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 |
|-----:|-----------------:|---------:|------------:|------------------:|-----------------:|
| 1945 |      0           |  600     |  0.358211   |     107.463       |    707.463       |
| 1946 |    707.463       |  618     | -0.0842915  |     -85.6792      |   1239.78        |
| 1947 |   1239.78        |  636.54  |  0.052      |      81.0188      |   1957.34        |
| 1948 |   1957.34        |  655.636 |  0.0570458  |     130.359       |   2743.34        |
| 1949 |   2743.34        |  675.305 |  0.183032   |     563.921       |   3982.56        |
| 1950 |   3982.56        |  695.564 |  0.308055   |    1333.99        |   6012.11        |
| 1951 |   6012.11        |  716.431 |  0.236785   |    1508.4         |   8236.94        |
| 1952 |   8236.94        |  737.924 |  0.18151    |    1562.06        |  10536.9         |
| 1953 |  10536.9         |  760.062 | -0.012082   |    -131.899       |  11165.1         |
| 1954 |  11165.1         |  782.864 |  0.525633   |    6074.49        |  18022.4         |
| 1955 |  18022.4         |  806.35  |  0.325973   |    6006.26        |  24835           |
| 1956 |  24835           |  830.54  |  0.0743951  |    1878.5         |  27544.1         |
| 1957 |  27544.1         |  855.457 | -0.104574   |   -2925.11        |  25474.4         |
| 1958 |  25474.4         |  881.12  |  0.4372     |   11330           |  37685.6         |
| 1959 |  37685.6         |  907.554 |  0.120565   |    4598.25        |  43191.4         |
| 1960 |  43191.4         |  934.78  |  0.00336535 |     146.927       |  44273.1         |
| 1961 |  44273.1         |  962.824 |  0.266377   |   11921.6         |  57157.5         |
| 1962 |  57157.5         |  991.709 | -0.0881146  |   -5080.1         |  53069.1         |
| 1963 |  53069.1         | 1021.46  |  0.226119   |   12115.4         |  66206           |
| 1964 |  66206           | 1052.1   |  0.164155   |   10954.4         |  78212.5         |
| 1965 |  78212.5         | 1083.67  |  0.123992   |    9764.94        |  89061.1         |
| 1966 |  89061.1         | 1116.18  | -0.0997095  |   -8935.88        |  81241.4         |
| 1967 |  81241.4         | 1149.66  |  0.23803    |   19474.7         | 101866           |
| 1968 | 101866           | 1184.15  |  0.108149   |   11080.7         | 114131           |
| 1969 | 114131           | 1219.68  | -0.0824137  |   -9456.18        | 105894           |
| 1970 | 105894           | 1256.27  |  0.0356114  |    3793.41        | 110944           |
| 1971 | 110944           | 1293.95  |  0.142212   |   15869.5         | 128107           |
| 1972 | 128107           | 1332.77  |  0.187554   |   24151.9         | 153592           |
| 1973 | 153592           | 1372.76  | -0.14308    |  -22074.2         | 132890           |
| 1974 | 132890           | 1413.94  | -0.259018   |  -34604.1         |  99700.2         |
| 1975 |  99700.2         | 1456.36  |  0.369951   |   37153.6         | 138310           |
| 1976 | 138310           | 1500.05  |  0.23831    |   33139.4         | 172950           |
| 1977 | 172950           | 1545.05  | -0.069797   |  -12125.3         | 162369           |
| 1978 | 162369           | 1591.4   |  0.0650928  |   10620.9         | 174582           |
| 1979 | 174582           |  819.572 |  0.185195   |   32407.5         | 207809           |
| 1980 | 207809           |  844.159 |  0.317352   |   66082.6         | 274736           |
| 1981 | 274736           |  869.483 | -0.0470239  |  -12939.6         | 262666           |
| 1982 | 262666           |  895.568 |  0.204191   |   53725.2         | 317286           |
| 1983 | 317286           |  922.435 |  0.223372   |   70975.8         | 389185           |
| 1984 | 389185           |  950.108 |  0.0614614  |   23949           | 414084           |
| 1985 | 414084           |  978.611 |  0.312351   |  129492           | 544555           |
| 1986 | 544555           | 1007.97  |  0.184946   |  100806           | 646369           |
| 1987 | 646369           | 1038.21  |  0.0581272  |   37601.8         | 685009           |
| 1988 | 685009           | 1069.36  |  0.165372   |  113370           | 799448           |
| 1989 | 799448           | 1101.44  |  0.314752   |  251801           |      1.05235e+06 |
| 1990 |      1.05235e+06 | 1134.48  | -0.0306445  |  -32266.2         |      1.02122e+06 |
| 1991 |      1.02122e+06 | 1168.51  |  0.302348   |  308941           |      1.33133e+06 |
| 1992 |      1.33133e+06 | 1203.57  |  0.0749373  |   99811.2         |      1.43234e+06 |
| 1993 |      1.43234e+06 | 1239.68  |  0.0996705  |  142824           |      1.57641e+06 |
| 1994 |      1.57641e+06 | 1276.87  |  0.0132592  |   20910.4         |      1.59859e+06 |
| 1995 |      1.59859e+06 | 1315.17  |  0.371952   |  594845           |      2.19475e+06 |
| 1996 |      2.19475e+06 | 1354.63  |  0.22681    |  497945           |      2.69405e+06 |
| 1997 |      2.69405e+06 |    0     |  0.331037   |  891830           |      3.58588e+06 |
| 1998 |      3.58588e+06 |    0     |  0.28338    |       1.01617e+06 |      4.60205e+06 |
| 1999 |      4.60205e+06 |    0     |  0.208854   |  961154           |      5.5632e+06  |
| 2000 |      5.5632e+06  |    0     | -0.0903182  | -502458           |      5.06075e+06 |
| 2001 |      5.06075e+06 |    0     | -0.118498   | -599686           |      4.46106e+06 |
| 2002 |      4.46106e+06 |    0     | -0.21966    | -979918           |      3.48114e+06 |
| 2003 |      3.48114e+06 |    0     |  0.283558   |  987105           |      4.46825e+06 |
| 2004 |      4.46825e+06 |    0     |  0.107428   |  480014           |      4.94826e+06 |
| 2005 |      4.94826e+06 |    0     |  0.0483448  |  239223           |      5.18748e+06 |
| 2006 |      5.18748e+06 |    0     |  0.156126   |  809899           |      5.99738e+06 |
| 2007 |      5.99738e+06 |    0     |  0.0548474  |  328940           |      6.32632e+06 |
| 2008 |      6.32632e+06 |    0     | -0.365523   |      -2.31242e+06 |      4.0139e+06  |
| 2009 |      4.0139e+06  |    0     |  0.259352   |       1.04102e+06 |      5.05492e+06 |
| 2010 |      5.05492e+06 |    0     |  0.148211   |  749194           |      5.80411e+06 |
| 2011 |      5.80411e+06 |    0     |  0.0209837  |  121792           |      5.9259e+06  |
| 2012 |      5.9259e+06  |    0     |  0.158906   |  941661           |      6.86756e+06 |
| 2013 |      6.86756e+06 |    0     |  0.321451   |       2.20758e+06 |      9.07515e+06 |
| 2014 |      9.07515e+06 |    0     |  0.134774   |       1.22309e+06 |      1.02982e+07 |



label: F=B+C+E
labeled_formulas: F=B+C+E
output: {"Arithmetic-Relationship": ["F=B+C+E"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_202163___xlsx_8a0ef7febdf92b8cceeea357a4ca78ff559c0401.xlsx___line_367