e-05
 de_dX_nickel-56                    6.263336935e+13           1.099646374e-05
 de_dX_neutron                      1.612682015e+14           1.099696415e-05
 de_dX_proton                       7.163573926e+13           7.963816484e-06
 dh_dX_hydrogen-1                   2.208836234e+20           2.982940319e-06
 dh_dX_helium-3                     4.880186719e+19           2.982940577e-06
 dh_dX_helium-4                     5.679959933e+19            2.98294069e-06
 dh_dX_carbon-12                    7.279505742e+19           2.982940612e-06
 dh_dX_nitrogen-14                  7.393759015e+19           2.982940608e-06
 dh_dX_oxygen-16                    7.479448969e+19           2.982940605e-06
 dh_dX_neon-20                      7.599414905e+19           2.982940601e-06
 dh_dX_magnesium-24                 7.679392195e+19           2.982940598e-06
 dh_dX_silicon-28                   7.736518832e+19           2.982940596e-06
 dh_dX_sulfur-32                     7.77936381e+19           2.982940595e-06
 dh_dX_argon-36                      7.81268768e+19           2.982940593e-06
 dh_dX_calcium-40                   7.839346777e+19           2.982940593e-06
 dh_dX_titanium-44                  7.861158765e+19           2.982940592e-06
 dh_dX_chromium-48                  7.879335422e+19           2.982940591e-06
 dh_dX_iron-52                      7.894715671e+19           2.982940591e-06
 dh_dX_iron-54                       7.90155137e+19           2.982940602e-06
 dh_dX_nickel-56                     7.90789874e+19           2.982940591e-06
 dh_dX_neutron                      2.208837497e+20           2.982939437e-06
 dh_dX_proton                       2.208836234e+20           2.982940319e-06
 Gamma_1                            0.0004564659514           0.0002671683353
 soundspeed                             4460444.231           1.496112105e-06
 Abar                                             0                         0
 Zbar                                             0                         0
 dp_dA                              4.882463921e+21            3.57698987e-07
 dp_dZ                              1.487885257e+21            1.25343828e-07
 de_dA                              1.128574092e+14            1.16597423e-05
 de_dZ                               1.38118338e+14           1.138328243e-05
 dp_de_rho                              603.7507143           1.908034674e-07
 dp_drho_e                          2.006708034e+19           2.989665516e-06

[zingale]

Here's a simulation (flame_wave) run with the original and new table -- they look basically the same.

original table:

new table:

[zingale]

This has been updated now to use approximate functions for log and pow, and seems to get better performance. It needs more testing, in particular, the comparison of test_eos shows large-ish errors for de_dX (but not dp_dX or dh_dX)

[zingale]

just tested this again on Perlmutter / CUDA, and the alternate EOS table is no faster. The logs we are doing for indexing into the table are not the rate-limiting step.