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 |
|----:|:---------------|---------:|----------:|---------:|----------:|---------:|----:|----:|----------:|------------:|----:|
|   1 | Accomack       |   273122 |   5101071 |  1070993 |   6445186 |    17080 |   0 |   0 |   3240894 | 7.08107e+06 |   1 |
|   2 | Albemarle      |  5289964 |   1191662 |        0 |   6481626 |        0 |   0 |   0 |       280 | 1.41026e+07 |   2 |
|   3 | Alleghany      |        0 |   2026562 |   785814 |   2812376 |    12385 |   0 |   0 |    761599 | 3.83982e+06 |   3 |
|   4 | Amelia         |     5008 |    163543 |   937716 |   1106267 |        0 |   0 |   0 |   3048988 | 2.48657e+06 |   4 |
|   5 | Amherst        |        0 |   2222680 |  1389251 |   3611931 |     8690 |   0 |   0 |    627981 | 5.63402e+06 |   5 |
|   6 | Appomattox     |        0 |   1330882 |   609017 |   1939899 |     9076 |   0 |   0 |    609881 | 2.86946e+06 |   6 |
|   7 | Arlington      | 66088436 |  15270051 | 14250196 |  95608683 | 12522086 |   0 |   0 |  29369215 | 5.72365e+06 |   7 |
|   9 | Bath           |        0 |   1160850 |   254271 |   1415121 |     4883 |   0 |   0 |    122997 | 2.64183e+06 |   9 |
|  10 | Bedford        |        0 |         0 |        0 |         0 |        0 |   0 |   0 |         0 | 0           |  10 |
|  11 | Bland          |     8512 |   1011935 |   636317 |   1656764 |     4883 |   0 |   0 |    950438 | 5.33524e+06 |  11 |
|  12 | Botetourt      |        0 |    779126 |   871107 |   1650233 |        0 |   0 |   0 |    251534 | 9.00178e+06 |  12 |
|  13 | Brunswick      |        0 |   1049215 |   955529 |   2004744 |     7416 |   0 |   0 |    602141 | 6.78137e+06 |  13 |
|  14 | Buchanan       |  4524186 |   3118642 |  1827189 |   9470017 |   321825 |   0 |   0 |    798171 | 5.20068e+06 |  14 |
|  15 | Buckingham     |     4949 |   1063371 |   684674 |   1752994 |     3054 |   0 |   0 |     19531 | 3.65804e+06 |  15 |
|  16 | Campbell       |    18596 |   3341160 |  2154151 |   5513907 |     5759 |   0 |   0 |   2060582 | 9.17818e+06 |  16 |
|  17 | Caroline       |   242658 |   1713676 |  2400319 |   4356653 |        0 |   0 |   0 |     43563 | 4.97025e+06 |  17 |
|  18 | Carroll        |        0 |    723466 |   805648 |   1529114 |     7903 |   0 |   0 |    870952 | 1.21489e+07 |  18 |
|  19 | Charles City   |        0 |    285894 |  1031472 |   1317366 |        0 |   0 |   0 |   3523284 | 1.9675e+06  |  19 |
|  22 | Clarke         |        0 |    240970 |   795805 |   1036775 |     4007 |   0 |   0 |         0 | 2.88711e+06 |  22 |
|  23 | Craig          |        0 |    511189 |   155002 |    666191 |     5904 |   0 |   0 |     13678 | 2.38879e+06 |  23 |
|  24 | Culpeper       |   371401 |   3029313 |  1185705 |   4586419 |        0 |   0 |   0 |   1750034 | 6.40989e+06 |  24 |
|  25 | Cumberland     |        0 |    964697 |   752804 |   1717501 |     4883 |   0 |   0 |       793 | 1.96448e+06 |  25 |
|  26 | Dickenson      |  1184462 |   1797379 |   603314 |   3585155 |  2163469 |   0 |   0 |    204621 | 3.50621e+06 |  26 |
|  27 | Dinwiddie      |    45399 |   1587826 |  3165776 |   4799001 |     6930 |   0 |   0 |    703080 | 8.02946e+06 |  27 |
|  28 | Essex          |        0 |    791643 |  1159200 |   1950843 |     5975 |   0 |   0 |    763952 | 2.4877e+06  |  28 |
|  29 | Fairfax        | 30518783 | 163396823 | 62050505 | 255966111 |   713645 |   0 |   0 | 158553543 | 7.46718e+07 |  29 |
|  30 | Fauquier       |        0 |  10243017 |  7071790 |  17314807 |    14078 |   0 |   0 |   3370335 | 1.28219e+07 |  30 |
|  31 | Floyd          |        0 |   1701976 |   280276 |   1982252 |     5489 |   0 |   0 |    166446 | 5.9596e+06  |  31 |
|  32 | Fluvanna       |        0 |    312225 |  2177271 |   2489496 |     6730 |   0 |   0 |    121286 | 3.07058e+06 |  32 |
|  33 | Franklin       |        0 |   3437968 |  1609928 |   5047896 |    12846 |   0 |   0 |   1326812 | 1.03616e+07 |  33 |
|  34 | Frederick      |   449224 |  10920323 |  2350728 |  13720275 |    13000 |   0 |   0 |   8251650 | 7.7762e+06  |  34 |
|  35 | Giles          |        0 |     83440 |  1013535 |   1096975 |     7946 |   0 |   0 |     34581 | 4.58154e+06 |  35 |
|  36 | Gloucester     |   322080 |     15464 |  2210877 |   2548421 |     7888 |   0 |   0 |    537113 | 2.45616e+06 |  36 |
|  37 | Goochland      |        0 |    924143 |  2345990 |   3270133 |     7500 |   0 |   0 |     63066 | 4.50995e+06 |  37 |
|  38 | Grayson        |        0 |   2153137 |   458955 |   2612092 |     5676 |   0 |   0 |   2835157 | 5.53135e+06 |  38 |
|  39 | Greene         |        0 |   1849856 |   829245 |   2679101 |     4883 |   0 |   0 |   1712614 | 2.93764e+06 |  39 |
|  40 | Greensville    |    26819 |   2261409 |   764485 |   3052713 |     7883 |   0 |   0 |   1279056 | 2.84383e+06 |  40 |
|  41 | Halifax        |        0 |   2559562 |   747324 |   3306886 |    14297 |   0 |   0 |    299903 | 1.03871e+07 |  41 |
|  43 | Henrico        | 50943624 |  19214833 | 10543854 |  80702311 | 49142660 |   0 |   0 |  14881017 | 9.91095e+06 |  43 |
|  45 | Highland       |        0 |    345461 |    47607 |    393068 |     5904 |   0 |   0 |    332005 | 2.74294e+06 |  45 |
|  46 | Isle of Wight  |  1894376 |   3557061 |  1573305 |   7024742 |     7641 |   0 |   0 |   1442663 | 6.50297e+06 |  46 |
|  47 | James City     |        0 |   2392824 |  7263492 |   9656316 |    10998 |   0 |   0 |   2357793 | 7.12923e+06 |  47 |
|  48 | King & Queen   |        0 |    417521 |   302384 |    719905 |        0 |   0 |   0 |   2706288 | 2.77442e+06 |  48 |
|  49 | King George    |   176107 |    617219 |  1949041 |   2742367 |        0 |   0 |   0 |         0 | 2.05582e+06 |  49 |
|  50 | King William   |        0 |   1375423 |   894080 |   2269503 |     6369 |   0 |   0 |   1797553 | 3.70341e+06 |  50 |
|  51 | Lancaster      |        0 |   1276219 |   286794 |   1563013 |        0 |   0 |   0 |      6270 | 1.93725e+06 |  51 |
|  52 | Lee            |        0 |         0 |        0 |         0 |        0 |   0 |   0 |         0 | 0           |  52 |
|  53 | Loudoun        | 14031655 |   8197378 | 13879132 |  36108165 |    57486 |   0 |   0 |   9622492 | 1.97211e+07 |  53 |
|  54 | Louisa         |        0 |   1846696 |  2506689 |   4353385 |     8069 |   0 |   0 |    331664 | 8.59678e+06 |  54 |
|  55 | Lunenburg      |        0 |    335635 |   218428 |    554063 |        0 |   0 |   0 |    359629 | 4.11455e+06 |  55 |
|  56 | Madison        |        0 |    548164 |   620309 |   1168473 |     5904 |   0 |   0 |    197183 | 2.89075e+06 |  56 |
|  57 | Mathews        |    17219 |    784333 |   614907 |   1416459 |        0 |   0 |   0 |    634869 | 2.5811e+06  |  57 |
|  58 | Mecklenburg    |        0 |   2038722 |   967627 |   3006349 |     9854 |   0 |   0 |    104328 | 9.26614e+06 |  58 |
|  59 | Middlesex      |        0 |    903970 |   666749 |   1570719 |     4883 |   0 |   0 |    548821 | 2.29248e+06 |  59 |
|  60 | Montgomery     |        0 |   3414115 |  3459255 |   6873370 |    25223 |   0 |   0 |   1866660 | 1.46363e+07 |  60 |
|  61 | Nelson         |        0 |   1855405 |   959613 |   2815018 |        0 |   0 |   0 |   1195705 | 5.28393e+06 |  61 |
|  62 | New Kent       |        0 |    823053 |   993464 |   1816517 |     5215 |   0 |   0 |     37913 | 3.2251e+06  |  62 |
|  64 | Northumberland |    18808 |   1267234 |   300018 |   1586060 |        0 |   0 |   0 |         0 | 2.70052e+06 |  64 |
|  65 | Nottoway       |        0 |   1313909 |   266712 |   1580621 |     4391 |   0 |   0 |    584955 | 3.15765e+06 |  65 |
|  66 | Orange         |        0 |   2920260 |  1319347 |   4239607 |     7355 |   0 |   0 |    575052 | 4.83993e+06 |  66 |
|  67 | Page           |        0 |   1756436 |   368707 |   2125143 |     9886 |   0 |   0 |   1943638 | 4.16385e+06 |  67 |
|  68 | Patrick        |        0 |    701117 |   818041 |   1519158 |     6875 |   0 |   0 |    276470 | 7.09988e+06 |  68 |
|  70 | Powhatan       |        0 |    500369 |  1903940 |   2404309 |     6250 |   0 |   0 |     65161 | 2.09817e+06 |  70 |
|  71 | Prince Edward  |        0 |   1536492 |   808615 |   2345107 |     5781 |   0 |   0 |    726586 | 3.74002e+06 |  71 |
|  72 | Prince George  |        0 |     42260 |  3480097 |   3522357 |     7367 |   0 |   0 |    494900 | 7.99373e+06 |  72 |
|  74 | Pulaski        |   -46666 |   4651616 |  1374122 |   5979072 |     9741 |   0 |   0 |   4587119 | 5.50378e+06 |  74 |
|  75 | Rappahannock   |        0 |    797951 |   331757 |   1129708 |     6938 |   0 |   0 |     10792 | 2.7438e+06  |  75 |
|  76 | Richmond       |     2651 |    770197 |   353211 |   1126059 |      500 |   0 |   0 |         0 | 2.71542e+06 |  76 |
|  77 | Roanoke        |        0 |   9510027 |  8663988 |  18174015 |     3000 |   0 |   0 |   3146161 | 2.06536e+07 |  77 |
|  79 | Rockingham     |  1427704 |   8142514 |  1908615 |  11478833 |        0 |   0 |   0 |   6733774 | 1.33903e+07 |  79 |
|  80 | Russell        |    94434 |   1817871 |   802883 |   2715188 |    10377 |   0 |   0 |    207823 | 5.22206e+06 |  80 |
|  81 | Scott          |      642 |   1334515 |   531093 |   1866250 |        0 |   0 |   0 |     41404 | 4.75546e+06 |  81 |
|  82 | Shenandoah     |        0 |   2882416 |  1342442 |   4224858 |    15823 |   0 |   0 |   1360485 | 9.08471e+06 |  82 |
|  83 | Smyth          |        0 |   1575372 |   508600 |   2083972 |     6882 |   0 |   0 |    855645 | 6.10249e+06 |  83 |
|  84 | Southampton    |    52487 |   1450758 |   760158 |   2263403 |    12086 |   0 |   0 |    949976 | 5.49887e+06 |  84 |
|  85 | Spotsylvania   |  3885621 |   5788890 |  4538947 |  14213458 |    21029 |   0 |   0 |   4807024 | 9.90248e+06 |  85 |
|  87 | Surry          |        0 |    916921 |   823012 |   1739933 |     4883 |   0 |   0 |      7694 | 1.37414e+07 |  87 |
|  88 | Sussex         |        0 |      5772 |   654769 |    660541 |     4693 |   0 |   0 |   5976642 | 5.02852e+06 |  88 |
|  89 | Tazewell       |  1175315 |   2768180 |  1104904 |   5048399 |        0 |   0 |   0 |    729561 | 5.17711e+06 |  89 |
|  90 | Warren         |  2002327 |   3691321 |  1500367 |   7194015 |    42153 |   0 |   0 |   1228912 | 2.93741e+06 |  90 |
|  91 | Washington     |        0 |   2168047 |  1253752 |   3421799 |    31557 |   0 |   0 |    532423 | 1.25525e+07 |  91 |
|  92 | Westmoreland   |     6535 |   1907325 |  1008036 |   2921896 |   330000 |   0 |   0 |    463523 | 2.29431e+06 |  92 |
|  93 | Wise           |   222177 |   3430692 |   884772 |   4537641 |    10800 |   0 |   0 |    441234 | 5.42128e+06 |  93 |
|  95 | York           |        0 |   5866202 |  6796616 |  12662818 |    10182 |   0 |   0 |   6838097 | 2.83647e+06 |  95 |



label: F=(C+D+E)
labeled_formulas: I=A*H
L=A+H
I=C*H
I=D*H
I=E*H
I=F*H
I=G*H
I=H*J
I=H*K
I=H*L
F=(C+D+E)
I=(A*C)*H
L=(C*H)+A
I=(A+D)*H
H=(A-L)*D
I=(A+E)*H
H=(A-L)*E
I=(A+F)*H
H=(A-L)*F
I=(A*G)*H
L=(G*H)+A
I=(A*H)*J
I=(A*H)*K
H=(A-L)*J
H=(A-L)*K
I=(C*D)*H
I=(C*E)*H
I=(C*F)*H
I=(C*G)*H
I=(C*H)*J
I=(C*H)*K
I=(C*H)*L
I=(D+E)*H
I=(D+F)*H
I=(D*G)*H
I=(D*H)*J
I=(D*H)*K
I=(D*H)*L
I=(E+F)*H
I=(E*G)*H
I=(E*H)*J
I=(E*H)*K
I=(E*H)*L
I=(F*G)*H
I=(F*H)*J
I=(F*H)*K
I=(F*H)*L
I=(G*H)*J
I=(G*H)*K
I=(G*H)*L
I=(H*J)*K
I=(H*J)*L
I=(H*K)*L
output: {"Arithmetic-Relationship": ["F=(C+D+E)"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_2316___xlsx_756f2de2f1b7907d0304e13012e31a03c6e2b186.xlsx___line_325