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 |
|:-------------|----------:|-----------:|-----------:|----:|-----------:|----:|----:|-----------:|----------:|------:|
| ALAMANCE     | 11226.8   |  44533.2   |  19637.1   |   0 |  2050.36   |   0 |   0 |  806.858   | 22038.9   |  2329 |
| ALEXANDER    |  2953.27  |  10884.6   |   5384.02  |   0 |   298.012  |   0 |   0 |  165.251   |  5037.28  |   798 |
| ALLEGHANY    |  1173.02  |   3506.16  |   2099.79  |   0 |    48.7671 |   0 |   0 |   40.1332  |  1317.47  |   267 |
| ANSON        |  2009.61  |   7464.02  |   4155.95  |   0 |   171.216  |   0 |   0 |   40.7479  |  3096.1   |   477 |
| ASHE         |  2548.7   |   8518.45  |   5454.59  |   0 |   174.1    |   0 |   0 |  132.691   |  2757.07  |   652 |
| AVERY        |  1396.63  |   5874.64  |   2630.08  |   0 |   231.436  |   0 |   0 |   28.9851  |  2984.14  |   481 |
| BEAUFORT     |  4537     |  20302     |  12625.6   |   0 |   806.417  |   0 |   0 |  350.57    |  6519.44  |  1013 |
| BERTIE       |  1162.14  |   5521.34  |   3321.6   |   0 |   155.024  |   0 |   0 |   24.9697  |  2019.75  |   246 |
| BLADEN       |  2901.53  |  15618     |   9590.32  |   0 |   560.417  |   0 |   0 |  257.277   |  5210.01  |   744 |
| BRUNSWICK    | 12981.6   |  42836.9   |  29163.8   |   0 |  2167.04   |   0 |   0 |  657.503   | 10848.5   |  2643 |
| BURKE        |  7461.16  |  34254.1   |  16857.5   |   0 |  1032.42   |   0 |   0 |  301.431   | 16062.7   |  2348 |
| CABARRUS     | 12386.9   |  51339.5   |  24012.3   |   0 |  4028.95   |   0 |   0 |  546.248   | 22752     |  3504 |
| CALDWELL     |  6940.88  |  29550     |  15012.3   |   0 |   989.471  |   0 |   0 |  209.415   | 13338.8   |  1949 |
| CAMDEN       |  1163.28  |   6381.82  |   4271.61  |   0 |   878.018  |   0 |   0 |   38.1622  |  1194.03  |   153 |
| CARTERET     |  9290.15  |  48302.4   |  32775.4   |   0 |  2592.09   |   0 |   0 |  809.032   | 12125.9   |  2223 |
| CASWELL      |  2282.76  |  11203.1   |   5215.32  |   0 |  1396.59   |   0 |   0 |  117.463   |  4473.68  |   510 |
| CATAWBA      | 11750.3   |  49756.6   |  23670.4   |   0 |  2454.09   |   0 |   0 |  594.276   | 23037.8   |  3573 |
| CHATHAM      |  5254.74  |  15841.6   |   7900.96  |   0 |  1409.53   |   0 |   0 |  456.666   |  6074.42  |   781 |
| CHEROKEE     |  2939.6   |  15205.7   |   8102.14  |   0 |   216.522  |   0 |   0 |  187.186   |  6699.89  |   845 |
| CHOWAN       |  1378.72  |   5876.57  |   3665.08  |   0 |   270.968  |   0 |   0 |   60.6992  |  1879.82  |   209 |
| CLAY         |  1142.86  |   5495.18  |   2928.19  |   0 |   107.002  |   0 |   0 |   43.3056  |  2416.69  |   352 |
| CLEVELAND    |  8758.63  |  35756.6   |  18569     |   0 |  1562.1    |   0 |   0 |  445.663   | 15179.8   |  2130 |
| COLUMBUS     |  4368.91  |  23414.4   |  14255.1   |   0 |   775.415  |   0 |   0 |  220.75    |  8163.13  |  1230 |
| CURRITUCK    |  3260.41  |  11047.1   |   7629.13  |   0 |  1493.97   |   0 |   0 |   96.9904  |  1827.01  |   317 |
| DARE         |  4168.24  |   7440.63  |   5224.37  |   0 |   255.588  |   0 |   0 |  224.48    |  1736.19  |   326 |
| DAVIDSON     | 13521.9   |  45769.9   |  21497.3   |   0 |  1966.56   |   0 |   0 |  693.186   | 21612.8   |  3407 |
| DAVIE        |  3251.49  |  12528.1   |   6218.55  |   0 |   481.988  |   0 |   0 |  398.903   |  5428.63  |   937 |
| DUPLIN       |  3767.9   |  20109.1   |  11742.6   |   0 |  1082.35   |   0 |   0 |  168.613   |  7115.48  |   988 |
| EDGECOMBE    |  3383.27  |  19657.5   |  12099.1   |   0 |   796.177  |   0 |   0 |  264.467   |  6497.66  |   916 |
| FRANKLIN     |  4342.07  |  19082.2   |   9397.75  |   0 |  1015.85   |   0 |   0 |  147.33    |  8521.24  |   879 |
| GASTON       | 17777.6   |  61187.9   |  30973     |   0 |  3276.62   |   0 |   0 |  886.526   | 26051.8   |  4226 |
| GATES        |   999.184 |   4620.68  |   2952.01  |   0 |   377.871  |   0 |   0 |   47.119   |  1243.69  |   181 |
| GRAHAM       |   761.631 |   3675.24  |   1806.22  |   0 |    68.8763 |   0 |   0 |   33.8859  |  1766.26  |   202 |
| GRANVILLE    |  4351.16  |  19621.9   |   9292.07  |   0 |  1315.97   |   0 |   0 |  116.498   |  8897.34  |   882 |
| GREENE       |  1678.34  |   7712.72  |   5002.14  |   0 |   476.155  |   0 |   0 |   51.8805  |  2182.54  |   306 |
| GUILFORD     | 33644.4   | 121854     |  62103.4   |   0 | 12019.6    |   0 |   0 | 2126.03    | 45604.6   |  6624 |
| HALIFAX      |  4117.93  |  21591.6   |  12657.2   |   0 |   609.051  |   0 |   0 |  120.19    |  8205.19  |   931 |
| HARNETT      | 11487.6   |  65533.6   |  39118.9   |   0 |  7135.48   |   0 |   0 |  711.262   | 18567.9   |  3005 |
| HAYWOOD      |  6862.36  |  30520.8   |  16982.2   |   0 |   985.88   |   0 |   0 |  292.547   | 12260.2   |  1631 |
| HENDERSON    | 11421.2   |  40573     |  20146.3   |   0 |  1084.41   |   0 |   0 | 1069.94    | 18272.3   |  2463 |
| HERTFORD     |  1653.74  |   9505.9   |   5337.57  |   0 |   390.435  |   0 |   0 |   17.4744  |  3760.43  |   347 |
| HOKE         |  4201.27  |  37832.9   |  23433.1   |   0 |  6283.75   |   0 |   0 |  346.867   |  7769.18  |  1584 |
| HYDE         |   353.463 |   1847.54  |   1307.03  |   0 |    67.8921 |   0 |   0 |   10.5893  |   462.028 |    55 |
| IREDELL      | 12562.4   |  44860.4   |  20794.1   |   0 |  2569.86   |   0 |   0 |  917.187   | 20579.3   |  3324 |
| JACKSON      |  3546.87  |  14494.5   |   8306.09  |   0 |   856.329  |   0 |   0 |  288.14    |  5043.97  |   772 |
| JOHNSTON     | 12886.2   |  51895.4   |  29182.5   |   0 |  3924.29   |   0 |   0 |  442.707   | 18345.9   |  2489 |
| JONES        |  1051.06  |   7453.36  |   4960.45  |   0 |   393.881  |   0 |   0 |   34.9296  |  2064.1   |   220 |
| LEE          |  5143.08  |  24858.1   |  15086.1   |   0 |  3093.85   |   0 |   0 |  281.084   |  6397.03  |  1065 |
| LENOIR       |  4973.64  |  29453.6   |  19070     |   0 |  1182.56   |   0 |   0 |  353.646   |  8847.43  |  1279 |
| LINCOLN      |  5664.49  |  21341.6   |   8966.75  |   0 |   806.914  |   0 |   0 |  120.23    | 11447.7   |  1632 |
| MC DOWELL    |  3857.79  |  26529.5   |  11879.7   |   0 |   486.148  |   0 |   0 |  145.579   | 14018.1   |  1567 |
| MACON        |  4178.43  |  18892.3   |   9633.66  |   0 |   349.128  |   0 |   0 |  134.572   |  8774.99  |  1242 |
| MADISON      |  1856.53  |   9134.86  |   4515.77  |   0 |   243.363  |   0 |   0 |   62.5452  |  4313.18  |   518 |
| MARTIN       |  1949.41  |  10307.1   |   6124.69  |   0 |   351.455  |   0 |   0 |   47.8253  |  3783.12  |   523 |
| MECKLENBURG  | 54792.3   | 195547     |  91990.7   |   0 | 23341.2    |   0 |   0 | 3712.12    | 76502.7   | 12003 |
| MITCHELL     |  1295.56  |   8423.91  |   4019.61  |   0 |   238.376  |   0 |   0 |   40.3912  |  4125.54  |   487 |
| MONTGOMERY   |  2216.84  |  11809.4   |   5424.46  |   0 |  1451.49   |   0 |   0 |  161.535   |  4771.92  |   644 |
| MOORE        | 10764.9   |  38790.5   |  25631.7   |   0 |  3327.47   |   0 |   0 | 1425.89    |  8405.49  |  1836 |
| NASH         |  8079.52  |  33954     |  19284.9   |   0 |  1454.69   |   0 |   0 |  355.047   | 12859.4   |  1512 |
| NORTHAMPTON  |  1599.14  |   8047.65  |   4504.7   |   0 |   164.088  |   0 |   0 |    9.82787 |  3369.04  |   380 |
| ONSLOW       | 17831.9   | 170218     | 128626     |   0 | 21942.6    |   0 |   0 | 1504.41    | 18144.6   |  4202 |
| ORANGE       |  7696.72  |  28052.8   |  12559.1   |   0 |  3116.27   |   0 |   0 |  925.227   | 11452.2   |  1187 |
| PAMLICO      |  1473.38  |   7737.28  |   4690.06  |   0 |   382.582  |   0 |   0 |  135.93    |  2528.71  |   319 |
| PASQUOTANK   |  4050.48  |  18906.2   |  12076.6   |   0 |  2725.17   |   0 |   0 |  389.174   |  3715.33  |   557 |
| PENDER       |  6163.35  |  22658.7   |  14787.2   |   0 |  2246.28   |   0 |   0 |  217.872   |  5407.28  |  1051 |
| PERQUIMANS   |  1589.59  |   5749.86  |   3630.35  |   0 |   425.068  |   0 |   0 |  106.798   |  1587.65  |   216 |
| PERSON       |  3407.16  |  16080.8   |   7224.06  |   0 |   530.796  |   0 |   0 |  161.572   |  8164.4   |   794 |
| PITT         | 10643.2   |  55073     |  34084.4   |   0 |  6314.1    |   0 |   0 |  638.243   | 14036.3   |  2128 |
| POLK         |  2727.61  |   7298.38  |   3967.29  |   0 |   187.735  |   0 |   0 |  169.541   |  2973.81  |   483 |
| RANDOLPH     | 11102     |  44373.1   |  22193.4   |   0 |  3672.5    |   0 |   0 |  453.081   | 18054.1   |  2529 |
| RICHMOND     |  3835.95  |  27299.2   |  16806.3   |   0 |   876.365  |   0 |   0 |  213.367   |  9403.18  |  1535 |
| ROBESON      |  8958.49  |  47129.7   |  27571.4   |   0 |  2761.1    |   0 |   0 |  453.96    | 16343.3   |  2599 |
| ROCKINGHAM   |  8813.93  |  27890.7   |  15278.2   |   0 |  1114.92   |   0 |   0 |  292.163   | 11205.5   |  1645 |
| RUTHERFORD   |  5686.23  |  33518.5   |  16365.6   |   0 |   773.974  |   0 |   0 |  190.489   | 16188.5   |  1945 |
| SAMPSON      |  4561.53  |  29296.9   |  17221     |   0 |   910.716  |   0 |   0 |  244.42    | 10920.8   |  1535 |
| SCOTLAND     |  3115.07  |  16169.6   |   9432.42  |   0 |  1841.93   |   0 |   0 |  191.313   |  4703.91  |   830 |
| STANLY       |  4967.34  |  18161.7   |   9024.84  |   0 |   708.044  |   0 |   0 |  219.054   |  8209.8   |  1235 |
| STOKES       |  4266.04  |  12496.7   |   7028.07  |   0 |   529.114  |   0 |   0 |   86.7372  |  4852.83  |   862 |
| SURRY        |  5517.13  |  26694     |  14686.8   |   0 |   739.517  |   0 |   0 |  393.228   | 10874.4   |  1911 |
| SWAIN        |  1359.4   |   7395.05  |   4551.52  |   0 |    70.6733 |   0 |   0 |   68.8664  |  2703.99  |   313 |
| TRANSYLVANIA |  3640.88  |  13099.6   |   7448.14  |   0 |   222.289  |   0 |   0 |  456.378   |  4972.79  |   793 |
| TYRRELL      |   347.184 |    820.636 |    535.451 |   0 |    26.3378 |   0 |   0 |   38.6873  |   220.159 |    38 |
| UNION        | 12034.2   |  40039.4   |  20336.2   |   0 |  3280.82   |   0 |   0 |  520.46    | 15901.9   |  2463 |
| VANCE        |  3000.7   |  17944.9   |   8406.03  |   0 |   630.702  |   0 |   0 |  116.027   |  8792.15  |   907 |
| WARREN       |  1695.18  |  10727.1   |   4793.29  |   0 |   189.796  |   0 |   0 |  461.594   |  5282.46  |   477 |
| WASHINGTON   |   971.605 |   4829.42  |   2827.59  |   0 |   216.058  |   0 |   0 |   52.2904  |  1733.48  |   192 |
| WATAUGA      |  3663.28  |  11803.6   |   6228.94  |   0 |  1579.68   |   0 |   0 |  187.55    |  3807.39  |   774 |
| WAYNE        | 13628.8   |  68696.8   |  46285.3   |   0 |  5955.6    |   0 |   0 |  873.231   | 15582.7   |  2567 |
| WILKES       |  5044.63  |  23388     |  13654.7   |   0 |   631.16   |   0 |   0 |  239.909   |  8862.23  |  1546 |
| WILSON       |  6078.63  |  30298.3   |  18695.9   |   0 |  1287.05   |   0 |   0 |  404.188   |  9911.2   |  1232 |
| YADKIN       |  3088.22  |  10952.3   |   5366.88  |   0 |   352.102  |   0 |   0 |  126.888   |  5106.4   |   866 |
| YANCEY       |  2004.78  |   9940.63  |   5615.04  |   0 |   114.394  |   0 |   0 |   34.9442  |  4176.25  |   551 |



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_119127___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_191