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 |
|:-------------|-----:|------------:|-------------:|
| ADAIR        |   17 |    53.6055  |    -36.6055  |
| ALLEN        |    6 |   184.984   |   -178.984   |
| ANDERSON     |   22 |   149.153   |   -127.153   |
| BALLARD      |   40 |   223.107   |   -183.107   |
| BARREN       |    9 |   195.732   |   -186.732   |
| BATH         |    7 |    72.0164  |    -65.0164  |
| BELL         |   12 |   107.162   |    -95.1616  |
| BOONE        |   50 |   129.373   |    -79.3726  |
| BOURBON      |   12 |   318.234   |   -306.234   |
| BOYD         |   32 |    77.1233  |    -45.1233  |
| BOYLE        |   18 |   129.953   |   -111.953   |
| BRACKEN      |   10 |    31.5697  |    -21.5697  |
| BREATHITT    |    5 |   103.005   |    -98.0055  |
| BRECKINRIDGE |   15 |   213.693   |   -198.693   |
| BULLITT      |   12 |  1040.61    |  -1028.61    |
| BUTLER       |    2 |   176.816   |   -174.816   |
| CALDWELL     |    5 |   380.321   |   -375.321   |
| CALLOWAY     |    8 |   253.553   |   -245.553   |
| CAMPBELL     |   35 |   180.233   |   -145.233   |
| CARLISLE     |    1 |   245.458   |   -244.458   |
| CARROLL      |   11 |    50.4274  |    -39.4274  |
| CARTER       |    5 |    96.6137  |    -91.6137  |
| CASEY        |    4 |   178.38    |   -174.38    |
| CHRISTIAN    |   13 |   349.186   |   -336.186   |
| CLARK        |   33 |   308.256   |   -275.256   |
| CLAY         |   10 |   122.764   |   -112.764   |
| CLINTON      |    8 |     9.27123 |     -1.27123 |
| CRITTENDEN   |   41 |   134.222   |    -93.2219  |
| CUMBERLAND   |    1 |    26.3014  |    -25.3014  |
| DAVIESS      |   48 |   260.356   |   -212.356   |
| EDMONSON     |    5 |   205.611   |   -200.611   |
| ELLIOTT      |    3 |    47.5918  |    -44.5918  |
| ESTILL       |    6 |    98.9205  |    -92.9205  |
| FAYETTE      |  173 |   454.905   |   -281.905   |
| FLEMING      |   10 |    66.9451  |    -56.9451  |
| FLOYD        |   32 |   185.164   |   -153.164   |
| FRANKLIN     |   51 |    88.1562  |    -37.1562  |
| FULTON       |    9 |    31.3726  |    -22.3726  |
| GALLATIN     |   12 |    67.0712  |    -55.0712  |
| GARRARD      |    7 |   254.436   |   -247.436   |
| GRANT        |   10 |   161.934   |   -151.934   |
| GRAVES       |   19 |   386.14    |   -367.14    |
| GRAYSON      |   18 |   239.205   |   -221.205   |
| GREEN        |    3 |    74.8082  |    -71.8082  |
| GREENUP      |   11 |    67.1836  |    -56.1836  |
| HANCOCK      |    1 |   167.649   |   -166.649   |
| HARDIN       |   27 |  1064.14    |  -1037.14    |
| HARLAN       |   12 |   113.66    |   -101.66    |
| HARRISON     |   27 |    67.0329  |    -40.0329  |
| HART         |    6 |   313.605   |   -307.605   |
| HENDERSON    |   18 |   258.778   |   -240.778   |
| HENRY        |   18 |   113.934   |    -95.9342  |
| HICKMAN      |    9 |   136.047   |   -127.047   |
| HOPKINS      |   17 |   359.271   |   -342.271   |
| JACKSON      |    8 |    77.9096  |    -69.9096  |
| JEFFERSON    |   92 |  1082.22    |   -990.225   |
| JESSAMINE    |   33 |   400.094   |   -367.094   |
| JOHNSON      |   23 |   110.449   |    -87.4493  |
| KENTON       |   38 |   213.359   |   -175.359   |
| KNOTT        |    1 |   224.244   |   -223.244   |
| KNOX         |   15 |    99.1562  |    -84.1562  |
| LARUE        |    3 |   297.999   |   -294.999   |
| LAUREL       |   31 |   112.995   |    -81.9945  |
| LAWRENCE     |   10 |    98.4521  |    -88.4521  |
| LEE          |   12 |    78.0164  |    -66.0164  |
| LESLIE       |    3 |   102.348   |    -99.3479  |
| LETCHER      |    7 |   139.523   |   -132.523   |
| LEWIS        |    4 |    79.8382  |    -75.8382  |
| LINCOLN      |   22 |   127.31    |   -105.31    |
| LIVINGSTON   |    5 |   211.036   |   -206.036   |
| LOGAN        |    6 |   155.329   |   -149.329   |
| LYON         |   11 |   171.211   |   -160.211   |
| MADISON      |   34 |   448.45    |   -414.45    |
| MAGOFFIN     |    6 |   139.707   |   -133.707   |
| MARION       |   12 |   164.84    |   -152.84    |
| MARSHALL     |   13 |   389.027   |   -376.027   |
| MARTIN       |    8 |   177.488   |   -169.488   |
| MASON        |    5 |    34.8903  |    -29.8903  |
| MCCRACKEN    |   36 |   293.099   |   -257.099   |
| MCCREARY     |    8 |    78.263   |    -70.263   |
| MCLEAN       |    0 |   440.926   |   -440.926   |
| MEADE        |   10 |   172.058   |   -162.058   |
| MENIFEE      |    3 |    45.6438  |    -42.6438  |
| MERCER       |    9 |   237.704   |   -228.704   |
| METCALFE     |    5 |   131.43    |   -126.43    |
| MONROE       |    2 |   101.132   |    -99.1315  |
| MONTGOMERY   |   23 |   109.789   |    -86.789   |
| MORGAN       |    8 |    96.7534  |    -88.7534  |
| MUHLENBERG   |    7 |   370.586   |   -363.586   |
| NELSON       |   27 |   331.794   |   -304.794   |
| NICHOLAS     |    6 |    72.1534  |    -66.1534  |
| OHIO         |   10 |   226.649   |   -216.649   |
| OLDHAM       |    4 |   876.929   |   -872.929   |
| OWEN         |    4 |    73.0959  |    -69.0959  |
| OWSLEY       |    9 |    86       |    -77       |
| PENDLETON    |    5 |   199.707   |   -194.707   |
| PERRY        |   20 |   113.145   |    -93.1452  |
| PIKE         |   27 |   142.732   |   -115.732   |
| POWELL       |    6 |    85.0493  |    -79.0493  |
| PULASKI      |   48 |   104.981   |    -56.9808  |
| ROBERTSON    |    3 |    61.0958  |    -58.0958  |
| ROCKCASTLE   |   12 |   124.055   |   -112.055   |
| ROWAN        |   19 |    60.6219  |    -41.6219  |
| RUSSELL      |    9 |    75.2164  |    -66.2164  |
| SCOTT        |   13 |   265.472   |   -252.472   |
| SHELBY       |    8 |   917.685   |   -909.685   |
| SIMPSON      |    1 |   119.247   |   -118.247   |
| SPENCER      |    9 |   909.595   |   -900.595   |
| TAYLOR       |   24 |   103.958   |    -79.9582  |
| TODD         |    6 |   163.156   |   -157.156   |
| TRIGG        |    9 |   325.384   |   -316.384   |
| TRIMBLE      |    1 |    69.1973  |    -68.1973  |
| UNION        |    9 |   133.762   |   -124.762   |
| WARREN       |   27 |   223.362   |   -196.362   |
| WASHINGTON   |    5 |   146.964   |   -141.964   |
| WAYNE        |   15 |    66.726   |    -51.726   |
| WEBSTER      |    6 |   347.047   |   -341.047   |
| WHITLEY      |   18 |    98.7534  |    -80.7534  |
| WOLFE        |    9 |    90.4384  |    -81.4384  |
| WOODFORD     |    8 |   405.744   |   -397.744   |
| Statewide    | 1886 | 25058       | -23172       |



label: D=B-C
labeled_formulas: D=B-C
output: {"Arithmetic-Relationship": ["D=B-C"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_216711___xlsx_e79043a777d3a93607c3a554d9f7d3717e2ba898.xlsx___line_714