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:
|    AE |     AF |     AG |         AH |         AI |         AJ |
|------:|-------:|-------:|-----------:|-----------:|-----------:|
|  1203 |   1647 |   2850 | 0.156072   | 0.0646999  | 0.0859366  |
|  2053 |   3364 |   5417 | 0.0772996  | 0.046457   | 0.0547338  |
|   102 |    166 |    268 | 0.019589   | 0.00976241 | 0.0120661  |
|   107 |    183 |    290 | 0.0344051  | 0.0191023  | 0.0228526  |
|   249 |    376 |    625 | 0.0311172  | 0.0154408  | 0.0193181  |
|    64 |    103 |    167 | 0.0174863  | 0.00910457 | 0.0111534  |
|  8053 |  23329 |  31382 | 0.228538   | 0.135327   | 0.151146   |
|  1450 |   1925 |   3375 | 0.0511843  | 0.0213479  | 0.0284805  |
|    37 |     64 |    101 | 0.0406593  | 0.0167495  | 0.0213486  |
|   519 |    705 |   1224 | 0.0282849  | 0.0124671  | 0.0163422  |
|     9 |     30 |     39 | 0.00656934 | 0.00550055 | 0.00571512 |
|   149 |    207 |    356 | 0.0184064  | 0.00826248 | 0.0107397  |
|   112 |    186 |    298 | 0.0286812  | 0.0137482  | 0.017093   |
|    38 |     57 |     95 | 0.00751731 | 0.00299323 | 0.00394224 |
|    94 |    194 |    288 | 0.0253301  | 0.0144399  | 0.0167969  |
|   322 |    596 |    918 | 0.0238979  | 0.0144073  | 0.016739   |
|   388 |    571 |    959 | 0.0519272  | 0.0270963  | 0.0335961  |
|   357 |    419 |    776 | 0.0519878  | 0.0180798  | 0.0258305  |
|    29 |     59 |     88 | 0.0198903  | 0.0101759  | 0.0121279  |
|    92 |    148 |    240 | 0.0284742  | 0.0158204  | 0.0190688  |
|  9294 |  14244 |  23538 | 0.096753   | 0.0599525  | 0.0705476  |
|   211 |    279 |    490 | 0.0600455  | 0.0265209  | 0.0349152  |
|    16 |     20 |     36 | 0.0130506  | 0.00504541 | 0.00693642 |
|  1655 |   2502 |   4157 | 0.124962   | 0.0748094  | 0.089036   |
|    85 |     96 |    181 | 0.0332421  | 0.0128085  | 0.0180064  |
|    34 |     52 |     86 | 0.0091447  | 0.00426754 | 0.00540778 |
|   311 |    363 |    674 | 0.0431046  | 0.0174637  | 0.0240706  |
|   135 |    214 |    349 | 0.0492701  | 0.0254429  | 0.0312976  |
| 55650 | 117497 | 173147 | 0.189472   | 0.142782   | 0.155063   |
|  1747 |   2431 |   4178 | 0.0972122  | 0.0514693  | 0.0640768  |
|   161 |    251 |    412 | 0.0445243  | 0.021521   | 0.0269651  |
|   318 |    442 |    760 | 0.0495559  | 0.0229324  | 0.0295823  |
|   635 |    789 |   1424 | 0.0474553  | 0.0184441  | 0.0253566  |
|  2238 |   2930 |   5168 | 0.10341    | 0.0517092  | 0.0659983  |
|    86 |    123 |    209 | 0.0208991  | 0.0093387  | 0.0120907  |
|   357 |    578 |    935 | 0.0393345  | 0.0208048  | 0.0253676  |
|   171 |    284 |    455 | 0.0357143  | 0.0167759  | 0.0209513  |
|   179 |    237 |    416 | 0.054457   | 0.0193533  | 0.0267817  |
|   343 |    438 |    781 | 0.0684631  | 0.0327037  | 0.0424387  |
|   150 |    285 |    435 | 0.0380807  | 0.0200267  | 0.0239406  |
|   254 |    333 |    587 | 0.0287851  | 0.0121457  | 0.0161971  |
|   831 |   1285 |   2116 | 0.0299492  | 0.0178185  | 0.021189   |
|  5480 |   9521 |  15001 | 0.0675892  | 0.042155   | 0.0488735  |
|  1384 |   1713 |   3097 | 0.0892097  | 0.0326547  | 0.0455629  |
|     6 |     12 |     18 | 0.015625   | 0.00619515 | 0.00775528 |
|   287 |    371 |    658 | 0.0324918  | 0.0140334  | 0.0186561  |
|  1154 |   1870 |   3024 | 0.0733723  | 0.0364657  | 0.0451283  |
|    84 |    100 |    184 | 0.0526316  | 0.0186951  | 0.0264939  |
|   353 |    437 |    790 | 0.0499363  | 0.0264608  | 0.0334973  |
|   135 |    189 |    324 | 0.0313225  | 0.0162581  | 0.0203326  |
|    40 |     78 |    118 | 0.0193611  | 0.00836461 | 0.0103591  |
|    68 |    338 |    406 | 0.0117892  | 0.0170543  | 0.0158674  |
| 14172 |  24404 |  38576 | 0.140259   | 0.115512   | 0.123518   |
|   316 |    446 |    762 | 0.0393622  | 0.0177512  | 0.0229843  |
|   163 |    307 |    470 | 0.0586331  | 0.0302941  | 0.0363946  |
|   103 |    133 |    236 | 0.0314312  | 0.0132589  | 0.0177337  |
|    40 |     64 |    104 | 0.0221607  | 0.00892235 | 0.0115839  |
|   264 |    542 |    806 | 0.0373409  | 0.0211248  | 0.024628   |
|    58 |    108 |    166 | 0.0291457  | 0.0120415  | 0.0151474  |
|   898 |   1638 |   2536 | 0.0375136  | 0.0232492  | 0.0268667  |
|   184 |    275 |    459 | 0.0579893  | 0.0232126  | 0.0305593  |
|   168 |    222 |    390 | 0.0373749  | 0.0159323  | 0.0211623  |
|   343 |    531 |    874 | 0.126802   | 0.0548327  | 0.0705465  |
|   146 |    236 |    382 | 0.0644307  | 0.0234499  | 0.0309813  |
|   207 |    402 |    609 | 0.0572297  | 0.0328539  | 0.0384154  |
|   470 |    669 |   1139 | 0.0560993  | 0.0266502  | 0.0340193  |
|   159 |    214 |    373 | 0.0279094  | 0.0116653  | 0.0155145  |
|   194 |    250 |    444 | 0.0493262  | 0.0171739  | 0.024013   |
|   527 |    805 |   1332 | 0.0354095  | 0.016556   | 0.0209744  |
|   191 |    311 |    502 | 0.0262724  | 0.0149692  | 0.0178992  |
|   184 |    341 |    525 | 0.0297158  | 0.0198533  | 0.0224666  |
|   723 |   1335 |   2058 | 0.0802175  | 0.0499775  | 0.0576067  |
| 31698 |  49762 |  81460 | 0.251272   | 0.180394   | 0.202636   |
|   177 |    255 |    432 | 0.0233941  | 0.00933861 | 0.0123882  |
|    91 |    137 |    228 | 0.0560345  | 0.0238302  | 0.0309236  |
|   180 |    330 |    510 | 0.100503   | 0.0442181  | 0.0551113  |
|  1031 |   1521 |   2552 | 0.0362505  | 0.0171405  | 0.0217788  |
|   246 |    424 |    670 | 0.0294611  | 0.0153351  | 0.0186116  |
|  4871 |   6870 |  11741 | 0.141689   | 0.0756192  | 0.093757   |
|   116 |    159 |    275 | 0.0176587  | 0.0071211  | 0.00951656 |
|   108 |    126 |    234 | 0.0217391  | 0.00691966 | 0.0100962  |
|  1123 |   1454 |   2577 | 0.110109   | 0.0457319  | 0.0613674  |
|   226 |    301 |    527 | 0.0299576  | 0.012204   | 0.0163624  |
|    95 |    108 |    203 | 0.0218743  | 0.0075912  | 0.0109316  |
|  3810 |   5468 |   9278 | 0.102145   | 0.0642561  | 0.0758025  |
|  4980 |   6895 |  11875 | 0.122563   | 0.0780604  | 0.0920821  |
|    26 |     60 |     86 | 0.0156627  | 0.0111152  | 0.0121848  |
|    83 |    185 |    268 | 0.0362445  | 0.0188833  | 0.0221726  |
|    97 |    199 |    296 | 0.00944867 | 0.00571642 | 0.0065664  |
|   553 |    765 |   1318 | 0.0545204  | 0.0278871  | 0.0350765  |
|   250 |    474 |    724 | 0.0205491  | 0.0110981  | 0.0131934  |
|   410 |    592 |   1002 | 0.106025   | 0.0435711  | 0.057408   |
|   173 |    298 |    471 | 0.0173052  | 0.00947385 | 0.0113625  |
|   111 |    169 |    280 | 0.0165154  | 0.00750644 | 0.00957756 |
|  1368 |   1745 |   3113 | 0.0614169  | 0.0315323  | 0.0401087  |
|  6221 |  16303 |  22524 | 0.240184   | 0.142927   | 0.160925   |
|    90 |    131 |    221 | 0.0213472  | 0.00961891 | 0.0123914  |
|   617 |   1606 |   2223 | 0.0756406  | 0.0454726  | 0.0511328  |
|  3971 |   5735 |   9706 | 0.0623724  | 0.0361732  | 0.0436796  |
|   510 |    735 |   1245 | 0.0486781  | 0.0225612  | 0.0289165  |
|    66 |     75 |    141 | 0.0287582  | 0.0119294  | 0.0164297  |
|   956 |   1651 |   2607 | 0.147531   | 0.0927216  | 0.107346   |
|   474 |    515 |    989 | 0.279481   | 0.0963337  | 0.140443   |
|  2372 |   3869 |   6241 | 0.0656282  | 0.0381961  | 0.0454102  |
|   555 |    925 |   1480 | 0.0883055  | 0.0567276  | 0.0655128  |
|   829 |   1471 |   2300 | 0.040376   | 0.026728   | 0.0304362  |
|  4844 |   7032 |  11876 | 0.412326   | 0.269704   | 0.314005   |
|  1776 |   2869 |   4645 | 0.402174   | 0.291062   | 0.32544    |
|  5079 |   8511 |  13590 | 0.10062    | 0.0653476  | 0.0751996  |
|  5050 |  11094 |  16144 | 0.0823092  | 0.0611411  | 0.0664901  |
|    24 |     44 |     68 | 0.0243408  | 0.0148048  | 0.0171804  |
|   417 |    799 |   1216 | 0.0549407  | 0.0321788  | 0.0375077  |
|  1044 |   1875 |   2919 | 0.0413564  | 0.0266748  | 0.0305542  |
|   147 |    238 |    385 | 0.0334852  | 0.0198036  | 0.0234642  |
|  3956 |   8847 |  12803 | 0.0823892  | 0.0566396  | 0.062694   |
|  1946 |   3399 |   5345 | 0.083462   | 0.0461094  | 0.0550849  |
|   996 |   1419 |   2415 | 0.0411247  | 0.0235066  | 0.0285512  |
| 11224 |  17763 |  28987 | 0.09588    | 0.0553483  | 0.0661813  |
|   343 |    598 |    941 | 0.098      | 0.0565859  | 0.0668894  |
|  1629 |   2412 |   4041 | 0.236945   | 0.124793   | 0.154219   |



label: AG=AE+AF
labeled_formulas: AG=AE+AF
output: {"Arithmetic-Relationship": ["AG=AE+AF"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_531021___xlsx_67926fbe75748411d6d181f66c57c07cdda5ef97.xlsx___line_420