Starting with polynomial:
P : 4*t^2 - 2
Extension levels are: 2 3 38
-------------------------------------------------
Trying to find an order 3 Kronrod extension for:
P1 : 4*t^2 - 2
Solvable: 1
-------------------------------------------------
Trying to find an order 38 Kronrod extension for:
P2 : 4*t^5 - 14*t^3 + 6*t
Solvable: 1
-------------------------------------------------
Ending with final polynomial:
P : 4*t^43 - 13538790826894975698918133043106201220877/8245215608705997557673176628566037678*t^41 + 5023791792361870677802820803790254245787457/16490431217411995115346353257132075356*t^39 - 372469002459309004732526088760107147733226209/10993620811607996743564235504754716904*t^37 + 18512296044115847598350252455393392225511764629/7329080541071997829042823669836477936*t^35 - 980382441175174613373656618407243249627195433575/7329080541071997829042823669836477936*t^33 + 38134282692972321006511099015851214870110432777225/7329080541071997829042823669836477936*t^31 - 2222537707512782775144512388699729710751150702543015/14658161082143995658085647339672955872*t^29 + 98125793794491892210931737999407051502733339600862315/29316322164287991316171294679345911744*t^27 - 6595384843580320032199012139318305705821375425569350015/117265288657151965264685178717383646976*t^25 + 168679028940576238039456860502846413774988645983683796375/234530577314303930529370357434767293952*t^23 - 3267536592708051596862914385435597251301077779264037212125/469061154628607861058740714869534587904*t^21 + 47505015657659265791461917760450352457800901649813670944875/938122309257215722117481429739069175808*t^19 - 63899222459136357344652567135794404148023728985129957864125/234530577314303930529370357434767293952*t^17 + 1996878894183585434068273480712932241394511529114593522525125/1876244618514431444234962859478138351616*t^15 - 11038395946094716623086447532106293222163105582003240115401875/3752489237028862888469925718956276703232*t^13 + 10434108225101370409727275397782870015412941494493299736768125/1876244618514431444234962859478138351616*t^11 - 413033826811243223127466192473359835332333449423217476712923125/60039827792461806215518811503300427251712*t^9 + 630277541799102361891880440491596969650443660214147650712910625/120079655584923612431037623006600854503424*t^7 - 544747462033753238361273889135041681063248387186664572957881875/240159311169847224862075246013201709006848*t^5 + 228698445490860556809413789831786310962768746103255596990228125/480318622339694449724150492026403418013696*t^3 - 15520386894389696408665506786559774420695104982384131697853125/480318622339694449724150492026403418013696*t
-------------------------------------------------
Computing nodes and weights
  current precision for roots: 53
  current precision for roots: 106
 current precision for weights: 53
Linear system for weights solvable: 0
  current precision for roots: 106
  current precision for roots: 212
 current precision for weights: 106
Linear system for weights solvable: 0
  current precision for roots: 212
  current precision for roots: 424
 current precision for weights: 212
Linear system for weights solvable: 1
  current precision for roots: 424
  current precision for roots: 848
 current precision for weights: 424
Linear system for weights solvable: 1
Sufficient bits for target precision reached
-------------------------------------------------
The nodes are:
| (-6.9598450133660276331 - 2.2788329394502507524e-964j)  +/-  (4.68e-246, 4.68e-246j)
| (4.6995841156440009006 + 1.8177540746657535913e-965j)  +/-  (2.58e-245, 2.58e-245j)
| (-5.1088620379766616608 + 1.6476121882925061897e-976j)  +/-  (3.22e-245, 3.22e-245j)
| (6.4504294320141249723 + 4.3171659240354621092e-989j)  +/-  (1.3e-245, 1.3e-245j)
| (-8.2125674721818823327 - 1.4801679940980026941e-1003j)  +/-  (8.5e-248, 8.5e-248j)
| (-6.4504294320141249723 + 2.2877120708043418109e-1000j)  +/-  (1.31e-245, 1.31e-245j)
| (6.9598450133660276331 - 3.442715625916473004e-1004j)  +/-  (4.68e-246, 4.68e-246j)
| (-2.0853487513071908308 - 2.6666252358399883784e-1018j)  +/-  (1.34e-248, 1.34e-248j)
| (7.5284717853982143351 - 1.1438538109976763208e-1014j)  +/-  (9.66e-247, 9.66e-247j)
| (-7.5284717853982143351 + 1.9974665172802929062e-1019j)  +/-  (9.21e-247, 9.21e-247j)
| (-5.9787644866883592882 - 6.3728797313817342504e-1017j)  +/-  (2.62e-245, 2.62e-245j)
| (2.0853487513071908308 + 3.8620423991469383246e-1026j)  +/-  (1.4e-248, 1.4e-248j)
| (5.5337905288433204533 - 1.0235218748362396608e-1023j)  +/-  (3.5e-245, 3.5e-245j)
| (-1.7320508075688772935 - 1.4955313817713771554e-1038j)  +/-  (2.09e-249, 2.09e-249j)
| (-3.9163500495168705986 + 4.8316647983185564852e-1032j)  +/-  (8.55e-246, 8.55e-246j)
| (3.5382993049917244002 - 1.5342247274458959296e-1032j)  +/-  (3.67e-246, 3.67e-246j)
| (4.3028497927860684077 + 8.9152210619402511688e-1034j)  +/-  (1.74e-245, 1.74e-245j)
| (-3.5382993049917244002 - 1.8607143693998111212e-1037j)  +/-  (3.39e-246, 3.39e-246j)
| (-0.79262148814810795328 + 9.3943682930141782159e-1043j)  +/-  (3.49e-251, 3.49e-251j)
| (-4.6995841156440009006 + 4.7233781422530410665e-1036j)  +/-  (2.61e-245, 2.61e-245j)
| (1.7320508075688772935 - 1.7769235165109374972e-1042j)  +/-  (2.35e-249, 2.35e-249j)
| (1.3807719845026063315 + 5.0053662178728639544e-1043j)  +/-  (3.86e-250, 3.86e-250j)
| (-4.3028497927860684077 + 5.6296387565553608788e-1037j)  +/-  (1.71e-245, 1.71e-245j)
| (-5.5337905288433204533 + 1.5198643326951454226e-1042j)  +/-  (3.57e-245, 3.57e-245j)
| (5.9787644866883592882 - 1.1182811328901587928e-1049j)  +/-  (2.39e-245, 2.39e-245j)
| (8.2125674721818823327 + 7.7511849012405503573e-1063j)  +/-  (8.47e-248, 8.47e-248j)
| (3.1672696690934046872 - 2.3474257050361570441e-1064j)  +/-  (1.17e-246, 1.17e-246j)
| (5.1088620379766616608 + 1.4700929338669614826e-1069j)  +/-  (3.46e-245, 3.46e-245j)
| (-2.8020838798949944333 + 5.0202849954302126697e-1080j)  +/-  (3.37e-247, 3.37e-247j)
| (-0.7071067811865475244 - 2.6589203499584158057e-1085j)  +/-  (1.49e-251, 1.49e-251j)
| (2.4417400540644662806 + 3.284269007957346697e-1080j)  +/-  (7.28e-248, 7.28e-248j)
| (1.0278731269292285042 + 9.1490370102947761642e-1085j)  +/-  (7.22e-251, 7.22e-251j)
| (0.34722846609708634099 - 7.8862234376450284503e-1088j)  +/-  (1.26e-253, 1.26e-253j)
| (3.9163500495168705986 + 1.5248100414227607323e-1081j)  +/-  (8.25e-246, 8.25e-246j)
| (-1.0278731269292285042 - 2.0190038674075748265e-1092j)  +/-  (6.62e-251, 6.62e-251j)
| (-2.4417400540644662806 + 1.6852199853628737721e-1089j)  +/-  (7.28e-248, 7.28e-248j)
| (0.7071067811865475244 + 5.207752858778033675e-1093j)  +/-  (1.39e-251, 1.39e-251j)
| (0.79262148814810795328 + 6.0097922017056821846e-1093j)  +/-  (3.44e-251, 3.44e-251j)
| (2.8020838798949944333 - 1.5314932641953666042e-1091j)  +/-  (3.76e-247, 3.76e-247j)
| (-1.3807719845026063315 + 3.6670707206728513359e-1094j)  +/-  (3.61e-250, 3.61e-250j)
| (-0.34722846609708634099 + 2.2765502539960280558e-1097j)  +/-  (1.06e-253, 1.06e-253j)
| (-3.1672696690934046872 + 5.100803507050073586e-1095j)  +/-  (1.12e-246, 1.12e-246j)
| (-4.6283643663952217624e-1221 - 2.2900934307570842077e-1221j)  +/-  (2.22e-1219, 2.22e-1219j)
-------------------------------------------------
The weights are:
| (2.7668016910326341335e-22 + 8.8232629209792063472e-983j)  +/-  (1.98e-92, 4.48e-214j)
| (5.8129323103963033009e-11 - 5.4290848232169835742e-975j)  +/-  (3.05e-86, 6.87e-208j)
| (1.0857977726372399784e-12 + 1.2397119801045447319e-977j)  +/-  (1.78e-87, 4.01e-209j)
| (2.3433145409096327762e-19 + 2.0055043820808093762e-980j)  +/-  (2.22e-92, 5e-214j)
| (2.2741178968319067287e-30 + 5.1978809479468578386e-987j)  +/-  (3.46e-96, 7.81e-218j)
| (2.3433145409096327762e-19 - 3.1449413917510564949e-981j)  +/-  (1.53e-91, 3.46e-213j)
| (2.7668016910326341335e-22 - 4.5750597987260451466e-982j)  +/-  (1.84e-94, 4.16e-216j)
| (0.0025862816694361694241 + 6.0186398994180476818e-972j)  +/-  (9.42e-77, 2.12e-198j)
| (8.3685693050790483142e-26 + 5.4225900311377467463e-984j)  +/-  (3.08e-96, 6.95e-218j)
| (8.3685693050790483142e-26 - 1.2561137308823994417e-984j)  +/-  (7.43e-95, 1.68e-216j)
| (7.7119614325224268536e-17 + 6.9570690285831925273e-980j)  +/-  (3.33e-91, 7.51e-213j)
| (0.0025862816694361694241 + 1.5621705142605427141e-971j)  +/-  (7.9e-80, 1.78e-201j)
| (1.2293079070645965307e-14 + 1.3173980335001552263e-977j)  +/-  (3.43e-92, 7.74e-214j)
| (0.0098892130188462243777 - 1.9540549981540611796e-971j)  +/-  (2.89e-77, 6.51e-199j)
| (4.7030639462551040684e-08 - 5.0900562996577241385e-975j)  +/-  (1.51e-87, 3.4e-209j)
| (7.7178860480090892783e-07 + 1.8933474185824236269e-973j)  +/-  (1.23e-87, 2.78e-209j)
| (2.0098488468246290991e-09 + 1.8791553373973257666e-974j)  +/-  (1.73e-89, 3.9e-211j)
| (7.7178860480090892783e-07 + 2.6686559368651899021e-974j)  +/-  (4.46e-87, 1e-208j)
| (-0.029891536805174519972 + 1.5822197104708649044e-969j)  +/-  (6.44e-80, 1.45e-201j)
| (5.8129323103963033009e-11 - 1.1239081970487587363e-976j)  +/-  (3.89e-90, 8.78e-212j)
| (0.0098892130188462243777 - 4.2353187810468137912e-971j)  +/-  (5.12e-82, 1.15e-203j)
| (0.02941484458537991661 + 1.2218870784544895423e-970j)  +/-  (3.73e-80, 8.4e-202j)
| (2.0098488468246290991e-09 + 8.2796831499098109565e-976j)  +/-  (1.93e-89, 4.35e-211j)
| (1.2293079070645965307e-14 - 1.0734450762408970047e-978j)  +/-  (2.94e-92, 6.63e-214j)
| (7.7119614325224268536e-17 - 5.8115175493921323301e-979j)  +/-  (4.16e-96, 9.38e-218j)
| (2.2741178968319067287e-30 - 1.9093349778508016394e-986j)  +/-  (8.34e-103, 1.88e-224j)
| (9.131054738037835216e-06 - 6.2442176445654268038e-973j)  +/-  (1.26e-90, 2.85e-212j)
| (1.0857977726372399784e-12 - 2.9719666623173696598e-976j)  +/-  (2.14e-94, 4.82e-216j)
| (7.9602234868012884237e-05 + 4.9051809106847402097e-973j)  +/-  (6.79e-91, 1.53e-212j)
| (0.14472168218664816689 - 1.5520301419537329164e-969j)  +/-  (5.25e-85, 1.18e-206j)
| (0.00052032744767408467761 - 5.6491690646198497138e-972j)  +/-  (1.64e-89, 3.7e-211j)
| (0.070592696122256454101 - 5.3313520463262375844e-970j)  +/-  (2.09e-86, 4.71e-208j)
| (0.17427167871350475071 + 5.436151291739724771e-970j)  +/-  (1.24e-86, 2.78e-208j)
| (4.7030639462551040684e-08 - 5.6002375982132295253e-974j)  +/-  (2.77e-92, 6.24e-214j)
| (0.070592696122256454101 - 3.417673721334474258e-970j)  +/-  (1.97e-87, 4.46e-209j)
| (0.00052032744767408467761 - 1.7859293358106615304e-972j)  +/-  (1.95e-90, 4.4e-212j)
| (0.14472168218664816689 - 2.1018336559751217344e-969j)  +/-  (6.74e-87, 1.51e-208j)
| (-0.029891536805174519972 + 2.2242549088290330435e-969j)  +/-  (6.49e-87, 1.46e-208j)
| (7.9602234868012884237e-05 + 1.9394280082797804303e-972j)  +/-  (9.31e-91, 2.09e-212j)
| (0.02941484458537991661 + 6.6690873015775519975e-971j)  +/-  (2.54e-89, 5.71e-211j)
| (0.17427167871350475071 + 4.6880722530053378395e-970j)  +/-  (1.22e-88, 2.73e-210j)
| (9.131054738037835216e-06 - 1.2161129846748248415e-973j)  +/-  (1.43e-92, 3.32e-214j)
| (0.19561051776700420173 - 4.3317433554765518295e-970j)  +/-  (7.4e-89, 1.58e-210j)
