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 |
|------:|----:|----:|----:|-----:|----:|----:|-----:|--------:|---------:|----:|--------:|---------:|----------:|----:|:----|----:|
| 42443 |  36 |  18 |  18 | 0.25 | 0.4 | 7.2 | 28.8 | 30      | 1        | 3.8 | 3.8     |  6       |  15       |   0 | NO  |   0 |
| 42444 |  36 |  18 |  18 | 0.26 | 0.4 | 7.2 | 28.8 | 28.5385 | 0.975783 | 3.9 | 3.80556 |  7.46154 |  19.4     |   0 | YES |   0 |
| 42446 |  36 |  18 |  18 | 0.28 | 0.4 | 7.2 | 28.8 | 34.6429 | 1        | 4   | 4       |  1.35714 |   3.8     |   0 | NO  |   0 |
| 42447 |  36 |  18 |  18 | 0.29 | 0.4 | 7.2 | 28.8 | 33.2635 | 1        | 4.2 | 4.2     |  2.73645 |   7.93571 |   0 | NO  |   0 |
| 42448 |  36 |  18 |  18 | 0.3  | 0.4 | 7.2 | 28.8 | 31.8635 | 1        | 3.9 | 3.9     |  4.13645 |  12.4094  |   0 | NO  |   0 |
| 42449 |  36 |  18 |  18 | 0.31 | 0.4 | 7.2 | 28.8 | 30.6055 | 1        | 3.9 | 3.9     |  5.39452 |  16.723   |   0 | NO  |   0 |
| 42450 |  36 |  18 |  18 | 0.32 | 0.4 | 7.2 | 28.8 | 29.3867 | 1        | 4.2 | 4.2     |  6.61327 |  21.1625  |   0 | NO  |   0 |
| 42451 |  36 |  18 |  18 | 0.33 | 0.4 | 7.2 | 28.8 | 28.114  | 0.936482 | 4.2 | 3.93322 |  7.88599 |  26.0238  |   0 | YES |   0 |
| 42453 |  36 |  18 |  18 | 0.35 | 0.4 | 7.2 | 28.8 | 33.1387 | 1        | 4.3 | 4.3     |  2.86131 |  10.0146  |   0 | NO  |   0 |
| 42454 |  36 |  18 |  18 | 0.36 | 0.4 | 7.2 | 28.8 | 31.9442 | 1        | 4.2 | 4.2     |  4.05576 |  14.6007  |   0 | NO  |   0 |
| 42455 |  36 |  18 |  18 | 0.37 | 0.4 | 7.2 | 28.8 | 30.8091 | 1        | 4.3 | 4.3     |  5.19089 |  19.2063  |   0 | NO  |   0 |
| 42456 |  36 |  18 |  18 | 0.38 | 0.4 | 7.2 | 28.8 | 29.6775 | 1        | 4.4 | 4.4     |  6.32247 |  24.0254  |   0 | NO  |   0 |
| 42457 |  36 |  18 |  18 | 0.39 | 0.4 | 7.2 | 28.8 | 28.5493 | 0.976789 | 4.5 | 4.39555 |  7.45068 |  29.0576  |   0 | YES |   0 |
| 42458 |  36 |  18 |  18 | 0.4  | 0.4 | 7.2 | 28.8 | 27.4504 | 0.87504  | 4.6 | 4.02519 |  8.54956 |  34.1983  |   0 | YES |   0 |
| 42460 |  36 |  18 |  18 | 0.42 | 0.4 | 7.2 | 28.8 | 31.4472 | 1        | 4.8 | 4.8     |  4.5528  |  19.1218  |   0 | NO  |   0 |
| 42461 |  36 |  18 |  18 | 0.43 | 0.4 | 7.2 | 28.8 | 30.3309 | 1        | 4.8 | 4.8     |  5.66908 |  24.3771  |   0 | NO  |   0 |
| 42462 |  36 |  18 |  18 | 0.44 | 0.4 | 7.2 | 28.8 | 29.24   | 1        | 5   | 5       |  6.75999 |  29.744   |   0 | NO  |   0 |
| 42463 |  36 |  18 |  18 | 0.45 | 0.4 | 7.2 | 28.8 | 28.1289 | 0.937861 | 4.9 | 4.59552 |  7.8711  |  35.42    |   0 | YES |   0 |
| 42464 |  36 |  18 |  18 | 0.46 | 0.4 | 7.2 | 28.8 | 27.1299 | 0.845358 | 5.2 | 4.39586 |  8.87013 |  40.8026  |   0 | YES |   0 |
| 42465 |  36 |  18 |  18 | 0.47 | 0.4 | 7.2 | 28.8 | 26.1946 | 0.758758 | 5.5 | 4.17317 |  9.80542 |  46.0855  |   0 | YES |   0 |
| 42467 |  36 |  18 |  18 | 0.49 | 0.4 | 7.2 | 28.8 | 29.3702 | 1        | 6.7 | 6.7     |  6.62976 |  32.4858  |   0 | NO  |   0 |
| 42468 |  36 |  18 |  18 | 0.5  | 0.4 | 7.2 | 28.8 | 28.0302 | 0.928726 | 6   | 5.57236 |  7.96976 |  39.8488  |   0 | YES |   0 |
| 42469 |  36 |  18 |  18 | 0.51 | 0.4 | 7.2 | 28.8 | 26.9376 | 0.827557 | 6.3 | 5.21361 |  9.06238 |  46.2181  |   0 | YES |   0 |
| 42470 |  36 |  18 |  18 | 0.52 | 0.4 | 7.2 | 28.8 | 25.935  | 0.734723 | 6.9 | 5.06959 | 10.065   |  52.338   |   0 | YES |   0 |
| 42471 |  36 |  18 |  18 | 0.53 | 0.4 | 7.2 | 28.8 | 24.9785 | 0.646155 | 7   | 4.52309 | 11.0215  |  58.4141  |   0 | YES |   0 |
| 42472 |  36 |  18 |  18 | 0.54 | 0.4 | 7.2 | 28.8 | 24.1409 | 0.568599 | 6.8 | 3.86647 | 11.8591  |  64.0393  |   0 | YES |   0 |
| 42474 |  36 |  18 |  18 | 0.56 | 0.4 | 7.2 | 28.8 | 27.1415 | 0.846433 | 4   | 3.38573 |  8.85852 |  49.6077  |   0 | YES |   0 |
| 42475 |  36 |  18 |  18 | 0.57 | 0.4 | 7.2 | 28.8 | 26.5475 | 0.791434 | 5.3 | 4.1946  |  9.45251 |  53.8793  |   0 | YES |   0 |
| 42476 |  36 |  18 |  18 | 0.58 | 0.4 | 7.2 | 28.8 | 25.8243 | 0.724471 | 6.2 | 4.49172 | 10.1757  |  59.0191  |   0 | YES |   0 |
| 42477 |  36 |  18 |  18 | 0.59 | 0.4 | 7.2 | 28.8 | 25.063  | 0.653979 | 6.3 | 4.12007 | 10.937   |  64.5284  |   0 | YES |   0 |
| 42478 |  36 |  18 |  18 | 0.6  | 0.4 | 7.2 | 28.8 | 24.3763 | 0.590398 | 6.3 | 3.71951 | 11.6237  |  69.7422  |   0 | YES |   0 |
| 42479 |  36 |  18 |  18 | 0.61 | 0.4 | 7.2 | 28.8 | 23.7665 | 0.533939 | 5.8 | 3.09685 | 12.2335  |  74.6241  |   0 | YES |   0 |
| 42481 |  36 |  18 |  18 | 0.63 | 0.4 | 7.2 | 28.8 | 26.6159 | 0.797772 | 4.5 | 3.58997 |  9.38406 |  59.1196  |   0 | YES |   0 |
| 42482 |  36 |  18 |  18 | 0.64 | 0.4 | 7.2 | 28.8 | 26.055  | 0.745834 | 6.2 | 4.62417 |  9.94499 |  63.648   |   0 | YES |   0 |
| 42483 |  36 |  18 |  18 | 0.65 | 0.4 | 7.2 | 28.8 | 25.3436 | 0.679962 | 7.1 | 4.82773 | 10.6564  |  69.2666  |   0 | YES |   0 |
| 42484 |  36 |  18 |  18 | 0.66 | 0.4 | 7.2 | 28.8 | 24.6121 | 0.612233 | 5.9 | 3.61218 | 11.3879  |  75.16    |   0 | YES |   0 |
| 42485 |  36 |  18 |  18 | 0.67 | 0.4 | 7.2 | 28.8 | 24.073  | 0.562314 | 6.1 | 3.43011 | 11.927   |  79.911   |   0 | YES |   0 |
| 42486 |  36 |  18 |  18 | 0.68 | 0.4 | 7.2 | 28.8 | 23.5686 | 0.515607 | 5.8 | 2.99052 | 12.4314  |  84.5338  |   0 | YES |   0 |
| 42488 |  36 |  18 |  18 | 0.7  | 0.4 | 7.2 | 28.8 | 26.098  | 0.749814 | 6.2 | 4.64885 |  9.90201 |  69.3141  |   0 | YES |   0 |
| 42489 |  36 |  18 |  18 | 0.71 | 0.4 | 7.2 | 28.8 | 25.4432 | 0.689187 | 6.3 | 4.34188 | 10.5568  |  74.9531  |   0 | YES |   0 |
| 42490 |  36 |  18 |  18 | 0.72 | 0.4 | 7.2 | 28.8 | 24.8402 | 0.63335  | 7.5 | 4.75013 | 11.1598  |  80.3507  |   0 | YES |   0 |
| 42491 |  36 |  18 |  18 | 0.73 | 0.4 | 7.2 | 28.8 | 24.1895 | 0.5731   | 8   | 4.5848  | 11.8105  |  86.2168  |   0 | YES |   0 |
| 42492 |  36 |  18 |  18 | 0.74 | 0.4 | 7.2 | 28.8 | 23.5699 | 0.515733 | 6.9 | 3.55856 | 12.4301  |  91.9826  |   0 | YES |   0 |
| 42493 |  36 |  18 |  18 | 0.75 | 0.4 | 7.2 | 28.8 | 23.0954 | 0.4718   | 5.9 | 2.78362 | 12.9046  |  96.7842  |   0 | YES |   0 |
| 42495 |  36 |  18 |  18 | 0.77 | 0.4 | 7.2 | 28.8 | 25.3533 | 0.680863 | 5.8 | 3.949   | 10.6467  |  81.9794  |   0 | YES |   0 |
| 42496 |  36 |  18 |  18 | 0.78 | 0.4 | 7.2 | 28.8 | 24.847  | 0.633985 | 6.1 | 3.86731 | 11.153   |  86.9931  |   0 | YES |   0 |
| 42497 |  36 |  18 |  18 | 0.79 | 0.4 | 7.2 | 28.8 | 24.3575 | 0.588658 | 7.1 | 4.17947 | 11.6425  |  91.9757  |   0 | YES |   0 |
| 42498 |  36 |  18 |  18 | 0.8  | 0.4 | 7.2 | 28.8 | 23.8351 | 0.540284 | 5.9 | 3.18768 | 12.1649  |  97.3194  |   0 | YES |   0 |
| 42499 |  36 |  18 |  18 | 0.81 | 0.4 | 7.2 | 28.8 | 23.4415 | 0.503845 | 6.1 | 3.07346 | 12.5585  | 101.724   |   0 | YES |   0 |
| 42500 |  36 |  18 |  18 | 0.82 | 0.4 | 7.2 | 28.8 | 23.0667 | 0.469141 | 6.1 | 2.86176 | 12.9333  | 106.053   |   0 | YES |   0 |
| 42502 |  36 |  18 |  18 | 0.84 | 0.4 | 7.2 | 28.8 | 25.1798 | 0.6648   | 7   | 4.6536  | 10.8202  |  90.8893  |   0 | YES |   0 |
| 42503 |  36 |  18 |  18 | 0.85 | 0.4 | 7.2 | 28.8 | 24.6324 | 0.614108 | 6.9 | 4.23734 | 11.3676  |  96.6249  |   0 | YES |   0 |
| 42504 |  36 |  18 |  18 | 0.86 | 0.4 | 7.2 | 28.8 | 24.1396 | 0.568486 | 5.5 | 3.12667 | 11.8604  | 101.999   |   0 | YES |   0 |
| 42505 |  36 |  18 |  18 | 0.87 | 0.4 | 7.2 | 28.8 | 23.7803 | 0.535209 | 6.9 | 3.69294 | 12.2197  | 106.312   |   0 | YES |   0 |
| 42506 |  36 |  18 |  18 | 0.88 | 0.4 | 7.2 | 28.8 | 23.3606 | 0.496352 | 6.8 | 3.3752  | 12.6394  | 111.227   |   0 | YES |   0 |
| 42507 |  36 |  18 |  18 | 0.89 | 0.4 | 7.2 | 28.8 | 22.9814 | 0.461238 | 5.5 | 2.53681 | 13.0186  | 115.866   |   0 | YES |   0 |
| 42509 |  36 |  18 |  18 | 0.91 | 0.4 | 7.2 | 28.8 | 24.9904 | 0.647256 | 7.3 | 4.72497 | 11.0096  | 100.188   |   0 | YES |   0 |
| 42510 |  36 |  18 |  18 | 0.92 | 0.4 | 7.2 | 28.8 | 24.4768 | 0.599702 | 6.1 | 3.65818 | 11.5232  | 106.014   |   0 | YES |   0 |
| 42511 |  36 |  18 |  18 | 0.93 | 0.4 | 7.2 | 28.8 | 24.0834 | 0.56328  | 5.1 | 2.87273 | 11.9166  | 110.824   |   0 | YES |   0 |
| 42512 |  36 |  18 |  18 | 0.94 | 0.4 | 7.2 | 28.8 | 23.7778 | 0.534983 | 6.2 | 3.31689 | 12.2222  | 114.889   |   0 | YES |   0 |
| 42513 |  36 |  18 |  18 | 0.95 | 0.4 | 7.2 | 28.8 | 23.4287 | 0.502654 | 6   | 3.01593 | 12.5713  | 119.428   |   0 | YES |   0 |
| 42514 |  36 |  18 |  18 | 0.94 | 0.4 | 7.2 | 28.8 | 23.1078 | 0.472947 | 6.5 | 3.07415 | 12.8922  | 121.186   |   0 | YES |   0 |
| 42516 |  36 |  18 |  18 | 0.92 | 0.4 | 7.2 | 28.8 | 24.962  | 0.644633 | 5.7 | 3.67441 | 11.038   | 101.549   |   0 | YES |   0 |
| 42517 |  36 |  18 |  18 | 0.91 | 0.4 | 7.2 | 28.8 | 24.5583 | 0.607246 | 5.2 | 3.15768 | 11.4417  | 104.12    |   0 | YES |   0 |
| 42518 |  36 |  18 |  18 | 0.9  | 0.4 | 7.2 | 28.8 | 24.2074 | 0.574759 | 5.6 | 3.21865 | 11.7926  | 106.133   |   0 | YES |   0 |
| 42519 |  36 |  18 |  18 | 0.89 | 0.4 | 7.2 | 28.8 | 23.8458 | 0.541274 | 6   | 3.24764 | 12.1542  | 108.173   |   0 | YES |   0 |
| 42520 |  36 |  18 |  18 | 0.88 | 0.4 | 7.2 | 28.8 | 23.4767 | 0.507102 | 6   | 3.04261 | 12.5233  | 110.205   |   0 | YES |   0 |
| 42521 |  36 |  18 |  18 | 0.87 | 0.4 | 7.2 | 28.8 | 23.127  | 0.47472  | 6.1 | 2.89579 | 12.873   | 111.995   |   0 | YES |   0 |
| 42523 |  36 |  18 |  18 | 0.85 | 0.4 | 7.2 | 28.8 | 25.2025 | 0.666901 | 6.5 | 4.33486 | 10.7975  |  91.7785  |   0 | YES |   0 |
| 42524 |  36 |  18 |  18 | 0.84 | 0.4 | 7.2 | 28.8 | 24.6865 | 0.619118 | 6.8 | 4.21001 | 11.3135  |  95.0336  |   0 | YES |   0 |
| 42525 |  36 |  18 |  18 | 0.83 | 0.4 | 7.2 | 28.8 | 24.1792 | 0.572153 | 6.5 | 3.71899 | 11.8208  |  98.1122  |   0 | YES |   0 |
| 42526 |  36 |  18 |  18 | 0.82 | 0.4 | 7.2 | 28.8 | 23.7257 | 0.530159 | 6.4 | 3.39302 | 12.2743  | 100.649   |   0 | YES |   0 |
| 42527 |  36 |  18 |  18 | 0.81 | 0.4 | 7.2 | 28.8 | 23.3068 | 0.491373 | 6.3 | 3.09565 | 12.6932  | 102.815   |   0 | YES |   0 |



label: D=B-C
G=D*F
L=K*J
M=B-I
H=B-D*F
labeled_formulas: D=B-C
G=D*F
L=K*J
M=B-I
Q=A*O
Q=B*O
O=C-D
Q=C*O
Q=D*O
Q=E*O
Q=F*O
Q=G*O
Q=H*O
Q=I*O
Q=J*O
Q=K*O
Q=L*O
Q=M*O
Q=N*O
H=B-D*F
Q=(A+B)*O
D=(A*O)+C
Q=(A+C)*O
Q=(A+D)*O
Q=(A+E)*O
Q=(A+F)*O
Q=(A+G)*O
Q=(A+H)*O
Q=(A+I)*O
Q=(A+J)*O
Q=(A+K)*O
Q=(A+L)*O
Q=(A+M)*O
Q=(A+N)*O
O=(B-C)-D
Q=(B+C)*O
Q=(B+D)*O
Q=(B+E)*O
Q=(B+F)*O
O=(B-G)-H
Q=(B+G)*O
Q=(B+H)*O
M=(B-I)+O
Q=(B+I)*O
Q=(B+J)*O
Q=(B+K)*O
Q=(B+L)*O
Q=(B+M)*O
Q=(B+N)*O
O=(C-D)*E
O=(C-D)*F
O=(C-D)*G
O=(C-D)*H
O=(C-D)*I
O=(C-D)*J
O=(C-D)*K
O=(C-D)*L
D=(M*O)+C
D=(N*O)+C
Q=(C+E)*O
O=(C*F)-G
Q=(C+F)*O
output: {"Arithmetic-Relationship": ["D=B-C", "G=D*F", "L=K*J", "M=B-I", "H=B-D*F"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_11770___xlsx_6cdbded4871ac96244fc6e0a2721697f83ca2bc1.xlsx___line_419