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 |
|:---------------------------------------------------|--------:|--------:|---------:|--------:|--------:|---------:|--------:|--------:|---------:|--------:|--------:|---------:|--------:|--------:|---------:|
| Alexandria Library                                 |  143464 |  607151 |  4.23208 |  140236 |  517219 |  3.6882  |  141738 |  509514 |  3.59    |  137947 |  492690 |  3.57159 |  136732 |  412317 |  3.01551 |
| Amherst County Public Library                      |   32496 |   90915 |  2.79773 |   32356 |   98619 |  3.04794 |   32839 |   87762 |  2.67    |   32341 |   88306 |  2.73047 |   32932 |  111638 |  3.38996 |
| Appomattox Regional Library System                 |   86625 |  233798 |  2.69897 |   86463 |  225867 |  2.6123  |   88929 |  209159 |  2.35    |   87829 |  196627 |  2.23875 |   85462 |  193027 |  2.25863 |
| Arlington Dept. of Libraries                       |  214373 |  637098 |  2.97191 |  208051 |  675633 |  3.24744 |  212038 |  675361 |  3.19    |  206822 |  694956 |  3.36016 |  201163 |  677282 |  3.36683 |
| Augusta County Library                             |   73792 |  210087 |  2.84702 |   73947 |  202957 |  2.74463 |   72114 |  203127 |  2.82    |   71586 |  188612 |  2.63476 |   70747 |  191218 |  2.70284 |
| Bedford Public Library System                      |   75516 |  148633 |  1.96823 |   75099 |  147284 |  1.9612  |   72940 |  147769 |  2.03    |   72629 |  206818 |  2.8476  |   72568 |  227206 |  3.13094 |
| Blackwater Regional Library                        |   81876 |  212608 |  2.59671 |   81727 |  213368 |  2.61074 |   81337 |  194546 |  2.39    |   81092 |  220700 |  2.7216  |   79808 |  200196 |  2.50847 |
| Blue Ridge Regional Library                        |   86403 |  289990 |  3.35625 |   86289 |  265530 |  3.07722 |   88106 |  303295 |  3.44    |   87473 |  304503 |  3.48111 |   89314 |  280817 |  3.14415 |
| Botetourt County Library                           |   33399 |  201827 |  6.04291 |   33209 |  216281 |  6.51272 |   33299 |  207373 |  6.23    |   32956 |  203720 |  6.18158 |   33141 |  190622 |  5.75185 |
| Bristol Public Library                             |   44482 |  284978 |  6.40659 |   44471 |  245251 |  5.51485 |   44241 |  199508 |  4.50957 |   44537 |  198250 |  4.45136 |   42022 |  182131 |  4.33418 |
| Buchanan County Public Library                     |   24006 |   97363 |  4.05578 |   24024 |   92512 |  3.85082 |   24592 |   96189 |  3.91    |   24312 |   98335 |  4.04471 |   23772 |  100781 |  4.23948 |
| Campbell County Public Library                     |   54759 |  169833 |  3.10146 |   54926 |  168940 |  3.07577 |   53237 |  181018 |  3.4     |   52595 |  231017 |  4.39238 |   52112 |  190116 |  3.64822 |
| Caroline Library, Inc.                             |   28826 |   79503 |  2.75803 |   28720 |   82610 |  2.87639 |   28245 |   79718 |  2.82    |   27838 |   78421 |  2.81705 |   26557 |   77040 |  2.90093 |
| Central Rappahannock Regional Library              |  298007 |  720770 |  2.41863 |  295192 |  732357 |  2.48095 |  288118 |  738204 |  2.56    |  284054 |  731110 |  2.57384 |  279397 |  693454 |  2.48197 |
| Central Virginia Regional Library                  |   40662 |   82220 |  2.02204 |   40643 |   77692 |  1.91157 |   38318 |   78632 |  2.05    |   38052 |   74390 |  1.95496 |   37597 |   70425 |  1.87315 |
| Charles P. Jones Memorial Library                  |   22271 |   70776 |  3.17794 |   22174 |   77685 |  3.50343 |   22842 |   74149 |  3.25    |   22716 |   76315 |  3.35953 |   23025 |   89271 |  3.87713 |
| Charlotte County Library                           |   12565 |   52683 |  4.19284 |   12586 |   52438 |  4.16638 |   12672 |   50349 |  3.97    |   12453 |   46946 |  3.76985 |   12696 |   46732 |  3.68084 |
| Chesapeake Public Library                          |  225898 |  526130 |  2.32906 |  222761 |  528886 |  2.37423 |  219960 |  565482 |  2.57    |  216622 |  559784 |  2.58415 |  216568 |  595878 |  2.75146 |
| Chesterfield County Public Library                 |  319641 |  865093 |  2.70645 |  317711 |  824040 |  2.59368 |  307594 |  797262 |  2.59    |  303538 |  828604 |  2.72982 |  298721 |  788374 |  2.63916 |
| Clifton Forge Public Library                       |    3884 |   63779 | 16.421   |    3884 |   59583 | 15.3406  |    3927 |   51064 | 13       |    4289 |   46327 | 10.8014  |    3982 |   57956 | 14.5545  |
| Colonial Heights Public Library                    |   17556 |   75455 |  4.29796 |   17420 |   70203 |  4.03002 |   17638 |   72785 |  4.13    |   17693 |   78815 |  4.45459 |   17463 |   69677 |  3.98998 |
| Craig County Public Library                        |    5241 |   23228 |  4.43198 |    5191 |   19237 |  3.70584 |    5211 |    7550 |  1.45    |    5242 |    8415 |  1.6053  |    5275 |    6959 |  1.31924 |
| Culpeper County Library                            |   47144 |  182036 |  3.86128 |   47040 |  149321 |  3.17434 |   47938 |  138997 |  2.9     |   47517 |  135479 |  2.85117 |   45505 |  124852 |  2.7437  |
| Cumberland County Public Library                   |   10100 |   31880 |  3.15644 |   10077 |   43143 |  4.28133 |    9995 |   41492 |  4.15    |    9820 |   40739 |  4.14857 |    9819 |   40598 |  4.13464 |
| Danville Public Library                            |   43332 |  144649 |  3.33816 |   42918 |  147627 |  3.43975 |   45232 |  158208 |  3.5     |   45152 |  158568 |  3.51187 |   45385 |  156343 |  3.44482 |
| Eastern Shore Public Library                       |   45768 |  146535 |  3.20169 |   45405 |  142908 |  3.14741 |   52960 |  140063 |  2.64    |   52222 |  139221 |  2.66595 |   52149 |  137651 |  2.63957 |
| Essex Public Library                               |   11287 |   39825 |  3.5284  |   11179 |   43916 |  3.92844 |   11003 |   40145 |  3.65    |   10732 |   35628 |  3.31979 |   10769 |   32642 |  3.03111 |
| Fairfax County Public Library                      | 1118889 | 2461272 |  2.19975 | 1106999 | 2425728 |  2.19126 | 1061174 | 2331580 |  2.2     | 1041304 | 2573962 |  2.47186 | 1035727 | 2612165 |  2.52206 |
| Fauquier County Public Library                     |   65780 |  255156 |  3.87893 |   65460 |  251229 |  3.8379  |   67702 |  248226 |  3.67    |   66801 |  241438 |  3.61429 |   65319 |  240753 |  3.6858  |
| Fluvanna County Public Library                     |   25989 |   78676 |  3.02728 |   25844 |   73266 |  2.83493 |   26652 |   67017 |  2.51    |   26389 |   66788 |  2.5309  |   26057 |   55238 |  2.11989 |
| Franklin County Public Library                     |   56300 |  163278 |  2.90014 |   56387 |  161586 |  2.86566 |   53450 |  152734 |  2.86    |   52841 |  141802 |  2.68356 |   52202 |  138260 |  2.64856 |
| Galax-Carroll Regional LIbrary                     |   36873 |   80530 |  2.18398 |   37100 |   85414 |  2.30226 |   37098 |   75882 |  2.05    |   36830 |   76967 |  2.08979 |   37042 |   68823 |  1.85797 |
| Gloucester Library                                 |   36987 |   97392 |  2.63314 |   36902 |   92850 |  2.51612 |   36647 |   97532 |  2.66    |   36109 |   95289 |  2.63893 |   36489 |  100520 |  2.7548  |
| Halifax County-South Boston Regional Library       |   36184 |  122373 |  3.38196 |   36207 |  118636 |  3.2766  |   36082 |  120928 |  3.35    |   35952 |  122349 |  3.40312 |   36691 |  121615 |  3.31457 |
| Hampton Public Library                             |  137372 |  271385 |  1.97555 |  137184 |  286170 |  2.08603 |  144749 |  285276 |  1.97    |  144204 |  289532 |  2.00779 |  145862 |  295356 |  2.0249  |
| Handley Library, The                               |  119534 |  340757 |  2.85071 |  121602 |  323858 |  2.66326 |  116195 |  319418 |  2.75    |  114630 |  302402 |  2.63807 |  113020 |  306508 |  2.71198 |
| Henrico County Public Library                      |  310742 |  708724 |  2.28075 |  308066 |  750210 |  2.43522 |  296490 |  892807 |  3.01    |  291767 |  832070 |  2.85183 |  289788 |  843284 |  2.91    |
| Heritage PublicLibrary                             |   26117 |   44095 |  1.68836 |   25832 |   59337 |  2.29703 |   24849 |   55165 |  2.22    |   24348 |   50800 |  2.08641 |   24160 |   47569 |  1.96892 |
| Highland County Public Library                     |    2315 |   34256 | 14.7974  |    2315 |   31737 | 13.7093  |    2273 |   28429 | 12.51    |    2282 |   20608 |  9.03067 |    2384 |   21005 |  8.81082 |
| Iris Brammer Library                               |    2029 |   38964 | 19.2035  |    2029 |   31433 | 15.4919  |    2149 |   30371 | 14.13    |    2111 |   29407 | 13.9304  |    2151 |   28754 | 13.3677  |
| J. Robert Jamerson Memorial Library                |   15002 |   47035 |  3.13525 |   15002 |   51410 |  3.42688 |   14649 |   57233 |  3.91    |   14414 |   55670 |  3.86222 |   14543 |   56953 |  3.91618 |
| James L. Hamner Public Library                     |   12841 |   71585 |  5.57472 |   12721 |   64958 |  5.10636 |   13147 |   59391 |  4.52    |   13003 |   58871 |  4.52749 |   12763 |   69093 |  5.41354 |
| Jefferson-Madison Regional Library                 |  212841 |  500342 |  2.35078 |  209773 |  487935 |  2.32601 |  202121 |  515100 |  2.55    |  198993 |  620269 |  3.11704 |  198938 |  609502 |  3.06378 |
| Lancaster County Public Library                    |   11463 |   48857 |  4.26215 |   11384 |   48626 |  4.27143 |   11790 |   48407 |  4.11    |   11754 |   47230 |  4.01821 |   11790 |   60169 |  5.10339 |
| Lewis Egerton Smoot Memorial Library               |   23333 |   81220 |  3.48091 |   23779 |   74144 |  3.11805 |   23891 |   68506 |  2.87    |   23029 |   63975 |  2.77802 |   22205 |   69889 |  3.14744 |
| Lonesome Pine Regional Library                     |  109835 |  514417 |  4.68354 |  110066 |  519733 |  4.72201 |  112064 |  580707 |  5.18    |  111162 |  560300 |  5.04039 |  108091 |  587893 |  5.43887 |
| Loudoun County Public Library                      |  324337 |  704276 |  2.17143 |  317035 |  654120 |  2.06324 |  298113 |  541146 |  1.82    |  287688 |  537588 |  1.86865 |  275596 |  592693 |  2.15059 |
| Lynchburg Public Library                           |   76448 |  146292 |  1.91361 |   75826 |  154441 |  2.03678 |   72371 |  159873 |  2.21    |   70734 |  155777 |  2.20229 |   69738 |  153611 |  2.20269 |
| Madison County Library, Inc.                       |   13424 |   52685 |  3.92469 |   13325 |   48711 |  3.65561 |   13950 |   46403 |  3.33    |   13832 |   43739 |  3.16216 |   13828 |   42152 |  3.04831 |
| Mary Riley Styles Public Library                   |   12567 |  132121 | 10.5133  |   12382 |  146208 | 11.8081  |   11711 |  150549 | 12.86    |   11399 |  142728 | 12.5211  |   11280 |  139391 | 12.3574  |
| Massanutten Regional Library (formerly Rockingham) |  151508 |  292591 |  1.93119 |  149716 |  317180 |  2.11854 |  147033 |  307858 |  2.09    |  144433 |  323412 |  2.23918 |  143279 |  311558 |  2.17448 |
| Mathews Memorial Library                           |    8937 |   96536 | 10.8018  |    8970 |   75869 |  8.45808 |    9501 |   53662 |  5.65    |    9418 |   47903 |  5.08632 |    9520 |   47091 |  4.94653 |
| Meherrin Regional Library                          |   35478 |   92565 |  2.60908 |   35595 |   97548 |  2.7405  |   36816 |  103310 |  2.81    |   36489 |  101710 |  2.78742 |   36166 |  100865 |  2.78895 |
| Middlesex County Public Library                    |   11065 |   67031 |  6.05793 |   10983 |   65286 |  5.94428 |   10412 |   61962 |  5.95    |   10277 |   61945 |  6.02754 |   10286 |   60566 |  5.8882  |
| Montgomery-Floyd Regional Library                  |  110480 |  338777 |  3.06641 |  109966 |  332730 |  3.02575 |  106813 |  313615 |  2.94    |  105611 |  288485 |  2.73158 |  103942 |  285973 |  2.75127 |
| Newport News Public Library System                 |  181027 |  375483 |  2.07418 |  180674 |  364766 |  2.01892 |  182591 |  350426 |  1.92    |  180978 |  378636 |  2.09217 |  182478 |  381763 |  2.0921  |
| Norfolk Public Library                             |  243985 |  609765 |  2.49919 |  242956 |  599518 |  2.4676  |  237764 |  621222 |  2.61    |  235092 |  620360 |  2.6388  |  235987 |  647230 |  2.74265 |
| Northumberland Public Library                      |   12449 |   40384 |  3.24396 |   12329 |   34423 |  2.79204 |   13401 |   35695 |  2.66    |   13232 |   36605 |  2.7664  |   13190 |   38136 |  2.89128 |
| Nottoway County Library                            |   15907 |   41929 |  2.63588 |   15852 |   38265 |  2.41389 |   15893 |   33746 |  2.12    |   15512 |   29639 |  1.91071 |   15625 |   32020 |  2.04928 |
| Orange County Public Library                       |   33715 |  160887 |  4.77197 |   33689 |  168771 |  5.00968 |   33885 |  164370 |  4.85    |   33105 |  143804 |  4.34388 |   32364 |  140320 |  4.33568 |
| Pamunkey Regional Library                          |  145664 |  358913 |  2.46398 |  145011 |  288081 |  1.98661 |  142924 |  316167 |  2.21    |  141388 |  316753 |  2.24031 |  140286 |  338904 |  2.41581 |
| Pearisburg Public Library                          |    2786 |   63558 | 22.8134  |    2786 |   61853 | 22.2014  |    2761 |   61385 | 22.23    |    2729 |   60327 | 22.1059  |    2766 |   59339 | 21.453   |
| Petersburg Public Library                          |   32948 |   72333 |  2.19537 |   32380 |  118029 |  3.64512 |   30513 |  134306 |  4.4     |   30489 |  130647 |  4.28505 |   31588 |  144090 |  4.56154 |
| Pittsylvania County Public Library                 |   62986 |  227605 |  3.61358 |   63535 |  212119 |  3.33862 |   62713 |  170731 |  2.72    |   62253 |  164366 |  2.64029 |   61731 |  164510 |  2.66495 |
| Poquoson Public Library                            |   12240 |  121478 |  9.92467 |   12162 |  107212 |  8.81533 |   11881 |   67707 |  5.7     |   11791 |   61995 |  5.25782 |   11948 |   58505 |  4.89664 |
| Portsmouth Public Library                          |   96368 |  240841 |  2.49918 |   95388 |  261539 |  2.74184 |   98124 |  243583 |  2.48    |   97599 |  257857 |  2.642   |   98543 |  281096 |  2.85252 |
| Powhatan County Public Library                     |   28290 |   64187 |  2.26889 |   28198 |   61652 |  2.1864  |   28106 |   58110 |  2.07    |   27550 |   54385 |  1.97405 |   27448 |   50728 |  1.84815 |
| Prince William Public Library System               |  468131 |  722554 |  1.54349 |  457789 |  747744 |  1.63338 |  437174 |  827720 |  1.89    |  429187 |  850153 |  1.98085 |  422416 |  872164 |  2.0647  |
| Pulaski County Library System                      |   34635 |   93602 |  2.70253 |   34857 |   91428 |  2.62295 |   34167 |   88838 |  2.6     |   34391 |   88492 |  2.57312 |   34997 |   71947 |  2.0558  |
| Radford Public Library                             |   16532 |  159955 |  9.67548 |   16418 |  140568 |  8.56182 |   15142 |   94778 |  6.26    |   15584 |   92198 |  5.9162  |   15593 |  103095 |  6.61162 |
| Rappahannock County Library                        |    7410 |   41193 |  5.55911 |    7381 |   41025 |  5.55819 |    7066 |   45923 |  6.5     |    6925 |   37654 |  5.4374  |    7193 |   36179 |  5.02975 |
| Richmond County Public Library                     |    9237 |   29507 |  3.19444 |    9263 |  123150 | 13.2948  |    9797 |   44594 |  4.55    |    9774 |   41644 |  4.26069 |    9094 |   75949 |  8.35155 |
| Richmond Public Library                            |  206238 |  555242 |  2.69224 |  204326 |  550317 |  2.69333 |  198102 |  547462 |  2.76    |  195463 |  612334 |  3.13274 |  195300 |  691107 |  3.53869 |
| Roanoke City Public Library                        |   97206 |  289979 |  2.98314 |   97061 |  295167 |  3.04105 |   92991 |  316726 |  3.41    |   92344 |  311495 |  3.3732  |   93504 |  319014 |  3.41177 |
| Roanoke County Public Library                      |   92687 |  471310 |  5.08496 |   92524 |  439494 |  4.75005 |   93304 |  451222 |  4.84    |   91197 |  426048 |  4.67173 |   90902 |  407244 |  4.48003 |
| Rockbridge Regional Library                        |   41030 |  193236 |  4.70963 |   40763 |  197451 |  4.84388 |   40486 |  192404 |  4.75    |   40303 |  187393 |  4.6496  |   40865 |  181924 |  4.45183 |
| Russell County Public Library                      |   29014 |   70339 |  2.42431 |   28881 |   68728 |  2.3797  |   29442 |   66678 |  2.26    |   28988 |   61212 |  2.11163 |   28795 |   56536 |  1.9634  |
| Salem Public Library                               |   24970 |  141044 |  5.64854 |   24797 |  165135 |  6.65947 |   25163 |  143944 |  5.72    |   24749 |  145306 |  5.87119 |   25040 |  150128 |  5.99553 |
| Samuels Public Library                             |   37688 |  151967 |  4.03224 |   37729 |  143414 |  3.80116 |   36472 |  103570 |  2.84    |   36101 |   91969 |  2.54755 |   35829 |  104599 |  2.91939 |
| Shenandoah County Library                          |   42114 |  225675 |  5.35867 |   42172 |  212959 |  5.04977 |   32091 |  193158 |  6.02    |   41252 |  163403 |  3.96109 |   40714 |  174567 |  4.28764 |
| Smyth-Bland Regional Library                       |   38852 |  165215 |  4.25242 |   38999 |  169950 |  4.3578  |   39213 |  168139 |  4.29    |   38919 |  163401 |  4.19849 |   39396 |  154448 |  3.9204  |
| Southside Regional                                 |   45903 |  157294 |  3.42666 |   45632 |  166384 |  3.64621 |   45571 |  167335 |  3.67    |   45344 |  172864 |  3.81228 |   46157 |  174520 |  3.78101 |
| Staunton Public Library                            |   24222 |  247021 | 10.1982  |   23737 |  245736 | 10.3524  |   23473 |  205035 |  8.73    |   23097 |  196181 |  8.49379 |   22937 |  188718 |  8.22767 |
| Suffolk Public Library System                      |   85692 |  406474 |  4.74343 |   85166 |  270435 |  3.17539 |   83006 |  215683 |  2.6     |   81907 |  216432 |  2.64241 |   81209 |  246964 |  3.04109 |
| Tazewell County Public Library                     |   45276 |  132180 |  2.91943 |   45079 |  131816 |  2.92411 |   44538 |  134428 |  3.02    |   44216 |  130235 |  2.94543 |   42995 |  130199 |  3.02824 |
| Virginia Beach Public Library                      |  441246 |  858784 |  1.94627 |  438207 |  886255 |  2.02246 |  434412 |  910470 |  2.1     |  431451 |  877015 |  2.03271 |  433033 |  922313 |  2.12989 |
| Washington County Public Library                   |   55179 |  193259 |  3.5024  |   54960 |  187865 |  3.41821 |   54443 |  153174 |  2.81    |   53223 |  151512 |  2.84674 |   52974 |  142418 |  2.68845 |
| Waynesboro Public Library                          |   21099 |  163678 |  7.75762 |   21039 |  162445 |  7.72114 |   20712 |  157292 |  7.59    |   20614 |  157876 |  7.65868 |   20405 |  159190 |  7.80152 |
| Williamsburg Regional Library                      |   83130 |  339869 |  4.0884  |   81670 |  337009 |  4.12647 |   77268 |  340523 |  4.41    |   75591 |  358976 |  4.74893 |   74983 |  350526 |  4.67474 |
| Wythe-Grayson Regional Library                     |   44861 |  147411 |  3.28595 |   44767 |  138019 |  3.08305 |   44272 |  145126 |  3.28    |   44071 |  139223 |  3.15906 |   44006 |  106364 |  2.41703 |
| York County Public Library                         |   65973 |  175865 |  2.66571 |   65695 |  175062 |  2.66477 |   65964 |  172377 |  2.61    |   64526 |  202615 |  3.14005 |   63184 |  182697 |  2.89151 |



label: D=C/B
G=F/E
M=L/K
P=O/N
labeled_formulas: D=C/B
G=F/E
M=L/K
P=O/N
output: {"Arithmetic-Relationship": ["D=C/B", "G=F/E", "M=L/K", "P=O/N"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_272096___xlsx_10f3c6da8e23032bc36e63f360dde15ab0630cb0.xlsx___line_72