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 |
|:---------------|----------:|-----------:|----------:|----:|-----------:|----:|----:|-----------:|----------:|-----:|
| ALCONA         |  1674.78  |   6937.72  |  4144.6   |   0 |    96.9487 |   0 |   0 |   50.6544  |  2645.52  |  510 |
| ALGER          |  1017.39  |   5148.14  |  3046.74  |   0 |   161.131  |   0 |   0 |   85.9934  |  1854.28  |  351 |
| ALLEGAN        |  8520.45  |  20814.2   | 10319.6   |   0 |  1161.07   |   0 |   0 |  453.69    |  8879.83  | 1400 |
| ALPENA         |  3050.07  |  11125.9   |  5460.46  |   0 |   392.337  |   0 |   0 |   79.9122  |  5193.18  |  953 |
| ANTRIM         |  2672.7   |   7109.46  |  4338.6   |   0 |   201.015  |   0 |   0 |  146.031   |  2423.82  |  487 |
| ARENAC         |  1659.29  |   6264.09  |  3193     |   0 |   160.104  |   0 |   0 |   79.2621  |  2831.72  |  376 |
| BARAGA         |   968.459 |   4372     |  1890.5   |   0 |    99.7448 |   0 |   0 |   20.3535  |  2361.4   |  302 |
| BARRY          |  5059.95  |  16320.1   |  7447.85  |   0 |   755.809  |   0 |   0 |  197.825   |  7918.63  |  959 |
| BAY            |  9240.92  |  29960.3   | 14996.9   |   0 |  1398.72   |   0 |   0 |  483.358   | 13081.3   | 1955 |
| BENZIE         |  1844.95  |   5409.93  |  3297.84  |   0 |   178.654  |   0 |   0 |  160.961   |  1772.47  |  380 |
| BERRIEN        | 12027.9   |  36932.3   | 17470.9   |   0 |  1982.62   |   0 |   0 |  642.817   | 16835.9   | 2929 |
| BRANCH         |  3734.51  |  12824.3   |  6378.67  |   0 |   409.243  |   0 |   0 |   51.8277  |  5984.59  |  776 |
| CALHOUN        | 11430.2   |  79220.9   | 29248.6   |   0 |  1911.8    |   0 |   0 |  791.787   | 47268.7   | 3221 |
| CASS           |  4611.02  |  11338     |  5651.73  |   0 |   464.383  |   0 |   0 |  194.573   |  5027.3   | 1143 |
| CHARLEVOIX     |  2830.49  |   5574.25  |  2908.21  |   0 |   165.961  |   0 |   0 |  254.074   |  2246     |  542 |
| CHEBOYGAN      |  2558.98  |  10171.5   |  6592.89  |   0 |   306.649  |   0 |   0 |   63.397   |  3208.61  |  715 |
| CHIPPEWA       |  3820.65  |  14796.3   |  8636.95  |   0 |   927.947  |   0 |   0 |  403.354   |  4828.04  |  928 |
| CLINTON        |  4980.52  |  10494.5   |  6173     |   0 |  1273.27   |   0 |   0 |  310.942   |  2737.29  |  551 |
| CRAWFORD       |  1790     |   5247.51  |  2741.39  |   0 |   175.355  |   0 |   0 |   55.1201  |  2275.64  |  391 |
| DELTA          |  4074.34  |  23471.1   | 10528.1   |   0 |   724.846  |   0 |   0 |  221.295   | 11996.8   | 1394 |
| EATON          |  8498.64  |  22316.6   | 12584.4   |   0 |  1998.8    |   0 |   0 |  444.298   |  7289.1   | 1155 |
| EMMET          |  3080.99  |   7552.21  |  4329.69  |   0 |   252.27   |   0 |   0 |  170.995   |  2799.26  |  564 |
| GENESEE        | 31231.4   |  86183.3   | 47648.7   |   0 |  5933.31   |   0 |   0 | 1618.19    | 30983.1   | 4064 |
| GLADWIN        |  2835.79  |  10677     |  5548.01  |   0 |   272.794  |   0 |   0 |   71.2949  |  4784.94  |  669 |
| GOGEBIC        |  2132.29  |   9995.65  |  4388.59  |   0 |   270.917  |   0 |   0 |  205.291   |  5130.85  |  791 |
| GRAND TRAVERSE |  7495.42  |  21694.4   | 12942.2   |   0 |  1589.53   |   0 |   0 |  638.297   |  6524.38  | 1452 |
| GRATIOT        |  3000.37  |  10976.8   |  5831.62  |   0 |   429.128  |   0 |   0 |   89.0372  |  4626.97  |  627 |
| HILLSDALE      |  3620.75  |  12784.6   |  6296.89  |   0 |   482.815  |   0 |   0 |  168.073   |  5836.83  |  807 |
| HOUGHTON       |  2934.56  |  13387.8   |  5948.87  |   0 |  1145.46   |   0 |   0 |  239.849   |  6053.61  |  891 |
| HURON          |  2886.27  |   9820.21  |  4803.66  |   0 |   255.487  |   0 |   0 |  253.267   |  4507.8   |  684 |
| INGHAM         | 15804.8   |  47136.1   | 24648.4   |   0 |  5909.3    |   0 |   0 | 1001.45    | 15576.9   | 2556 |
| IONIA          |  4388.07  |  12221.7   |  6214.83  |   0 |  1065.81   |   0 |   0 |  212.31    |  4728.75  |  674 |
| IOSCO          |  3862.58  |  13513.7   |  7447.94  |   0 |   252.627  |   0 |   0 |  187.021   |  5626.08  | 1015 |
| IRON           |  1368.97  |  11281.8   |  4210.3   |   0 |    93.1382 |   0 |   0 |   74.0375  |  6904.32  |  669 |
| ISABELLA       |  3932.33  |  12089.9   |  6164.64  |   0 |  1549.63   |   0 |   0 |  179.78    |  4195.89  |  718 |
| JACKSON        | 12409.9   |  41814.1   | 22237     |   0 |  2136.71   |   0 |   0 |  799.344   | 16641.1   | 2650 |
| KALKASKA       |  1830.77  |   6253.61  |  3467.56  |   0 |   163.95   |   0 |   0 |   64.2003  |  2557.9   |  413 |
| KENT           | 35893.9   | 110885     | 57403.9   |   0 | 10601.9    |   0 |   0 | 2763.84    | 40114.9   | 7570 |
| KEWEENAW       |   303.208 |    919.242 |   452.811 |   0 |    10.2714 |   0 |   0 |   19.5616  |   436.598 |  115 |
| LAKE           |  1314.73  |   5578.28  |  3199.98  |   0 |    94.4968 |   0 |   0 |   20.814   |  2262.99  |  339 |
| LAPEER         |  7321.27  |  16147.1   |  9868.34  |   0 |  1069.01   |   0 |   0 |  282.325   |  4927.42  | 1046 |
| LEELANAU       |  2549.92  |   5255.68  |  3405.91  |   0 |   221.745  |   0 |   0 |  140.235   |  1487.8   |  447 |
| LENAWEE        |  8515.94  |  28750.9   | 14425.4   |   0 |  1507.1    |   0 |   0 |  295.75    | 12522.7   | 1666 |
| LIVINGSTON     | 12693.7   |  32178.9   | 15791.9   |   0 |  2393.41   |   0 |   0 | 1208.04    | 12785.5   | 1642 |
| LUCE           |   752.565 |   2189.66  |  1457.39  |   0 |    45.4589 |   0 |   0 |   16.2612  |   670.551 |  103 |
| MACKINAC       |  1099.35  |   4734.74  |  2817.73  |   0 |   215.425  |   0 |   0 |   38.3898  |  1663.19  |  291 |
| MACOMB         | 59070.9   | 157229     | 88749.1   |   0 | 12217.1    |   0 |   0 | 3771.98    | 52490.5   | 7641 |
| MANISTEE       |  2533.93  |   8570.88  |  4890.53  |   0 |   239.943  |   0 |   0 |  117.563   |  3322.85  |  612 |
| MARQUETTE      |  6797.45  |  33969.7   | 19988.9   |   0 |  2334.74   |   0 |   0 |  308.43    | 11337.6   | 1832 |
| MASON          |  3043.46  |   8290.98  |  4804.17  |   0 |   464.839  |   0 |   0 |  104.126   |  2917.84  |  465 |
| MECOSTA        |  3833.96  |   9201.3   |  5348.56  |   0 |  1045.26   |   0 |   0 |   57.8729  |  2749.61  |  578 |
| MENOMINEE      |  2729.18  |  14005.3   |  6841.13  |   0 |   380.905  |   0 |   0 |   98.2677  |  6685.02  | 1025 |
| MIDLAND        |  6203.61  |  20642.2   |  9743.11  |   0 |  1103.68   |   0 |   0 |  393.282   |  9402.13  | 1232 |
| MISSAUKEE      |  1359.05  |   5449.76  |  2955.89  |   0 |   199.347  |   0 |   0 |   56.048   |  2238.48  |  297 |
| MONROE         | 12160.3   |  32509.7   | 15572.5   |   0 |  1977.91   |   0 |   0 |  599.571   | 14359.7   | 1686 |
| MONTCALM       |  5079.71  |  13478.3   |  7759.48  |   0 |   669.207  |   0 |   0 |  201.283   |  4848.36  |  859 |
| MONTMORENCY    |  1081.14  |   4625.49  |  2696.46  |   0 |    89.1128 |   0 |   0 |   38.4733  |  1801.45  |  315 |
| MUSKEGON       | 14623.2   |  39518.6   | 23053.9   |   0 |  2557.36   |   0 |   0 |  720.664   | 13186.6   | 2846 |
| NEWAYGO        |  4333.37  |  13313.8   |  7658.43  |   0 |   590.708  |   0 |   0 |   94.5242  |  4970.13  |  948 |
| OCEANA         |  2517.8   |   5801.64  |  3814.64  |   0 |   263.463  |   0 |   0 |   86.5903  |  1636.95  |  395 |
| OGEMAW         |  2532.61  |   9966.44  |  4854.67  |   0 |   215.453  |   0 |   0 |  122.694   |  4773.62  |  672 |
| ONTONAGON      |   993.108 |   5975.3   |  2765.89  |   0 |   104.723  |   0 |   0 |    7.06749 |  3097.62  |  411 |
| OSCEOLA        |  2166.79  |   7185.45  |  3864.68  |   0 |   274.44   |   0 |   0 |   74.5918  |  2971.73  |  469 |
| OSCODA         |  1133.67  |   4712.22  |  2555.1   |   0 |   100.478  |   0 |   0 |   41.822   |  2014.82  |  291 |
| OTSEGO         |  2321.32  |   6814.17  |  3959.85  |   0 |   156.254  |   0 |   0 |  139.762   |  2558.31  |  484 |
| OTTAWA         | 16295.9   |  32837.4   | 17290.7   |   0 |  3894.93   |   0 |   0 |  944.292   | 10707.5   | 2566 |
| PRESQUE ISLE   |  1943.92  |   5086.09  |  2929.01  |   0 |   196.38   |   0 |   0 |   58.5895  |  1902.1   |  449 |
| ROSCOMMON      |  2854     |  11191.4   |  5647.14  |   0 |   333.126  |   0 |   0 |  239.654   |  4971.44  |  701 |
| ST. CLAIR      | 14128.6   |  37333.8   | 21485.9   |   0 |  2188.02   |   0 |   0 |  808.199   | 12851.7   | 2755 |
| ST. JOSEPH     |  4508.4   |  11986.7   |  5625.62  |   0 |   731.758  |   0 |   0 |  301.533   |  5327.82  |  898 |
| SANILAC        |  3501.14  |  12281.5   |  6329.71  |   0 |   534.22   |   0 |   0 |  164.644   |  5252.97  |  976 |
| SCHOOLCRAFT    |  1033.27  |   4016.41  |  2161.37  |   0 |    48.3526 |   0 |   0 |   14.5404  |  1792.15  |  304 |
| SHIAWASSEE     |  5705.55  |  16805.6   |  8855.36  |   0 |  1269.63   |   0 |   0 |  269.891   |  6410.74  |  910 |
| TUSCOLA        |  4703.1   |  15783.8   |  7570.06  |   0 |   736.831  |   0 |   0 |  177.637   |  7299.31  |  963 |
| VAN BUREN      |  6218.87  |  18323.1   |  9210.21  |   0 |  1086.38   |   0 |   0 |  326.862   |  7699.69  | 1066 |
| WEXFORD        |  2766.27  |  10437.5   |  5696.15  |   0 |   641.379  |   0 |   0 |   50.873   |  4049.06  |  657 |



labeled_formulas: G=B*E
H=B*E
H=B*G
G=C*E
H=C*E
H=C*G
G=D*E
H=D*E
H=D*G
G=E*F
H=E*F
G=E*H
G=E*I
G=E*J
G=E*K
H=E*I
H=E*J
H=E*K
H=F*G
H=G*I
H=G*J
H=G*K
G=(B+C)*E
H=(B+C)*E
H=(B+C)*G
G=(B+D)*E
H=(B+D)*E
H=(B+D)*G
G=(B*E)*F
H=(B*E)*F
G=(B*E)*H
G=(B*E)*I
G=(B*E)*J
G=(B*E)*K
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
G=(C+D)*E
H=(C+D)*E
H=(C+D)*G
G=(C*E)*F
H=(C*E)*F
G=(C*E)*H
G=(C*E)*I
G=(C*E)*J
G=(C*E)*K
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
G=(D*E)*F
H=(D*E)*F
G=(D*E)*H
G=(D*E)*I
G=(D*E)*J
G=(D*E)*K
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
G=(E*F)*H
G=(E*F)*I
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119122___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_186