Task: Arithmetic-Relationship (source dataset: Auto-Relate)

You are given a table with numerical values in multiple columns. Your task is to analyze the data and identify semantic relationships, expressed in arithmetic relationships, that exist between columns.

These relationships may involve any combination of the following operations: addition (+), subtraction (−), multiplication (*), and division (/). The expressions can involve two or more columns and may be nested (e.g., B = (A + C) / D).

Please only identify arithmetic relationships that hold semantically based on the data, and do not return relationships that are likely spurious, e.g., (A = B * 0), where A is an all 0 column. Please only return minimal atomic relationships, e.g., if A = B + C and D = E + F, then there is no need to return A = B + C + E + F - D.

No explanation, return all the formula(s) in JSON format: {"Arithmetic-Relationship": [<list-of-formulas>]}. If no relationship is found, return {"Arithmetic-Relationship": []}.

Input Table:
| A              |                B |           C |               D |          E |                F |           G |                H |           I |                J |          K |                L |          M |           N |         O |              P |           Q |           R |           S | T    |
|:---------------|-----------------:|------------:|----------------:|-----------:|-----------------:|------------:|-----------------:|------------:|-----------------:|-----------:|-----------------:|-----------:|------------:|----------:|---------------:|------------:|------------:|------------:|:-----|
| Alabama        |      0           | 0           |     7.61468e+07 | 0.19784    |      1.67401e+06 | 0.00434931  | 281213           | 0.000730629 |      2.94618e+07 | 0.0765457  |      1.17874e+08 | 0.306254   | 1.549e+08   | 0.402451  |    4.55322e+06 | 0.0118299   | 3.84891e+08 | 3.84891e+08 | True |
| Arizona        |      0           | 0           |     5.23671e+07 | 0.142188   | 270524           | 0.000734531 |      0           | 0           |      2.16969e+07 | 0.0589118  |      6.834e+07   | 0.185558   | 2.12462e+08 | 0.576879  |    1.31588e+07 | 0.035729    | 3.68295e+08 | 3.68295e+08 | True |
| Arkansas       |      0           | 0           |     5.508e+07   | 0.251607   | 284560           | 0.00129988  |  70851.5         | 0.000323652 |      2.89287e+07 | 0.132147   |      6.20202e+07 | 0.28331    | 6.14701e+07 | 0.280797  |    1.10582e+07 | 0.0505143   | 2.18913e+08 | 2.18913e+08 | True |
| California     |      0           | 0           |     5.6767e+08  | 0.408504   |      3.62585e+06 | 0.00260922  |      2.98907e+06 | 0.00215098  |      1.15608e+08 | 0.0831937  |      3.75789e+08 | 0.270424   | 2.81747e+08 | 0.202749  |    4.22016e+07 | 0.0303689   | 1.38963e+09 | 1.38963e+09 | True |
| Connecticut    |      0           | 0           |     1.67625e+08 | 0.238965   |      4.05645e+08 | 0.578283    | 546655           | 0.000779306 |      3.46969e+07 | 0.0494636  |      6.244e+07   | 0.0890138  | 2.83549e+07 | 0.0404225 |    2.15566e+06 | 0.00307309  | 7.01464e+08 | 7.01464e+08 | True |
| Florida        |      0           | 0           |     1.91913e+07 | 0.018282   | 824938           | 0.000785853 | 252593           | 0.000240625 |      2.71029e+07 | 0.0258188  |      1.72808e+08 | 0.16462    | 8.26592e+08 | 0.787429  |    2.96443e+06 | 0.00282397  | 1.04974e+09 | 1.04974e+09 | True |
| Georgia        |  94114.7         | 0.000118577 |     2.34925e+08 | 0.295986   | 916041           | 0.00115413  | 556622           | 0.000701297 |      7.45803e+07 | 0.0939649  |      2.27602e+08 | 0.286759   | 2.47128e+08 | 0.311361  |    7.90148e+06 | 0.0099552   | 7.93704e+08 | 7.93704e+08 | True |
| Hawaii         |      0           | 0           | 72312.3         | 0.00127387 |      0           | 0           |      0           | 0           | 350308           | 0.00617111 | 275238           | 0.00484866 | 5.60601e+07 | 0.98757   | 7760.03        | 0.000136703 | 5.67657e+07 | 5.67657e+07 | True |
| Illinois       |      1.39529e+06 | 0.00132335  |     6.84789e+08 | 0.64948    |      4.1311e+06  | 0.00391809  | 408705           | 0.000387631 |      1.05077e+08 | 0.0996586  |      1.5155e+08  | 0.143736   | 1.04488e+08 | 0.0991    |    2.52653e+06 | 0.00239625  | 1.05437e+09 | 1.05437e+09 | True |
| Kansas         |      0           | 0           |     1.02154e+08 | 0.423619   | 481576           | 0.00199704  |  21818           | 9.04766e-05 |      3.18472e+07 | 0.132067   |      5.5256e+07  | 0.22914    | 4.65599e+07 | 0.193078  |    4.82501e+06 | 0.0200087   | 2.41145e+08 | 2.41145e+08 | True |
| Louisiana      |      0           | 0           |     6.22477e+07 | 0.194645   |  78712.8         | 0.00024613  |  13319.9         | 4.16507e-05 |      1.184e+07   | 0.037023   |      9.74158e+07 | 0.304613   | 1.42875e+08 | 0.446761  |    5.331e+06   | 0.0166697   | 3.19801e+08 | 3.19801e+08 | True |
| Maryland       | 110934           | 0.000194386 |     1.5759e+08  | 0.276138   |      1.21102e+08 | 0.212202    |      1.65765e+06 | 0.00290464  |      3.61429e+07 | 0.0633318  |      1.70597e+08 | 0.29893    | 7.58138e+07 | 0.132845  |    7.6779e+06  | 0.0134537   | 5.70691e+08 | 5.70691e+08 | True |
| Massachusetts  |      0           | 0           |     3.9159e+08  | 0.344362   |      5.66049e+08 | 0.49778     |      1.29786e+06 | 0.00114133  |      4.55518e+07 | 0.0400579  |      9.49464e+07 | 0.0834952  | 3.25249e+07 | 0.0286022 |    5.18753e+06 | 0.00456188  | 1.13715e+09 | 1.13715e+09 | True |
| Michigan       |      1.39499e+06 | 0.00118896  |     7.28726e+08 | 0.6211     |      2.7459e+07  | 0.0234036   | 855344           | 0.000729018 |      1.64645e+08 | 0.140328   |      1.68994e+08 | 0.144035   | 6.63913e+07 | 0.0565859 |    1.4817e+07  | 0.0126287   | 1.17328e+09 | 1.17328e+09 | True |
| Minnesota      | 254304           | 0.000538084 |     1.83424e+08 | 0.38811    |      5.65809e+07 | 0.11972     | 384531           | 0.000813635 |      1.12448e+08 | 0.23793    |      9.24145e+07 | 0.195541   | 1.26898e+07 | 0.0268505 |    1.44132e+07 | 0.0304971   | 4.72609e+08 | 4.72609e+08 | True |
| Mississippi    |      0           | 0           |     3.25802e+07 | 0.152683   |  12363.3         | 5.79392e-05 | 202870           | 0.000950728 |      2.86678e+07 | 0.134348   |      5.80519e+07 | 0.272054   | 9.10361e+07 | 0.426631  |    2.83268e+06 | 0.0132751   | 2.13384e+08 | 2.13384e+08 | True |
| Missouri       |      0           | 0           |     2.08496e+08 | 0.37662    |      2.93821e+06 | 0.00530748  | 241673           | 0.00043655  |      8.17863e+07 | 0.147736   |      1.27174e+08 | 0.229723   | 1.01317e+08 | 0.183016  |    3.16436e+07 | 0.0571599   | 5.53597e+08 | 5.53597e+08 | True |
| Nevada         |      0           | 0           |     5.31599e+07 | 0.28477    |      1.81943e+06 | 0.00974644  |  85741.6         | 0.000459306 |      1.02978e+07 | 0.055164   |      6.9502e+07  | 0.372313   | 4.98204e+07 | 0.266881  |    1.99117e+06 | 0.0106664   | 1.86676e+08 | 1.86676e+08 | True |
| New Jersey     |      0           | 0           |     4.60133e+08 | 0.480278   |      2.3553e+08  | 0.245842    | 652621           | 0.000681193 |      3.51047e+07 | 0.0366416  |      1.32449e+08 | 0.138248   | 9.31641e+07 | 0.0972429 |    1.02218e+06 | 0.00106693  | 9.58056e+08 | 9.58056e+08 | True |
| New York       |   9496.34        | 3.40124e-06 |     1.13502e+09 | 0.406522   |      1.16283e+09 | 0.416484    |      1.71731e+07 | 0.00615077  |      1.17961e+08 | 0.0422492  |      2.15614e+08 | 0.077225   | 1.24946e+08 | 0.0447511 |    1.84674e+07 | 0.00661434  | 2.79203e+09 | 2.79203e+09 | True |
| North Carolina |      1.17387e+06 | 0.00161065  |     9.69121e+07 | 0.132972   |      5.75962e+07 | 0.0790268   |      9.36884e+06 | 0.0128548   |      1.01873e+08 | 0.139778   |      2.41663e+08 | 0.331582   | 1.99195e+08 | 0.273313  |    2.10361e+07 | 0.0288633   | 7.28818e+08 | 7.28818e+08 | True |
| Ohio           |      1.49888e+06 | 0.00151023  |     5.53177e+08 | 0.557365   |      7.07083e+07 | 0.0712437   |      2.27807e+06 | 0.00229532  |      1.01451e+08 | 0.102219   |      1.50298e+08 | 0.151436   | 1.04567e+08 | 0.105358  |    8.50871e+06 | 0.00857313  | 9.92486e+08 | 9.92486e+08 | True |
| Oklahoma       |      0           | 0           |     9.58676e+07 | 0.31253    | 420102           | 0.00136954  |  60453.6         | 0.00019708  |      2.46892e+07 | 0.0804873  |      9.57937e+07 | 0.312289   | 8.403e+07   | 0.273939  |    5.88581e+06 | 0.0191879   | 3.06747e+08 | 3.06747e+08 | True |
| Oregon         |      0           | 0           |     5.84334e+07 | 0.25512    |      1.40294e+07 | 0.0612522   |      1.32644e+06 | 0.00579121  |      1.07846e+07 | 0.0470852  |      1.1698e+08  | 0.510734   | 5.7018e+06  | 0.024894  |    2.17875e+07 | 0.0951238   | 2.29043e+08 | 2.29043e+08 | True |
| Pennsylvania   | 621681           | 0.0004103   |     5.46864e+08 | 0.360922   |      6.25889e+08 | 0.413077    |      8.40302e+06 | 0.00554586  |      8.34804e+07 | 0.0550957  |      1.36431e+08 | 0.0900426  | 9.66146e+07 | 0.0637641 |    1.68831e+07 | 0.0111426   | 1.51519e+09 | 1.51519e+09 | True |
| South Carolina |  13707.3         | 4.11481e-05 |     4.30392e+07 | 0.1292     |      9.86148e+06 | 0.0296034   |      2.12919e+06 | 0.00639165  |      2.51596e+07 | 0.075527   |      1.08894e+08 | 0.326891   | 1.39488e+08 | 0.418733  |    4.53486e+06 | 0.0136133   | 3.3312e+08  | 3.3312e+08  | True |
| Tennessee      |      1.19651e+06 | 0.00267305  |     8.52291e+07 | 0.190404   |      3.53307e+06 | 0.00789298  |      1.82556e+06 | 0.00407836  |      2.9837e+07  | 0.0666568  |      1.81573e+08 | 0.405639   | 1.26097e+08 | 0.281705  |    1.83304e+07 | 0.0409505   | 4.47622e+08 | 4.47622e+08 | True |
| Texas          |  68455           | 4.05235e-05 |     2.56472e+08 | 0.151825   | 171383           | 0.000101454 |  58183.5         | 3.44431e-05 |      8.15205e+07 | 0.048258   |      5.8015e+08  | 0.343433   | 7.51178e+08 | 0.444677  |    1.96471e+07 | 0.0116305   | 1.68927e+09 | 1.68927e+09 | True |
| Utah           |      0           | 0           |     9.30503e+07 | 0.632976   |      1.09981e+06 | 0.00748147  |   7551.62        | 5.137e-05   |      6.60131e+06 | 0.0449055  |      3.18535e+07 | 0.216684   | 1.20795e+07 | 0.0821712 |    2.31243e+06 | 0.0157303   | 1.47004e+08 | 1.47004e+08 | True |
| Virginia       | 812380           | 0.00121352  |     1.05486e+08 | 0.157572   |      1.38358e+08 | 0.206676    |      5.43803e+06 | 0.00812322  |      5.01736e+07 | 0.0749483  |      2.33643e+08 | 0.34901    | 1.03352e+08 | 0.154386  |    3.21807e+07 | 0.0480709   | 6.69443e+08 | 6.69443e+08 | True |
| Washington     |      0           | 0           |     1.04472e+08 | 0.279926   |      1.99884e+07 | 0.0535575   | 177668           | 0.000476048 |      2.91897e+07 | 0.0782117  |      1.87221e+08 | 0.501645   | 7.33036e+06 | 0.0196412 |    2.48344e+07 | 0.0665419   | 3.73214e+08 | 3.73214e+08 | True |
| West Virginia  |      0           | 0           |     4.35922e+07 | 0.254774   |      9.44599e+06 | 0.0552069   | 926527           | 0.00541507  |      1.66848e+07 | 0.097514   |      6.27529e+07 | 0.366758   | 1.92102e+07 | 0.112274  |    1.8489e+07  | 0.108058    | 1.71102e+08 | 1.71102e+08 | True |
| Wisconsin      | 367293           | 0.000709773 |     2.29248e+08 | 0.443008   |      3.86126e+07 | 0.0746168   | 346661           | 0.000669904 |      9.78324e+07 | 0.189056   |      9.82618e+07 | 0.189886   | 3.07498e+07 | 0.0594223 |    2.2061e+07  | 0.0426317   | 5.17479e+08 | 5.17479e+08 | True |



label: C=B/S
E=D/S
G=F/S
I=H/S
K=J/S
M=L/S
O=N/S
Q=P/S
labeled_formulas: C=B/S
E=D/S
G=F/S
I=H/S
K=J/S
M=L/S
O=N/S
Q=P/S
S=G+R
S=K+R
S=Q+R
B=(E+R)*C
B=(E+S)*C
B=(G+R)*C
B=(G+S)*C
B=(K+R)*C
B=(K+S)*C
B=(M+R)*C
B=(M+S)*C
B=(Q+R)*C
B=(Q+S)*C
output: {"Arithmetic-Relationship": ["C=B/S", "E=D/S", "G=F/S", "I=H/S", "K=J/S", "M=L/S", "O=N/S", "Q=P/S"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_333462___xlsx_f2c933e511428c8a30fd112b44c49391edfa2c9a.xlsx___line_843