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 |
|----:|:--------------------------------------|:---------|------------:|-----------------:|-----------------:|----------:|------------:|-----------------:|
|   1 | Accomack                              | Eastern  | 3.52789e+07 |      2.61296e+07 | 259944           |     -0.07 | 6.16685e+07 | 259944           |
|   3 | Albemarle                             | Piedmont | 4.21966e+07 |      3.75198e+07 |      4.1079e+06  |   1489.34 | 8.38258e+07 |      4.10939e+06 |
|   5 | Alleghany/Covington                   | Piedmont | 2.23779e+07 |      1.93677e+07 | 697849           |    199.99 | 4.24437e+07 | 698049           |
|   7 | Amelia                                | Central  | 1.11444e+07 |      8.77097e+06 | 184640           |     -0.01 | 2.01e+07    | 184640           |
|   9 | Amherst                               | Piedmont | 2.8467e+07  |      2.32685e+07 | 430538           |     -0.07 | 5.2166e+07  | 430538           |
|  11 | Appomattox                            | Piedmont | 1.63886e+07 |      1.36246e+07 | 550671           |     -0.13 | 3.05639e+07 | 550671           |
|  13 | Arlington                             | Northern | 7.00179e+07 |      5.79447e+07 |      3.49873e+06 | 455806    | 1.31917e+08 |      3.95454e+06 |
|  15 | Augusta/Staunton/Waynesboro           | Piedmont | 8.56243e+07 |      7.093e+07   |      3.21068e+06 |  27442.2  | 1.59792e+08 |      3.23812e+06 |
|  17 | Bath                                  | Piedmont | 3.07323e+06 |      2.5724e+06  |  74360.7         |      0.01 | 5.71999e+06 |  74360.7         |
|  19 | Bedford                               | Piedmont | 4.72357e+07 |      3.91156e+07 |      1.05787e+06 |    -15.44 | 8.74091e+07 |      1.05786e+06 |
|  21 | Bland                                 | Western  | 4.64223e+06 |      4.02874e+06 | 102353           |     -0.02 | 8.77332e+06 | 102353           |
|  23 | Botetourt                             | Piedmont | 1.69195e+07 |      1.4674e+07  | 559437           |     -0.01 | 3.21529e+07 | 559437           |
|  25 | Brunswick                             | Eastern  | 2.30228e+07 |      1.77948e+07 | 257845           |      0    | 4.10755e+07 | 257845           |
|  27 | Buchanan                              | Western  | 2.904e+07   |      2.20928e+07 | 702073           |     -0.12 | 5.18349e+07 | 702073           |
|  29 | Buckingham                            | Central  | 1.68301e+07 |      1.35332e+07 | 402777           |     -0.02 | 3.07661e+07 | 402777           |
|  31 | Campbell                              | Piedmont | 4.76108e+07 |      3.8018e+07  | 955483           |    791.24 | 8.65851e+07 | 956274           |
|  33 | Caroline                              | Central  | 2.42933e+07 |      1.76095e+07 | 733171           |   1173.01 | 4.26371e+07 | 734344           |
|  35 | Carroll                               | Western  | 3.00831e+07 |      2.37046e+07 | 644542           |      0.01 | 5.44322e+07 | 644542           |
|  36 | Charles City                          | Central  | 6.07781e+06 |      4.54569e+06 |  78498.9         |      0    | 1.0702e+07  |  78498.9         |
|  37 | Charlotte                             | Piedmont | 1.29119e+07 |      1.01892e+07 | 282762           |     -0.14 | 2.33838e+07 | 282762           |
|  41 | Chesterfield/Colonial Heights         | Central  | 2.09041e+08 |      1.62357e+08 |      5.1368e+06  |   -382.65 | 3.76535e+08 |      5.13642e+06 |
|  43 | Clarke                                | Northern | 6.08401e+06 |      5.3332e+06  | 322846           |      7.86 | 1.17401e+07 | 322854           |
|  45 | Craig                                 | Piedmont | 3.67611e+06 |      3.01814e+06 | 184125           |     -0.08 | 6.87837e+06 | 184124           |
|  47 | Culpeper                              | Northern | 3.75376e+07 |      3.01066e+07 |      1.76373e+06 |     -8.43 | 6.9408e+07  |      1.76372e+06 |
|  49 | Cumberland                            | Central  | 1.16554e+07 |      9.12706e+06 | 339591           |     -0.02 | 2.11221e+07 | 339591           |
|  51 | Dickenson                             | Western  | 2.01004e+07 |      1.59465e+07 | 431152           |    461.65 | 3.64786e+07 | 431613           |
|  53 | Dinwiddie                             | Eastern  | 2.60469e+07 |      2.0059e+07  | 784355           |   7498.6  | 4.68977e+07 | 791854           |
|  57 | Essex                                 | Central  | 1.17589e+07 |      8.56294e+06 | 378435           |     91.18 | 2.07004e+07 | 378526           |
|  59 | Fairfax Co./Fairfax City/Falls Church | Northern | 3.98437e+08 |      3.14957e+08 |      1.96552e+07 |  21081    | 7.33071e+08 |      1.96763e+07 |
|  61 | Fauquier                              | Northern | 3.13187e+07 |      2.75412e+07 |      2.57574e+06 |  24831.3  | 6.14605e+07 |      2.60058e+06 |
|  63 | Floyd                                 | Western  | 1.30961e+07 |      1.03532e+07 | 116707           |     -0.06 | 2.3566e+07  | 116707           |
|  65 | Fluvanna                              | Central  | 1.22294e+07 |      1.07881e+07 |      1.06947e+06 |     -0.02 | 2.40869e+07 |      1.06947e+06 |
|  67 | Franklin Co.                          | Piedmont | 4.39249e+07 |      3.57326e+07 |      1.55744e+06 |     -0.15 | 8.1215e+07  |      1.55744e+06 |
|  69 | Frederick                             | Northern | 4.00596e+07 |      3.13046e+07 |      1.42007e+06 |   4197.4  | 7.27884e+07 |      1.42427e+06 |
|  71 | Giles                                 | Western  | 1.66811e+07 |      1.41187e+07 | 519878           |     -0.21 | 3.13197e+07 | 519878           |
|  73 | Gloucester                            | Eastern  | 2.42718e+07 |      1.88905e+07 | 415776           |     -0.08 | 4.35781e+07 | 415776           |
|  75 | Goochland                             | Central  | 8.67327e+06 |      7.16243e+06 | 804181           | 205723    | 1.68456e+07 |      1.0099e+06  |
|  77 | Grayson                               | Western  | 1.64325e+07 |      1.25034e+07 | 158734           |     -0.03 | 2.90945e+07 | 158734           |
|  79 | Greene                                | Northern | 1.27001e+07 |      1.00738e+07 | 593727           |     -0.03 | 2.33677e+07 | 593727           |
|  81 | Greensville/Emporia                   | Eastern  | 2.47326e+07 |      1.84514e+07 | 180149           |     -0.03 | 4.33642e+07 | 180149           |
|  83 | Halifax/South Boston                  | Piedmont | 4.32979e+07 |      3.44569e+07 | 802448           |     -0.15 | 7.85573e+07 | 802447           |
|  85 | Hanover                               | Central  | 4.1366e+07  |      3.3794e+07  |      1.86367e+06 |   -432.26 | 7.70233e+07 |      1.86324e+06 |
|  87 | Henrico                               | Central  | 2.12058e+08 |      1.60474e+08 |      5.09794e+06 |   8937.11 | 3.77639e+08 |      5.10688e+06 |
|  89 | Henry/Martinsville                    | Piedmont | 9.42734e+07 |      6.9577e+07  | 351853           |     -0.08 | 1.64202e+08 | 351853           |
|  91 | Highland                              | Piedmont | 1.19177e+06 | 993514           |   3656.46        |     -0.01 | 2.18894e+06 |   3656.45        |
|  93 | Isle Of Wight                         | Eastern  | 2.34573e+07 |      1.76174e+07 | 112141           |    792.95 | 4.11876e+07 | 112933           |
|  95 | James City                            | Eastern  | 2.40457e+07 |      1.66228e+07 | 454601           |    763.88 | 4.11238e+07 | 455365           |
|  97 | King & Queen                          | Central  | 6.30107e+06 |      4.85904e+06 | 243312           |    374.69 | 1.14038e+07 | 243686           |
|  99 | King George                           | Northern | 1.38232e+07 |      1.0145e+07  | 665618           |     -0.05 | 2.46338e+07 | 665618           |
| 101 | King William                          | Central  | 1.07202e+07 |      8.31788e+06 | 333929           |      0    | 1.9372e+07  | 333929           |
| 103 | Lancaster                             | Central  | 8.67558e+06 |      6.59195e+06 | 425827           |      0    | 1.56934e+07 | 425827           |
| 105 | Lee                                   | Western  | 3.62928e+07 |      2.81141e+07 | 419008           |   1674.56 | 6.48276e+07 | 420683           |
| 107 | Loudoun                               | Northern | 8.72097e+07 |      6.9601e+07  |      2.98529e+06 |   8730.77 | 1.59805e+08 |      2.99402e+06 |
| 109 | Louisa                                | Northern | 2.68018e+07 |      2.17477e+07 |      1.47324e+06 |     -0.09 | 5.00227e+07 |      1.47324e+06 |
| 111 | Lunenburg                             | Central  | 1.28226e+07 |      1.01846e+07 | 194736           |     -0.03 | 2.32019e+07 | 194736           |
| 113 | Madison                               | Northern | 9.76345e+06 |      9.69102e+06 |      1.25455e+06 |    999.89 | 2.071e+07   |      1.25555e+06 |
| 115 | Mathews                               | Eastern  | 7.04679e+06 |      5.80808e+06 | 124398           |     -0.01 | 1.29793e+07 | 124398           |
| 117 | Mecklenburg                           | Piedmont | 3.25869e+07 |      2.62188e+07 | 565771           |    468.17 | 5.93719e+07 | 566240           |
| 119 | Middlesex                             | Central  | 9.99592e+06 |      7.7592e+06  | 201475           |      0.32 | 1.79566e+07 | 201475           |
| 121 | Montgomery                            | Western  | 4.37664e+07 |      3.31057e+07 | 345114           |     -0.15 | 7.72173e+07 | 345114           |
| 125 | Nelson                                | Piedmont | 1.33891e+07 |      1.05094e+07 | 318652           |     -0.03 | 2.42171e+07 | 318652           |
| 127 | New Kent                              | Central  | 8.28762e+06 |      6.83026e+06 | 527747           |     -0.01 | 1.56456e+07 | 527747           |
| 131 | Northampton                           | Eastern  | 1.93281e+07 |      1.53939e+07 | 136709           |      0    | 3.48587e+07 | 136709           |
| 133 | Northumberland                        | Central  | 9.38764e+06 |      6.94339e+06 | 192477           |    249.9  | 1.65237e+07 | 192727           |
| 135 | Nottoway                              | Central  | 1.77229e+07 |      1.29072e+07 | 225318           |      0    | 3.08554e+07 | 225318           |
| 137 | Orange                                | Northern | 2.18147e+07 |      1.77572e+07 |      1.00056e+06 |     -0.1  | 4.05725e+07 |      1.00056e+06 |
| 139 | Page                                  | Northern | 1.95582e+07 |      1.47156e+07 | 406277           |      0    | 3.46801e+07 | 406277           |
| 141 | Patrick                               | Western  | 1.79505e+07 |      1.34273e+07 | 146055           |     -0.11 | 3.15238e+07 | 146055           |
| 143 | Pittsylvania                          | Piedmont | 6.04466e+07 |      4.93137e+07 |      1.19745e+06 |   3190.96 | 1.10961e+08 |      1.20064e+06 |
| 145 | Powhatan                              | Central  | 9.61008e+06 |      8.42939e+06 |      1.02437e+06 |  52116.3  | 1.9116e+07  |      1.07649e+06 |
| 147 | Prince Edward                         | Central  | 2.14205e+07 |      1.7211e+07  | 295474           |     -0.03 | 3.8927e+07  | 295474           |
| 149 | Prince George                         | Eastern  | 1.72551e+07 |      1.30323e+07 | 575735           |     -0.04 | 3.08631e+07 | 575735           |
| 153 | Prince William                        | Northern | 2.0328e+08  |      1.52455e+08 |      3.79838e+06 |     -0.16 | 3.59533e+08 |      3.79838e+06 |
| 155 | Pulaski                               | Western  | 3.35388e+07 |      2.77521e+07 |      1.3085e+06  |   3298.21 | 6.26027e+07 |      1.3118e+06  |
| 157 | Rappahannock                          | Northern | 2.92258e+06 |      2.73323e+06 | 565386           |   4071.51 | 6.22526e+06 | 569457           |
| 159 | Richmond Co.                          | Central  | 8.27726e+06 |      6.09818e+06 |  76087.5         |     -0.08 | 1.44515e+07 |  76087.4         |
| 161 | Roanoke Co./Salem                     | Piedmont | 6.21106e+07 |      5.42757e+07 |      3.71195e+06 |  15103.5  | 1.20113e+08 |      3.72705e+06 |
| 163 | Rockbridge/Buena Vista/Lexington      | Piedmont | 2.48448e+07 |      2.15558e+07 | 972001           |     -0.11 | 4.73725e+07 | 972001           |
| 165 | Rockingham/Harrisonburg               | Northern | 7.03362e+07 |      5.90772e+07 |      3.73159e+06 |     -0.26 | 1.33145e+08 |      3.73159e+06 |
| 167 | Russell                               | Western  | 3.15527e+07 |      2.39506e+07 | 498355           |     -0.33 | 5.60017e+07 | 498355           |
| 169 | Scott                                 | Western  | 2.35267e+07 |      1.85757e+07 | 322224           |     -0.09 | 4.24247e+07 | 322223           |
| 171 | Shenandoah                            | Northern | 3.1367e+07  |      2.52393e+07 |      1.07666e+06 |  -8232.52 | 5.76747e+07 |      1.06843e+06 |
| 173 | Smyth                                 | Western  | 3.48287e+07 |      2.59128e+07 | 360175           |    999.86 | 6.11026e+07 | 361175           |
| 175 | Southampton                           | Eastern  | 1.78503e+07 |      1.3579e+07  | 233172           |     -0.05 | 3.16625e+07 | 233172           |
| 177 | Spotsylvania                          | Northern | 7.5426e+07  |      6.10159e+07 |      4.39201e+06 |     -0.16 | 1.40834e+08 |      4.39201e+06 |
| 179 | Stafford                              | Northern | 6.12009e+07 |      4.7826e+07  |      2.29759e+06 |     -0.08 | 1.11324e+08 |      2.29759e+06 |
| 181 | Surry                                 | Eastern  | 5.98994e+06 |      4.36852e+06 |  48050.3         |    150    | 1.04067e+07 |  48200.3         |
| 183 | Sussex                                | Eastern  | 1.20835e+07 |      9.59871e+06 | 168622           |    212.36 | 2.1851e+07  | 168835           |
| 185 | Tazewell                              | Western  | 4.34718e+07 |      3.21025e+07 | 591544           |  32120.8  | 7.61979e+07 | 623665           |
| 187 | Warren                                | Northern | 2.8635e+07  |      2.181e+07   | 722060           |     -0.11 | 5.11671e+07 | 722060           |
| 191 | Washington                            | Western  | 4.19797e+07 |      3.20468e+07 | 683663           |   9718.78 | 7.47198e+07 | 693382           |
| 193 | Westmoreland                          | Central  | 1.59862e+07 |      1.1574e+07  | 406919           |   1499.99 | 2.79687e+07 | 408419           |
| 195 | Wise                                  | Western  | 4.85421e+07 |      3.69191e+07 | 517172           |     -0.19 | 8.59784e+07 | 517172           |
| 197 | Wythe                                 | Western  | 2.60038e+07 |      2.13497e+07 | 570644           |    704.92 | 4.79249e+07 | 571349           |
| 199 | York/Poquoson                         | Eastern  | 2.21749e+07 |      1.76553e+07 | 583262           |    656.97 | 4.04141e+07 | 583919           |
| 510 | Alexandria                            | Northern | 7.18605e+07 |      5.8284e+07  |      4.24644e+06 |   1310.29 | 1.34392e+08 |      4.24775e+06 |
| 520 | Bristol                               | Western  | 2.23492e+07 |      1.69551e+07 | 737482           |     -0.25 | 4.00417e+07 | 737482           |
| 540 | Charlottesville                       | Piedmont | 3.71742e+07 |      3.46301e+07 |      2.24418e+06 |  -2412.5  | 7.40461e+07 |      2.24177e+06 |
| 550 | Chesapeake                            | Eastern  | 1.3859e+08  |      1.02978e+08 |      1.62133e+06 |  -2765.66 | 2.43186e+08 |      1.61856e+06 |
| 590 | Danville                              | Piedmont | 8.17022e+07 |      6.3558e+07  | 993260           |     -0.13 | 1.46254e+08 | 993260           |
| 620 | Franklin                              | Eastern  | 1.61746e+07 |      1.20162e+07 |  70813.6         |      0    | 2.82617e+07 |  70813.6         |
| 630 | Fredericksburg                        | Northern | 2.48592e+07 |      1.92454e+07 | 609668           |     -0.16 | 4.47143e+07 | 609668           |
| 640 | Galax                                 | Western  | 1.13016e+07 |      8.62582e+06 | 130611           |     -0.05 | 2.0058e+07  | 130611           |
| 650 | Hampton                               | Eastern  | 1.26443e+08 |      9.88347e+07 |      1.63686e+06 |   2400.89 | 2.26916e+08 |      1.63926e+06 |
| 670 | Hopewell                              | Central  | 3.53074e+07 |      2.55965e+07 | 824617           |     -0.1  | 6.17286e+07 | 824617           |
| 680 | Lynchburg                             | Piedmont | 8.26854e+07 |      6.69917e+07 |      2.04109e+06 |     -0.47 | 1.51718e+08 |      2.04109e+06 |
| 683 | Manassas                              | Northern | 2.64903e+07 |      1.82336e+07 | 622453           |   1842.94 | 4.53481e+07 | 624296           |
| 685 | Manassas Park                         | Northern | 8.41748e+06 |      6.19117e+06 | 418941           |      0    | 1.50276e+07 | 418941           |
| 700 | Newport News                          | Eastern  | 1.76011e+08 |      1.24139e+08 |      1.88411e+06 |     93.59 | 3.02034e+08 |      1.8842e+06  |
| 710 | Norfolk                               | Eastern  | 2.55687e+08 |      1.88237e+08 |      2.71417e+06 |   1901.31 | 4.4664e+08  |      2.71607e+06 |
| 720 | Norton                                | Western  | 5.0171e+06  |      3.65918e+06 |  77883.4         |     -0.03 | 8.75416e+06 |  77883.4         |
| 730 | Petersburg                            | Central  | 6.69418e+07 |      5.289e+07   |      2.13292e+06 |  46198    | 1.22011e+08 |      2.17912e+06 |
| 740 | Portsmouth                            | Eastern  | 1.30331e+08 |      9.55366e+07 |      1.05997e+06 |   8115.06 | 2.26936e+08 |      1.06809e+06 |
| 750 | Radford                               | Western  | 9.87932e+06 |      8.51001e+06 | 329845           |     -0.09 | 1.87192e+07 | 329845           |
| 760 | Richmond                              | Central  | 3.0946e+08  |      2.44383e+08 |      8.4752e+06  | 335955    | 5.62654e+08 |      8.81116e+06 |
| 770 | Roanoke                               | Piedmont | 1.37204e+08 |      1.08429e+08 |      3.5769e+06  |   1643.76 | 2.49211e+08 |      3.57854e+06 |
| 800 | Suffolk                               | Eastern  | 8.14455e+07 |      6.28699e+07 | 348761           |  13140.9  | 1.44677e+08 | 361902           |
| 810 | Virginia Beach                        | Eastern  | 2.27183e+08 |      1.73801e+08 |      4.88314e+06 |  61216.8  | 4.05928e+08 |      4.94436e+06 |
| 830 | Williamsburg                          | Eastern  | 5.92482e+06 |      3.89817e+06 | 130071           |  23369.7  | 9.97642e+06 | 153441           |
| 840 | Winchester                            | Northern | 2.77443e+07 |      2.19562e+07 | 966415           |  13860.2  | 5.06807e+07 | 980275           |



label: I=F+G
labeled_formulas: I=F+G
output: {"Arithmetic-Relationship": ["I=F+G"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_518630___xlsx_3092d67527a3e1cad7a3b0bdd14ce703917ebf64.xlsx___line_234