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             |
|:-----------------------------|------------:|---------:|------------:|-------------:|----:|----:|----:|----:|:--------------|
| Cons Dance/Drama             | -58.4574    | 1.35769  | -61.1184    | -55.7963     |   0 |   1 |   0 |   0 | Other         |
| Guildhall School Music/Drama | -51.7997    | 1.86011  | -55.4455    | -48.1539     |   0 |   1 |   0 |   0 | Other         |
| Trinity Cons Music/Dance     | -39.8776    | 1.93232  | -43.665     | -36.0903     |   0 |   1 |   0 |   0 | Other         |
| Leeds City                   | -32.3912    | 2.09712  | -36.5015    | -28.2808     |   0 |   1 |   0 |   0 | Other         |
| L'pool Inst Perf Arts        | -32.1438    | 1.78965  | -35.6515    | -28.6361     |   0 |   1 |   0 |   0 | Other         |
| Central Sch Speech/Drama     | -25.8013    | 2.46624  | -30.6351    | -20.9674     |   0 |   1 |   0 |   0 | Other         |
| Rose Bruford                 | -24.011     | 2.47318  | -28.8584    | -19.1636     |   0 |   1 |   0 |   0 | Other         |
| Royal Northern C Music       | -21.4082    | 2.74251  | -26.7835    | -16.0328     |   0 |   1 |   0 |   0 | Other         |
| Bangor                       | -17.0683    | 0.719155 | -18.4779    | -15.6588     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Glyndwr                      | -14.6395    | 2.06586  | -18.6886    | -10.5904     |   0 |   1 |   0 |   0 | Other         |
| U Campus Suffolk             | -14.6367    | 1.56556  | -17.7052    | -11.5682     |   0 |   1 |   0 |   0 | Other         |
| Glamorgan                    | -14.4729    | 0.765432 | -15.9731    | -12.9727     |   0 |   0 |   1 |   0 | Post 1992 uni |
| Wales Trinity St David       | -13.584     | 1.06295  | -15.6673    | -11.5006     |   0 |   1 |   0 |   0 | Other         |
| Bolton                       | -13.4762    | 1.06823  | -15.57      | -11.3825     |   0 |   1 |   0 |   0 | Other         |
| U Arts London                | -12.545     | 1.43503  | -15.3577    |  -9.73235    |   0 |   1 |   0 |   0 | Other         |
| London Met                   | -12.3574    | 0.985676 | -14.2893    | -10.4255     |   0 |   0 |   1 |   0 | Post 1992 uni |
| UC Falmouth                  | -12.1857    | 1.46408  | -15.0553    |  -9.3161     |   0 |   1 |   0 |   0 | Other         |
| L'pool Hope                  | -11.5508    | 1.02815  | -13.566     |  -9.53566    |   0 |   1 |   0 |   0 | Other         |
| West London                  | -10.632     | 1.55242  | -13.6747    |  -7.58928    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Derby                        | -10.5392    | 0.722953 | -11.9561    |  -9.12217    |   0 |   1 |   0 |   0 | Other         |
| UC Birmingham                | -10.5352    | 1.44269  | -13.3629    |  -7.70756    |   0 |   1 |   0 |   0 | Other         |
| Bradford                     | -10.4701    | 0.767178 | -11.9737    |  -8.96639    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Aberdeen                     | -10.3632    | 1.4467   | -13.1987    |  -7.52767    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Huddersfield                 | -10.3296    | 0.664147 | -11.6313    |  -9.02783    |   0 |   0 |   1 |   0 | Post 1992 uni |
| East London                  | -10.0457    | 1.27524  | -12.5452    |  -7.54626    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Winchester                   |  -9.54857   | 1.04633  | -11.5994    |  -7.49776    |   0 |   1 |   0 |   0 | Other         |
| Prifysgol Aberystwyth        |  -9.51132   | 0.631822 | -10.7497    |  -8.27295    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Middlesex                    |  -8.73601   | 1.16096  | -11.0115    |  -6.46053    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Norwich UC Arts              |  -8.67649   | 2.76638  | -14.0986    |  -3.25439    |   0 |   1 |   0 |   0 | Other         |
| Writtle C                    |  -8.46683   | 2.25854  | -12.8936    |  -4.0401     |   0 |   1 |   0 |   0 | Other         |
| Chester                      |  -8.39286   | 0.742321 |  -9.84781   |  -6.93792    |   0 |   1 |   0 |   0 | Other         |
| Staffordshire                |  -8.06609   | 0.64566  |  -9.33158   |  -6.80059    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Chichester                   |  -7.69486   | 1.35838  | -10.3573    |  -5.03243    |   0 |   1 |   0 |   0 | Other         |
| Cumbria                      |  -7.63682   | 0.915162 |  -9.43053   |  -5.8431     |   0 |   1 |   0 |   0 | Other         |
| St George's Hospital         |  -7.33159   | 3.74236  | -14.6666    |   0.00343475 |   0 |   1 |   0 |   0 | Other         |
| Teesside                     |  -7.09621   | 0.815357 |  -8.69431   |  -5.49811    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Cental Lancashire            |  -6.87625   | 0.569671 |  -7.9928    |  -5.75969    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Lincoln                      |  -6.81816   | 0.628847 |  -8.0507    |  -5.58562    |   0 |   1 |   0 |   0 | Other         |
| York St John UC              |  -6.81465   | 1.00074  |  -8.7761    |  -4.8532     |   0 |   1 |   0 |   0 | Other         |
| De Montfort                  |  -6.60584   | 0.627862 |  -7.83645   |  -5.37523    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Royal Agricultural C         |  -6.40659   | 2.19012  | -10.6992    |  -2.11395    |   0 |   1 |   0 |   0 | Other         |
| Northampton                  |  -6.31115   | 0.766107 |  -7.81272   |  -4.80958    |   0 |   1 |   0 |   0 | Other         |
| Manchester Met               |  -6.19606   | 0.50507  |  -7.186     |  -5.20612    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Bath Spa                     |  -6.13482   | 1.09001  |  -8.27123   |  -3.99841    |   0 |   1 |   0 |   0 | Other         |
| Heythrop College             |  -5.7376    | 2.60719  | -10.8477    |  -0.627506   |   0 |   1 |   0 |   0 | Other         |
| St Mark & St John            |  -5.6561    | 1.41744  |  -8.43429   |  -2.87791    |   0 |   1 |   0 |   0 | Other         |
| Goldsmiths                   |  -5.50291   | 1.20263  |  -7.86006   |  -3.14577    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Arts Inst Bournemouth        |  -5.4012    | 2.39427  | -10.094     |  -0.708423   |   0 |   1 |   0 |   0 | Other         |
| Edge Hill                    |  -5.22234   | 0.939714 |  -7.06418   |  -3.3805     |   0 |   1 |   0 |   0 | Other         |
| Cardiff Met                  |  -4.94055   | 1.14746  |  -7.18958   |  -2.69152    |   0 |   1 |   0 |   0 | Other         |
| Salford                      |  -4.82818   | 0.6543   |  -6.11061   |  -3.54575    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Worcester                    |  -4.63575   | 1.11851  |  -6.82804   |  -2.44346    |   0 |   1 |   0 |   0 | Other         |
| Plymouth                     |  -4.52088   | 0.555594 |  -5.60984   |  -3.43192    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Gloucestershire              |  -4.50383   | 0.842998 |  -6.15611   |  -2.85156    |   0 |   1 |   0 |   0 | Other         |
| Sussex                       |  -4.39001   | 0.740278 |  -5.84096   |  -2.93907    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Anglia Ruskin                |  -4.34416   | 0.786406 |  -5.88552   |  -2.8028     |   0 |   0 |   1 |   0 | Post 1992 uni |
| Bedfordshire                 |  -4.31033   | 1.10417  |  -6.4745    |  -2.14616    |   0 |   1 |   0 |   0 | Other         |
| Canterbury Christchuch       |  -4.30239   | 0.8998   |  -6.066     |  -2.53878    |   0 |   1 |   0 |   0 | Other         |
| U Creative Arts              |  -4.11852   | 2.17576  |  -8.38301   |   0.145984   |   0 |   1 |   0 |   0 | Other         |
| L'pool John Moores           |  -4.11833   | 0.643605 |  -5.3798    |  -2.85687    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Buckingham New               |  -4.09915   | 1.01832  |  -6.09506   |  -2.10324    |   0 |   1 |   0 |   0 | Other         |
| Leeds Met                    |  -4.09647   | 0.597641 |  -5.26785   |  -2.9251     |   0 |   1 |   0 |   0 | Other         |
| Bishop Grosseteste           |  -4.07232   | 2.0607   |  -8.1113    |  -0.0333505  |   0 |   1 |   0 |   0 | Other         |
| SOAS                         |  -3.97406   | 1.72454  |  -7.35416   |  -0.593962   |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| London South Bank            |  -3.72452   | 1.21881  |  -6.11338   |  -1.33566    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Essex                        |  -3.6342    | 0.660941 |  -4.92964   |  -2.33875    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Greenwich                    |  -3.37719   | 0.925257 |  -5.19069   |  -1.56368    |   0 |   0 |   1 |   0 | Post 1992 uni |
| U West England               |  -3.27311   | 0.603309 |  -4.4556    |  -2.09063    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Hull                         |  -3.00396   | 0.619808 |  -4.21879   |  -1.78914    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Sunderland                   |  -2.90586   | 0.880226 |  -4.6311    |  -1.18061    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Stirling                     |  -2.7993    | 1.64608  |  -6.02561   |   0.427005   |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Leeds Trinity                |  -2.44023   | 1.18613  |  -4.76505   |  -0.115411   |   0 |   1 |   0 |   0 | Other         |
| Wolverhampton                |  -2.33649   | 0.822641 |  -3.94886   |  -0.724112   |   0 |   0 |   1 |   0 | Post 1992 uni |
| Birmingham City              |  -1.94058   | 0.777504 |  -3.46449   |  -0.416675   |   0 |   0 |   1 |   0 | Post 1992 uni |
| Swansea                      |  -1.53298   | 0.744298 |  -2.99181   |  -0.0741588  |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Sheffield Hallam             |  -1.44917   | 0.627707 |  -2.67948   |  -0.218863   |   0 |   0 |   1 |   0 | Post 1992 uni |
| Portsmouth                   |  -1.36648   | 0.583232 |  -2.50961   |  -0.223342   |   0 |   0 |   1 |   0 | Post 1992 uni |
| UEA                          |  -0.683094  | 0.676748 |  -2.00952   |   0.643333   |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| S'ton Solent                 |  -0.680404  | 0.820721 |  -2.28902   |   0.928209   |   0 |   1 |   0 |   0 | Other         |
| Keele                        |  -0.433266  | 0.748597 |  -1.90052   |   1.03398    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Kent                         |  -0.292272  | 0.635301 |  -1.53746   |   0.952918   |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Coventry                     |  -0.209086  | 0.71028  |  -1.60123   |   1.18306    |   0 |   0 |   1 |   0 | Post 1992 uni |
| St Mary's UC Twickeham       |  -0.193323  | 1.2705   |  -2.68351   |   2.29686    |   0 |   1 |   0 |   0 | Other         |
| Leicester                    |   0.0643489 | 0.692614 |  -1.29317   |   1.42187    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Newman                       |   0.2965    | 1.72597  |  -3.0864    |   3.6794     |   0 |   1 |   0 |   0 | Other         |
| Lancaster                    |   0.451366  | 0.671318 |  -0.864416  |   1.76715    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Northumbria                  |   0.549404  | 0.657699 |  -0.739686  |   1.83849    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Roehampton                   |   0.985181  | 1.23006  |  -1.42574   |   3.3961     |   0 |   1 |   0 |   0 | Other         |
| Nottingham Trent             |   1.07341   | 0.539247 |   0.0164858 |   2.13033    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Westminster                  |   1.20082   | 1.01561  |  -0.789773  |   3.19142    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Brighton                     |   1.35728   | 0.751868 |  -0.116378  |   2.83095    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Royal Holloway               |   1.36509   | 0.886429 |  -0.372308  |   3.10249    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Hertfordshire                |   1.76679   | 0.686374 |   0.421498  |   3.11209    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Harper Adams                 |   1.782     | 2.24865  |  -2.62535   |   6.18935    |   0 |   1 |   0 |   0 | Other         |
| Kingston                     |   2.74832   | 0.770917 |   1.23732   |   4.25932    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Ravensbourne                 |   4.64549   | 1.95456  |   0.814552  |   8.47643    |   0 |   1 |   0 |   0 | Other         |
| Bournemouth                  |   4.95752   | 0.74262  |   3.50198   |   6.41305    |   0 |   0 |   1 |   0 | Post 1992 uni |
| Oxford Brookes               |   6.12668   | 0.719117 |   4.71721   |   7.53615    |   0 |   0 |   1 |   0 | Post 1992 uni |
| City                         |   6.87789   | 1.17933  |   4.5664    |   9.18938    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Brunel                       |   7.215     | 0.814571 |   5.61844   |   8.81156    |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Herriot-Watt                 |   8.99861   | 3.19436  |   2.73767   |  15.2596     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Reading                      |   9.66924   | 0.677571 |   8.3412    |  10.9973     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Surrey                       |  10.556     | 1.08437  |   8.43061   |  12.6813     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Aston                        |  14.0817    | 1.0126   |  12.097     |  16.0664     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Loughborough                 |  20.0421    | 0.711754 |  18.6471    |  21.4371     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| Bath                         |  22.1913    | 1.14039  |  19.9562    |  24.4265     |   0 |   0 |   0 |   1 | Pre 1992 uni  |
| St Andrews                   |  24.5013    | 1.62564  |  21.3151    |  27.6876     |   0 |   0 |   0 |   1 | Pre 1992 uni  |



labeled_formulas: F=G*H
F=G*I
F=H*I
F=(B*G)*H
F=(B*G)*I
F=(B*H)*I
F=(C*G)*H
F=(C*G)*I
F=(C*H)*I
F=(D*G)*H
F=(D*G)*I
F=(D*H)*I
F=(E*G)*H
F=(E*G)*I
F=(E*H)*I
F=(G+H)*I
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_595502___xlsx_d216e3f0abe4f7e0e87b8d0eec5ad915579c5ab5.xlsx___line_731