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 |
|:----------------------------------------------------|--------:|--------:|------------:|-------------:|-------------:|
| Scottish Water                                      |   -14.5 |   -94.5 |   -92.9204  |  -78.4204    |  5.4083      |
| Digital Strategy                                    |    50.9 |   116   |   114.061   |   63.161     |  1.24088     |
| Other Transport Policy, Projects and Agency         |    65.9 |   117.7 |   115.733   |   49.8325    |  0.756184    |
| Miscellaneous                                       |    31.2 |    38.2 |    37.5615  |    6.36146   |  0.203893    |
| National Records of Scotland                        |    23.2 |    28.4 |    27.9253  |    4.72527   |  0.203675    |
| Revenue Scotland                                    |     4.3 |     5.2 |     5.11308 |    0.813078  |  0.189088    |
| Motorways and Trunk Roads                           |   694.8 |   820.3 |   806.588   |  111.788     |  0.160892    |
| Community Justice Services                          |    26.7 |    30.7 |    30.1868  |    3.48682   |  0.130593    |
| Other Expenditure                                   |    23.3 |    24.9 |    24.4838  |    1.18378   |  0.0508058   |
| Ferry Services in Scotland                          |   187.1 |   198.6 |   195.28    |    8.18024   |  0.0437212   |
| Health                                              | 12288.5 | 12977.2 | 12760.3     |  471.775     |  0.0383916   |
| Learning                                            |   191.3 |   201.5 |   198.132   |    6.83176   |  0.0357123   |
| EU Support and Related Services                     |   172.5 |   178.8 |   175.811   |    3.31121   |  0.0191954   |
| Climate Change                                      |    19.4 |    19.8 |    19.469   |    0.0690265 |  0.00355807  |
| Housing                                             |   694.4 |   707.2 |   695.379   |    0.978564  |  0.00140922  |
| Scottish Courts & Tribunals Service                 |    87.4 |    88.9 |    87.414   |    0.0139626 |  0.000159756 |
| Social Security                                     |    73.2 |    74.3 |    73.058   |   -0.141986  | -0.0019397   |
| Historic Environment  Scotland                      |    44.6 |    45   |    44.2478  |   -0.352212  | -0.00789714  |
| Scottish Police Authority (SPA)                     |  1063.9 |  1069.6 |  1051.72    |  -12.1793    | -0.0114477   |
| Scottish Further & Higher Education Funding Council |  1647.2 |  1652.1 |  1624.48    |  -22.7162    | -0.0137908   |
| Concessionary Fares and Bus Services                |   260.6 |   261.3 |   256.932   |   -3.66785   | -0.0140746   |
| Criminal Injuries Compensation                      |    17.5 |    17.5 |    17.2075  |   -0.292527  | -0.0167158   |
| Office of the Scottish Charity Regulator            |     3   |     3   |     2.94985 |   -0.0501475 | -0.0167158   |
| Judiciary                                           |    40.5 |    40.5 |    39.823   |   -0.676991  | -0.0167158   |
| Equalities                                          |    20.3 |    20.3 |    19.9607  |   -0.339331  | -0.0167158   |
| Third Sector                                        |    24.5 |    24.5 |    24.0905  |   -0.409538  | -0.0167158   |
| Police and Fire Pensions                            |   350.6 |   350.6 |   344.739   |   -5.86057   | -0.0167158   |
| Environmental and Rural services                    |   180.9 |   180.3 |   177.286   |   -3.61386   | -0.0199771   |
| Social Justice and Regeneration                     |    56.9 |    56.5 |    55.5556  |   -1.34444   | -0.0236282   |
| Administration                                      |   197.7 |   193   |   189.774   |   -7.92616   | -0.0400918   |
| Planning                                            |     4.2 |     4.1 |     4.03147 |   -0.168535  | -0.0401274   |
| Food Standards Scotland                             |    15.7 |    15.3 |    15.0442  |   -0.655752  | -0.0417677   |
| Children & Families                                 |   120.4 |   117.2 |   115.241   |   -5.1591    | -0.0428496   |
| Scottish Public Pensions Agency                     |  3560.1 |  3461.9 |  3404.03    | -156.069     | -0.0438382   |
| Fair Work, Skills & Training**                      |   261.6 |   252.5 |   248.279   |  -13.3207    | -0.0509203   |
| Marine & Fisheries                                  |    55.3 |    53.2 |    52.3107  |   -2.98928   | -0.0540557   |
| Forestry Commission Scotland                        |    64.1 |    61.3 |    60.2753  |   -3.82468   | -0.0596674   |
| Scottish Fire and Rescue Service                    |   309.2 |   294.7 |   289.774   |  -19.4262    | -0.0628272   |
| Air Services in Scotland                            |    63.4 |    60.3 |    59.292   |   -4.10796   | -0.0647944   |
| Local Government                                    | 10756.7 | 10152.3 |  9982.6     | -774.104     | -0.0719648   |
| Legal Aid                                           |   146.8 |   136.9 |   134.612   |  -12.1884    | -0.0830272   |
| Rail Services in Scotland                           |   808.3 |   751.3 |   738.741   |  -69.5586    | -0.0860554   |
| Scottish Housing Regulator                          |     4   |     3.7 |     3.63815 |   -0.361849  | -0.0904621   |
| Enterprise, Energy & Tourism                        |   469   |   428.7 |   421.534   |  -47.4661    | -0.101207    |
| Research, Analysis & Other Services                 |    73.8 |    67   |    65.88    |   -7.91996   | -0.107317    |
| Culture                                             |   170.2 |   154.1 |   151.524   |  -18.6759    | -0.109729    |
| Europe and External Affairs                         |    17.9 |    16.2 |    15.9292  |   -1.9708    | -0.1101      |
| Higher Education Student Support                    |   970.3 |   876.3 |   861.652   | -108.648     | -0.111974    |
| Scottish Prison Service                             |   396.2 |   357.2 |   351.229   |  -44.9709    | -0.113506    |
| Advanced Learning & Science                         |     7.7 |     6.7 |     6.588   |   -1.112     | -0.144415    |
| Parliamentary Business & Government Strategy        |    75.9 |    63.8 |    62.7335  |  -13.1665    | -0.173471    |
| Governance, Elections & Reform                      |     2.5 |     2.1 |     2.0649  |   -0.435103  | -0.174041    |
| Police Central Government                           |    98.1 |    80.4 |    79.056   |  -19.044     | -0.194128    |
| Accountant in Bankruptcy                            |     1.8 |     1.2 |     1.17994 |   -0.620059  | -0.344477    |
| Sport                                               |    71.8 |    45.8 |    45.0344  |  -26.7656    | -0.37278     |
| Safer and Stronger Communities                      |    40.3 |     7.1 |     6.98132 |  -33.3187    | -0.826766    |



label: E=D-B
F=E/B
labeled_formulas: E=D-B
F=E/B
output: {"Arithmetic-Relationship": ["E=D-B", "F=E/B"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_444485___xlsx_d863e4b1d4d4ff9082fcd3e99825ab0eed47f229.xlsx___line_565