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 |   K |       L |        M |   N |   O |   P |        Q |   R |   S |   T |         U |   V |   W |   X |         Y |   Z |   AA |   AB |        AC |      AD |        AE |        AF |        AG |      AH |      AI |
|:------|:----|-----:|----------:|----------:|----------:|----------:|----------:|----------:|---------:|----:|--------:|---------:|----:|----:|----:|---------:|----:|----:|----:|----------:|----:|----:|----:|----------:|----:|-----:|-----:|----------:|--------:|----------:|----------:|----------:|--------:|--------:|
| NES21 | NES | 2021 | 13.641    |  69.86    |  73.289   |  83.501   | 143.149   | 156.79    | 0.202    |   0 |  3.819  | 1.189    | 0.1 |   0 |   0 | 1.292    |   0 |   0 |   0 | 0.038     |   0 |   0 |   0 | 0.296     |   0 |    0 |    0 | 0         | 15.016  |  84.782   |  93.003   |  99.798   | 177.785 | 192.801 |
| NES22 | NES | 2022 | 13.555    |  67.124   |  78.028   |  80.679   | 145.152   | 158.707   | 0.202    |   0 |  3.827  | 1.443    | 0.1 |   0 |   0 | 1.567    |   0 |   0 |   0 | 0.046     |   0 |   0 |   0 | 0.359     |   0 |    0 |    0 | 0         | 14.911  | 102.033   | 101.685   | 116.944   | 203.718 | 218.629 |
| NES23 | NES | 2023 | 13.467    |  76.298   |  87.626   |  89.765   | 163.924   | 177.391   | 0.202    |   0 |  3.834  | 1.457    | 0.1 |   0 |   0 | 1.583    |   0 |   0 |   0 | 0.046     |   0 |   0 |   0 | 0.363     |   0 |    0 |    0 | 0         | 14.823  | 126.581   | 102.213   | 141.404   | 228.794 | 243.617 |
| NES24 | NES | 2024 | 13.382    | 115.917   |  89.939   | 129.299   | 205.856   | 219.238   | 0.202    |   0 |  3.842  | 1.471    | 0.1 |   0 |   0 | 1.599    |   0 |   0 |   0 | 0.047     |   0 |   0 |   0 | 0.366     |   0 |    0 |    0 | 0         | 14.738  | 189.75    | 103.08    | 204.488   | 292.83  | 307.568 |
| NES25 | NES | 2025 | 13.299    |  88.377   |  67.543   | 101.676   | 155.92    | 169.219   | 0.202    |   0 |  3.85   | 1.485    | 0.1 |   0 |   0 | 1.614    |   0 |   0 |   0 | 0.047     |   0 |   0 |   0 | 0.37      |   0 |    0 |    0 | 0         | 14.655  | 157.327   |  80.272   | 171.982   | 237.599 | 252.254 |
| NWT21 | NWT | 2021 | 68.0885   | 191.934   | 165.165   | 260.023   | 357.099   | 425.188   | 0.127108 |   0 | 14.6034 | 4.24074  | 0   |   0 |   0 | 7.53211  |   0 |   0 |   0 | 0.70062   |   0 |   0 |   0 | 1.9156    |   0 |    0 |    0 | 0.386112  | 76.8732 | 308.151   | 172.082   | 385.024   | 480.233 | 557.106 |
| NWT22 | NWT | 2022 | 70.3664   | 204.639   | 168.002   | 275.006   | 372.641   | 443.007   | 0.127731 |   0 | 14.7578 | 3.14232  | 0   |   0 |   0 | 5.58118  |   0 |   0 |   0 | 0.522913  |   0 |   0 |   0 | 1.43053   |   0 |    0 |    0 | 0.288307  | 79.2476 | 332.365   | 176.469   | 411.613   | 508.834 | 588.082 |
| NWT23 | NWT | 2023 | 65.7172   | 202.306   | 166.506   | 268.023   | 368.812   | 434.53    | 0.128044 |   0 | 14.9139 | 0.191702 | 0   |   0 |   0 | 0.339986 |   0 |   0 |   0 | 0.0331644 |   0 |   0 |   0 | 0.0908113 |   0 |    0 |    0 | 0.0182986 | 74.6218 | 289.893   | 176.835   | 364.515   | 466.728 | 541.35  |
| NWT24 | NWT | 2024 | 60.8429   | 248.689   | 166.72    | 309.532   | 415.409   | 476.252   | 0.127731 |   0 | 22.6713 | 0.167572 | 0   |   0 |   0 | 0.29704  |   0 |   0 |   0 | 0.029676  |   0 |   0 |   0 | 0.0813474 |   0 |    0 |    0 | 0.016388  | 69.7561 | 456.323   | 202.863   | 526.079   | 659.186 | 728.942 |
| NWT25 | NWT | 2025 | 63.0299   | 222.976   | 166.73    | 286.005   | 389.706   | 452.736   | 0.127731 |   0 | 22.438  | 0.106619 | 0   |   0 |   0 | 0.188993 |   0 |   0 |   0 | 0.0193373 |   0 |   0 |   0 | 0.0531014 |   0 |    0 |    0 | 0.0106938 | 71.9517 | 366.849   | 220.139   | 438.801   | 586.988 | 658.94  |
| SWB21 | SWB | 2021 | 17.778    |  74.399   |  66.317   |  92.177   | 140.716   | 158.494   | 1.59     |   0 |  8.277  | 1.346    | 0   |   0 |   0 | 2.723    |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0.731     |   0 |    0 |    0 | 0.079     | 19.74   | 113.061   |  78.891   | 132.801   | 191.952 | 211.692 |
| SWB22 | SWB | 2022 | 17.98     |  74.645   |  66.421   |  92.625   | 141.066   | 159.046   | 1.59     |   0 |  9.549  | 1.359    | 0   |   0 |   0 | 2.75     |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0.739     |   0 |    0 |    0 | 0.08      | 20.143  | 109.758   |  78.548   | 129.901   | 188.306 | 208.449 |
| SWB23 | SWB | 2023 | 18.094    |  70.318   |  62.54    |  88.412   | 132.858   | 150.952   | 1.59     |   0 | 10.686  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 20.808  | 102.364   |  70.087   | 123.172   | 172.451 | 193.259 |
| SWB24 | SWB | 2024 | 18.04     |  71.337   |  62.281   |  89.377   | 133.618   | 151.658   | 1.59     |   0 | 10.744  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 20.478  |  99.836   |  71.291   | 120.314   | 171.127 | 191.605 |
| SWB25 | SWB | 2025 | 17.988    |  72.352   |  59.954   |  90.34    | 132.306   | 150.294   | 1.59     |   0 | 10.673  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 19.954  |  89.86    |  68.441   | 109.814   | 158.301 | 178.255 |
| WSH21 | WSH | 2021 | 23.757    | 108.924   |  94.361   | 132.681   | 203.285   | 227.042   | 0.199    |   0 |  9.689  | 0.41     | 0   |   0 |   0 | 0.622    |   0 |   0 |   0 | 0.023     |   0 |   0 |   0 | 0.126     |   0 |    0 |    0 | 0.044     | 25.701  | 141.555   | 199.991   | 167.256   | 341.546 | 367.247 |
| WSH22 | WSH | 2022 | 23.198    | 108.841   |  97.831   | 132.039   | 206.672   | 229.87    | 0.199    |   0 |  9.782  | 0.402    | 0   |   0 |   0 | 0.61     |   0 |   0 |   0 | 0.022     |   0 |   0 |   0 | 0.123     |   0 |    0 |    0 | 0.045     | 25.306  | 145.575   | 130.241   | 170.881   | 275.816 | 301.122 |
| WSH23 | WSH | 2023 | 23.457    | 111.839   |  98.491   | 135.296   | 210.33    | 233.787   | 0.199    |   0 |  9.87   | 0.395    | 0   |   0 |   0 | 0.597    |   0 |   0 |   0 | 0.022     |   0 |   0 |   0 | 0.122     |   0 |    0 |    0 | 0.044     | 25.556  | 155.826   | 129.495   | 181.382   | 285.321 | 310.877 |
| WSH24 | WSH | 2024 | 22.682    | 109.95    |  94.557   | 132.632   | 204.507   | 227.189   | 0.199    |   0 |  9.901  | 0.386    | 0   |   0 |   0 | 0.586    |   0 |   0 |   0 | 0.021     |   0 |   0 |   0 | 0.118     |   0 |    0 |    0 | 0.044     | 24.784  | 155.219   | 119.789   | 180.003   | 275.008 | 299.792 |
| WSH25 | WSH | 2025 | 22.016    | 106.967   |  93.71    | 128.983   | 200.677   | 222.693   | 0.199    |   0 |  9.916  | 0.379    | 0   |   0 |   0 | 0.574    |   0 |   0 |   0 | 0.021     |   0 |   0 |   0 | 0.117     |   0 |    0 |    0 | 0.041     | 24.112  | 154.291   | 118.384   | 178.403   | 272.675 | 296.787 |
| WSX21 | WSX | 2021 | 21.9509   |  88.5011  |  77.7131  | 110.452   | 166.214   | 188.165   | 0.10201  |   0 |  9.1485 | 2.07403  | 0   |   0 |   0 | 3.08197  |   0 |   0 |   0 | 0.775342  |   0 |   0 |   0 | 0.484589  |   0 |    0 |    0 | 0.155068  | 23.6129 | 197.963   | 112.001   | 221.576   | 309.964 | 333.577 |
| WSX22 | WSX | 2022 | 21.2272   |  91.527   |  83.3056  | 112.754   | 174.833   | 196.06    | 0.10201  |   0 |  9.1485 | 2.08047  | 0   |   0 |   0 | 3.09153  |   0 |   0 |   0 | 0.778082  |   0 |   0 |   0 | 0.486301  |   0 |    0 |    0 | 0.155616  | 22.8892 | 166.143   | 122.205   | 189.033   | 288.349 | 311.238 |
| WSX23 | WSX | 2023 | 23.4212   |  87.4622  |  91.1606  | 110.883   | 178.623   | 202.044   | 0.10201  |   0 |  9.1485 | 2.0861   | 0   |   0 |   0 | 3.0999   |   0 |   0 |   0 | 0.779726  |   0 |   0 |   0 | 0.487329  |   0 |    0 |    0 | 0.155945  | 25.7085 | 178.357   | 107.202   | 204.066   | 285.56  | 311.268 |
| WSX24 | WSX | 2024 | 21.2192   | 101.837   |  84.6873  | 123.056   | 186.524   | 207.743   | 0.10201  |   0 |  9.1485 | 2.09414  | 0   |   0 |   0 | 3.11186  |   0 |   0 |   0 | 0.783014  |   0 |   0 |   0 | 0.489384  |   0 |    0 |    0 | 0.156603  | 23.9233 | 185.095   |  95.1101  | 209.019   | 280.205 | 304.129 |
| WSX25 | WSX | 2025 | 23.0753   |  98.266   |  86.6127  | 121.341   | 184.879   | 207.954   | 0.10201  |   0 |  9.1485 | 2.09897  | 0   |   0 |   0 | 3.11903  |   0 |   0 |   0 | 0.784658  |   0 |   0 |   0 | 0.490411  |   0 |    0 |    0 | 0.156932  | 27.2382 | 173.385   |  93.2633  | 200.623   | 266.648 | 293.886 |
| YKY21 | YKY | 2021 | 69.475    | 199.22    | 182.025   | 268.695   | 381.245   | 450.72    | 0        |   0 | 10.774  | 2.094    | 0   |   0 |   0 | 2.682    |   0 |   0 |   0 | 0.275     |   0 |   0 |   0 | 0.974     |   0 |    0 |    0 | 0.035     | 70.662  | 396.707   | 230.162   | 467.369   | 626.869 | 697.531 |
| YKY22 | YKY | 2022 | 70.104    | 215.147   | 151.826   | 285.251   | 366.973   | 437.077   | 0        |   0 | 10.853  | 2.094    | 0   |   0 |   0 | 2.682    |   0 |   0 |   0 | 0.275     |   0 |   0 |   0 | 0.974     |   0 |    0 |    0 | 0.035     | 71.64   | 438.826   | 191.012   | 510.466   | 629.838 | 701.478 |
| YKY23 | YKY | 2023 | 59.126    | 195.767   | 133.897   | 254.893   | 329.664   | 388.79    | 0        |   0 | 10.953  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 68.708  | 367.513   | 167.954   | 436.221   | 535.467 | 604.175 |
| YKY24 | YKY | 2024 | 53.111    | 170.848   | 117.49    | 223.959   | 288.338   | 341.449   | 0        |   0 | 11.047  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 94.87   | 281.044   | 150.919   | 375.914   | 431.963 | 526.833 |
| YKY25 | YKY | 2025 | 52.54     | 137.33    | 115.277   | 189.87    | 252.607   | 305.147   | 0        |   0 | 11.146  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         | 66.139  | 188.575   | 145.437   | 254.714   | 334.012 | 400.151 |
| SVH21 | SVH | 2021 | 56.9292   | 212.574   | 214.576   | 269.504   | 427.151   | 484.08    | 0.699015 |   0 | 29.1178 | 4.68951  | 0   |   0 |   0 | 9.21675  |   0 |   0 |   0 | 0.0270445 |   0 |   0 |   0 | 2.36909   |   0 |    0 |    0 | 1.69839   | 65.5743 | 285.774   | 224.504   | 351.348   | 510.278 | 575.852 |
| SVH22 | SVH | 2022 | 55.245    | 223.821   | 205.049   | 279.066   | 428.87    | 484.115   | 0.703175 |   0 | 17.1671 | 4.64658  | 0   |   0 |   0 | 9.13237  |   0 |   0 |   0 | 0.0267969 |   0 |   0 |   0 | 2.34741   |   0 |    0 |    0 | 1.68284   | 65.8585 | 336.897   | 215.205   | 402.755   | 552.102 | 617.961 |
| SVH23 | SVH | 2023 | 53.3831   | 226.176   | 188.849   | 279.559   | 415.025   | 468.409   | 0.707165 |   0 | 17.3837 | 4.60252  | 0   |   0 |   0 | 9.04578  |   0 |   0 |   0 | 0.0265428 |   0 |   0 |   0 | 2.32515   |   0 |    0 |    0 | 1.66689   | 65.0803 | 354.743   | 205.46    | 419.823   | 560.203 | 625.284 |
| SVH24 | SVH | 2024 | 50.9287   | 221.57    | 184.935   | 272.499   | 406.505   | 457.434   | 0.710949 |   0 | 17.3021 | 4.55939  | 0   |   0 |   0 | 8.96101  |   0 |   0 |   0 | 0.0262941 |   0 |   0 |   0 | 2.30336   |   0 |    0 |    0 | 1.65127   | 62.7271 | 351.087   | 202.035   | 413.814   | 553.122 | 615.849 |
| SVH25 | SVH | 2025 | 45.8024   | 205.226   | 176.44    | 251.029   | 381.666   | 427.469   | 0.714622 |   0 | 18.6825 | 4.51843  | 0   |   0 |   0 | 8.8805   |   0 |   0 |   0 | 0.0260578 |   0 |   0 |   0 | 2.28266   |   0 |    0 |    0 | 1.63643   | 56.0965 | 303.422   | 191.133   | 359.518   | 494.555 | 550.652 |
| SVE21 | SVE | 2021 | 56.3602   | 210.106   | 213.367   | 266.467   | 423.473   | 479.833   | 0.699015 |   0 | 29.0528 | 4.68951  | 0   |   0 |   0 | 9.21675  |   0 |   0 |   0 | 0.0270445 |   0 |   0 |   0 | 2.36909   |   0 |    0 |    0 | 1.69839   | 64.8763 | 283.143   | 223.269   | 348.019   | 506.413 | 571.289 |
| SVE22 | SVE | 2022 | 54.6775   | 221.271   | 203.844   | 275.948   | 425.115   | 479.792   | 0.703175 |   0 | 17.0981 | 4.64658  | 0   |   0 |   0 | 9.13237  |   0 |   0 |   0 | 0.0267969 |   0 |   0 |   0 | 2.34741   |   0 |    0 |    0 | 1.68284   | 65.1325 | 333.93    | 213.975   | 399.063   | 547.905 | 613.038 |
| SVE23 | SVE | 2023 | 52.8155   | 223.646   | 187.648   | 276.462   | 411.295   | 464.11    | 0.707165 |   0 | 17.3107 | 4.60252  | 0   |   0 |   0 | 9.04578  |   0 |   0 |   0 | 0.0265428 |   0 |   0 |   0 | 2.32515   |   0 |    0 |    0 | 1.66689   | 64.3543 | 351.572   | 204.234   | 415.926   | 555.805 | 620.16  |
| SVE24 | SVE | 2024 | 50.3609   | 219.044   | 183.742   | 269.405   | 402.785   | 453.146   | 0.710949 |   0 | 17.2271 | 4.55939  | 0   |   0 |   0 | 8.96101  |   0 |   0 |   0 | 0.0262941 |   0 |   0 |   0 | 2.30336   |   0 |    0 |    0 | 1.65127   | 62.0011 | 346.794   | 200.816   | 408.795   | 547.61  | 609.611 |
| SVE25 | SVE | 2025 | 45.2345   | 202.7     | 175.244   | 247.934   | 377.943   | 423.178   | 0.714622 |   0 | 18.6075 | 4.51843  | 0   |   0 |   0 | 8.8805   |   0 |   0 |   0 | 0.0260578 |   0 |   0 |   0 | 2.28266   |   0 |    0 |    0 | 1.63643   | 55.3705 | 300.704   | 189.911   | 356.074   | 490.615 | 545.986 |
| HDD21 | HDD | 2021 |  0.569017 |   2.46813 |   1.20936 |   3.03715 |   3.67749 |   4.24651 | 0        |   0 |  0.065  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         |  0.698  |   2.63042 |   1.23458 |   3.32842 |   3.865 |   4.563 |
| HDD22 | HDD | 2022 |  0.567511 |   2.54978 |   1.20508 |   3.1173  |   3.75486 |   4.32237 | 0        |   0 |  0.069  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         |  0.726  |   2.96663 |   1.23037 |   3.69263 |   4.197 |   4.923 |
| HDD23 | HDD | 2023 |  0.567667 |   2.52966 |   1.20123 |   3.09733 |   3.73089 |   4.29856 | 0        |   0 |  0.073  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         |  0.726  |   3.17139 |   1.22661 |   3.89739 |   4.398 |   5.124 |
| HDD24 | HDD | 2024 |  0.567818 |   2.52628 |   1.19345 |   3.0941  |   3.71972 |   4.28754 | 0        |   0 |  0.075  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         |  0.726  |   4.29309 |   1.21891 |   5.01909 |   5.512 |   6.238 |
| HDD25 | HDD | 2025 |  0.567903 |   2.52674 |   1.19631 |   3.09464 |   3.72305 |   4.29095 | 0        |   0 |  0.075  | 0        | 0   |   0 |   0 | 0        |   0 |   0 |   0 | 0         |   0 |   0 |   0 | 0         |   0 |    0 |    0 | 0         |  0.726  |   2.71814 |   1.22186 |   3.44414 |   3.94  |   4.666 |



label: AG=AD+AE
AH=AE+AF
AI=AD+AE+AF
labeled_formulas: AG=AD+AE
AH=AE+AF
O=C*K
P=C*K
S=C*K
T=C*K
W=C*K
X=C*K
AA=C*K
AB=C*K
R=C*O
V=C*O
Z=C*O
R=C*P
V=C*P
Z=C*P
S=C*R
T=C*R
V=C*R
W=C*R
X=C*R
Z=C*R
AA=C*R
AB=C*R
V=C*S
Z=C*S
V=C*T
Z=C*T
W=C*V
X=C*V
Z=C*V
AA=C*V
AB=C*V
Z=C*W
Z=C*X
AA=C*Z
AB=C*Z
O=D*K
P=D*K
S=D*K
T=D*K
W=D*K
X=D*K
AA=D*K
AB=D*K
R=D*O
V=D*O
Z=D*O
R=D*P
V=D*P
Z=D*P
S=D*R
AI=AD+AE+AF
O=(C+D)*K
P=(C+D)*K
S=(C+D)*K
T=(C+D)*K
W=(C+D)*K
X=(C+D)*K
AA=(C+D)*K
AB=(C+D)*K
R=(C+D)*O
V=(C+D)*O
Z=(C+D)*O
R=(C+D)*P
V=(C+D)*P
Z=(C+D)*P
S=(C+D)*R
T=(C+D)*R
V=(C+D)*R
W=(C+D)*R
X=(C+D)*R
Z=(C+D)*R
AA=(C+D)*R
AB=(C+D)*R
V=(C+D)*S
Z=(C+D)*S
V=(C+D)*T
Z=(C+D)*T
W=(C+D)*V
X=(C+D)*V
Z=(C+D)*V
AA=(C+D)*V
AB=(C+D)*V
Z=(C+D)*W
Z=(C+D)*X
AA=(C+D)*Z
AB=(C+D)*Z
O=(C+E)*K
P=(C+E)*K
S=(C+E)*K
T=(C+E)*K
W=(C+E)*K
X=(C+E)*K
AA=(C+E)*K
AB=(C+E)*K
R=(C+E)*O
V=(C+E)*O
Z=(C+E)*O
R=(C+E)*P
V=(C+E)*P
Z=(C+E)*P
S=(C+E)*R
output: {"Arithmetic-Relationship": ["AG=AD+AE", "AH=AE+AF", "AI=AD+AE+AF"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_690039___xlsx_aa995ce0a336d8fb36c0bfdb6e027a2f362c967a.xlsx___line_517