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 |
|:-----------|----------:|----------:|-----------:|----:|-------------:|----:|----:|-----------:|----------:|-----:|
| ANDERSON   |  7056.22  |  22241.6  |  13570.8   |   0 |  1366.04     |   0 |   0 |  300.045   |  7004.74  | 1263 |
| BEDFORD    |  3130.07  |  15567.7  |   6683     |   0 |   511.379    |   0 |   0 |  235.788   |  8137.56  |  959 |
| BENTON     |  1644.36  |   8628.37 |   4617.15  |   0 |   218.063    |   0 |   0 |   28.0035  |  3765.15  |  474 |
| BLEDSOE    |   819.716 |   4438.21 |   2447.51  |   0 |   111.257    |   0 |   0 |   11.5964  |  1867.85  |  260 |
| BLOUNT     | 11589.6   |  35805.6  |  23467.1   |   0 |  2142.88     |   0 |   0 |  769.472   |  9426.1   | 1777 |
| BRADLEY    |  7705.85  |  27814.4  |  17662.8   |   0 |  1798.38     |   0 |   0 |  425.06    |  7928.12  | 1514 |
| CAMPBELL   |  3471.01  |  20687.1  |  14850.6   |   0 |   400.659    |   0 |   0 |  149.161   |  5286.65  |  933 |
| CANNON     |  1101.55  |   6955.7  |   2698.97  |   0 |   177.629    |   0 |   0 |   20.6028  |  4058.49  |  392 |
| CARROLL    |  2390.97  |   8806.05 |   5272.15  |   0 |   321.944    |   0 |   0 |   45.8537  |  3166.1   |  448 |
| CARTER     |  6081.86  |  49302.5  |  21283.2   |   0 |   892.687    |   0 |   0 |  303.57    | 26823.1   | 2859 |
| CHEATHAM   |  3007.56  |  14136.2  |   6548.42  |   0 |   945.861    |   0 |   0 |   66.7576  |  6575.14  |  666 |
| CHESTER    |  1424.87  |   5146.85 |   3135.38  |   0 |   250.918    |   0 |   0 |   56.3984  |  1704.15  |  259 |
| CLAIBORNE  |  2464.9   |  14622.1  |   9625.66  |   0 |   524.479    |   0 |   0 |  218.859   |  4253.08  |  599 |
| CLAY       |   537.508 |   3728.99 |   1881.86  |   0 |    71.1282   |   0 |   0 |    7.68641 |  1768.32  |  199 |
| COCKE      |  3169.63  |  18698.9  |  10340.4   |   0 |   380.973    |   0 |   0 |  225.499   |  7752.03  | 1064 |
| COFFEE     |  4448.45  |  24941.9  |  11340.7   |   0 |   942.835    |   0 |   0 |  172.658   | 12485.8   | 1552 |
| CROCKETT   |  1096.47  |   4464.2  |   2763.16  |   0 |   142.094    |   0 |   0 |   36.8279  |  1522.11  |  222 |
| CUMBERLAND |  6556.83  |  26315.8  |  14050.4   |   0 |   532.001    |   0 |   0 |  550.689   | 11182.7   | 1845 |
| DECATUR    |  1123.15  |   4972.58 |   2931.89  |   0 |    91.6169   |   0 |   0 |   74.4634  |  1874.6   |  264 |
| DE KALB    |  1590.38  |   6426.76 |   3150.06  |   0 |    99.9354   |   0 |   0 |   75.4143  |  3101.35  |  363 |
| DICKSON    |  4849.33  |  19376.3  |  10176.4   |   0 |   889.665    |   0 |   0 |  165.729   |  8144.45  |  917 |
| DYER       |  2863.6   |  11804.5  |   6786.11  |   0 |   348.116    |   0 |   0 |  278.899   |  4391.4   |  729 |
| FAYETTE    |  2929.41  |  11833.4  |   5789.8   |   0 |   683.837    |   0 |   0 |  174.252   |  5185.51  |  610 |
| FENTRESS   |  1012.69  |   7249.48 |   4419.43  |   0 |   195.171    |   0 |   0 |   17.3634  |  2617.52  |  494 |
| FRANKLIN   |  3831.23  |  17879.7  |   9587.71  |   0 |   539.215    |   0 |   0 |  333.969   |  7418.8   | 1074 |
| GIBSON     |  4040.16  |  13579    |   8108.2   |   0 |   592.516    |   0 |   0 |  144.786   |  4733.54  |  832 |
| GILES      |  2530.55  |  10831.5  |   5898.98  |   0 |   351.113    |   0 |   0 |   98.7727  |  4482.61  |  547 |
| GRAINGER   |  2065.67  |   6657.56 |   4179.37  |   0 |   150.082    |   0 |   0 |   32.184   |  2295.92  |  439 |
| GREENE     |  5482.95  |  34834.9  |  17334.9   |   0 |   685.445    |   0 |   0 |  235.417   | 16579.2   | 2180 |
| GRUNDY     |   783.211 |   4987    |   2200.19  |   0 |   109.543    |   0 |   0 |   23.8629  |  2653.41  |  299 |
| HAMBLEN    |  5561.39  |  22405.4  |  13334.1   |   0 |   750.766    |   0 |   0 |   87.0296  |  8233.46  | 1496 |
| HANCOCK    |   401.918 |   2412.41 |   1382.48  |   0 |    56.577    |   0 |   0 |   16.212   |   957.141 |  131 |
| HARDEMAN   |  2500.8   |   7553.21 |   3869.19  |   0 |   186.37     |   0 |   0 |   82.0867  |  3415.57  |  472 |
| HARDIN     |  2825.34  |   9641.62 |   5938.94  |   0 |   165.415    |   0 |   0 |   97.2446  |  3440.02  |  594 |
| HAWKINS    |  4772.29  |  25562.6  |  14447.3   |   0 |   654.935    |   0 |   0 |   79.6513  | 10380.7   | 1749 |
| HAYWOOD    |   966.501 |   4939.68 |   2784.11  |   0 |   241.913    |   0 |   0 |   22.2016  |  1891.46  |  257 |
| HENDERSON  |  2015.12  |   7643.57 |   4139.94  |   0 |   233.343    |   0 |   0 |  246.316   |  3023.97  |  433 |
| HENRY      |  3149.92  |  14948    |   8962.17  |   0 |   565.945    |   0 |   0 |  119.202   |  5300.7   |  720 |
| HICKMAN    |  1900.16  |   7518.27 |   3265.56  |   0 |   272.515    |   0 |   0 |   42.1426  |  3938.06  |  433 |
| HOUSTON    |   841.135 |   4177.54 |   2351.36  |   0 |   124.942    |   0 |   0 |   23.0598  |  1678.18  |  203 |
| HUMPHREYS  |  1930.52  |   7951.62 |   3981.46  |   0 |   325.177    |   0 |   0 |   56.7135  |  3588.27  |  375 |
| JACKSON    |   868.79  |   4851.95 |   2427.64  |   0 |   172.667    |   0 |   0 |   15.4058  |  2236.24  |  326 |
| JEFFERSON  |  4462.88  |  18852.9  |  11394.6   |   0 |   604.979    |   0 |   0 |  118.963   |  6734.36  | 1196 |
| JOHNSON    |  1849.46  |  11262.7  |   5489.61  |   0 |    94.2232   |   0 |   0 |   55.2154  |  5623.67  |  739 |
| LAKE       |   417.783 |   1772.97 |    785.889 |   0 |     0.286468 |   0 |   0 |   10.0514  |   976.742 |   95 |
| LAUDERDALE |  2158.68  |   6417.73 |   3410.45  |   0 |   145.46     |   0 |   0 |   24.7748  |  2837.05  |  435 |
| LAWRENCE   |  2682.04  |  14313.4  |   7906.93  |   0 |   439.919    |   0 |   0 |  116.772   |  5849.78  |  768 |
| LEWIS      |   840.957 |   4369.44 |   2264.15  |   0 |   104.535    |   0 |   0 |   16.8307  |  1983.93  |  268 |
| LINCOLN    |  2938.5   |  11212.8  |   5931.93  |   0 |   506.299    |   0 |   0 |  160.604   |  4613.97  |  523 |
| LOUDON     |  4683.6   |  12974.4  |   9235.06  |   0 |   561.999    |   0 |   0 |  257.62    |  2919.77  |  784 |
| MC MINN    |  4104.42  |  19425.2  |  12634.7   |   0 |   738.317    |   0 |   0 |  359.595   |  5692.61  |  982 |
| MC NAIRY   |  2424.49  |   9365.8  |   5887.72  |   0 |   227.226    |   0 |   0 |   89.6681  |  3161.18  |  571 |
| MACON      |  1558.93  |   6599.08 |   3278.04  |   0 |   206.475    |   0 |   0 |   58.5778  |  3055.99  |  342 |
| MADISON    |  7223.29  |  24911.9  |  15218.6   |   0 |  1369.43     |   0 |   0 |  382.981   |  7940.83  | 1371 |
| MARION     |  2140.31  |  10503.3  |   6408.92  |   0 |   341.499    |   0 |   0 |  109.15    |  3643.72  |  542 |
| MARSHALL   |  1938.71  |   9708.67 |   4433.54  |   0 |   297.575    |   0 |   0 |   40.5675  |  4936.99  |  563 |
| MAURY      |  6586.22  |  22379.7  |  10912.1   |   0 |  1160.71     |   0 |   0 |  276.062   | 10030.8   | 1184 |
| MEIGS      |  1389.36  |   4482.27 |   2711.56  |   0 |   195.149    |   0 |   0 |   33.0764  |  1542.48  |  254 |
| MONROE     |  3189.5   |  13992.6  |   9428.04  |   0 |   503.479    |   0 |   0 |  134.665   |  3926.47  |  749 |
| MONTGOMERY | 22182.5   | 184613    | 118448     |   0 | 28858.9      |   0 |   0 | 1889.41    | 35416.5   | 6224 |
| MOORE      |   587.925 |   2914.49 |   1362.27  |   0 |   115.105    |   0 |   0 |   21.2129  |  1415.9   |  136 |
| MORGAN     |  1811.71  |   7985.53 |   5105.48  |   0 |   191.924    |   0 |   0 |   54.3824  |  2633.75  |  419 |
| OBION      |  2496.63  |   9927.6  |   5747.6   |   0 |   391.625    |   0 |   0 |   98.7764  |  3689.61  |  695 |
| OVERTON    |  1382.75  |   8792.69 |   4612.79  |   0 |   265.555    |   0 |   0 |   32.1508  |  3882.19  |  588 |
| PERRY      |   754.779 |   3235.91 |   1890.24  |   0 |    32.8971   |   0 |   0 |   47.2985  |  1265.47  |  185 |
| PICKETT    |   418.398 |   2716.25 |   1590.54  |   0 |    65.6925   |   0 |   0 |    7.62383 |  1052.39  |  173 |
| POLK       |  1270.67  |   4951.57 |   2882.94  |   0 |   120.331    |   0 |   0 |   20.1484  |  1928.15  |  331 |
| PUTNAM     |  5063.59  |  24048    |  13265.8   |   0 |  1596.09     |   0 |   0 |  285.777   |  8900.27  | 1674 |
| RHEA       |  2655.46  |  10443.9  |   6943.37  |   0 |   470.086    |   0 |   0 |   81.0639  |  2949.37  |  573 |
| ROANE      |  4893.75  |  18854.2  |  11973     |   0 |   746.809    |   0 |   0 |  270.338   |  5864.09  | 1168 |
| ROBERTSON  |  4728.42  |  21010.4  |  11348.9   |   0 |  1399.27     |   0 |   0 |  181.974   |  8080.28  |  963 |
| SCOTT      |  1354.16  |   8747.87 |   6058.62  |   0 |   220.704    |   0 |   0 |  129.834   |  2338.71  |  463 |
| SEQUATCHIE |  1060.36  |   5887.94 |   4003.38  |   0 |   176.489    |   0 |   0 |   67.429   |  1640.64  |  319 |
| SEVIER     |  7734.32  |  30778.1  |  18257.9   |   0 |   902.574    |   0 |   0 |  288.423   | 11329.2   | 1800 |
| SMITH      |  1560.74  |   5641.06 |   2640.71  |   0 |   245.323    |   0 |   0 |   23.2724  |  2731.76  |  312 |
| STEWART    |  2054.24  |  11569.9  |   8283.03  |   0 |   409.178    |   0 |   0 |  128.256   |  2749.4   |  454 |
| SULLIVAN   | 15548.9   |  89102.6  |  48268.1   |   0 |  2299.75     |   0 |   0 |  713.027   | 37821.7   | 5299 |
| SUMNER     | 11807.7   |  43678.1  |  21910.4   |   0 |  2816.93     |   0 |   0 |  582.312   | 18368.5   | 2213 |
| TIPTON     |  5991.75  |  26119.5  |  14687.6   |   0 |  2780.03     |   0 |   0 |  268.567   |  8383.27  | 1198 |
| TROUSDALE  |   662.159 |   3320.98 |   1665.85  |   0 |   107.712    |   0 |   0 |   34.9575  |  1512.46  |  155 |
| UNICOI     |  2075.53  |  13952.7  |   6019.5   |   0 |   271.691    |   0 |   0 |   56.2038  |  7605.28  |  869 |
| UNION      |  1394.13  |   5426.49 |   3565.15  |   0 |   266.335    |   0 |   0 |   30.2988  |  1564.71  |  306 |
| VAN BUREN  |   535.378 |   3225.69 |   1533.23  |   0 |    86.7647   |   0 |   0 |    3.13619 |  1602.56  |  182 |
| WARREN     |  2860.72  |  15399.8  |   6584.24  |   0 |   243.429    |   0 |   0 |  189.896   |  8382.24  | 1079 |
| WAYNE      |  1239.79  |   5511.87 |   2917.12  |   0 |   157.278    |   0 |   0 |   28.3751  |  2409.1   |  280 |
| WEAKLEY    |  2765.89  |   9451.86 |   4862.96  |   0 |   626.016    |   0 |   0 |   69.3992  |  3893.49  |  561 |
| WHITE      |  1961.09  |   9941.91 |   5025.61  |   0 |   364.914    |   0 |   0 |   86.252   |  4465.14  |  714 |
| WILLIAMSON | 10201.7   |  27535.7  |  14354.7   |   0 |  2382.9      |   0 |   0 | 1177.1     |  9621.05  | 1289 |
| WILSON     |  8962.97  |  39339.6  |  17786.1   |   0 |  2536.68     |   0 |   0 | 1123.5     | 17893.3   | 1940 |



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_119142___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_206