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 |
|----:|:-------------------|-------:|---------:|---------:|----------:|---------:|---------:|
|   1 | 01-AGRA            | 117.24 |  51.694  |  49.68   |   2.01401 |  67.56   | 0.96104  |
|   2 | 02-ALIGARH         | 103.79 |  57.8583 |  55.69   |   2.16826 |  48.1    | 0.962525 |
|   3 | 03-ALLAHABAD       | 202.24 | 167.649  | 167.57   |   0.07926 |  34.67   | 0.999527 |
|   4 | 04-AMBEDKAR NAGAR  |  89.71 |  47.6916 |  47.51   |   0.18159 |  42.2    | 0.996192 |
|   5 | 05-AURAIYA         |  65.6  |  33.1653 |  33.16   |   0.00535 |  32.44   | 0.999839 |
|   6 | 06-AZAMGARH        | 179.09 |  82.5902 |  81.89   |   0.70017 |  97.2    | 0.991522 |
|   7 | 07-BADAUN          | 130.83 | 118.034  |  63.7    |  54.3335  |  67.13   | 0.539677 |
|   8 | 08-BAGHPAT         |  43.04 |  21.6364 |  21.25   |   0.3864  |  21.79   | 0.982141 |
|   9 | 09-BAHRAICH        | 199.74 | 176.442  |  92.84   |  83.6021  | 106.9    | 0.526178 |
|  10 | 10-BALLIA          | 131.91 |  64.96   |  64.96   |   0       |  66.95   | 1        |
|  11 | 11-BALRAMPUR       | 119.38 |  34.8734 |  34.8734 |   0       |  84.5066 | 1        |
|  12 | 12-BANDA           | 109.88 |  55.0501 |  54.58   |   0.47008 |  55.3    | 0.991461 |
|  13 | 13-BARABANKI       | 133.9  | 121.362  |  24.0291 |  97.3331  | 109.871  | 0.197995 |
|  14 | 14-BAREILY         | 146.68 |  86.0547 |  70.4872 |  15.5675  |  76.1928 | 0.819097 |
|  15 | 15-BASTI           | 110.53 |  50.1464 |  49.58   |   0.56641 |  60.95   | 0.988705 |
|  16 | 16-BHADOHI         |  65.76 |  47.9366 |  28.26   |  19.6766  |  37.5    | 0.589528 |
|  17 | 17-BIJNOUR         | 118.93 |  58.8108 |  57.83   |   0.98081 |  61.1    | 0.983323 |
|  18 | 18-BULANDSHAHAR    | 130.06 |  65.0006 |  60.6718 |   4.32879 |  69.3882 | 0.933404 |
|  19 | 19-CHANDAULI       | 103.77 |  50.6242 |  50.62   |   0.00421 |  53.15   | 0.999917 |
|  20 | 20-CHITRAKOOT      |  75.02 |  42.3426 |  30.4054 |  11.9372  |  44.6146 | 0.71808  |
|  21 | 21-AMETHI          |  75.18 |  33.2519 |  33.25   |   0.00186 |  41.93   | 0.999944 |
|  22 | 22-DEORIA          | 135.05 |  55.2632 |  54.35   |   0.91315 |  80.7    | 0.983476 |
|  23 | 23-ETAH            |  77.4  |  50.5693 |  37.96   |  12.6093  |  39.44   | 0.750654 |
|  24 | 24-FAIZABAD        | 104.52 |  50.4732 |  50.47   |   0.00317 |  54.05   | 0.999937 |
|  25 | 25-FARRUKHABAD     |  89.97 |  83.4731 |  47.1    |  36.3731  |  42.87   | 0.564253 |
|  26 | 26-FATEHPUR        | 128.94 |  55.0267 |  55.0267 |   0       |  73.9133 | 1        |
|  27 | 27-FIROZABAD       |  79.72 |  35.2954 |  34.65   |   0.64538 |  45.07   | 0.981715 |
|  28 | 28-G.B. NAGAR      |  44.93 |  22.3277 |  21.08   |   1.24766 |  23.85   | 0.94412  |
|  29 | 29-GHAZIPUR        | 147.08 | 138.476  |   0      | 138.476   | 147.08   | 0        |
|  30 | 30-GHAZIYABAD      |  45.73 |  21.8581 |  21.45   |   0.40814 |  24.28   | 0.981328 |
|  31 | 31-GONDA           | 161.54 |  75.6008 |  70.93   |   4.67084 |  90.61   | 0.938217 |
|  32 | 32-GORAKHPUR       | 154.24 | 111.559  |  65.74   |  45.819   |  88.5    | 0.589285 |
|  33 | 33-HAMEERPUR       |  56.84 |  43.5487 |  24.91   |  18.6387  |  31.93   | 0.572004 |
|  34 | 34-HARDOI          | 221.22 |  93.9184 |  93.9    |   0.01835 | 127.32   | 0.999805 |
|  35 | 35-HATHRAS         |  59.82 |  45.8445 |  26.06   |  19.7845  |  33.76   | 0.568443 |
|  36 | 36-ITAWAH          |  66.74 |  31.0029 |  30.7    |   0.30289 |  36.04   | 0.99023  |
|  37 | 37-J.P. NAGAR      |  61.61 |  28.2035 |  27.87   |   0.33349 |  33.74   | 0.988176 |
|  38 | 38-JALAUN          |  66.8  |  28.783  |   8.096  |  20.687   |  58.704  | 0.281277 |
|  39 | 39-JAUNPUR         | 200.73 |  27.5009 |  27.5    |   0.0009  | 173.23   | 0.999967 |
|  40 | 40-JHANSI          |  66.71 |  33.3789 |  32.66   |   0.71893 |  34.05   | 0.978462 |
|  41 | 41-KANNAUJ         |  78.51 |  37.8091 |  37.8091 |   0       |  40.7009 | 1        |
|  42 | 42-KANPUR DEHAT    |  68.03 |  57.0341 |  11.1008 |  45.9333  |  56.9292 | 0.194635 |
|  43 | 43-KANPUR NAGAR    |  87.75 |  73.6209 |  46.75   |  26.8709  |  41      | 0.63501  |
|  44 | 44-KAAS GANJ       |  65.53 |  59.3851 |  38.62   |  20.7651  |  26.91   | 0.650331 |
|  45 | 45-KAUSHAMBI       |  73.32 |  41.3391 |  34.17   |   7.16915 |  39.15   | 0.826577 |
|  46 | 46-KUSHINAGAR      | 157.43 |  84.6419 |  83.83   |   0.8119  |  73.6    | 0.990408 |
|  47 | 47-LAKHIMPUR KHERI | 224.72 | 112.55   | 112.55   |   0       | 112.17   | 1        |
|  48 | 48-LALITPUR        |  71.01 |  43.2115 |  34.14   |   9.07148 |  36.87   | 0.790068 |
|  49 | 49-LUCKNOW         | 102.98 |  47.1832 |  46.66   |   0.52324 |  56.32   | 0.98891  |
|  50 | 50-MAHOBA          |  56.01 |  24.6884 |  23.71   |   0.97845 |  32.3    | 0.960368 |
|  51 | 51-MAHRAJGANJ      | 121.86 |  90.8007 |  55.9    |  34.9007  |  65.96   | 0.615634 |
|  52 | 52-MAINPURI        |  74    |  61.5629 |  30.85   |  30.7129  |  43.15   | 0.501113 |
|  53 | 53-MATHURA         |  80.51 |  55.1187 |  41.05   |  14.0687  |  39.46   | 0.744756 |
|  54 | 54-MAU             |  94.42 |  88.2784 |  47.87   |  40.4084  |  46.55   | 0.542262 |
|  55 | 55-MEERUT          |  75.95 |  41.0793 |  14.73   |  26.3493  |  61.22   | 0.358575 |
|  56 | 56-MIRZAPUR        | 141.41 |  62.98   |  62.98   |   0       |  78.43   | 1        |
|  57 | 57-MORADABAD       |  85.47 |  48.502  |  38.5752 |   9.92675 |  46.8948 | 0.795333 |
|  58 | 58-MUZAFFARNAGAR   |  65.27 |  34.3869 |  32.19   |   2.19689 |  33.08   | 0.936113 |
|  59 | 59-PILIBHIT        |  82.88 |  57.0817 |  36.87   |  20.2117  |  46.01   | 0.645916 |
|  60 | 60-PRATAPGARH      | 138.77 |  59.8403 |  59.73   |   0.11031 |  79.04   | 0.998157 |
|  61 | 61-RAI BAREILY     | 116.56 |  77.9489 |  50.428  |  27.521   |  66.132  | 0.646936 |
|  62 | 62-RAMPUR          |  76.69 |  37.0704 |  37.06   |   0.01041 |  39.63   | 0.999719 |
|  63 | 63-SAHARANPUR      | 105.19 |  53.7187 |  42.95   |  10.7687  |  62.24   | 0.799535 |
|  64 | 64-SANTKABIR NAGAR |  72.96 |  33.4522 |  33.15   |   0.30218 |  39.81   | 0.990967 |
|  65 | 65-SHAHJAHANPUR    | 158.74 | 106.076  |  74      |  32.076   |  84.74   | 0.697613 |
|  66 | 66-SHRAWASTI       |  56.86 |  23.7924 |   8.5083 |  15.2841  |  48.3517 | 0.357605 |
|  67 | 67-SIDDHARTHNAGAR  | 154.71 |  73.082  |  67.912  |   5.17    |  86.798  | 0.929258 |
|  68 | 68-SITAPUR         | 218.31 | 115.229  | 105.9    |   9.32891 | 112.41   | 0.91904  |
|  69 | 69-SONBHADRA       | 119.99 |  52.1442 |  52.13   |   0.01421 |  67.86   | 0.999727 |
|  70 | 70-SULTANPUR       | 132.09 |  89.7103 |  56.21   |  33.5003  |  75.88   | 0.626573 |
|  71 | 71-UNNAO           | 123.6  |  57.8549 |  55.6    |   2.25488 |  68      | 0.961025 |
|  72 | 72-VARANASI        | 118.94 |  77.5815 |  77.21   |   0.3715  |  41.73   | 0.995211 |
|  73 | 73-SAMBHAL         | 100.61 |  48.0755 |  47.03   |   1.04548 |  53.58   | 0.978253 |
|  74 | 74-HAPUR           |  35.77 |  35.7433 |  16      |  19.7433  |  19.77   | 0.447637 |
|  75 | 75-SHAMLI          |  43.06 |  19.706  |  19.26   |   0.44601 |  23.8    | 0.977367 |



label: F=D-E
G=C-E
labeled_formulas: F=D-E
G=C-E
output: {"Arithmetic-Relationship": ["F=D-E", "G=C-E"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_377345___xlsx_5e696bd67948a6dabbd8fc6de2fbfac425952fd0.xlsx___line_198