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  | 3.55723e+07 |      2.4655e+07  | 250113           |     -0.07 | 6.04774e+07 | 250113           |
|   3 | Albemarle                             | Piedmont | 4.05858e+07 |      3.41e+07    |      4.00244e+06 |     -0.39 | 7.86882e+07 |      4.00244e+06 |
|   5 | Alleghany/Covington                   | Piedmont | 2.06156e+07 |      1.6798e+07  | 544399           |    411.6  | 3.79583e+07 | 544811           |
|   7 | Amelia                                | Central  | 1.06507e+07 |      7.88815e+06 | 103816           |      0    | 1.86427e+07 | 103816           |
|   9 | Amherst                               | Piedmont | 2.62303e+07 |      2.09693e+07 | 358640           |     -0.03 | 4.75582e+07 | 358640           |
|  11 | Appomattox                            | Piedmont | 1.4806e+07  |      1.14336e+07 | 326838           |     -0.04 | 2.65664e+07 | 326838           |
|  13 | Arlington                             | Northern | 6.24626e+07 |      5.17709e+07 |      3.76462e+06 | 471681    | 1.1847e+08  |      4.2363e+06  |
|  15 | Augusta/Staunton/Waynesboro           | Piedmont | 8.34633e+07 |      6.67834e+07 |      2.92718e+06 |  21909.6  | 1.53196e+08 |      2.94909e+06 |
|  17 | Bath                                  | Piedmont | 2.50983e+06 |      1.97579e+06 |  51130.6         |     71.52 | 4.53682e+06 |  51202.1         |
|  19 | Bedford                               | Piedmont | 4.27284e+07 |      3.34362e+07 | 517298           |     -0.17 | 7.66819e+07 | 517297           |
|  21 | Bland                                 | Western  | 4.13171e+06 |      3.57126e+06 | 105559           |     -0.08 | 7.80852e+06 | 105559           |
|  23 | Botetourt                             | Piedmont | 1.40957e+07 |      1.20106e+07 | 471401           |     -0.04 | 2.65777e+07 | 471401           |
|  25 | Brunswick                             | Eastern  | 2.10002e+07 |      1.53942e+07 | 191406           |      0    | 3.65858e+07 | 191406           |
|  27 | Buchanan                              | Western  | 2.6082e+07  |      1.93743e+07 | 479705           |     -0.13 | 4.5936e+07  | 479705           |
|  29 | Buckingham                            | Central  | 1.58571e+07 |      1.18138e+07 | 335231           |     -0.05 | 2.80061e+07 | 335231           |
|  31 | Campbell                              | Piedmont | 4.60617e+07 |      3.5274e+07  | 788185           |   9017.83 | 8.21329e+07 | 797202           |
|  33 | Caroline                              | Central  | 2.30685e+07 |      1.62028e+07 | 861521           |     -0.07 | 4.01328e+07 | 861521           |
|  35 | Carroll                               | Western  | 2.96606e+07 |      2.22718e+07 | 552804           |     -0.09 | 5.24852e+07 | 552804           |
|  36 | Charles City                          | Central  | 5.38119e+06 |      3.69274e+06 |  51386           |      0    | 9.12532e+06 |  51386           |
|  37 | Charlotte                             | Piedmont | 1.26981e+07 |      9.82104e+06 | 243588           |     -0.08 | 2.27627e+07 | 243587           |
|  41 | Chesterfield/Colonial Heights         | Central  | 1.92478e+08 |      1.41436e+08 |      3.70366e+06 | 157693    | 3.37775e+08 |      3.86135e+06 |
|  43 | Clarke                                | Northern | 6.18961e+06 |      5.34787e+06 | 263873           |      0    | 1.18014e+07 | 263873           |
|  45 | Craig                                 | Piedmont | 3.16541e+06 |      2.54678e+06 | 116138           |     -0.03 | 5.82833e+06 | 116138           |
|  47 | Culpeper                              | Northern | 3.4599e+07  |      2.69956e+07 |      1.76697e+06 |   6436.87 | 6.3368e+07  |      1.77341e+06 |
|  49 | Cumberland                            | Central  | 1.18624e+07 |      8.71673e+06 | 211842           |     -0.02 | 2.0791e+07  | 211842           |
|  51 | Dickenson                             | Western  | 1.77321e+07 |      1.39975e+07 | 504957           |   1083.55 | 3.22356e+07 | 506040           |
|  53 | Dinwiddie                             | Eastern  | 2.46108e+07 |      1.7705e+07  | 593170           |   1402.87 | 4.29104e+07 | 594573           |
|  57 | Essex                                 | Central  | 1.18946e+07 |      7.94507e+06 | 325203           |     -0.01 | 2.01648e+07 | 325203           |
|  59 | Fairfax Co./Fairfax City/Falls Church | Northern | 3.59928e+08 |      2.87254e+08 |      1.84105e+07 |  60503.5  | 6.65653e+08 |      1.8471e+07  |
|  61 | Fauquier                              | Northern | 2.91392e+07 |      2.44597e+07 |      2.02831e+06 |   -237.57 | 5.56269e+07 |      2.02807e+06 |
|  63 | Floyd                                 | Western  | 1.16422e+07 |      9.19025e+06 | 191785           |     -0.06 | 2.10242e+07 | 191785           |
|  65 | Fluvanna                              | Central  | 1.12735e+07 |      9.63663e+06 | 873145           |      0    | 2.17833e+07 | 873145           |
|  67 | Franklin Co.                          | Piedmont | 4.31628e+07 |      3.3542e+07  |      1.40768e+06 |  34795.1  | 7.81473e+07 |      1.44248e+06 |
|  69 | Frederick                             | Northern | 3.80203e+07 |      2.80136e+07 | 760710           |   6016.97 | 6.68006e+07 | 766727           |
|  71 | Giles                                 | Western  | 1.47903e+07 |      1.21726e+07 | 412447           |    524.73 | 2.73759e+07 | 412972           |
|  73 | Gloucester                            | Eastern  | 2.38819e+07 |      1.78917e+07 | 540096           |   1226.4  | 4.2315e+07  | 541322           |
|  75 | Goochland                             | Central  | 8.36423e+06 |      6.57798e+06 | 464167           |  72849.8  | 1.54792e+07 | 537017           |
|  77 | Grayson                               | Western  | 1.6e+07     |      1.19425e+07 | 197206           |     -0.05 | 2.81398e+07 | 197206           |
|  79 | Greene                                | Northern | 1.18517e+07 |      9.35203e+06 | 506360           |     -0.08 | 2.17101e+07 | 506360           |
|  81 | Greensville/Emporia                   | Eastern  | 2.41029e+07 |      1.77081e+07 | 229734           |     -0.05 | 4.20408e+07 | 229734           |
|  83 | Halifax/South Boston                  | Piedmont | 4.11931e+07 |      3.22035e+07 | 634723           |     -0.11 | 7.40313e+07 | 634723           |
|  85 | Hanover                               | Central  | 3.73789e+07 |      2.95894e+07 |      1.69477e+06 |     -0.18 | 6.8663e+07  |      1.69477e+06 |
|  87 | Henrico                               | Central  | 1.95269e+08 |      1.40949e+08 |      3.36319e+06 |  15862.7  | 3.39598e+08 |      3.37906e+06 |
|  89 | Henry/Martinsville                    | Piedmont | 8.62283e+07 |      6.06582e+07 | 398686           |     -0.1  | 1.47285e+08 | 398686           |
|  91 | Highland                              | Piedmont | 1.21808e+06 | 993467           |  14807.2         |      0    | 2.22635e+06 |  14807.2         |
|  93 | Isle Of Wight                         | Eastern  | 2.25199e+07 |      1.62441e+07 | 202059           |   1566.39 | 3.89676e+07 | 203625           |
|  95 | James City                            | Eastern  | 2.29715e+07 |      1.5179e+07  | 294371           |    812.85 | 3.84456e+07 | 295184           |
|  97 | King & Queen                          | Central  | 6.48062e+06 |      4.75538e+06 | 149748           |      7.99 | 1.13858e+07 | 149756           |
|  99 | King George                           | Northern | 1.38344e+07 |      9.6797e+06  | 807879           |     -0.04 | 2.4322e+07  | 807879           |
| 101 | King William                          | Central  | 9.23088e+06 |      6.38152e+06 | 385808           |      0    | 1.59982e+07 | 385808           |
| 103 | Lancaster                             | Central  | 9.18921e+06 |      6.69863e+06 | 432373           |   3431.98 | 1.63236e+07 | 435805           |
| 105 | Lee                                   | Western  | 3.38613e+07 |      2.51721e+07 | 314082           |     -0.2  | 5.93474e+07 | 314081           |
| 107 | Loudoun                               | Northern | 7.17229e+07 |      5.67023e+07 |      2.58066e+06 |  14749.9  | 1.31021e+08 |      2.59541e+06 |
| 109 | Louisa                                | Northern | 2.48514e+07 |      1.89914e+07 | 961410           |     -0.14 | 4.48041e+07 | 961409           |
| 111 | Lunenburg                             | Central  | 1.28322e+07 |      9.94352e+06 | 234902           |     -0.08 | 2.30106e+07 | 234902           |
| 113 | Madison                               | Northern | 9.34376e+06 |      8.74925e+06 | 931240           |    999.86 | 1.90252e+07 | 932240           |
| 115 | Mathews                               | Eastern  | 6.84555e+06 |      5.41047e+06 | 164541           |   2784.6  | 1.24233e+07 | 167326           |
| 117 | Mecklenburg                           | Piedmont | 3.27365e+07 |      2.53899e+07 | 427861           |     -0.05 | 5.85543e+07 | 427861           |
| 119 | Middlesex                             | Central  | 9.40721e+06 |      6.93234e+06 | 206158           |     -0.02 | 1.65457e+07 | 206158           |
| 121 | Montgomery                            | Western  | 4.32444e+07 |      3.13498e+07 | 317418           |     -0.15 | 7.49116e+07 | 317418           |
| 125 | Nelson                                | Piedmont | 1.27991e+07 |      9.49921e+06 | 237460           |    278.88 | 2.2536e+07  | 237739           |
| 127 | New Kent                              | Central  | 6.99926e+06 |      5.45688e+06 | 439547           |     -0.01 | 1.28957e+07 | 439547           |
| 131 | Northampton                           | Eastern  | 1.77216e+07 |      1.33751e+07 | 150287           |     -0.03 | 3.1247e+07  | 150287           |
| 133 | Northumberland                        | Central  | 8.68503e+06 |      5.90448e+06 |   4072.08        |    499.97 | 1.45941e+07 |   4572.05        |
| 135 | Nottoway                              | Central  | 1.77623e+07 |      1.24418e+07 | 319699           |      0    | 3.05239e+07 | 319699           |
| 137 | Orange                                | Northern | 2.14721e+07 |      1.67953e+07 | 746873           |  60870.4  | 3.90751e+07 | 807743           |
| 139 | Page                                  | Northern | 1.90182e+07 |      1.33479e+07 | 208644           |     -0.02 | 3.25747e+07 | 208644           |
| 141 | Patrick                               | Western  | 1.69596e+07 |      1.20514e+07 |  30128.6         |     -0.05 | 2.90412e+07 |  30128.6         |
| 143 | Pittsylvania                          | Piedmont | 5.76761e+07 |      4.45139e+07 |      1.00004e+06 |   3062.95 | 1.03193e+08 |      1.0031e+06  |
| 145 | Powhatan                              | Central  | 8.74614e+06 |      7.28785e+06 | 861502           |  34999.4  | 1.69305e+07 | 896502           |
| 147 | Prince Edward                         | Central  | 2.07562e+07 |      1.58651e+07 | 224202           |     -0.05 | 3.68454e+07 | 224202           |
| 149 | Prince George                         | Eastern  | 1.47886e+07 |      1.06249e+07 | 500779           |     -0.04 | 2.59143e+07 | 500779           |
| 153 | Prince William                        | Northern | 1.79213e+08 |      1.30076e+08 |      2.74555e+06 |      3.83 | 3.12035e+08 |      2.74555e+06 |
| 155 | Pulaski                               | Western  | 3.21402e+07 |      2.52818e+07 |      1.12768e+06 |   6859.74 | 5.85565e+07 |      1.13454e+06 |
| 157 | Rappahannock                          | Northern | 3.19364e+06 |      2.99896e+06 | 681478           |    578.59 | 6.87465e+06 | 682057           |
| 159 | Richmond Co.                          | Central  | 7.83856e+06 |      5.41607e+06 | 162975           |     -0.03 | 1.34176e+07 | 162975           |
| 161 | Roanoke Co./Salem                     | Piedmont | 5.79999e+07 |      4.86747e+07 |      2.83031e+06 |  40730.9  | 1.09546e+08 |      2.87105e+06 |
| 163 | Rockbridge/Buena Vista/Lexington      | Piedmont | 2.38641e+07 |      1.98977e+07 |      1.0485e+06  |    649.93 | 4.48109e+07 |      1.04915e+06 |
| 165 | Rockingham/Harrisonburg               | Northern | 6.56998e+07 |      5.47197e+07 |      3.39393e+06 |     -0.2  | 1.23813e+08 |      3.39393e+06 |
| 167 | Russell                               | Western  | 3.0874e+07  |      2.22017e+07 | 282487           |     -0.13 | 5.33582e+07 | 282487           |
| 169 | Scott                                 | Western  | 2.15395e+07 |      1.61007e+07 | 280223           |     -0.12 | 3.79204e+07 | 280223           |
| 171 | Shenandoah                            | Northern | 2.90948e+07 |      2.16059e+07 | 624803           |     -0.1  | 5.13255e+07 | 624803           |
| 173 | Smyth                                 | Western  | 3.26811e+07 |      2.32524e+07 | 261352           |    499.83 | 5.61954e+07 | 261852           |
| 175 | Southampton                           | Eastern  | 1.61789e+07 |      1.1695e+07  | 180989           |      0.04 | 2.80549e+07 | 180989           |
| 177 | Spotsylvania                          | Northern | 6.98569e+07 |      5.35625e+07 |      3.88395e+06 |     38.31 | 1.27303e+08 |      3.88399e+06 |
| 179 | Stafford                              | Northern | 5.50333e+07 |      4.05014e+07 |      1.95949e+06 |     -0.03 | 9.74942e+07 |      1.95949e+06 |
| 181 | Surry                                 | Eastern  | 5.77695e+06 |      4.09027e+06 | 114540           |      0    | 9.98176e+06 | 114540           |
| 183 | Sussex                                | Eastern  | 1.19223e+07 |      9.54465e+06 | 272969           |     -0.05 | 2.17399e+07 | 272969           |
| 185 | Tazewell                              | Western  | 4.09809e+07 |      2.94169e+07 | 522714           |  36617.7  | 7.09571e+07 | 559331           |
| 187 | Warren                                | Northern | 2.73646e+07 |      1.99993e+07 | 655965           |     -0.14 | 4.80199e+07 | 655965           |
| 191 | Washington                            | Western  | 4.02705e+07 |      2.95825e+07 | 645665           |   7199.78 | 7.05059e+07 | 652865           |
| 193 | Westmoreland                          | Central  | 1.55835e+07 |      1.12654e+07 | 567774           |     -0.05 | 2.74167e+07 | 567773           |
| 195 | Wise                                  | Western  | 4.74384e+07 |      3.50679e+07 | 650212           |     -0.18 | 8.31565e+07 | 650212           |
| 197 | Wythe                                 | Western  | 2.54117e+07 |      2.0073e+07  | 572945           |    481.21 | 4.60581e+07 | 573427           |
| 199 | York/Poquoson                         | Eastern  | 2.02465e+07 |      1.54358e+07 | 483596           |    927.98 | 3.61668e+07 | 484524           |
| 510 | Alexandria                            | Northern | 6.2321e+07  |      5.03037e+07 |      3.67861e+06 | -47441.6  | 1.16256e+08 |      3.63117e+06 |
| 520 | Bristol                               | Western  | 2.21681e+07 |      1.55784e+07 | 470872           |     -0.19 | 3.82174e+07 | 470872           |
| 540 | Charlottesville                       | Piedmont | 3.61928e+07 |      3.33302e+07 |      2.43483e+06 |  10273.3  | 7.1968e+07  |      2.4451e+06  |
| 550 | Chesapeake                            | Eastern  | 1.30477e+08 |      9.24013e+07 |      1.29549e+06 |  -1741.68 | 2.24172e+08 |      1.29374e+06 |
| 590 | Danville                              | Piedmont | 7.57572e+07 |      5.65875e+07 | 882253           |     -0.13 | 1.33227e+08 | 882252           |
| 620 | Franklin                              | Eastern  | 1.56678e+07 |      1.12176e+07 | 138070           |     -0.02 | 2.70235e+07 | 138070           |
| 630 | Fredericksburg                        | Northern | 2.49787e+07 |      1.88099e+07 | 545582           |     -0.18 | 4.43342e+07 | 545582           |
| 640 | Galax                                 | Western  | 1.09581e+07 |      7.92055e+06 |  77346.5         |     -0.07 | 1.8956e+07  |  77346.4         |
| 650 | Hampton                               | Eastern  | 1.23814e+08 |      9.30914e+07 |      1.67756e+06 |   6839.03 | 2.1859e+08  |      1.6844e+06  |
| 670 | Hopewell                              | Central  | 3.37097e+07 |      2.47965e+07 | 944206           |     -0.11 | 5.94505e+07 | 944206           |
| 680 | Lynchburg                             | Piedmont | 8.26419e+07 |      6.39856e+07 |      1.54151e+06 |  -1865.97 | 1.48167e+08 |      1.53965e+06 |
| 683 | Manassas                              | Northern | 2.38901e+07 |      1.63731e+07 | 661575           |    388.9  | 4.09252e+07 | 661964           |
| 685 | Manassas Park                         | Northern | 7.65985e+06 |      5.86781e+06 | 371874           |     -0.02 | 1.38995e+07 | 371874           |
| 700 | Newport News                          | Eastern  | 1.7611e+08  |      1.18426e+08 |      1.7751e+06  |   7587.77 | 2.96318e+08 |      1.78269e+06 |
| 710 | Norfolk                               | Eastern  | 2.52361e+08 |      1.74043e+08 |      2.63279e+06 |   -633.49 | 4.29036e+08 |      2.63216e+06 |
| 720 | Norton                                | Western  | 4.79203e+06 |      3.28077e+06 |  61553.5         |    524.95 | 8.13487e+06 |  62078.4         |
| 730 | Petersburg                            | Central  | 6.52143e+07 |      4.9325e+07  |      1.57423e+06 |   -671.77 | 1.16113e+08 |      1.57356e+06 |
| 740 | Portsmouth                            | Eastern  | 1.27665e+08 |      9.14447e+07 |      1.08921e+06 |   2076.18 | 2.20201e+08 |      1.09129e+06 |
| 750 | Radford                               | Western  | 9.71443e+06 |      7.90374e+06 | 321796           |     -0.04 | 1.794e+07   | 321796           |
| 760 | Richmond                              | Central  | 3.12294e+08 |      2.30643e+08 |      5.48114e+06 | 446370    | 5.48865e+08 |      5.92751e+06 |
| 770 | Roanoke                               | Piedmont | 1.36432e+08 |      1.03415e+08 |      3.60341e+06 |   3655.42 | 2.43454e+08 |      3.60706e+06 |
| 800 | Suffolk                               | Eastern  | 7.75762e+07 |      5.69639e+07 | 383986           |  14679.9  | 1.34939e+08 | 398666           |
| 810 | Virginia Beach                        | Eastern  | 2.07914e+08 |      1.53338e+08 |      4.33746e+06 |  65365.6  | 3.65655e+08 |      4.40282e+06 |
| 830 | Williamsburg                          | Eastern  | 5.88808e+06 |      3.95692e+06 | 120932           |  27696.2  | 9.99363e+06 | 148628           |
| 840 | Winchester                            | Northern | 2.47584e+07 |      1.83122e+07 | 557788           |  10101.3  | 4.36384e+07 | 567890           |



label: I=F+G
labeled_formulas: I=F+G
output: {"Arithmetic-Relationship": ["I=F+G"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_155182___xlsx_626ce64c9360feed315a74c45b0da2ce9c633089.xlsx___line_427