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:
|   CA |   CB | CC               |               CD |      CE |     CF |       CG |       CH |   CI |       CJ |         CK |
|-----:|-----:|:-----------------|-----------------:|--------:|-------:|---------:|---------:|-----:|---------:|-----------:|
|    1 |    1 | ABINGTON         | 395931           |  357235 |  38696 |      0   |  11367.6 |    0 |  50063.6 |  37195.3   |
|    2 |    2 | ACTON            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|    4 |    4 | ADAMS            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|    6 |    6 | ALFORD           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   10 |   10 | ARLINGTON        | 216988           |  152340 |  64648 |  19812   |      0   |    0 |  84460   |  62140.8   |
|   11 |   11 | ASHBURNHAM       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   12 |   12 | ASHBY            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   13 |   13 | ASHFIELD         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   15 |   15 | ATHOL            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   16 |   16 | ATTLEBORO        |      3.99799e+06 | 3614356 | 383634 |      0   | 167583   |    0 | 551217   | 368756     |
|   18 |   18 | AVON             | 393982           |  328712 |  65270 |  89312.4 |      0   |    0 | 154582   |  62738.7   |
|   19 |   19 | AYER             |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   21 |   21 | BARRE            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   22 |   22 | BECKET           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   23 |   23 | BEDFORD          |  16046           |    6928 |   9118 |      0   |   6858.8 |    0 |  15976.8 |   8764.39  |
|   24 |   24 | BELCHERTOWN      | 597225           |  587742 |   9483 |  56328   |      0   |    0 |  65811   |   9115.23  |
|   26 |   26 | BELMONT          |  37894           |   31986 |   5908 |   1638.6 |      0   |    0 |   7546.6 |   5678.88  |
|   27 |   27 | BERKLEY          |  12437           |       0 |  12437 |      0   |      0   |    0 |  12437   |  11954.7   |
|   28 |   28 | BERLIN           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   29 |   29 | BERNARDSTON      |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   32 |   32 | BLACKSTONE       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   33 |   33 | BLANDFORD        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   34 |   34 | BOLTON           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   36 |   36 | BOURNE           |      2.04528e+06 | 1893265 | 152015 |      0   |  95662.4 |    0 | 247677   | 146120     |
|   37 |   37 | BOXBOROUGH       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   38 |   38 | BOXFORD          |  18484           |   14971 |   3513 |      0   |   7159.6 |    0 |  10672.6 |   3376.76  |
|   39 |   39 | BOYLSTON         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   41 |   41 | BREWSTER         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   42 |   42 | BRIDGEWATER      |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   43 |   43 | BRIMFIELD        |  61192           |   45583 |  15609 |  19516.2 |      0   |    0 |  35125.2 |  15003.7   |
|   45 |   45 | BROOKFIELD       |  83433           |   63230 |  20203 |  30961.2 |      0   |    0 |  51164.2 |  19419.5   |
|   46 |   46 | BROOKLINE        |  49658           |   45793 |   3865 |      0   |   3881.6 |    0 |   7746.6 |   3715.11  |
|   47 |   47 | BUCKLAND         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   50 |   50 | CANTON           | 255262           |  209142 |  46120 |      0   |  32213.2 |    0 |  78333.2 |  44331.4   |
|   51 |   51 | CARLISLE         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   52 |   52 | CARVER           |      1.07706e+06 |  809143 | 267914 | 141947   |      0   |    0 | 409861   | 257524     |
|   53 |   53 | CHARLEMONT       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   54 |   54 | CHARLTON         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   55 |   55 | CHATHAM          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   56 |   56 | CHELMSFORD       |      1.4806e+06  | 1441859 |  38746 |      0   |  38080.4 |    0 |  76826.4 |  37243.4   |
|   58 |   58 | CHESHIRE         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   59 |   59 | CHESTER          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   60 |   60 | CHESTERFIELD     |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   62 |   62 | CHILMARK         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   63 |   63 | CLARKSBURG       |  41469           |       0 |  41469 |      0   |      0   |    0 |  41469   |  39860.8   |
|   66 |   66 | COLRAIN          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   67 |   67 | CONCORD          |  38201           |   37354 |    847 |      0   |   7810.8 |    0 |   8657.8 |    814.152 |
|   68 |   68 | CONWAY           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   69 |   69 | CUMMINGTON       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   70 |   70 | DALTON           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   74 |   74 | DEERFIELD        | 124782           |  105455 |  19327 |      0   |  11618.8 |    0 |  30945.8 |  18577.5   |
|   75 |   75 | DENNIS           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   76 |   76 | DIGHTON          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   77 |   77 | DOUGLAS          |  13970           |       0 |  13970 |      0   |      0   |    0 |  13970   |  13428.2   |
|   78 |   78 | DOVER            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   79 |   79 | DRACUT           |      3.07798e+06 | 2602434 | 475546 |      0   |  26563.2 |    0 | 502109   | 457103     |
|   80 |   80 | DUDLEY           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   81 |   81 | DUNSTABLE        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   82 |   82 | DUXBURY          | 146590           |  140930 |   5660 |      0   |      0   |    0 |   5660   |   5440.49  |
|   83 |   83 | EAST BRIDGEWATER | 181170           |  116810 |  64360 |  13251.6 |      0   |    0 |  77611.6 |  61864     |
|   84 |   84 | EAST BROOKFIELD  |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   85 |   86 | EASTHAM          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   89 |   89 | EDGARTOWN        | 896448           |  818853 |  77595 |      0   |  23106.4 |    0 | 100701   |  74585.7   |
|   90 |   90 | EGREMONT         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   91 |   91 | ERVING           |  78975           |   64722 |  14253 |      0   |      0   |    0 |  14253   |  13700.2   |
|   92 |   92 | ESSEX            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|   93 |   93 | EVERETT          |      8.2234e+06  | 8120845 | 102554 |      0   | 260344   |    0 | 362898   |  98576.8   |
|  102 |  102 | FREETOWN         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  103 |  103 | GARDNER          | 369962           |  308217 |  61745 |  48828   |      0   |    0 | 110573   |  59350.4   |
|  104 |  104 | AQUINNAH         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  106 |  106 | GILL             |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  107 |  107 | GLOUCESTER       |  14497           |   13863 |    634 |   8317.8 |      0   |    0 |   8951.8 |    609.412 |
|  108 |  108 | GOSHEN           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  109 |  109 | GOSNOLD          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  112 |  112 | GRANVILLE        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  113 |  113 | GREAT BARRINGTON |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  114 |  114 | GREENFIELD       |      1.37136e+06 | 1206995 | 164369 | 102038   |      0   |    0 | 266407   | 157994     |
|  115 |  115 | GROTON           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  116 |  116 | GROVELAND        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  117 |  117 | HADLEY           | 585640           |  580537 |   5103 |      0   |      0   |    0 |   5103   |   4905.1   |
|  118 |  118 | HALIFAX          |  39456           |   28190 |  11266 |      0   |  18186.8 |    0 |  29452.8 |  10829.1   |
|  119 |  119 | HAMILTON         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  120 |  120 | HAMPDEN          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  121 |  121 | HANCOCK          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  122 |  122 | HANOVER          | 491008           |  421560 |  69448 |  54335.4 |      0   |    0 | 123783   |  66754.7   |
|  123 |  123 | HANSON           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  124 |  124 | HARDWICK         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  125 |  125 | HARVARD          | 400525           |  312833 |  87692 |  46765.2 |      0   |    0 | 134457   |  84291.1   |
|  126 |  126 | HARWICH          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  129 |  129 | HAWLEY           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  130 |  130 | HEATH            |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  131 |  131 | HINGHAM          | 234243           |  164674 |  69569 |  34867.8 |      0   |    0 | 104437   |  66871     |
|  132 |  132 | HINSDALE         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  134 |  134 | HOLDEN           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  135 |  135 | HOLLAND          |  95262           |   69427 |  25835 |      0   |  24585.6 |    0 |  50420.6 |  24833.1   |
|  138 |  138 | HOPEDALE         | 118993           |   99652 |  19341 |  44505.6 |      0   |    0 |  63846.6 |  18590.9   |
|  140 |  140 | HUBBARDSTON      |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  142 |  142 | HULL             | 550650           |  520024 |  30626 |      0   |      0   |    0 |  30626   |  29438.3   |
|  143 |  143 | HUNTINGTON       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  144 |  144 | IPSWICH          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  146 |  146 | LAKEVILLE        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  147 |  147 | LANCASTER        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  148 |  148 | LANESBOROUGH     |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  150 |  150 | LEE              |      0           |       0 |      0 |      0   |    451.6 |    0 |    451.6 |      0     |
|  153 |  153 | LEOMINSTER       |      1.05251e+06 |  916134 | 136374 |      0   |   3290.4 |    0 | 139664   | 131085     |
|  155 |  155 | LEXINGTON        | 112212           |   34053 |  78159 |      0   |   2306.4 |    0 |  80465.4 |  75127.8   |
|  156 |  156 | LEYDEN           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  157 |  157 | LINCOLN          |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  166 |  166 | MANCHESTER       |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  167 |  167 | MANSFIELD        |      1.18545e+06 | 1057693 | 127758 |      0   |   7162.4 |    0 | 134920   | 122803     |
|  168 |  168 | MARBLEHEAD       |      2.07448e+06 | 1593396 | 481082 |      0   |      0   |    0 | 481082   | 462425     |
|  169 |  169 | MARION           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  171 |  171 | MARSHFIELD       | 470251           |  303058 | 167193 |      0   |  24054.4 |    0 | 191247   | 160709     |
|  172 |  172 | MASHPEE          | 976803           |  868272 | 108531 |  47113.2 |      0   |    0 | 155644   | 104322     |
|  175 |  175 | MEDFIELD         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  178 |  178 | MELROSE          |      3.14775e+06 | 2739760 | 407988 | 251995   |      0   |    0 | 659983   | 392165     |
|  179 |  179 | MENDON           |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  180 |  180 | MERRIMAC         |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  183 |  183 | MIDDLEFIELD      |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |
|  188 |  188 | MILLVILLE        |      0           |       0 |      0 |      0   |      0   |    0 |      0   |      0     |



labeled_formulas: CF=CD-CE
CI=(CD-CE)-CF
CI=(CF*CG)*CH
CI=(CG*CH)*CK
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_683762___xlsx_7bcfd6da22c8caf60b9f51defe0e3b19ce8ee549.xlsx___line_449