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 |
|----:|:----------------------------------------------|----------:|---------:|---------:|----------:|---------:|
|   1 | Los Angeles-Long Beach-Anaheim CA             | 170460    | 777048   | -78.0632 | 13109.9   | 13.0024  |
|   2 | San Francisco-Oakland-Hayward CA              | 162305    | 605467   | -73.1933 |  4696.9   | 34.5559  |
|   3 | Seattle-Tacoma-Bellevue WA                    | 138869    | 410664   | -66.1842 |  4018.6   | 34.5567  |
|   4 | Portland-Vancouver-Hillsboro OR-WA            |  67649.3  | 213886   | -68.3713 |  2510.26  | 26.9491  |
|   5 | San Diego-Carlsbad CA                         |  64063.8  | 239046   | -73.2003 |  3332.43  | 19.2244  |
|   6 | San Jose-Sunnyvale-Santa Clara CA             |  59939.7  | 247714   | -75.8029 |  1971.16  | 30.4083  |
|   7 | Sacramento--Roseville--Arden-Arcade CA        |  52895.2  | 178160   | -70.3103 |  2374.75  | 22.274   |
|   8 | Phoenix-Mesa-Scottsdale AZ                    |  51860.5  | 171534   | -69.7667 |  5059.91  | 10.2493  |
|   9 | New York-Newark-Jersey City NY-NJ-PA          |  50264.6  | 179873   | -72.0555 | 19124.4   |  2.6283  |
|  10 | Chicago-Naperville-Elgin IL-IN-WI             |  40697.9  | 135075   | -69.8701 |  9406.64  |  4.32651 |
|  11 | Dallas-Fort Worth-Arlington TX                |  37265.2  | 123234   | -69.7606 |  7694.14  |  4.84332 |
|  12 | Denver-Aurora-Lakewood CO                     |  36644.4  | 119357   | -69.2986 |  2991.23  | 12.2506  |
|  13 | Minneapolis-St. Paul-Bloomington MN-WI        |  35054.2  |  78563.1 | -55.3808 |  3657.48  |  9.58427 |
|  14 | Riverside-San Bernardino-Ontario CA           |  33496.2  | 160614   | -79.1449 |  4678.37  |  7.1598  |
|  15 | Las Vegas-Henderson-Paradise NV               |  28777.4  |  89653.9 | -67.9017 |  2315.96  | 12.4257  |
|  16 | Anchorage AK                                  |  28262.7  |  61597.9 | -54.1175 |   397.308 | 71.1354  |
|  17 | Salt Lake City UT                             |  24739.1  |  68663.7 | -63.9706 |  1240.03  | 19.9504  |
|  18 | Washington-Arlington-Alexandria DC-VA-MD-WV   |  24582    | 103301   | -76.2035 |  6324.63  |  3.88672 |
|  19 | Houston-The Woodlands-Sugar Land TX           |  21619.7  |  80372.8 | -73.1007 |  7154.48  |  3.02184 |
|  20 | Boston-Cambridge-Newton MA-NH                 |  18257    |  58947.1 | -69.0282 |  4878.21  |  3.74255 |
|  21 | Provo-Orem UT                                 |  15354.4  |  40412.6 | -62.0058 |   663.181 | 23.1527  |
|  22 | Atlanta-Sandy Springs-Roswell GA              |  14616.7  |  58245.7 | -74.9051 |  6087.76  |  2.40099 |
|  23 | Santa Rosa CA                                 |  13710.8  |  42996.1 | -68.1115 |   489.819 | 27.9915  |
|  24 | Boise City ID                                 |  12472.4  |  36152   | -65.5002 |   770.353 | 16.1905  |
|  25 | Ogden-Clearfield UT                           |  12346.8  |  34661.4 | -64.3787 |   691.359 | 17.8588  |
|  26 | Austin-Round Rock TX                          |  12102.7  |  44983.4 | -73.0952 |  2295.3   |  5.27281 |
|  27 | Philadelphia-Camden-Wilmington PA-NJ-DE-MD    |  12097.7  |  47415.5 | -74.4859 |  6107.91  |  1.98066 |
|  28 | Detroit-Warren-Dearborn MI                    |  12004.9  |  38790.9 | -69.0522 |  4304.14  |  2.78916 |
|  29 | Spokane-Spokane Valley WA                     |  11579.6  |  35466.3 | -67.3504 |   574.585 | 20.153   |
|  30 | Oxnard-Thousand Oaks-Ventura CA               |  11449.9  |  56166.8 | -79.6144 |   841.387 | 13.6084  |
|  31 | Kansas City MO-KS                             |  10801.9  |  32635.1 | -66.9011 |  2173.21  |  4.97046 |
|  32 | Reno NV                                       |  10118.5  |  29060   | -65.1808 |  6173.01  |  1.63915 |
|  32 | Miami-Fort Lauderdale-West Palm Beach FL      |   9330.24 |  33403.9 | -72.0684 |   481.289 | 19.386   |
|  34 | Stockton-Lodi CA                              |   9163.38 |  35466.3 | -74.1632 |   767.967 | 11.932   |
|  35 | St. Louis MO-IL                               |   9085.07 |  31528.2 | -71.1843 |  2805.47  |  3.23834 |
|  36 | Vallejo-Fairfield CA                          |   9065.62 |  34889.1 | -74.0159 |   446.935 | 20.284   |
|  37 | San Antonio-New Braunfels TX                  |   8229.57 |  31063.9 | -73.5076 |  2590.73  |  3.17654 |
|  38 | Santa Cruz-Watsonville CA                     |   8025.95 |  29433.6 | -72.732  |   269.925 | 29.734   |
|  39 | Bremerton-Silverdale WA                       |   7762.4  |  22354.4 | -65.2758 |   272.787 | 28.4559  |
|  40 | Baltimore-Columbia-Towson MD                  |   7318.44 |  29712.7 | -75.3693 |  2800.19  |  2.61355 |
|  41 | Salem OR                                      |   6899.63 |  21457.2 | -67.8447 |   436.948 | 15.7905  |
|  42 | Tucson AZ                                     |   6783.17 |  25598.3 | -73.5015 |  1061.17  |  6.39213 |
|  43 | Tampa-St. Petersburg-Clearwater FL            |   6500.37 |  22740.3 | -71.4147 |  3243.96  |  2.00384 |
|  44 | Orlando-Kissimmee-Sanford FL                  |   6465.51 |  21813.3 | -70.3599 |  2639.37  |  2.44964 |
|  45 | Colorado Springs CO                           |   6368.8  |  23314.6 | -72.6832 |   753.839 |  8.44849 |
|  46 | Fresno CA                                     |   6265.52 |  27047.7 | -76.8353 |  1000.92  |  6.25978 |
|  47 | Olympia-Tumwater WA                           |   6203.51 |  19970.2 | -68.9362 |   294.074 | 21.0951  |
|  48 | Milwaukee-Waukesha-West Allis WI              |   6186.84 |  17161.2 | -63.9487 |  1577.68  |  3.92149 |
|  49 | Virginia Beach-Norfolk-Newport News VA-NC     |   6123.89 |  18966.9 | -67.7127 |   451.231 | 13.5715  |
|  50 | Eugene OR                                     |   5953.37 |  19979.2 | -70.2022 |   382.986 | 15.5446  |
|  51 | Bend-Redmond OR                               |   5740.35 |  17091.6 | -66.4143 |   201.769 | 28.4501  |
|  52 | Indianapolis-Carmel-Anderson IN               |   5673.28 |  20568.1 | -72.4171 |  2091.02  |  2.71317 |
|  53 | Bellingham WA                                 |   5569.6  |  16666.5 | -66.5821 |   231.016 | 24.1092  |
|  54 | Modesto CA                                    |   5519.88 |  20065.5 | -72.4907 |   550.081 | 10.0347  |
|  55 | Nashville-Davidson--Murfreesboro--Franklin TN |   5420.32 |  20387.8 | -73.4139 |  1961.23  |  2.76373 |
|  56 | Cincinnati OH-KY-IN                           |   5411.85 |  19510.9 | -72.2624 |  2232.91  |  2.42368 |
|  57 | Santa Maria-Santa Barbara CA                  |   5405.67 |  24135.6 | -77.6029 |   444.766 | 12.154   |
|  58 | Albuquerque NM                                |   5188.28 |  20075.4 | -74.156  |   923.63  |  5.61727 |
|  59 | Omaha-Council Bluffs NE-IA                    |   5186.8  |  14158.1 | -63.3652 |   954.27  |  5.43536 |
|  60 | Boulder CO                                    |   5135.19 |  17934.8 | -71.3675 |   327.171 | 15.6958  |



label: G=C/F
labeled_formulas: G=C/F
output: {"Arithmetic-Relationship": ["G=C/F"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_646939___xlsx_51dbdd9f699b41ff307e28f4e3137dd1f450d287.xlsx___line_338