(* --------------------- Code --------------------- *)

(* Uniform distribution of coefficients *)

rlog[tau_] := 
   Module[{ans=1},
      While[NumberQ[ans],
         ans=Log2[RandomInteger[{1, 2^tau}]/RandomInteger[{1, 2^tau}]]];
      ans]
      
rpoly[nu_, tau_, t_] :=
   Sum[RandomInteger[{-2^(tau-1), 2^(tau-1)}] t^i, {i, 0, nu}]
   
rprob1[d_, nu_, tau_, x_] :=
   With[{t=rlog[tau]}, Sum[rpoly[nu, tau, t] x^i, {i, 0, d}]]

rtest[d_, nu_, tau_, n_] :=
   Module[{tt, ll, ans, p},
      tt=Table[0, {n}];
      ll=Table[0, {n}];
      Do[p=rprob1[d, nu, tau, x];
         ans=TimeConstrained[
                AbsoluteTiming[System`Private`DCMLIsol[p, x]], 
                3600];
         If[!ListQ[ans], Return[$Aborted, Module]];
         tt[[i]]=ans[[1]];
         ll[[i]]=Length[ans[[2]]],
         {i, n}];
      ans={{d, nu, tau}, {(Plus@@tt)/n, Min[tt], Max[tt]}, N[(Plus@@ll)/n]};
      Print[ans];
      ans]

runtests10[] :=
   Module[{dd, nn},
      dd={10, 20, 50, 100, 200, 500, 1000};
      nn={10, 20, 50, 100, 200, 500, 1000};
      Table[rtest[dd[[i]], nn[[j]], 10, 10], {i, 7}, {j, 7}]]

runtests1000[] :=
   Module[{dd, nn},
      dd={10, 20, 50, 100, 200, 500, 1000};
      nn={10, 20, 50, 100, 200, 500, 1000};
      Table[rtest[dd[[i]], nn[[j]], 1000, 10], {i, 7}, {j, 7}]]

(* Gaussian distribution of coefficients *)
      
rpoly2[nu_, sigma_, t_] :=
   Sum[Round[RandomVariate[NormalDistribution[0, sigma]]] t^i, {i, 0, nu}]
   
rprob2[d_, nu_, x_] :=
   Sum[rpoly2[nu, Sqrt[Binomial[d, i]], Log2[3]] x^i, {i, 0, d}]

rtest2[d_, nu_, n_] :=
   Module[{tt, ll, ans, p},
      tt=Table[0, {n}];
      ll=Table[0, {n}];
      Do[p=rprob2[d, nu, x];
         ans=TimeConstrained[
                AbsoluteTiming[System`Private`DCMLIsol[p, x]], 
                3600];
         If[!ListQ[ans], Return[$Aborted, Module]];
         tt[[i]]=ans[[1]];
         ll[[i]]=Length[ans[[2]]],
         {i, n}];
      ans={{d, nu}, {(Plus@@tt)/n, Min[tt], Max[tt]}, N[(Plus@@ll)/n]};
      Print[ans];
      ans]

runtests2[dd_] :=
   Module[{nn},
      nn={10, 20, 50, 100, 200, 500, 1000};
      Table[rtest2[dd, nn[[i]], 100], {i, 7}]]

(* --------------------- Test results --------------------- *)


(*     

Uniform distribution of coefficients:

In[2]:= tau10=runtests10[];
{{10, 10, 10}, {0.006102, 0.004487, 0.007966}, 2.2}
{{10, 20, 10}, {0.010722, 0.009310, 0.012259}, 1.4}
{{10, 50, 10}, {0.027507, 0.024752, 0.031766}, 2.}
{{10, 100, 10}, {0.060153, 0.055679, 0.067662}, 2.2}
{{10, 200, 10}, {0.121906, 0.108434, 0.135277}, 2.}
{{10, 500, 10}, {0.357743, 0.336965, 0.410570}, 2.}
{{10, 1000, 10}, {0.856813, 0.710344, 0.969133}, 1.8}
{{20, 10, 10}, {0.015057, 0.012605, 0.018066}, 2.6}
{{20, 20, 10}, {0.024852, 0.021472, 0.031192}, 2.4}
{{20, 50, 10}, {0.057545, 0.053802, 0.059851}, 2.8}
{{20, 100, 10}, {0.110276, 0.099363, 0.121870}, 2.6}
{{20, 200, 10}, {0.234979, 0.215494, 0.271148}, 2.8}
{{20, 500, 10}, {0.677973, 0.545825, 0.750333}, 2.2}
{{20, 1000, 10}, {1.528529, 1.401648, 1.626550}, 3.}
{{50, 10, 10}, {0.041671, 0.035298, 0.051128}, 3.2}
{{50, 20, 10}, {0.068013, 0.052015, 0.084304}, 3.2}
{{50, 50, 10}, {0.141542, 0.131862, 0.151214}, 2.6}
{{50, 100, 10}, {0.272052, 0.241628, 0.293075}, 2.8}
{{50, 200, 10}, {0.580774, 0.525108, 0.645822}, 2.6}
{{50, 500, 10}, {1.608304, 1.438432, 1.756491}, 3.4}
{{50, 1000, 10}, {3.560268, 3.318446, 3.960426}, 3.2}
{{100, 10, 10}, {0.115961, 0.071205, 0.152400}, 3.4}
{{100, 20, 10}, {0.164283, 0.134389, 0.208516}, 3.}
{{100, 50, 10}, {0.339361, 0.289312, 0.399740}, 3.4}
{{100, 100, 10}, {0.640332, 0.561805, 0.711043}, 3.8}
{{100, 200, 10}, {1.187615, 1.009189, 1.311924}, 4.}
{{100, 500, 10}, {3.135120, 2.833715, 3.484829}, 4.}
{{100, 1000, 10}, {7.650408, 7.153960, 8.125055}, 3.4}
{{200, 10, 10}, {0.496115, 0.329754, 0.638446}, 4.4}
{{200, 20, 10}, {0.516051, 0.348935, 0.714264}, 3.2}
{{200, 50, 10}, {0.900138, 0.780196, 1.215373}, 3.}
{{200, 100, 10}, {1.646894, 1.484047, 2.123404}, 5.}
{{200, 200, 10}, {2.755469, 2.449282, 3.227248}, 4.}
{{200, 500, 10}, {6.414754, 5.313379, 7.439842}, 3.4}
{{200, 1000, 10}, {16.696399, 13.498766, 18.642335}, 3.2}
{{500, 10, 10}, {3.426162, 0.946272, 5.530551}, 4.4}
{{500, 20, 10}, {4.529804, 2.571537, 8.835584}, 4.6}
{{500, 50, 10}, {5.297568, 3.655016, 9.453382}, 4.}
{{500, 100, 10}, {6.520840, 5.000045, 8.858889}, 4.}
{{500, 200, 10}, {10.383512, 7.806817, 12.986310}, 4.6}
{{500, 500, 10}, {21.544832, 20.377791, 23.802310}, 3.6}
{{500, 1000, 10}, {54.634395, 47.927819, 63.569638}, 3.8}
{{1000, 10, 10}, {25.534824, 20.155322, 35.406964}, 5.4}
{{1000, 20, 10}, {23.093777, 10.553334, 32.037060}, 4.8}
{{1000, 50, 10}, {27.688104, 18.111625, 40.049109}, 4.}
{{1000, 100, 10}, {36.825698, 23.942069, 46.451517}, 5.2}
{{1000, 200, 10}, {45.699433, 34.187532, 56.917885}, 4.8}
{{1000, 500, 10}, {79.919694, 64.559750, 105.000482}, 5.4}
{{1000, 1000, 10}, {173.398711, 153.574059, 202.836302}, 5.4}

In[2]:= tau1000=runtests1000[];
{{10, 10, 1000}, {0.006020, 0.005217, 0.007064}, 2.6}
{{10, 20, 1000}, {0.011108, 0.008175, 0.014582}, 2.2}
{{10, 50, 1000}, {0.027578, 0.025147, 0.031383}, 2.4}
{{10, 100, 1000}, {0.054037, 0.048247, 0.058999}, 1.6}
{{10, 200, 1000}, {0.119941, 0.106534, 0.146227}, 1.8}
{{10, 500, 1000}, {0.361789, 0.318300, 0.415038}, 2.8}
{{10, 1000, 1000}, {0.882925, 0.661186, 0.988297}, 2.}
{{20, 10, 1000}, {0.015076, 0.013659, 0.016626}, 2.}
{{20, 20, 1000}, {0.026338, 0.021889, 0.031594}, 2.2}
{{20, 50, 1000}, {0.059639, 0.053618, 0.064673}, 2.4}
{{20, 100, 1000}, {0.116083, 0.108486, 0.123745}, 2.2}
{{20, 200, 1000}, {0.237420, 0.216605, 0.265617}, 2.4}
{{20, 500, 1000}, {0.809320, 0.720199, 0.928179}, 2.6}
{{20, 1000, 1000}, {1.650910, 1.536338, 1.784954}, 2.2}
{{50, 10, 1000}, {0.044850, 0.037475, 0.054952}, 2.8}
{{50, 20, 1000}, {0.072020, 0.058102, 0.080530}, 3.2}
{{50, 50, 1000}, {0.156523, 0.144641, 0.174518}, 3.6}
{{50, 100, 1000}, {0.298656, 0.278561, 0.318383}, 3.2}
{{50, 200, 1000}, {0.670996, 0.642266, 0.727172}, 2.8}
{{50, 500, 1000}, {1.735774, 1.328309, 2.102796}, 2.8}
{{50, 1000, 1000}, {3.980888, 3.630943, 4.465523}, 2.8}
{{100, 10, 1000}, {0.136054, 0.102847, 0.171813}, 3.4}
{{100, 20, 1000}, {0.199668, 0.147777, 0.256655}, 3.8}
{{100, 50, 1000}, {0.356184, 0.304577, 0.431141}, 2.}
{{100, 100, 1000}, {0.759103, 0.709646, 0.800144}, 3.4}
{{100, 200, 1000}, {1.369057, 1.267260, 1.465971}, 3.4}
{{100, 500, 1000}, {3.407342, 2.810532, 3.807396}, 3.8}
{{100, 1000, 1000}, {7.776168, 7.343931, 8.287376}, 3.6}
{{200, 10, 1000}, {0.442014, 0.180531, 0.614924}, 3.6}
{{200, 20, 1000}, {0.604513, 0.427640, 0.700839}, 4.6}
{{200, 50, 1000}, {0.984861, 0.879915, 1.039115}, 3.4}
{{200, 100, 1000}, {1.622085, 1.441258, 1.769162}, 3.8}
{{200, 200, 1000}, {2.842575, 2.590887, 3.146688}, 3.6}
{{200, 500, 1000}, {7.245375, 5.528330, 8.147614}, 4.2}
{{200, 1000, 1000}, {17.854574, 15.509972, 19.353331}, 4.4}
{{500, 10, 1000}, {4.295497, 2.755585, 5.801147}, 5.}
{{500, 20, 1000}, {4.480439, 2.492373, 6.502616}, 4.4}
{{500, 50, 1000}, {5.946167, 3.639854, 10.144421}, 4.4}
{{500, 100, 1000}, {7.553519, 5.554299, 9.131769}, 4.6}
{{500, 200, 1000}, {12.590375, 10.409884, 15.732497}, 5.6}
{{500, 500, 1000}, {25.419753, 22.781632, 28.361278}, 4.8}
{{500, 1000, 1000}, {60.146103, 51.364588, 67.427157}, 5.4}
{{1000, 10, 1000}, {20.546604, 14.497562, 29.779969}, 3.8}
{{1000, 20, 1000}, {30.419806, 23.441143, 55.524203}, 5.8}
{{1000, 50, 1000}, {30.381643, 18.039105, 39.651833}, 5.}
{{1000, 100, 1000}, {34.803853, 18.972426, 47.275029}, 5.4}
{{1000, 200, 1000}, {44.663525, 34.453089, 61.967860}, 5.2}
{{1000, 500, 1000}, {81.369925, 69.383327, 100.720222}, 4.4}
{{1000, 1000, 1000}, {183.407411, 159.958797, 222.286819}, 5.2}


tau10=runtests10[];
tau1000=runtests1000[];

tau10=Flatten[tau10, 1];
tau1000=Flatten[tau1000, 1];

tau10={{{10, 10, 10}, 
  {0.00610240000000000009858225347159077501`3.8061005846455362,
   0.00448699999999999966926456096416586661`3.6725559828126983,
   0.00796600000000000085798035343032097444`3.9218402153529337}, 2.2},
 {{10, 20, 10}, {0.01072170000000000015388801344329294807`4.050863564502494,
   0.00931000000000000056676885407114241389`3.989549594260967,
   0.01225900000000000079902751082272516214`4.10905495832379}, 1.4},
 {{10, 50, 10}, {0.0275067000000000002474243032679623866`4.460038404150725,
   0.02475199999999999969646502506748220185`4.414210209634916,
   0.03176600000000000256772381135306204669`4.522562444842798}, 2.},
 {{10, 100, 10},
  {0.06015290000000000011914913500277179992`4.7998564830043255,
   0.05567899999999999932631666865745501127`4.76629133998303,
   0.06766199999999999992184029906638897955`4.850944744096743}, 2.2},
 {{10, 200, 10}, {0.12190600000000000047162274086076649837`5.106624994638489,
   0.10843400000000000260591548340016743168`5.0557653919503105,
   0.13527700000000000835065350202057743445`5.151823876749465}, 2.},
 {{10, 500, 10}, {0.35774320000000000563389335184183437377`5.574171300699432,
   0.33696500000000001451283537790004629642`5.548184707040415,
   0.41056999999999999051425447760266251862`5.633987125945838}, 2.},
 {{10, 1000, 10},
  {0.85681309999999999416075979752349667251`5.953486011176041,
   0.71034399999999997543653762477333657444`5.872068629775374,
   0.96913300000000002221156591986073181033`6.006983295286482}, 1.8},
 {{20, 10, 10}, {0.01505700000000000007144285163462882338`4.198338363681343,
   0.01260499999999999988509191695129629807`4.121142763292089,
   0.01806599999999999872635214615002041683`4.277461919176027}, 2.6},
 {{20, 20, 10}, {0.02485200000000000013222756223285614396`4.415961258220702,
   0.02147200000000000150168766310798673658`4.352472411768522,
   0.03119200000000000097544194943566253642`4.514643135459553}, 2.4},
 {{20, 50, 10}, {0.05754460000000000130526700559130404145`4.780604488852116,
   0.05380200000000000259969823446226655506`4.751398333422212,
   0.05985100000000000142197364993990049697`4.797671324343754}, 2.8},
 {{20, 100, 10}, {0.11027619999999999922390969686603057198`5.063081705647482,
   0.09936300000000000687272461163956904784`5.017824608670271,
   0.12187000000000000610178574333986034617`5.106496724411212}, 2.6},
 {{20, 200, 10}, {0.23497880000000000433324487403297098353`5.391628594877582,
   0.21549399999999999111288673248054692522`5.354035095882327,
   0.27114800000000000013145040611561853439`5.45380631868703}, 2.8},
 {{20, 500, 10}, {0.6779726000000000030176749987731454894`5.851812055658099,
   0.54582500000000000461852778244065120816`5.757653336723062,
   0.75033300000000002771827212200150825083`5.895853960626417}, 2.2},
 {{20, 1000, 10},
  {1.52852870000000000683115786159760318696`6.204873510829537,
   1.40164800000000000501643171446630731225`6.167238874947654,
   1.62654999999999994031441019615158438683`6.231867331275817}, 3.},
 {{50, 10, 10}, {0.04167050000000000128941302079965680605`4.640428624822532,
   0.03529800000000000298472357940227084327`4.568350012050715,
   0.05112799999999999983613108156532689463`4.729258717820123}, 3.2},
 {{50, 20, 10}, {0.06801279999999999964499508564585994464`4.853190567842286,
   0.052014999999999998736566197976571857`4.736728516103457,
   0.08430400000000000393374222085185465403`4.9464480945083364}, 3.2},
 {{50, 50, 10}, {0.14154180000000000094750873813609359786`5.17148462754658,
   0.13186200000000000698285873568238457665`5.1407195718432055,
   0.15121399999999998731148309616401093081`5.2001919150352665}, 2.6},
 {{50, 100, 10}, {0.27205209999999999115338766841887263581`5.455251995901227,
   0.2416280000000000094395602445729309693`5.403747172454288,
   0.29307499999999997442046151263639330864`5.4875786869377325}, 2.8},
 {{50, 200, 10}, {0.58077360000000000095354835139005444944`5.784606779864727,
   0.52510800000000001919175929288030602038`5.7408485480766585,
   0.64582200000000000716937620381941087544`5.830712748503757}, 2.6},
 {{50, 500, 10}, {1.60830400000000000027000623958883807063`6.226968055356262,
   1.43843199999999993288213318010093644261`6.178489249285043,
   1.75649100000000002452793523843865841627`6.265245842152701}, 3.4},
 {{50, 1000, 10},
  {3.56026829999999994136317127413349226117`6.572082640696008,
   3.31844599999999978479081619298085570335`6.541534668181345,
   3.96042600000000000193267624126747250557`6.618341816250272}, 3.2},
 {{100, 10, 10}, {0.11596140000000000058744120678966282867`5.084913363224091,
   0.07120500000000000440092406961412052624`4.873110404053666,
   0.15240000000000000768274333040608325973`5.2035848802832065}, 3.4},
 {{100, 20, 10}, {0.16428260000000000362607721626773127355`5.236191480828159,
   0.13438900000000000845012948502699146047`5.148963635607485,
   0.20851600000000000689759360739117255434`5.339739298465212}, 3.},
 {{100, 50, 10}, {0.33936149999999999926636462532769655809`5.551262484066334,
   0.28931200000000001359978796244831755757`5.48196636084292,
   0.39973999999999998422595126612577587366`5.622377521409865}, 3.4},
 {{100, 100, 10},
  {0.64033160000000000078301809480763040483`5.827004847818133,
   0.56180499999999999882760448599583469331`5.770185513333411,
   0.71104299999999998060928874110686592758`5.872495778563596}, 3.8},
 {{100, 200, 10},
  {1.18761539999999998773461129530915059149`6.095275733806575,
   1.00918899999999989169907621544552966952`6.024572421410005,
   1.31192400000000009008260803966550156474`6.138508590288291}, 4.},
 {{100, 500, 10},
  {3.13512040000000005157687610335415229201`6.516854137250685,
   2.83371500000000020591528482327703386545`6.472956082355711,
   3.48482899999999995443999978306237608194`6.562781385475935}, 4.},
 {{100, 1000, 10},
  {7.65040779999999989158254720678087323904`6.9042844988344685,
   7.15395999999999965268671076046302914619`6.875146420824775,
   8.12505499999999969418240652885288000107`6.930426222760157}, 3.4},
 {{200, 10, 10}, {0.49611540000000000083524298588599776849`5.716182621531724,
   0.32975399999999999156585772652761079371`5.538789985632648,
   0.63844599999999995798560803450527600944`5.825724083643695}, 4.4},
 {{200, 20, 10},
  {0.51605079999999999840198938727553468198`5.7332923689257225,
   0.34893499999999999516830939683131873608`5.563344446913766,
   0.71426400000000001000444171950221061707`5.874458674848453}, 3.2},
 {{200, 50, 10}, {0.90013840000000001628777113182877656072`5.97490920242479,
   0.78019600000000000061817218011128716171`5.9128036326694176,
   1.21537300000000003663558345579076558352`6.105309497371506}, 3.},
 {{200, 100, 10},
  {1.64689419999999995258832541367155499756`6.2372656133406545,
   1.48404699999999989401544553402345627546`6.192047568614426,
   2.1234039999999998471480466832872480154`6.347632544411497}, 5.},
 {{200, 200, 10},
  {2.75546880000000000698889834893634542823`6.460795411179834,
   2.44928200000000018121681932825595140457`6.4096387041181915,
   3.22724799999999989452703630377072840929`6.529432253611659}, 4.},
 {{200, 500, 10},
  {6.41475380000000008351435099029913544655`6.827779905969791,
   5.31337900000000029621105568367056548595`6.745970708247931,
   7.43984199999999962216179483220912516117`6.892163625807121}, 3.4},
 {{200, 1000, 10},
  {16.69639879999999951820655041956342756748`7.243222722754061,
   13.49876599999999982060217007528990507126`7.150893982227544,
   18.6423349999999992121502145892009139061`7.291100221195907}, 3.2},
 {{500, 10, 10}, {3.42616220000000004652918050851440057158`6.555407832591398,
   0.94627200000000000201794136955868452787`5.996615902873655,
   5.53055099999999999482724888366647064686`6.763368314808333}, 4.4},
 {{500, 20, 10}, {4.52980390000000006089919679652666673064`6.676679314632119,
   2.57153700000000018377477317699231207371`6.430792690733244,
   8.83558400000000077056938607711344957352`6.966835173358435}, 4.6},
 {{500, 50, 10},
  {5.29756799999999983263876401906600221992`6.7446764533364005,
   3.65501599999999982060217007528990507126`6.583489195721657,
   9.45338199999999950762230582768097519875`6.996187120854882}, 4.},
 {{500, 100, 10},
  {6.52083969999999997213535607443191111088`6.834903437477094,
   5.0000450000000000727595761418342590332`6.719573826248391,
   8.85888899999999956946794554824009537697`6.967979173380721}, 4.},
 {{500, 200, 10},
  {10.38351179999999986947045726992655545473`7.036944174058113,
   7.80681699999999967332087180693633854389`6.913073912444276,
   12.98630999999999957594809529837220907211`7.1340856791105605}, 4.6},
 {{500, 500, 10},
  {21.54483179999999968856627674540504813194`7.35394302119034,
   20.37779099999999843362274987157434225082`7.3297570169683395,
   23.80230999999999852434484637342393398285`7.3972190204021295}, 3.6},
 {{500, 1000, 10},
  {54.63439499999999924284566077403724193573`7.758066051511089,
   47.92781899999999950523488223552703857422`7.701187579746558,
   63.56963799999999764622771181166172027588`7.823849651608774}, 3.8},
 {{1000, 10, 10},
  {25.53482380000000055986220104387030005455`7.42773277868435,
   20.15532200000000173645275935996323823929`7.324989654082071,
   35.40696400000000210184225579723715782166`7.5696886026874335}, 5.4},
 {{1000, 20, 10},
  {23.0937769999999993331130099250003695488`7.384094881108551,
   10.55333399999999954843588056974112987518`7.043989596521199,
   32.03705999999999676219886168837547302246`7.526252567871557}, 4.8},
 {{1000, 50, 10},
  {27.68810410000000103991624200716614723206`7.462893132401962,
   18.11162500000000008526512829121202230453`7.278557330842312,
   40.04910900000000140153133543208241462708`7.6231927717601105}, 4.},
 {{1000, 100, 10},
  {36.8256980999999992576476870453916490078`7.586750901736696,
   23.94206900000000004524736141320317983627`7.399761591365753,
   46.45151700000000261070454143919050693512`7.687599814902526}, 5.2},
 {{1000, 200, 10},
  {45.69943290000000004624780558515340089798`7.68051072407313,
   34.18753199999999736746758571825921535492`7.554467663462927,
   56.91788499999999828560248715803027153015`7.775848667132229}, 4.8},
 {{1000, 500, 10},
  {79.91969400000000121053744805976748466492`7.923253725655132,
   64.55974999999999397459760075435042381287`7.830561753242853,
   105.00048200000000520049070473760366439819`8.041791205963465}, 5.4},
 {{1000, 1000, 10},
  {173.39871089999999469455360667780041694641`8.25964577775143,
   153.57405900000000542604539077728986740112`8.20691777621077,
   202.83630199999998922066879458725452423096`8.327745597409285}, 5.4}};

tau1000={{{10, 10, 1000},
  {0.00602010000000000001715849684558179433`3.800203618671582,
   0.00521700000000000028294033782572114433`3.738020750001999,
   0.00706399999999999996663779811001404596`3.869650603849136}, 2.6},
 {{10, 20, 1000},
  {0.01110829999999999977144948815066527459`4.066247513434655,
   0.00817499999999999997501998194593397784`3.9330876746119476,
   0.01458199999999999947608575467938862857`4.18441700717895}, 2.2},
 {{10, 50, 1000},
  {0.02757759999999999968312014431148782023`4.461156381351879,
   0.0251469999999999992146282323801642633`4.421086095070828,
   0.0313830000000000011728396032140153693`4.517294370094925}, 2.4},
 {{10, 100, 1000},
  {0.0540370000000000003104183576851937687`4.7532911433481315,
   0.04824699999999999822097862534064915963`4.704070227354222,
   0.05899900000000000283018053437444905285`4.791444563935965}, 1.6},
 {{10, 200, 1000},
  {0.11994070000000000147943879369449859951`5.099566492425792,
   0.1065340000000000036939340475328208413`5.048088146931096,
   0.14622699999999999587174670523381792009`5.185627483359598}, 1.8},
 {{10, 500, 1000},
  {0.36178870000000000750617346056969836354`5.579054911391353,
   0.31830000000000002735589532676385715604`5.523436551900628,
   0.41503800000000001801936377887614071369`5.638687774894827}, 2.8},
 {{10, 1000, 1000},
  {0.88292519999999999891215907155128661543`5.966523825687026,
   0.66118600000000005145750492374645546079`5.840923562493741,
   0.98829699999999998105693066463572904468`6.015487390334889}, 2.},
 {{20, 10, 1000}, {0.0150758999999999997176924892983151949`4.198883161346643,
   0.01365899999999999926691973684000913636`4.156018818307476,
   0.01662599999999999841882036832885205513`4.241387689445499}, 2.},
 {{20, 20, 1000},
  {0.02633769999999999965323294048857860616`4.4411777598005395,
   0.02188899999999999873567801955687173177`4.360825834536814,
   0.0315939999999999970858866049638891127`4.520204527090887}, 2.2},
 {{20, 50, 1000},
  {0.05963949999999999945776707477307354566`4.796133907078837,
   0.05361799999999999899458202889945823699`4.749910523650782,
   0.06467299999999999438049513855730765499`4.8313229204031245}, 2.4},
 {{20, 100, 1000},
  {0.1160833999999999990859755882866011234`5.085370033071243,
   0.10848599999999999909938708242407301441`5.055973609850842,
   0.12374499999999999388933247246313840151`5.113127573278938}, 2.2},
 {{20, 200, 1000},
  {0.23741959999999999453290655537784914486`5.396116482238875,
   0.2166049999999999919886306543048704043`5.356268390718265,
   0.26561699999999999199928879534127190709`5.444855780549113}, 2.4},
 {{20, 500, 1000},
  {0.80932040000000001755964262883935589343`5.92872040078174,
   0.72019900000000003359446054673753678799`5.878052427295197,
   0.92817899999999997628918890768545679748`5.9882316515844725}, 2.6},
 {{20, 1000, 1000},
  {1.65091049999999999187139110290445387363`6.238323442987168,
   1.53633799999999998142641288723098114133`6.207086685867279,
   1.78495399999999992957100403145886957645`6.2722269416810095}, 2.2},
 {{50, 10, 1000},
  {0.04484970000000000137085898188615828985`4.6723594556702075,
   0.03747500000000000136557432028894254472`4.594341554799941,
   0.05495200000000000084554585555451922119`4.760583416648002}, 2.8},
 {{50, 20, 1000},
  {0.07201989999999999919610971232941665221`4.878052427295197,
   0.058102000000000000867750316047022352`4.784790995308755,
   0.08053000000000000435651514862911426462`4.926557612372051}, 3.2},
 {{50, 50, 1000},
  {0.15652259999999999784403570401991601102`5.215176966642401,
   0.14464099999999999179856047248904360458`5.180891328989595,
   0.17451800000000000645528075438051018864`5.262440140549135}, 3.6},
 {{50, 100, 1000},
  {0.29865610000000000212594386539421975613`5.495771302844162,
   0.27856100000000000305178105008963029832`5.465520226119876,
   0.3183830000000000270965472282114205882`5.523549783892693}, 3.2},
 {{50, 200, 1000},
  {0.67099630000000000107363007373351138085`5.847320038673631,
   0.64226600000000000356692453351570293307`5.828314845408599,
   0.72717200000000004056488478454411961138`5.882237061161842}, 2.8},
 {{50, 500, 1000},
  {1.73577400000000003910116674887831322849`6.260093092102644,
   1.32830899999999996197175278211943805218`6.143899028511022,
   2.10279600000000010950884643534664064646`6.343397055499143}, 2.8},
 {{50, 1000, 1000},
  {3.98088840000000003804814241448184475303`6.620579916047458,
   3.6309429999999998095461251068627461791`6.580619344517007,
   4.46552300000000013113776731188409030437`6.67047224385301}, 2.8},
 {{100, 10, 1000},
  {0.13605390000000000122470922292450268287`5.154310908672898,
   0.10284699999999999397992667127255117521`5.0327915413235145,
   0.17181299999999999350563939515268430114`5.255655934327044}, 3.4},
 {{100, 20, 1000},
  {0.19966760000000000063735683397680986673`5.320907511035595,
   0.1477769999999999916973081326432293281`5.190206759040939,
   0.25665500000000002200906124016910325736`5.42994964262551}, 3.8},
 {{100, 50, 1000},
  {0.3561839999999999950563989159491029568`5.572274320075073,
   0.30457699999999998663469114035251550376`5.5042970179584945,
   0.43114099999999999646149717591470107436`5.655219237982465}, 2.},
 {{100, 100, 1000},
  {0.7591034999999999866027167172433109954`5.900900907112413,
   0.7096459999999999990194510246510617435`5.8716416724407665,
   0.80014399999999996637711774383205920458`5.923768066243584}, 3.4},
 {{100, 200, 1000},
  {1.36905729999999998813819956922088749707`6.157021538588952,
   1.26726000000000005307754236127948388457`6.1234656402239365,
   1.46597099999999991304377999767893925309`6.18672529240837}, 3.4},
 {{100, 500, 1000},
  {3.40734210000000001805631200113566592336`6.553015652566204,
   2.81053199999999980818188305420335382223`6.469388447702854,
   3.80739599999999978052755977842025458813`6.60122796258436}, 3.8},
 {{100, 1000, 1000},
  {7.77616830000000005540528036362957209349`6.9113655647567835,
   7.34393100000000043081627154606394469738`6.88652850107248,
   8.28737600000000007582912076031789183617`6.939016956602723}, 3.6},
 {{200, 10, 1000},
  {0.44201360000000000094466656719305319712`5.666035545330271,
   0.18053099999999999702993136452278122306`5.277151701088934,
   0.61492400000000002613376182125648483634`5.809421356827055}, 3.6},
 {{200, 20, 1000},
  {0.60451319999999999499351588383433409035`5.802005701726324,
   0.42764000000000002010835942201083526015`5.651678234154069,
   0.70083899999999998975397375033935531974`5.866218174552548}, 4.6},
 {{200, 50, 1000},
  {0.98486139999999999794511040818179026246`6.013975029614286,
   0.87991500000000000270006239588838070631`5.965040634504901,
   1.03911500000000001087130385712953284383`6.037263527344642}, 3.4},
 {{200, 100, 1000},
  {1.62208489999999996289403725313604809344`6.230673494743271,
   1.44125799999999992806465343164745718241`6.179341644230228,
   1.76916199999999990133403571235248818994`6.268367515824608}, 3.8},
 {{200, 200, 1000},
  {2.84257529999999998260307165764970704913`6.474311891280148,
   2.59088699999999994005861481127794831991`6.4340483851822245,
   3.14668800000000015160139810177497565746`6.518453597217419}, 3.6},
 {{200, 500, 1000},
  {7.24537490000000001799662641133181750774`6.88066077552144,
   5.52833000000000041040948417503386735916`6.763193872555582,
   8.14761400000000080012796388473361730576`6.93163035903228}, 4.2},
 {{200, 1000, 1000},
  {17.85457400000000003359446054673753678799`7.272349405912239,
   15.50997199999999942576778266811743378639`7.211210927066378,
   19.35333100000000072782313509378582239151`7.307355637694928}, 4.4},
 {{500, 10, 1000},
  {4.29549690000000006229186055861646309495`6.653613323227029,
   2.75558499999999995111465977970510721207`6.4608137252884035,
   5.80114700000000027557689463719725608826`6.784113783829099}, 5.},
 {{500, 20, 1000},
  {4.48043939999999989431955782492877915502`6.671920520946889,
   2.49237300000000017163870325020980089903`6.417212951154226,
   6.50261599999999972965269989799708127975`6.833688021584759}, 4.4},
 {{500, 50, 1000},
  {5.94616739999999981414191552175907418132`6.794837044834196,
   3.6398540000000001448654529667692258954`6.581683877075379,
   10.14442099999999946646767057245597243309`7.026827177691777}, 4.4},
 {{500, 100, 1000},
  {7.5535192000000000334125616063829511404`6.898749250743282,
   5.55429900000000031923264032229781150818`6.765229168340039,
   9.13176900000000024704149836907163262367`6.981154830191755}, 4.6},
 {{500, 200, 1000},
  {12.59037489999999976930666889529675245285`7.12063857544286,
   10.40988399999999991507593222195282578468`7.038045803362631,
   15.73249700000000039779024518793448805809`7.217397570884365}, 5.6},
 {{500, 500, 1000},
  {25.4197532000000002483375283190980553627`7.425771242959513,
   22.78163199999999832812136446591466665268`7.378184745542765,
   28.36127799999999865576683077961206436157`7.473325710165917}, 4.8},
 {{500, 1000, 1000},
  {60.14610329999999933647814032156020402908`7.799807409126282,
   51.36458799999999769170244690030813217163`7.731263722228489,
   67.42715699999999401370587293058633804321`7.849434761741055}, 5.4},
 {{1000, 10, 1000},
  {20.54660449999999976000708556966856122017`7.333339974588766,
   14.49756200000000028182967071188613772392`7.181894887999433,
   29.77996900000000124464349937625229358673`7.494524154609282}, 3.8},
 {{1000, 20, 1000},
  {30.41980610000000027071109798271209001541`7.503756354753064,
   23.44114300000000028489921533036977052689`7.390578697523728,
   55.52420299999999997453414835035800933838`7.765082246638158}, 5.8},
 {{1000, 50, 1000},
  {30.38164289999999922997631074395030736923`7.503211168096466,
   18.03910499999999927922544884495437145233`7.276814899744774,
   39.65183300000000343743522535078227519989`7.618863181689492}, 5.},
 {{1000, 100, 1000},
  {34.80385319999999929052592051448300480843`7.562227241446327,
   18.97242599999999868032318772748112678528`7.298722780860355,
   47.27502900000000352065399056300520896912`7.695231717206773}, 5.4},
 {{1000, 200, 1000},
  {44.66352480000000042537067201919853687286`7.670552907446828,
   34.45308899999999852070686756633222103119`7.557828079305939,
   61.96786000000000171894498635083436965942`7.812766411749154}, 5.2},
 {{1000, 500, 1000},
  {81.36992499999999921556081972084939479828`7.931063828979044,
   69.3833269999999942001522867940366268158`7.861855034140214,
   100.72022200000000680120137985795736312866`8.023716587619987}, 4.4},
 {{1000, 1000, 1000},
  {183.40741119999999853007466299459338188171`8.284016794113262,
   159.9587970000000041181920096278190612793`8.22460804256065,
   222.28681900000000837280822452157735824585`8.367513624267934}, 5.2}};

{d, nu, tau}

logdata10 = {Log2[#[[1, 1]]], Log2[#[[1, 2]]], Log2[#[[2, 1]]]}&/@tau10;
logdata1000 = {Log2[#[[1, 1]]], Log2[#[[1, 2]]], Log2[#[[2, 1]]]}&/@tau1000;

Fit[logdata10, {1, x, y}, {x, y}]
-14.56 + 1.391 x + 0.816 y
Fit[logdata1000, {1, x, y}, {x, y}]
-14.56 + 1.400 x + 0.821 y


Gaussian distribution of coefficients (runtests2):

tau={{{10, 10}, {0.003649, 0.001959, 0.011429}, 3.2},
{{10, 20}, {0.005288, 0.003415, 0.011800}, 3.06},
{{10, 50}, {0.012546, 0.009647, 0.024996}, 3.28},
{{10, 100}, {0.028359, 0.022825, 0.066087}, 3.14},
{{10, 200}, {0.072058, 0.065638, 0.109591}, 3.3},
{{10, 500}, {0.290117, 0.246546, 0.391322}, 3.1},
{{10, 1000}, {0.992683, 0.751464, 1.210348}, 3.22},
{{20, 10}, {0.012866, 0.008655, 0.017460}, 4.4},
{{20, 20}, {0.022269, 0.017665, 0.027470}, 4.18},
{{20, 50}, {0.050360, 0.045219, 0.060892}, 4.56},
{{20, 100}, {0.108834, 0.090885, 0.145132}, 4.48},
{{20, 200}, {0.239452, 0.198089, 0.343603}, 4.66},
{{20, 500}, {0.902403, 0.715687, 1.221316}, 4.28},
{{20, 1000}, {2.067790, 1.554802, 2.630937}, 4.14},
{{50, 10}, {0.079914, 0.042715, 0.126415}, 7.46},
{{50, 20}, {0.117502, 0.086191, 0.184580}, 7.22},
{{50, 50}, {0.190565, 0.149076, 0.277305}, 6.74},
{{50, 100}, {0.406157, 0.301625, 0.533294}, 6.96},
{{50, 200}, {0.794419, 0.660836, 1.074711}, 7.12},
{{50, 500}, {2.344101, 1.751442, 2.908548}, 7.06},
{{50, 1000}, {5.334312, 4.087481, 7.318953}, 6.86},
{{100, 10}, {0.308733, 0.196984, 0.493275}, 9.92},
{{100, 20}, {0.383922, 0.219574, 0.572284}, 10.12},
{{100, 50}, {0.595907, 0.393973, 0.809013}, 9.98},
{{100, 100}, {1.059363, 0.795960, 1.554766}, 10.12},
{{100, 200}, {2.029937, 1.649458, 2.585539}, 9.9},
{{100, 500}, {5.070653, 3.474233, 6.849705}, 10.44},
{{100, 1000}, {11.422124, 7.779208, 14.886385}, 10.02},
{{200, 10}, {1.747921, 1.093731, 2.458029}, 13.98},
{{200, 20}, {2.193705, 1.333294, 3.029945}, 14.02},
{{200, 50}, {2.494700, 1.540472, 3.272282}, 13.78},
{{200, 100}, {4.096523, 2.639179, 6.117161}, 14.36},
{{200, 200}, {6.561018, 3.825825, 9.334200}, 13.98},
{{200, 500}, {9.421427, 7.120093, 12.919331}, 14.24},
{{200, 1000}, {18.840447, 15.664496, 29.616474}, 13.92},
{{500, 10}, {32.389055, 19.047558, 49.812016}, 22.92},
{{500, 20}, {32.873718, 15.424353, 49.047758}, 22.5},
{{500, 50}, {34.375926, 19.031484, 47.698995}, 22.46},
{{500, 100}, {35.929646, 22.866555, 52.542512}, 22.1},
{{500, 200}, {39.857706, 20.933021, 63.913423}, 21.92},
{{500, 500}, {51.721074, 32.429875, 81.010824}, 22.72},
{{500, 1000}, {88.514765, 61.977931, 169.923654}, 22.8}};

logdata = {Log2[#[[1, 1]]], Log2[#[[1, 2]]], Log2[#[[2, 1]]]}&/@tau;
Fit[logdata, {1, x, y}, {x, y}]
-15.829 + 1.7183 x + 0.793607 y

*)