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 |
|-------:|:--------------------------|---------:|----------:|--------:|
| 100240 | Autauga County            |  8773.5  |  48287155 | 5503.75 |
| 100270 | Baldwin County            | 29555    | 144906987 | 4902.97 |
| 100300 | Barbour County            |   646.7  |   2990560 | 4624.34 |
| 100360 | Bibb County               |  3052.15 |  19371462 | 6346.83 |
| 100420 | Blount County             |  7612.05 |  44501693 | 5846.22 |
| 100480 | Bullock County            |  1386.5  |   8358327 | 6028.36 |
| 100510 | Butler County             |  2825.3  |  15824613 | 5601.04 |
| 100540 | Calhoun County            |  7817.05 |  47869517 | 6123.73 |
| 100600 | Chambers County           |  3259.7  |  18213201 | 5587.39 |
| 100630 | Cherokee County           |  3686.15 |  21398029 | 5804.98 |
| 100660 | Chilton County            |  7396.95 |  42280015 | 5715.87 |
| 100690 | Choctaw County            |  1070.75 |   5197308 | 4853.89 |
| 100720 | Clarke County             |  2230.8  |  12648077 | 5669.75 |
| 100750 | Clay County               |  1794.95 |  11049112 | 6155.67 |
| 100780 | Cleburne County           |  2436.1  |  15171092 | 6227.61 |
| 100810 | Coffee County             |  2569.95 |  14301358 | 5564.84 |
| 100840 | Colbert County            |  2498.65 |  13027470 | 5213.8  |
| 100870 | Conecuh County            |  1579.85 |  10874621 | 6883.32 |
| 100900 | Coosa County              |   780.25 |   3455984 | 4429.33 |
| 100930 | Covington County          |  2755.1  |  16313072 | 5921.05 |
| 100960 | Crenshaw County           |  2140.15 |  12271200 | 5733.8  |
| 101020 | Cullman County            |  9185.05 |  53858883 | 5863.76 |
| 101050 | Dale County               |  3381.4  |  20284382 | 5998.81 |
| 101110 | Dallas County             |  2738.3  |  17498036 | 6390.11 |
| 101140 | DeKalb County             |  8332.75 |  51464102 | 6176.12 |
| 101290 | Elmore County             | 11495.6  |  61453074 | 5345.79 |
| 101350 | Escambia County           |  4074.35 |  22246006 | 5460.01 |
| 101380 | Etowah County             |  8269.55 |  49850028 | 6028.14 |
| 101470 | Fayette County            |  2126    |  12654895 | 5952.44 |
| 101590 | Franklin County           |  3490.65 |  21554420 | 6174.9  |
| 101660 | Geneva County             |  2538.85 |  15723305 | 6193.08 |
| 101680 | Greene County             |   913.05 |   5006997 | 5483.81 |
| 101710 | Hale County               |  2270.95 |  13669194 | 6019.15 |
| 101740 | Henry County              |  2399.95 |  14232613 | 5930.38 |
| 101770 | Houston County            |  6177.6  |  33410070 | 5408.26 |
| 101830 | Jackson County            |  4971.1  |  28374604 | 5707.91 |
| 101920 | Jefferson County          | 34576.1  | 193459257 | 5595.17 |
| 101950 | Lamar County              |  2190.55 |  13160486 | 6007.85 |
| 102010 | Lauderdale County         |  7678.15 |  43959686 | 5725.3  |
| 102040 | Lawrence County           |  4599.7  |  26815098 | 5829.75 |
| 102070 | Lee County                |  9065.7  |  50723328 | 5595.08 |
| 102100 | Limestone County          | 12638.2  |  63916459 | 5057.38 |
| 102160 | Lowndes County            |  1306.05 |   7625627 | 5838.69 |
| 102190 | Macon County              |  1897.15 |  11519743 | 6072.13 |
| 102220 | Madison County            | 18753.7  | 105780375 | 5640.51 |
| 102250 | Marengo County            |   912.15 |   5135982 | 5630.63 |
| 102310 | Marion County             |  3190.85 |  18585384 | 5824.59 |
| 100006 | Marshall County           |  5528.45 |  32356456 | 5852.72 |
| 102370 | Mobile County             | 51299.8  | 273470885 | 5330.84 |
| 102400 | Monroe County             |  3056.6  |  17585517 | 5753.29 |
| 102430 | Montgomery County         | 27503.9  | 147258770 | 5354.11 |
| 102480 | Morgan County             |  7271.7  |  38538076 | 5299.73 |
| 102670 | Perry County              |  1123.15 |   6455816 | 5747.96 |
| 102730 | Pickens County            |  2311.45 |  14345659 | 6206.35 |
| 102790 | Pike County               |  2102.3  |  12345885 | 5872.56 |
| 102820 | Randolph County           |  1987    |  11668672 | 5872.51 |
| 102880 | Russell County            |  3396.95 |  18768350 | 5525.06 |
| 103062 | Saint Clair County        |  9151.9  |  52636129 | 5751.39 |
| 103030 | Shelby County             | 20438.5  | 111255519 | 5443.42 |
| 103090 | Sumter County             |  1138.25 |   6372522 | 5598.53 |
| 103180 | Talladega County          |  6827.3  |  37108153 | 5435.26 |
| 103210 | Tallapoosa County         |  2684.55 |  12798311 | 4767.4  |
| 103390 | Tuscaloosa County         | 18165.5  | 100306154 | 5521.81 |
| 103450 | Walker County             |  6938.7  |  40211377 | 5795.23 |
| 103480 | Washington County         |  2520.75 |  12946856 | 5136.11 |
| 103510 | Wilcox County             |  1324.1  |   7795244 | 5887.2  |
| 103580 | Winston County            |  2186.95 |  12819596 | 5861.86 |
| 100005 | Albertville City          |  5722.85 |  32123354 | 5613.17 |
| 100030 | Alexander City            |  2853.45 |  15921063 | 5579.58 |
| 100190 | Alabaster City            |  6076.5  |  34303316 | 5645.24 |
| 100060 | Andalusia City            |  1724.2  |   9813128 | 5691.41 |
| 100090 | Anniston City             |  1738.75 |   8726387 | 5018.77 |
| 100100 | Arab City                 |  2509.7  |  15181987 | 6049.32 |
| 100120 | Athens City               |  4533.25 |  25844311 | 5701.06 |
| 100180 | Attalla City              |  1440.35 |   9011232 | 6256.28 |
| 100210 | Auburn City               |  8953.55 |  44421650 | 4961.34 |
| 100330 | Bessemer City             |  3229.1  |  18483155 | 5723.93 |
| 100390 | Birmingham City           | 20818.8  | 110772236 | 5320.79 |
| 100012 | Boaz City                 |  2337.45 |  13842471 | 5922.04 |
| 100450 | Brewton City              |  1145.4  |   6800326 | 5937.08 |
| 100188 | Chickasaw City            |  2691.3  |   7479273 | 2779.06 |
| 100990 | Cullman City              |  3163.4  |  16302784 | 5153.56 |
| 101080 | Daleville City            |  1084.85 |   6150070 | 5669.05 |
| 101170 | Decatur City              |  8520.35 |  46048271 | 5404.5  |
| 101200 | Demopolis City            |  2052.95 |  12546738 | 6111.57 |
| 101230 | Dothan City               |  8049.6  |  43562792 | 5411.8  |
| 101260 | Elba City                 |   612.3  |   3728277 | 6088.97 |
| 101320 | Enterprise City           |  6531.35 |  38000866 | 5818.23 |
| 101410 | Eufaula City              |  6375.65 |  31325122 | 4913.24 |
| 101440 | Fairfield City            |  1534.15 |   9784294 | 6377.66 |
| 101530 | Florence City             |  4476.75 |  24302245 | 5428.55 |
| 101560 | Fort Payne City           |  3353.5  |  18224242 | 5434.39 |
| 101620 | Gadsden City              |  4824.95 |  28270859 | 5859.31 |
| 101640 | Geneva City               |  1231.9  |   7714999 | 6262.68 |
| 100202 | Gulf Shores City          |  2156.3  |   6639694 | 3079.21 |
| 101690 | Guntersville City         |  1767.8  |   9961801 | 5635.14 |
| 101720 | Haleyville City           |  1573.9  |   9463688 | 6012.89 |
| 101730 | Hartselle City            |  3412.2  |  20474243 | 6000.31 |
| 101760 | Homewood City             |  4177.15 |  19526132 | 4674.51 |
| 100007 | Hoover City               | 13430.4  |  66955694 | 4985.4  |
| 101800 | Huntsville City           | 22869.2  | 115310106 | 5042.15 |
| 101860 | Jacksonville City         |  1620.95 |   9163671 | 5653.27 |
| 101890 | Jasper City               |  2557.6  |  14784959 | 5780.79 |
| 101980 | Lanett City               |   943.15 |   5580250 | 5916.61 |
| 100011 | Leeds City                |  1951.65 |  10936328 | 5603.63 |
| 102130 | Linden City               |   478.4  |   3140419 | 6564.42 |
| 100008 | Madison City              | 11513    |  61693198 | 5358.59 |
| 102350 | Midfield City             |  1013.4  |   6109338 | 6028.56 |
| 102490 | Mountain Brook City       |  4375.6  |  20897759 | 4775.98 |
| 102520 | Muscle Shoals City        |  2771.5  |  16197677 | 5844.37 |
| 100194 | Pelham City               |  3401.55 |  16562114 | 4868.99 |
| 102550 | Oneonta City              |  1429.15 |   8663571 | 6062.04 |
| 102580 | Opelika City              |  4688.3  |  23757924 | 5067.49 |
| 102610 | Opp City                  |  1227.95 |   7612844 | 6199.64 |
| 102635 | Oxford City               |  4050.1  |  21961713 | 5422.51 |
| 102640 | Ozark City                |  1947    |  11803754 | 6062.53 |
| 102650 | Pell City                 |  3942.55 |  23297277 | 5909.19 |
| 102700 | Phenix City               |  7073.6  |  42486633 | 6006.37 |
| 102760 | Piedmont City             |  1096.4  |   6577761 | 5999.42 |
| 100195 | Pike Road City            |  2407.15 |  11627294 | 4830.32 |
| 100185 | Saraland City             |  3117.55 |  17870299 | 5732.16 |
| 102850 | Roanoke City              |  1394.5  |   8432448 | 6046.93 |
| 102910 | Russellville City         |  2442.05 |  14956940 | 6124.75 |
| 102940 | Scottsboro City           |  2388.3  |  13734370 | 5750.69 |
| 102970 | Selma City                |  2674.2  |  16409141 | 6136.09 |
| 103000 | Sheffield City            |   979.1  |   5688829 | 5810.26 |
| 103120 | Sylacauga City            |  2011    |  11265028 | 5601.7  |
| 103150 | Talladega City            |  1803.55 |  11137943 | 6175.57 |
| 103240 | Tallassee City            |  1614.45 |   9584462 | 5936.67 |
| 100189 | Satsuma City              |  1481.35 |   9136148 | 6167.45 |
| 103270 | Tarrant City              |  1183.6  |   7454015 | 6297.75 |
| 103300 | Thomasville City          |  1064.4  |   6608456 | 6208.62 |
| 103330 | Troy City                 |  1645.6  |   9247811 | 5619.72 |
| 103360 | Tuscaloosa City           | 10275.6  |  50334040 | 4898.4  |
| 103420 | Tuscumbia City            |  1429.75 |   8725792 | 6103.02 |
| 103430 | Vestavia Hills City       |  7027.65 |  36967013 | 5260.22 |
| 103540 | Winfield City             |  1242    |   7560583 | 6087.43 |
| 100013 | Trussville City           |  4802.05 |  25433735 | 5296.43 |
| 100197 | Accel Academy             |   328.55 |   2028574 | 6174.32 |
| 100199 | University Charter School |   478    |   2623594 | 5488.69 |
| 100203 | Lead Academy              |   453.15 |   2207578 | 4871.63 |
| 100201 | Legacy Prep               |   252.7  |    981497 | 3884.04 |
| 100204 | I3 Academy                |   412.3  |   2896167 | 7024.42 |



label: E=D/C
labeled_formulas: E=D/C
output: {"Arithmetic-Relationship": ["E=D/C"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_733921___xlsx_cbb8798c98f56e7a14f804853ae7e879bffd1911.xlsx___line_557