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 |
|:------------|----------:|----------:|---------:|----:|----------:|----:|----:|-----------:|---------:|-----:|
| AUTAUGA     |  5879.97  |  30140.8  | 18890.8  |   0 |  3554.79  |   0 |   0 |  345.08    |  7350.14 | 1196 |
| BALDWIN     | 20596.3   |  63240.4  | 38748.1  |   0 |  3736.14  |   0 |   0 | 1523.35    | 19232.8  | 3854 |
| BARBOUR     |  2831.7   |  13492.2  |  7912.56 |   0 |   384.701 |   0 |   0 |  171.197   |  5023.74 |  607 |
| BIBB        |  1422.56  |   8679.72 |  4635.62 |   0 |   335.768 |   0 |   0 |   36.6241  |  3671.71 |  506 |
| BLOUNT      |  4371.17  |  18181.4  | 10239.8  |   0 |   766.026 |   0 |   0 |  153.646   |  7021.99 |  966 |
| BULLOCK     |  1047.84  |   6387.24 |  2801.59 |   0 |   149.472 |   0 |   0 |   17.966   |  3418.21 |  278 |
| BUTLER      |  2002.19  |   9929.71 |  5311.86 |   0 |   508.097 |   0 |   0 |   53.509   |  4056.25 |  442 |
| CALHOUN     | 12600.4   |  65484.9  | 43411.5  |   0 |  2845.32  |   0 |   0 |  928.558   | 18299.6  | 3310 |
| CHAMBERS    |  3217.45  |  17582.6  |  9575.69 |   0 |   488.087 |   0 |   0 |   89.6448  |  7429.2  |  842 |
| CHEROKEE    |  2224.87  |  10096.5  |  6216.93 |   0 |   174.117 |   0 |   0 |  176.318   |  3529.12 |  475 |
| CHILTON     |  3282.79  |  19154.6  | 11171.1  |   0 |   784.654 |   0 |   0 |  210.788   |  6988.15 |  815 |
| CHOCTAW     |   976.627 |   6794.5  |  4236.12 |   0 |   188.671 |   0 |   0 |   59.6647  |  2310.04 |  347 |
| CLARKE      |  1889.72  |  10220.1  |  5474.71 |   0 |   368.009 |   0 |   0 |  187.331   |  4190.02 |  497 |
| CLAY        |  1210.54  |   6973.84 |  4100.15 |   0 |   154.997 |   0 |   0 |  145.022   |  2573.67 |  366 |
| CLEBURNE    |  1190.26  |   4709.03 |  2642.22 |   0 |   164.101 |   0 |   0 |  254.768   |  1647.94 |  265 |
| COFFEE      |  6218.68  |  39646.2  | 27815.8  |   0 |  3624.03  |   0 |   0 |  590.199   |  7616.22 | 1573 |
| COLBERT     |  4590.2   |  24146.9  | 16047.2  |   0 |   932.589 |   0 |   0 |  126.981   |  7040.17 | 1283 |
| CONECUH     |  1271.18  |   6650.43 |  4252.22 |   0 |   198.096 |   0 |   0 |   61.0163  |  2139.11 |  263 |
| COOSA       |  1057.94  |   4667.38 |  2356.63 |   0 |   114.395 |   0 |   0 |    7.32264 |  2189.04 |  204 |
| COVINGTON   |  3491.83  |  18768.5  | 12429.4  |   0 |   671.071 |   0 |   0 |  252.407   |  5415.55 |  805 |
| CRENSHAW    |  1136.96  |   5935.41 |  3583.22 |   0 |   254.711 |   0 |   0 |   40.1932  |  2057.29 |  282 |
| CULLMAN     |  7740.7   |  29028.4  | 18297.7  |   0 |  1509.88  |   0 |   0 |  267.492   |  8953.35 | 1370 |
| DALE        |  6871.02  |  42519.8  | 30398.3  |   0 |  3431.1   |   0 |   0 |  455.201   |  8235.2  | 1817 |
| DALLAS      |  2972.92  |  18078.6  |  9413.36 |   0 |   746.208 |   0 |   0 |  127.162   |  7791.82 |  930 |
| DE KALB     |  4567.37  |  19274.9  | 12054.8  |   0 |   672.735 |   0 |   0 |   94.6728  |  6452.7  | 1004 |
| ELMORE      |  8663.5   |  48585    | 29802.3  |   0 |  4009.89  |   0 |   0 |  560.597   | 14212.2  | 1905 |
| ESCAMBIA    |  3131.24  |  16106.8  | 10052.8  |   0 |   895.299 |   0 |   0 |  148.178   |  5010.52 |  853 |
| ETOWAH      |  9292.31  |  48810.3  | 29754.8  |   0 |  1273.8   |   0 |   0 |  373.895   | 17407.9  | 2464 |
| FAYETTE     |  1388.55  |  11955.4  |  5543.06 |   0 |   293.864 |   0 |   0 |   65.3836  |  6053.09 |  588 |
| FRANKLIN    |  1804.16  |   7935.07 |  5337.17 |   0 |   221.134 |   0 |   0 |   52.6217  |  2324.14 |  419 |
| GENEVA      |  2558.58  |  17104.7  | 12114.8  |   0 |   597.546 |   0 |   0 |  276.112   |  4116.24 |  898 |
| GREENE      |   578.055 |   4634.06 |  2540.27 |   0 |   130.095 |   0 |   0 |    6.86727 |  1956.83 |  230 |
| HALE        |  1016.41  |  10386.2  |  4400.33 |   0 |   217.903 |   0 |   0 |  166.869   |  5601.09 |  481 |
| HENRY       |  1807.88  |   6864.92 |  5100.86 |   0 |   358.018 |   0 |   0 |   70.8053  |  1335.24 |  445 |
| HOUSTON     |  8960.86  |  45357.9  | 30719.4  |   0 |  2798.31  |   0 |   0 |  684.277   | 11155.8  | 2849 |
| JACKSON     |  4022.05  |  14440    |  8037.06 |   0 |   467.513 |   0 |   0 |  139.011   |  5796.37 |  769 |
| LAMAR       |  1333.06  |   6439.31 |  3763.15 |   0 |   113.449 |   0 |   0 |   64.2226  |  2498.5  |  459 |
| LAUDERDALE  |  7171.43  |  29306.1  | 19426.9  |   0 |  1436.86  |   0 |   0 |  381.318   |  8061.08 | 1706 |
| LAWRENCE    |  2219.2   |   9661.96 |  5760.66 |   0 |   321.534 |   0 |   0 |  174.324   |  3405.44 |  467 |
| LEE         | 10336.4   |  59765.5  | 34707.5  |   0 |  7229.56  |   0 |   0 |  776.894   | 17051.5  | 2372 |
| LIMESTONE   |  6418.12  |  28479.8  | 17643.3  |   0 |  2752.81  |   0 |   0 |  420.341   |  7663.26 | 1233 |
| LOWNDES     |   702.539 |   4536.04 |  2350.06 |   0 |   225.19  |   0 |   0 |   15.598   |  1945.19 |  209 |
| MADISON     | 31731.7   | 135366    | 87509.5  |   0 | 16153.6   |   0 |   0 | 2685.81    | 29016.8  | 6235 |
| MARENGO     |  1579.98  |  14938.2  |  7251.88 |   0 |   402.18  |   0 |   0 |  239.338   |  7044.78 |  678 |
| MARION      |  2355.39  |   9954.94 |  5557.4  |   0 |   336.167 |   0 |   0 |   39.834   |  4021.54 |  622 |
| MARSHALL    |  6358.9   |  29280.5  | 17498.4  |   0 |  1131.87  |   0 |   0 |  793.149   |  9857.01 | 1468 |
| MONROE      |  1912.62  |   7283.45 |  4793.31 |   0 |   383.544 |   0 |   0 |   38.3476  |  2068.26 |  377 |
| MORGAN      |  9918.12  |  36706.1  | 23672.8  |   0 |  2343.08  |   0 |   0 |  534.383   | 10155.8  | 1752 |
| PERRY       |   645.018 |   4756.27 |  3208.89 |   0 |   138.609 |   0 |   0 |   16.208   |  1392.56 |  166 |
| PICKENS     |  1749.78  |  13449.4  |  7174.79 |   0 |   405.836 |   0 |   0 |   25.5054  |  5843.28 |  709 |
| PIKE        |  2247.8   |  12604.9  |  6839.17 |   0 |  1208.72  |   0 |   0 |  143.878   |  4413.16 |  588 |
| RANDOLPH    |  1818.05  |  10596.3  |  5995.8  |   0 |   263.329 |   0 |   0 |  114.417   |  4222.75 |  459 |
| SAINT CLAIR |  7321.38  |  28238.7  | 14776.4  |   0 |  1357.35  |   0 |   0 |  290.78    | 11814.2  | 1447 |
| SHELBY      | 15178.1   |  50171    | 27642.1  |   0 |  4148.57  |   0 |   0 | 1147.56    | 17232.8  | 2476 |
| SUMTER      |   803.839 |   6048.9  |  3354.79 |   0 |   292.312 |   0 |   0 |    9.56751 |  2392.24 |  266 |
| TALLADEGA   |  6700.34  |  29659.5  | 16031.4  |   0 |  1187.41  |   0 |   0 |  184.945   | 12255.8  | 1690 |
| TALLAPOOSA  |  4087.64  |  23471.5  | 14354.6  |   0 |   496.726 |   0 |   0 |  214.889   |  8405.29 | 1111 |
| WALKER      |  6686.87  |  36577.8  | 21996.8  |   0 |   818.881 |   0 |   0 |  203.184   | 13558.9  | 1690 |
| WASHINGTON  |  1166.43  |   5748.05 |  3769.64 |   0 |   239.23  |   0 |   0 |   40.0249  |  1699.16 |  314 |
| WILCOX      |   601.123 |   3921.65 |  2373.72 |   0 |   141.259 |   0 |   0 |   16.8933  |  1389.78 |  184 |
| WINSTON     |  2079.07  |  10227.4  |  5754.47 |   0 |   219.349 |   0 |   0 |   33.7475  |  4219.82 |  509 |



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_119101___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_165