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 |
|:------------------------|----------:|----------:|-----------:|----:|-----------:|----:|----:|-----------:|----------:|-----:|
| ACCOMACK                |  4232.88  |   9925.65 |   5341.55  |   0 |   859.561  |   0 |   0 |  278.315   |  3446.22  |  596 |
| ALBEMARLE               |  7305.17  |  15966.9  |   8318.26  |   0 |  2954.82   |   0 |   0 | 1126.82    |  3567     |  788 |
| ALLEGHANY               |  1952.72  |   7695.92 |   3016.12  |   0 |   170.644  |   0 |   0 |   42.9054  |  4466.25  |  544 |
| AMELIA                  |  1358.62  |   5274.07 |   2041.89  |   0 |   255.635  |   0 |   0 |   36.8466  |  2939.7   |  321 |
| AMHERST                 |  2932.78  |   9233.57 |   4842.59  |   0 |   332.672  |   0 |   0 |  172.975   |  3885.33  |  678 |
| APPOMATTOX              |  1482.98  |   4555.12 |   2805.64  |   0 |   191.09   |   0 |   0 |   56.155   |  1502.24  |  262 |
| ARLINGTON               | 13317     |  45737.4  |  23905.7   |   0 | 12642.4    |   0 |   0 | 2088.59    |  7100.75  | 1111 |
| AUGUSTA                 |  7252.46  |  20263.5  |  11336.4   |   0 |  1166.47   |   0 |   0 |  305.592   |  7455.02  | 1394 |
| BATH                    |   512.174 |   1490.69 |   1013.15  |   0 |    46.9489 |   0 |   0 |    5.60629 |   424.988 |   75 |
| BEDFORD                 |  7805.37  |  28651.5  |  14145.3   |   0 |  1066.55   |   0 |   0 |  528.004   | 12911.6   | 1691 |
| BLAND                   |   841.153 |   3109.13 |   1870.27  |   0 |    88.4757 |   0 |   0 |   11.7414  |  1138.64  |  177 |
| BOTETOURT               |  3960.32  |  17245.9  |   6701.26  |   0 |   497.687  |   0 |   0 |  258.036   |  9788.9   |  914 |
| BRUNSWICK               |  1866.94  |   8113.53 |   3549.54  |   0 |   214.198  |   0 |   0 |   23.4799  |  4326.31  |  394 |
| BUCHANAN                |  1251.53  |   9047.98 |   6296.07  |   0 |   252.361  |   0 |   0 |   24.0364  |  2475.52  |  352 |
| BUCKINGHAM              |  1385.74  |   4756.79 |   2997.41  |   0 |   179.909  |   0 |   0 |  120.889   |  1458.58  |  200 |
| CAMPBELL                |  4981.57  |  19114.6  |   9298.09  |   0 |  1354.05   |   0 |   0 |  148.07    |  8314.42  | 1310 |
| CAROLINE                |  3027.61  |  12002.8  |   5546.23  |   0 |  1117.1    |   0 |   0 |  127.037   |  5212.42  |  570 |
| CARROLL                 |  2559.57  |  12556.3  |   6319.05  |   0 |   462.705  |   0 |   0 |  157.485   |  5617.05  | 1033 |
| CHARLES CITY            |   743.779 |   3217    |   1398.42  |   0 |   234.755  |   0 |   0 |   22.5427  |  1561.29  |  168 |
| CHARLOTTE               |   833.783 |   5503.32 |   2774.91  |   0 |    92.4294 |   0 |   0 |   34.8011  |  2601.18  |  312 |
| CHESTERFIELD            | 30671.2   | 121165    |  56908.6   |   0 | 12836.7    |   0 |   0 | 1850.02    | 49570     | 5668 |
| CLARKE                  |  1733.89  |   4593.92 |   1779.53  |   0 |   256.231  |   0 |   0 |  158.875   |  2399.28  |  264 |
| CRAIG                   |   520.561 |   3776.16 |   1675.89  |   0 |   693.997  |   0 |   0 |    3.10956 |  1403.17  |  154 |
| CUMBERLAND              |   606.003 |   4367.02 |   1342.44  |   0 |   157.614  |   0 |   0 |   40.9131  |  2826.05  |  281 |
| DICKENSON               |  1214.82  |  10019.8  |   5448.27  |   0 |  1309.32   |   0 |   0 |   55.4172  |  3206.81  |  425 |
| DINWIDDIE               |  2356.55  |  15120    |   7505.5   |   0 |  1025.8    |   0 |   0 |   84.7876  |  6503.93  |  660 |
| ESSEX                   |   900.553 |   3673.79 |   1762.56  |   0 |   258.482  |   0 |   0 |   44.1369  |  1608.62  |  224 |
| FAIRFAX                 | 89858.4   | 260838    | 162238     |   0 | 63143.4    |   0 |   0 | 9538.71    | 25917.9   | 4984 |
| FAUQUIER                |  7712.17  |  12641.2  |   7780.87  |   0 |  1557.26   |   0 |   0 |  382.191   |  2920.92  |  498 |
| FLOYD                   |  1351.82  |   6714.5  |   2565.9   |   0 |   817.12   |   0 |   0 |   44.3766  |  3287.1   |  398 |
| FLUVANNA                |  2581.67  |   6845.75 |   3779.84  |   0 |   646.551  |   0 |   0 |  332.718   |  2086.64  |  377 |
| FRANKLIN                |  6043.74  |  26603.9  |  11770.9   |   0 |   841.185  |   0 |   0 |  241.535   | 13750.3   | 1540 |
| FREDERICK               |  7623.63  |  23847.5  |  10754.9   |   0 |  1901.84   |   0 |   0 |  502.826   | 10687.9   | 1561 |
| GILES                   |  1735.2   |   7698.33 |   3890.55  |   0 |   382.55   |   0 |   0 |   72.7945  |  3352.43  |  410 |
| GLOUCESTER              |  5858.7   |  19792    |  11799     |   0 |  3040.26   |   0 |   0 |  210.897   |  4741.82  |  626 |
| GOOCHLAND               |  1419.01  |   4925.37 |   2169.6   |   0 |   261.44   |   0 |   0 |  138.7     |  2355.63  |  297 |
| GRAYSON                 |  1542.91  |   7103.47 |   3663.42  |   0 |   272.848  |   0 |   0 |   67.701   |  3099.5   |  605 |
| GREENE                  |  2169.47  |   4406.75 |   2117.32  |   0 |   566.944  |   0 |   0 |   44.1726  |  1678.31  |  221 |
| GREENSVILLE             |  1406.11  |   4671.78 |   2348.04  |   0 |    82.6803 |   0 |   0 |   26.8591  |  2214.19  |  217 |
| HALIFAX                 |  3295.8   |  13558.5  |   7418.53  |   0 |   388.754  |   0 |   0 |  193.493   |  5557.69  |  874 |
| HANOVER                 | 10223.8   |  19979.5  |   9131.75  |   0 |  1986.22   |   0 |   0 |  757.24    |  8104.26  | 1125 |
| HENRICO                 | 24613.7   |  80181.2  |  32910.2   |   0 |  8223.31   |   0 |   0 | 2264.94    | 36782.8   | 4457 |
| HENRY                   |  4904.47  |  22685.7  |   9126.63  |   0 |   640.651  |   0 |   0 |  205.34    | 12713.1   | 1704 |
| HIGHLAND                |   356.501 |   1050.68 |    746.789 |   0 |    39.0609 |   0 |   0 |   23.3206  |   241.509 |   53 |
| ISLE OF WIGHT           |  3695.12  |  17204.9  |  10432.2   |   0 |  2995.97   |   0 |   0 |  389.959   |  3386.75  |  564 |
| JAMES CITY              |  8464.62  |  28981.4  |  17972.4   |   0 |  4420.08   |   0 |   0 | 1010.14    |  5578.74  |  960 |
| KING AND QUEEN          |   679.254 |   2218.57 |   1340.35  |   0 |   135.166  |   0 |   0 |    9.02721 |   734.033 |  113 |
| KING GEORGE             |  3222.17  |   9773.39 |   5407.09  |   0 |  1946.92   |   0 |   0 |  481.583   |  1937.8   |  337 |
| KING WILLIAM            |  1553.4   |   4483.75 |   2237.25  |   0 |   369.123  |   0 |   0 |   16.8823  |  1860.49  |  228 |
| LANCASTER               |  1746.74  |   3778    |   2023.74  |   0 |    93.2454 |   0 |   0 |  117.829   |  1543.19  |  213 |
| LEE                     |  2187.6   |  12663.7  |   7187.78  |   0 |   222.089  |   0 |   0 |   24.6856  |  5229.17  |  714 |
| LOUDOUN                 | 22042.8   |  45105.6  |  25555.5   |   0 | 10793.6    |   0 |   0 | 2078.82    |  6677.63  | 1155 |
| LOUISA                  |  3278.5   |   9032.19 |   4695.32  |   0 |   614.688  |   0 |   0 |  156.063   |  3566.11  |  464 |
| LUNENBURG               |  1716.48  |   5907.42 |   2845.81  |   0 |   183.503  |   0 |   0 |   99.9054  |  2778.2   |  359 |
| MADISON                 |  1290.79  |   3045.6  |   1824.81  |   0 |   153.993  |   0 |   0 |   85.5009  |   981.304 |  139 |
| MATHEWS                 |  1424.8   |   4932.19 |   2363.91  |   0 |  1430.19   |   0 |   0 |   61.5076  |  1076.58  |  162 |
| MECKLENBURG             |  3123.76  |  14393.9  |   6758.1   |   0 |   281.922  |   0 |   0 |  294.197   |  7059.72  |  821 |
| MIDDLESEX               |  1427.43  |   4456.13 |   2415.24  |   0 |   221.078  |   0 |   0 |   89.5751  |  1730.24  |  251 |
| MONTGOMERY              |  6176.99  |  23859.1  |   9845.48  |   0 |  3519.31   |   0 |   0 |  327.769   | 10166.5   | 1142 |
| NELSON                  |  1525.81  |   4985.06 |   2750.86  |   0 |   120.878  |   0 |   0 |   85.5824  |  2027.74  |  277 |
| NEW KENT                |  1820.91  |   7075.67 |   3944.9   |   0 |   589.874  |   0 |   0 |   86.191   |  2454.7   |  326 |
| NORTHAMPTON             |  1193.86  |   3449.53 |   1981.17  |   0 |   273.965  |   0 |   0 |  217.411   |   976.984 |  194 |
| NORTHUMBERLAND          |  1706.39  |   3993.77 |   1913.33  |   0 |   129.871  |   0 |   0 |  189.728   |  1760.83  |  257 |
| NOTTOWAY                |  1594.1   |   7209.94 |   3433.64  |   0 |   343.145  |   0 |   0 |   60.7779  |  3372.38  |  419 |
| ORANGE                  |  3696.33  |  11358.7  |   6302.09  |   0 |  1416.22   |   0 |   0 |  336.357   |  3304.05  |  528 |
| PAGE                    |  2304.42  |   8142.44 |   3359.72  |   0 |   273.337  |   0 |   0 |   39.6477  |  4469.74  |  547 |
| PATRICK                 |  1808.33  |   7343.69 |   3139.37  |   0 |   184.089  |   0 |   0 |   73.9309  |  3946.3   |  594 |
| PITTSYLVANIA            |  5614.01  |  25689.5  |  12299.9   |   0 |   850.117  |   0 |   0 |  331.739   | 12207.7   | 1789 |
| POWHATAN                |  3047.28  |   7352.44 |   3630.22  |   0 |   558.337  |   0 |   0 |  112.925   |  3050.96  |  413 |
| PRINCE EDWARD           |  1570.1   |   7846.04 |   3770.94  |   0 |   471.4    |   0 |   0 |  133.511   |  3470.19  |  427 |
| PRINCE GEORGE           |  4616.62  |  27740.7  |  15799.8   |   0 |  3925.72   |   0 |   0 |  123.964   |  7891.22  | 1117 |
| PULASKI                 |  3671.1   |  13593.4  |   6574.01  |   0 |   728.383  |   0 |   0 |  117.318   |  6173.65  |  842 |
| RAPPAHANNOCK            |   743.709 |   2015.31 |    903.95  |   0 |   123.273  |   0 |   0 |  113.536   |   874.55  |   98 |
| RICHMOND                |  1071.1   |   3769.24 |   1498.42  |   0 |    73.6837 |   0 |   0 |   30.211   |  2166.92  |  174 |
| ROANOKE                 |  8980.35  |  43045.4  |  19528.5   |   0 |  1613.67   |   0 |   0 |  759.539   | 21143.6   | 2432 |
| ROCKBRIDGE              |  1771.54  |  10511.2  |   4097.18  |   0 |   379.299  |   0 |   0 |  145.451   |  5889.23  |  556 |
| ROCKINGHAM              |  5980.78  |  16286.3  |   7918.86  |   0 |  1264.8    |   0 |   0 |  522.565   |  6580.08  | 1375 |
| RUSSELL                 |  2509.5   |   7814.23 |   4922.21  |   0 |   219.684  |   0 |   0 |  228.543   |  2443.78  |  463 |
| SCOTT                   |  2257.46  |   8585.43 |   5488.26  |   0 |   175.49   |   0 |   0 |   20.9842  |  2900.7   |  491 |
| SHENANDOAH              |  3961.98  |  15154.9  |   7193.94  |   0 |   467.836  |   0 |   0 |  360.113   |  7133     | 1014 |
| SMYTH                   |  3199.9   |  13294.9  |   7132.18  |   0 |   344.464  |   0 |   0 |   83.2724  |  5735     |  908 |
| SOUTHAMPTON             |  1919.77  |   6889.04 |   3745.31  |   0 |   726.997  |   0 |   0 |  440.332   |  1976.4   |  274 |
| SPOTSYLVANIA            | 13837.9   |  49529.9  |  29036.6   |   0 |  8892.91   |   0 |   0 |  900.112   | 10700.3   | 1992 |
| STAFFORD                | 19081.8   |  75140.4  |  45905.2   |   0 | 20748.9    |   0 |   0 |  761.102   |  7725.13  | 1772 |
| SURRY                   |   705.345 |   2554.33 |   1284.39  |   0 |   141.438  |   0 |   0 |    7.58062 |  1120.93  |  144 |
| SUSSEX                  |  1221.98  |   4434.4  |   2072.53  |   0 |   203.667  |   0 |   0 |   73.9294  |  2084.28  |  246 |
| TAZEWELL                |  3718.49  |  26281    |  15284.8   |   0 |   639.556  |   0 |   0 |  167.467   | 10189.1   | 1326 |
| WARREN                  |  3721.58  |  13669.9  |   5878.7   |   0 |   928.32   |   0 |   0 |  394.176   |  6468.69  |  741 |
| WASHINGTON              |  4702.43  |  22029.5  |  11558.6   |   0 |   627.443  |   0 |   0 |  242.262   |  9601.18  | 1512 |
| WESTMORELAND            |  1868.08  |   5735.1  |   2779.49  |   0 |   366.659  |   0 |   0 |  199.259   |  2389.7   |  315 |
| WISE                    |  3045.47  |  17079.7  |  10421.6   |   0 |   678.532  |   0 |   0 |   50.505   |  5929     | 1014 |
| WYTHE                   |  2874.05  |   9834.41 |   4945.97  |   0 |   301.22   |   0 |   0 |   42.5311  |  4544.69  |  693 |
| YORK                    |  9772.05  |  42559.4  |  28267.7   |   0 |  8443.43   |   0 |   0 |  785.672   |  5062.58  |  998 |
| BEDFORD (CITY)          |   756.75  |   2577.81 |    683.932 |   0 |    47.8468 |   0 |   0 |   16.7057  |  1829.32  |  203 |
| BRISTOL (CITY)          |  1534.94  |   7959.9  |   4214.15  |   0 |   130.105  |   0 |   0 |  164.656   |  3450.98  |  583 |
| BUENA VISTA (CITY)      |   617.071 |   2146.22 |    799.824 |   0 |   192.729  |   0 |   0 |   27.3813  |  1126.29  |  157 |
| CHARLOTTESVILLE (CITY)  |  2851.31  |   5905.51 |   2726.56  |   0 |  1450.83   |   0 |   0 |  599.411   |  1128.7   |  229 |
| CHESAPEAKE CITY (CITY)  | 30820.3   | 138526    |  81233.1   |   0 | 36507.4    |   0 |   0 | 1597.34    | 19188.5   | 3533 |
| COLONIAL HEIGHTS (CITY) |  1855.49  |  12264.5  |   7101.36  |   0 |  1094.64   |   0 |   0 |  293.886   |  3774.59  |  445 |
| COVINGTON (CITY)        |   762.088 |   3524.41 |   1743.05  |   0 |   119.804  |   0 |   0 |    4.39164 |  1657.17  |  209 |
| EMPORIA (CITY)          |   373.521 |   2086.76 |    708.061 |   0 |    31.5212 |   0 |   0 |   11.5113  |  1335.67  |  111 |
| FAIRFAX (CITY)          |  2211.87  |   5197.57 |   2981.47  |   0 |  1309.93   |   0 |   0 |  227.162   |   679.006 |  112 |
| FALLS CHURCH (CITY)     |  1376.42  |   2441.27 |   1550.82  |   0 |   533.486  |   0 |   0 |  144.76    |   212.208 |   54 |
| FRANKLIN (CITY)         |   882.112 |   3728.02 |   1483.86  |   0 |   351.457  |   0 |   0 |  277.28    |  1615.42  |  141 |
| FREDERICKSBURG (CITY)   |  2103.4   |   7897.42 |   3934.83  |   0 |  1809.62   |   0 |   0 |  135.887   |  2017.08  |  342 |
| GALAX (CITY)            |   543.357 |   3018.45 |   1011.63  |   0 |   200.564  |   0 |   0 |   38.5894  |  1767.66  |  223 |
| HARRISONBURG (CITY)     |  1967.27  |   6351.45 |   2519     |   0 |  1134.17   |   0 |   0 |  243.256   |  2455.02  |  408 |
| HOPEWELL (CITY)         |  3220.2   |  15267.9  |   7968.19  |   0 |  1409.52   |   0 |   0 |  107.92    |  5782.3   |  735 |
| LEXINGTON (CITY)        |   525.879 |   1361.51 |    799.56  |   0 |   109.517  |   0 |   0 |   51.927   |   400.51  |   73 |
| LYNCHBURG (CITY)        |  5610.51  |  19982.8  |   8849.43  |   0 |  1917.74   |   0 |   0 |  254.088   |  8961.58  | 1412 |
| MANASSAS (CITY)         |  3874.08  |   6707.06 |   3462.88  |   0 |  1681.43   |   0 |   0 |  197.527   |  1365.23  |  197 |
| MANASSAS PARK (CITY)    |   965.939 |   2324.36 |   1335.93  |   0 |   517.461  |   0 |   0 |   36.2493  |   434.722 |   79 |
| MARTINSVILLE (CITY)     |  1548.85  |   5428.75 |   1942.98  |   0 |   109.95   |   0 |   0 |   67.684   |  3308.14  |  389 |
| NEWPORT NEWS (CITY)     | 25824.3   | 140584    |  74664.7   |   0 | 30050.6    |   0 |   0 | 2180.09    | 33688.6   | 4571 |
| NORFOLK (CITY)          | 29196     | 146763    |  69779.4   |   0 | 42082.7    |   0 |   0 | 2097.25    | 32804     | 4559 |
| NORTON (CITY)           |   350.529 |   1592.67 |   1084.37  |   0 |    40.9116 |   0 |   0 |   29.3533  |   438.036 |  106 |
| PETERSBURG (CITY)       |  4210.1   |  24539.2  |  12011     |   0 |  2109.1    |   0 |   0 |  176.153   | 10242.9   | 1155 |
| POQUOSON (CITY)         |  1709.71  |   6327.04 |   4276.12  |   0 |   805.996  |   0 |   0 |  138.246   |  1106.68  |  144 |
| PORTSMOUTH (CITY)       | 11790.4   |  58882.3  |  30442.5   |   0 | 12632.1    |   0 |   0 |  667.164   | 15140.5   | 2040 |
| RADFORD (CITY)          |  1180.14  |   3944.27 |   1877.02  |   0 |   542.941  |   0 |   0 |   29.2208  |  1495.09  |  238 |
| STAUNTON (CITY)         |  2605.55  |   8052.97 |   4144.22  |   0 |   575.424  |   0 |   0 |  220.008   |  3113.32  |  542 |
| SUFFOLK (CITY)          | 10468.5   |  53839.2  |  30028.6   |   0 | 13227.8    |   0 |   0 |  518.329   | 10064.6   | 1495 |
| VIRGINIA BEACH (CITY)   | 69290.6   | 315641    | 174509     |   0 | 98144.6    |   0 |   0 | 5908.25    | 37079     | 6971 |
| WAYNESBORO (CITY)       |  2407.72  |   6611.96 |   3751.71  |   0 |   399.927  |   0 |   0 |  168.638   |  2291.69  |  420 |
| WILLIAMSBURG (CITY)     |  1012.28  |   7569.68 |   5006.85  |   0 |  1378.16   |   0 |   0 |  225.72    |   958.955 |  136 |
| WINCHESTER (CITY)       |  2413.49  |   8456.92 |   2985.09  |   0 |   863.488  |   0 |   0 |  163.494   |  4444.86  |  539 |



labeled_formulas: H=B*E
H=B*G
H=C*E
H=C*G
H=D*E
H=D*G
H=E*F
H=E*I
H=E*J
H=E*K
H=F*G
H=G*I
H=G*J
H=G*K
H=(B+C)*E
H=(B+C)*G
H=(B+D)*E
H=(B+D)*G
H=(B*E)*F
H=(B*E)*I
H=(B*E)*J
H=(B*E)*K
H=(B+F)*G
H=(B*G)*I
H=(B*G)*J
H=(B*G)*K
H=(C+D)*E
H=(C+D)*G
H=(C*E)*F
H=(C*E)*I
H=(C*E)*J
H=(C*E)*K
H=(C+F)*G
H=(C*G)*I
H=(C*G)*J
H=(C*G)*K
H=(D*E)*F
H=(D*E)*I
H=(D*E)*J
H=(D*E)*K
H=(D+F)*G
H=(D*G)*I
H=(D*G)*J
H=(D*G)*K
H=(E*F)*I
H=(E*F)*J
H=(E*F)*K
H=(E*I)*J
H=(E*I)*K
H=(E*J)*K
H=(F*G)*I
H=(F*G)*J
H=(F*G)*K
H=(G*I)*J
H=(G*I)*K
H=(G*J)*K
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119145___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_209