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 |
|:------------------------------------|----------:|-------:|----:|------------:|------------:|-----------:|------------:|------------:|-----------:|------------:|------------:|------------:|------------:|------------:|----------:|------------:|------------:|---------:|------------:|------------:|---------:|------------:|------------:|
| Rice-01 IAC600, purple              |  83830000 | 1.2325 |   1 | 1.0332e+08  | 1.0332e+08  |  354900000 | 4.37414e+08 | 4.37414e+08 |          0 | 0           | 0           |  3727000000 | 4.59353e+09 | 4.59353e+09 | 245400000 | 3.02456e+08 | 3.02456e+08 | 57800000 | 7.12385e+07 | 7.12385e+07 | 55510000 | 6.84161e+07 | 6.84161e+07 |
| Rice-02                             |  95480000 | 1.2325 |   1 | 1.17679e+08 | 1.17679e+08 |  407500000 | 5.02244e+08 | 5.02244e+08 |          0 | 0           | 0           |  5018000000 | 6.18468e+09 | 6.18468e+09 | 234300000 | 2.88775e+08 | 2.88775e+08 | 79840000 | 9.84028e+07 | 9.84028e+07 | 58820000 | 7.24956e+07 | 7.24956e+07 |
| Rice-03                             |  96740000 | 1.2325 |   1 | 1.19232e+08 | 1.19232e+08 |  461400000 | 5.68676e+08 | 5.68676e+08 |          0 | 0           | 0           |  4159000000 | 5.12597e+09 | 5.12597e+09 | 236900000 | 2.91979e+08 | 2.91979e+08 | 62760000 | 7.73517e+07 | 7.73517e+07 | 56580000 | 6.97348e+07 | 6.97348e+07 |
| Rice-04 Cocodrie, light brown       | 220700000 | 1.1061 |   1 | 2.44116e+08 | 2.44116e+08 |  838700000 | 9.27686e+08 | 9.27686e+08 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-05                             | 279200000 | 1.1061 |   1 | 3.08823e+08 | 3.08823e+08 |  919800000 | 1.01739e+09 | 1.01739e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-06                             | 256900000 | 1.1061 |   1 | 2.84157e+08 | 2.84157e+08 |  876000000 | 9.68944e+08 | 9.68944e+08 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-07 Paray Kinta Kabaras, purple |  71850000 | 1.2286 |   1 | 8.82749e+07 | 8.82749e+07 |  192400000 | 2.36383e+08 | 2.36383e+08 |  718100000 | 8.82258e+08 | 8.82258e+08 |  1843000000 | 2.26431e+09 | 2.26431e+09 |  98980000 | 1.21607e+08 | 1.21607e+08 |        0 | 0           | 0           | 83950000 | 1.03141e+08 | 1.03141e+08 |
| Rice-08                             |  60760000 | 1.2286 |   1 | 7.46497e+07 | 7.46497e+07 |  200900000 | 2.46826e+08 | 2.46826e+08 |  649700000 | 7.98221e+08 | 7.98221e+08 |  2065000000 | 2.53706e+09 | 2.53706e+09 |  80920000 | 9.94183e+07 | 9.94183e+07 |        0 | 0           | 0           | 87940000 | 1.08043e+08 | 1.08043e+08 |
| Rice-09                             |  71550000 | 1.2286 |   1 | 8.79063e+07 | 8.79063e+07 |  197400000 | 2.42526e+08 | 2.42526e+08 |  644500000 | 7.91833e+08 | 7.91833e+08 |  2379000000 | 2.92284e+09 | 2.92284e+09 | 104600000 | 1.28512e+08 | 1.28512e+08 |        0 | 0           | 0           | 99640000 | 1.22418e+08 | 1.22418e+08 |
| Rice-10 Timonchette 3, red          | 135000000 | 1.3626 |   1 | 1.83951e+08 | 1.83951e+08 |  404600000 | 5.51308e+08 | 5.51308e+08 |  373900000 | 5.09476e+08 | 5.09476e+08 |           0 | 0           | 0           |  49660000 | 6.76667e+07 | 6.76667e+07 | 42260000 | 5.75835e+07 | 5.75835e+07 |        0 | 0           | 0           |
| Rice-11                             | 134700000 | 1.3626 |   1 | 1.83542e+08 | 1.83542e+08 |  504900000 | 6.87977e+08 | 6.87977e+08 |  359500000 | 4.89855e+08 | 4.89855e+08 |           0 | 0           | 0           |  51360000 | 6.99831e+07 | 6.99831e+07 | 42250000 | 5.75698e+07 | 5.75698e+07 |        0 | 0           | 0           |
| Rice-12                             | 118300000 | 1.3626 |   1 | 1.61196e+08 | 1.61196e+08 |  417200000 | 5.68477e+08 | 5.68477e+08 |  380900000 | 5.19014e+08 | 5.19014e+08 |           0 | 0           | 0           |  62470000 | 8.51216e+07 | 8.51216e+07 | 43910000 | 5.98318e+07 | 5.98318e+07 |        0 | 0           | 0           |
| Rice-13 242, purple                 |  37110000 | 1.1527 |   1 | 4.27767e+07 | 4.27767e+07 |  147700000 | 1.70254e+08 | 1.70254e+08 |  593500000 | 6.84127e+08 | 6.84127e+08 |  2623000000 | 3.02353e+09 | 3.02353e+09 |  98040000 | 1.13011e+08 | 1.13011e+08 |        0 | 0           | 0           | 76320000 | 8.79741e+07 | 8.79741e+07 |
| Rice-14                             |  36700000 | 1.1527 |   1 | 4.23041e+07 | 4.23041e+07 |  182300000 | 2.10137e+08 | 2.10137e+08 |  650300000 | 7.49601e+08 | 7.49601e+08 |  2604000000 | 3.00163e+09 | 3.00163e+09 |  89730000 | 1.03432e+08 | 1.03432e+08 |        0 | 0           | 0           | 87720000 | 1.01115e+08 | 1.01115e+08 |
| Rice-15                             |  48850000 | 1.1527 |   1 | 5.63094e+07 | 5.63094e+07 |  168000000 | 1.93654e+08 | 1.93654e+08 |  599100000 | 6.90583e+08 | 6.90583e+08 |  2510000000 | 2.89328e+09 | 2.89328e+09 |  92750000 | 1.06913e+08 | 1.06913e+08 |        0 | 0           | 0           | 74760000 | 8.61759e+07 | 8.61759e+07 |
| Rice-16 Kataktara, light brown      | 182200000 | 1.2338 |   1 | 2.24798e+08 | 2.24798e+08 | 1056000000 | 1.30289e+09 | 1.30289e+09 |          0 | 0           | 0           |           0 | 0           | 0           |  31640000 | 3.90374e+07 | 3.90374e+07 |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-17                             | 183800000 | 1.2338 |   1 | 2.26772e+08 | 2.26772e+08 | 1082000000 | 1.33497e+09 | 1.33497e+09 |          0 | 0           | 0           |           0 | 0           | 0           |  32490000 | 4.00862e+07 | 4.00862e+07 |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-18                             | 178800000 | 1.2338 |   1 | 2.20603e+08 | 2.20603e+08 | 1105000000 | 1.36335e+09 | 1.36335e+09 |          0 | 0           | 0           |           0 | 0           | 0           |  41980000 | 5.17949e+07 | 5.17949e+07 |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-19 WC 320, purple              | 110500000 | 1.2691 |   1 | 1.40236e+08 | 1.40236e+08 |  283100000 | 3.59282e+08 | 3.59282e+08 |          0 | 0           | 0           | 12360000000 | 1.56861e+10 | 1.56861e+10 | 131800000 | 1.67267e+08 | 1.67267e+08 | 39800000 | 5.05102e+07 | 5.05102e+07 | 13110000 | 1.66379e+07 | 1.66379e+07 |
| Rice-20                             |  92900000 | 1.2691 |   1 | 1.17899e+08 | 1.17899e+08 |  267400000 | 3.39357e+08 | 3.39357e+08 |          0 | 0           | 0           | 12600000000 | 1.59907e+10 | 1.59907e+10 | 133400000 | 1.69298e+08 | 1.69298e+08 | 39050000 | 4.95584e+07 | 4.95584e+07 | 14810000 | 1.87954e+07 | 1.87954e+07 |
| Rice-21                             |  98770000 | 1.2691 |   1 | 1.25349e+08 | 1.25349e+08 |  290100000 | 3.68166e+08 | 3.68166e+08 |          0 | 0           | 0           | 11360000000 | 1.4417e+10  | 1.4417e+10  | 135700000 | 1.72217e+08 | 1.72217e+08 | 41880000 | 5.31499e+07 | 5.31499e+07 | 15760000 | 2.0001e+07  | 2.0001e+07  |
| Rice-22 CH 434-94, brown            | 121100000 | 1.174  |   2 | 1.42171e+08 | 2.84343e+08 |  980500000 | 1.15111e+09 | 2.30221e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-23                             | 138600000 | 1.174  |   2 | 1.62716e+08 | 3.25433e+08 |  940200000 | 1.10379e+09 | 2.20759e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-24                             | 136200000 | 1.174  |   2 | 1.59899e+08 | 3.19798e+08 | 1015000000 | 1.19161e+09 | 2.38322e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-25 Kakani 2, red               |  81270000 | 1.1658 |   2 | 9.47446e+07 | 1.89489e+08 |  438600000 | 5.1132e+08  | 1.02264e+09 | 2030000000 | 2.36657e+09 | 4.73315e+09 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-26                             |  85680000 | 1.1658 |   2 | 9.98857e+07 | 1.99771e+08 |  462200000 | 5.38833e+08 | 1.07767e+09 | 1880000000 | 2.1917e+09  | 4.38341e+09 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-27                             |  91540000 | 1.1658 |   2 | 1.06717e+08 | 2.13435e+08 |  492900000 | 5.74623e+08 | 1.14925e+09 | 1865000000 | 2.17422e+09 | 4.34843e+09 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-28 WC 3777, light brown        |  96530000 | 1.0482 |   2 | 1.01183e+08 | 2.02365e+08 |  320000000 | 3.35424e+08 | 6.70848e+08 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-29                             | 108300000 | 1.0482 |   2 | 1.1352e+08  | 2.2704e+08  |  323100000 | 3.38673e+08 | 6.77347e+08 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-30                             |  78010000 | 1.0482 |   2 | 8.17701e+07 | 1.6354e+08  |  328900000 | 3.44753e+08 | 6.89506e+08 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-34Dular, brown                 | 184500000 | 1.2993 |   2 | 2.39721e+08 | 4.79442e+08 | 1068000000 | 1.38765e+09 | 2.7753e+09  |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-35                             | 188500000 | 1.2993 |   2 | 2.44918e+08 | 4.89836e+08 | 1227000000 | 1.59424e+09 | 3.18848e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-36                             | 185500000 | 1.2993 |   2 | 2.4102e+08  | 4.8204e+08  | 1082000000 | 1.40584e+09 | 2.81169e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-46 Bengal, light brown         | 100800000 | 1.1044 |   2 | 1.11324e+08 | 2.22647e+08 |  470900000 | 5.20062e+08 | 1.04012e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-47                             | 116600000 | 1.1044 |   2 | 1.28773e+08 | 2.57546e+08 |  459500000 | 5.07472e+08 | 1.01494e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-48                             | 107300000 | 1.1044 |   2 | 1.18502e+08 | 2.37004e+08 |  483400000 | 5.33867e+08 | 1.06773e+09 |          0 | 0           | 0           |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-49 IITA119, red                |  71620000 | 1.2345 |   2 | 8.84149e+07 | 1.7683e+08  |  194900000 | 2.40604e+08 | 4.81208e+08 |  470900000 | 5.81326e+08 | 1.16265e+09 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-50                             |  62120000 | 1.2345 |   2 | 7.66871e+07 | 1.53374e+08 |  184400000 | 2.27642e+08 | 4.55284e+08 |  494600000 | 6.10584e+08 | 1.22117e+09 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-51                             |  74060000 | 1.2345 |   2 | 9.14271e+07 | 1.82854e+08 |  206700000 | 2.55171e+08 | 5.10342e+08 |  454800000 | 5.61451e+08 | 1.1229e+09  |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-55 DV 123, brown               | 115300000 | 1.1027 |   2 | 1.27141e+08 | 2.54283e+08 |  683500000 | 7.53695e+08 | 1.50739e+09 |   61600000 | 6.79263e+07 | 1.35853e+08 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-56                             | 102900000 | 1.1027 |   2 | 1.13468e+08 | 2.26936e+08 |  695200000 | 7.66597e+08 | 1.53319e+09 |   53190000 | 5.86526e+07 | 1.17305e+08 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |
| Rice-57                             |  99010000 | 1.1027 |   2 | 1.09178e+08 | 2.18357e+08 |  700600000 | 7.72552e+08 | 1.5451e+09  |   67420000 | 7.4344e+07  | 1.48688e+08 |           0 | 0           | 0           |         0 | 0           | 0           |        0 | 0           | 0           |        0 | 0           | 0           |



label: E=B*C
F=E*D
H=G*C
I=H*D
N=M*C
O=N*D
Q=P*C
R=Q*D
T=S*C
U=T*D
W=V*C
X=W*D
labeled_formulas: E=B*C
F=E*D
H=G*C
I=H*D
N=M*C
O=N*D
Q=P*C
R=Q*D
T=S*C
U=T*D
W=V*C
X=W*D
R=C*P
U=C*S
X=C*V
P=(B/E)*R
S=(B/E)*U
V=(B/E)*X
P=(B/F)*Q
S=(B/F)*T
V=(B/F)*W
output: {"Arithmetic-Relationship": ["E=B*C", "F=E*D", "H=G*C", "I=H*D", "N=M*C", "O=N*D", "Q=P*C", "R=Q*D", "T=S*C", "U=T*D", "W=V*C", "X=W*D"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_272147___xlsx_20571df37b25a5baa97c4e681f63643b229a1e6a.xlsx___line_93