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| """ | |
| Check that the output from irrational functions is accurate for | |
| high-precision input, from 5 to 200 digits. The reference values were | |
| verified with Mathematica. | |
| """ | |
| import time | |
| from mpmath import * | |
| precs = [5, 15, 28, 35, 57, 80, 100, 150, 200] | |
| # sqrt(3) + pi/2 | |
| a = \ | |
| "3.302847134363773912758768033145623809041389953497933538543279275605"\ | |
| "841220051904536395163599428307109666700184672047856353516867399774243594"\ | |
| "67433521615861420725323528325327484262075464241255915238845599752675" | |
| # e + 1/euler**2 | |
| b = \ | |
| "5.719681166601007617111261398629939965860873957353320734275716220045750"\ | |
| "31474116300529519620938123730851145473473708966080207482581266469342214"\ | |
| "824842256999042984813905047895479210702109260221361437411947323431" | |
| # sqrt(a) | |
| sqrt_a = \ | |
| "1.817373691447021556327498239690365674922395036495564333152483422755"\ | |
| "144321726165582817927383239308173567921345318453306994746434073691275094"\ | |
| "484777905906961689902608644112196725896908619756404253109722911487" | |
| # sqrt(a+b*i).real | |
| sqrt_abi_real = \ | |
| "2.225720098415113027729407777066107959851146508557282707197601407276"\ | |
| "89160998185797504198062911768240808839104987021515555650875977724230130"\ | |
| "3584116233925658621288393930286871862273400475179312570274423840384" | |
| # sqrt(a+b*i).imag | |
| sqrt_abi_imag = \ | |
| "1.2849057639084690902371581529110949983261182430040898147672052833653668"\ | |
| "0629534491275114877090834296831373498336559849050755848611854282001250"\ | |
| "1924311019152914021365263161630765255610885489295778894976075186" | |
| # log(a) | |
| log_a = \ | |
| "1.194784864491089550288313512105715261520511949410072046160598707069"\ | |
| "4336653155025770546309137440687056366757650909754708302115204338077595203"\ | |
| "83005773986664564927027147084436553262269459110211221152925732612" | |
| # log(a+b*i).real | |
| log_abi_real = \ | |
| "1.8877985921697018111624077550443297276844736840853590212962006811663"\ | |
| "04949387789489704203167470111267581371396245317618589339274243008242708"\ | |
| "014251531496104028712866224020066439049377679709216784954509456421" | |
| # log(a+b*i).imag | |
| log_abi_imag = \ | |
| "1.0471204952840802663567714297078763189256357109769672185219334169734948"\ | |
| "4265809854092437285294686651806426649541504240470168212723133326542181"\ | |
| "8300136462287639956713914482701017346851009323172531601894918640" | |
| # exp(a) | |
| exp_a = \ | |
| "27.18994224087168661137253262213293847994194869430518354305430976149"\ | |
| "382792035050358791398632888885200049857986258414049540376323785711941636"\ | |
| "100358982497583832083513086941635049329804685212200507288797531143" | |
| # exp(a+b*i).real | |
| exp_abi_real = \ | |
| "22.98606617170543596386921087657586890620262522816912505151109385026"\ | |
| "40160179326569526152851983847133513990281518417211964710397233157168852"\ | |
| "4963130831190142571659948419307628119985383887599493378056639916701" | |
| # exp(a+b*i).imag | |
| exp_abi_imag = \ | |
| "-14.523557450291489727214750571590272774669907424478129280902375851196283"\ | |
| "3377162379031724734050088565710975758824441845278120105728824497308303"\ | |
| "6065619788140201636218705414429933685889542661364184694108251449" | |
| # a**b | |
| pow_a_b = \ | |
| "928.7025342285568142947391505837660251004990092821305668257284426997"\ | |
| "361966028275685583421197860603126498884545336686124793155581311527995550"\ | |
| "580229264427202446131740932666832138634013168125809402143796691154" | |
| # (a**(a+b*i)).real | |
| pow_a_abi_real = \ | |
| "44.09156071394489511956058111704382592976814280267142206420038656267"\ | |
| "67707916510652790502399193109819563864568986234654864462095231138500505"\ | |
| "8197456514795059492120303477512711977915544927440682508821426093455" | |
| # (a**(a+b*i)).imag | |
| pow_a_abi_imag = \ | |
| "27.069371511573224750478105146737852141664955461266218367212527612279886"\ | |
| "9322304536553254659049205414427707675802193810711302947536332040474573"\ | |
| "8166261217563960235014674118610092944307893857862518964990092301" | |
| # ((a+b*i)**(a+b*i)).real | |
| pow_abi_abi_real = \ | |
| "-0.15171310677859590091001057734676423076527145052787388589334350524"\ | |
| "8084195882019497779202452975350579073716811284169068082670778986235179"\ | |
| "0813026562962084477640470612184016755250592698408112493759742219150452"\ | |
| # ((a+b*i)**(a+b*i)).imag | |
| pow_abi_abi_imag = \ | |
| "1.2697592504953448936553147870155987153192995316950583150964099070426"\ | |
| "4736837932577176947632535475040521749162383347758827307504526525647759"\ | |
| "97547638617201824468382194146854367480471892602963428122896045019902" | |
| # sin(a) | |
| sin_a = \ | |
| "-0.16055653857469062740274792907968048154164433772938156243509084009"\ | |
| "38437090841460493108570147191289893388608611542655654723437248152535114"\ | |
| "528368009465836614227575701220612124204622383149391870684288862269631" | |
| # sin(1000*a) | |
| sin_1000a = \ | |
| "-0.85897040577443833776358106803777589664322997794126153477060795801"\ | |
| "09151695416961724733492511852267067419573754315098042850381158563024337"\ | |
| "216458577140500488715469780315833217177634490142748614625281171216863" | |
| # sin(a+b*i) | |
| sin_abi_real = \ | |
| "-24.4696999681556977743346798696005278716053366404081910969773939630"\ | |
| "7149215135459794473448465734589287491880563183624997435193637389884206"\ | |
| "02151395451271809790360963144464736839412254746645151672423256977064" | |
| sin_abi_imag = \ | |
| "-150.42505378241784671801405965872972765595073690984080160750785565810981"\ | |
| "8314482499135443827055399655645954830931316357243750839088113122816583"\ | |
| "7169201254329464271121058839499197583056427233866320456505060735" | |
| # cos | |
| cos_a = \ | |
| "-0.98702664499035378399332439243967038895709261414476495730788864004"\ | |
| "05406821549361039745258003422386169330787395654908532996287293003581554"\ | |
| "257037193284199198069707141161341820684198547572456183525659969145501" | |
| cos_1000a = \ | |
| "-0.51202523570982001856195696460663971099692261342827540426136215533"\ | |
| "52686662667660613179619804463250686852463876088694806607652218586060613"\ | |
| "951310588158830695735537073667299449753951774916401887657320950496820" | |
| # tan | |
| tan_a = \ | |
| "0.162666873675188117341401059858835168007137819495998960250142156848"\ | |
| "639654718809412181543343168174807985559916643549174530459883826451064966"\ | |
| "7996119428949951351938178809444268785629011625179962457123195557310" | |
| tan_abi_real = \ | |
| "6.822696615947538488826586186310162599974827139564433912601918442911"\ | |
| "1026830824380070400102213741875804368044342309515353631134074491271890"\ | |
| "467615882710035471686578162073677173148647065131872116479947620E-6" | |
| tan_abi_imag = \ | |
| "0.9999795833048243692245661011298447587046967777739649018690797625964167"\ | |
| "1446419978852235960862841608081413169601038230073129482874832053357571"\ | |
| "62702259309150715669026865777947502665936317953101462202542168429" | |
| def test_hp(): | |
| for dps in precs: | |
| mp.dps = dps + 8 | |
| aa = mpf(a) | |
| bb = mpf(b) | |
| a1000 = 1000*mpf(a) | |
| abi = mpc(aa, bb) | |
| mp.dps = dps | |
| assert (sqrt(3) + pi/2).ae(aa) | |
| assert (e + 1/euler**2).ae(bb) | |
| assert sqrt(aa).ae(mpf(sqrt_a)) | |
| assert sqrt(abi).ae(mpc(sqrt_abi_real, sqrt_abi_imag)) | |
| assert log(aa).ae(mpf(log_a)) | |
| assert log(abi).ae(mpc(log_abi_real, log_abi_imag)) | |
| assert exp(aa).ae(mpf(exp_a)) | |
| assert exp(abi).ae(mpc(exp_abi_real, exp_abi_imag)) | |
| assert (aa**bb).ae(mpf(pow_a_b)) | |
| assert (aa**abi).ae(mpc(pow_a_abi_real, pow_a_abi_imag)) | |
| assert (abi**abi).ae(mpc(pow_abi_abi_real, pow_abi_abi_imag)) | |
| assert sin(a).ae(mpf(sin_a)) | |
| assert sin(a1000).ae(mpf(sin_1000a)) | |
| assert sin(abi).ae(mpc(sin_abi_real, sin_abi_imag)) | |
| assert cos(a).ae(mpf(cos_a)) | |
| assert cos(a1000).ae(mpf(cos_1000a)) | |
| assert tan(a).ae(mpf(tan_a)) | |
| assert tan(abi).ae(mpc(tan_abi_real, tan_abi_imag)) | |
| # check that complex cancellation is avoided so that both | |
| # real and imaginary parts have high relative accuracy. | |
| # abs_eps should be 0, but has to be set to 1e-205 to pass the | |
| # 200-digit case, probably due to slight inaccuracy in the | |
| # precomputed input | |
| assert (tan(abi).real).ae(mpf(tan_abi_real), abs_eps=1e-205) | |
| assert (tan(abi).imag).ae(mpf(tan_abi_imag), abs_eps=1e-205) | |
| mp.dps = 460 | |
| assert str(log(3))[-20:] == '02166121184001409826' | |
| mp.dps = 15 | |
| # Since str(a) can differ in the last digit from rounded a, and I want | |
| # to compare the last digits of big numbers with the results in Mathematica, | |
| # I made this hack to get the last 20 digits of rounded a | |
| def last_digits(a): | |
| r = repr(a) | |
| s = str(a) | |
| #dps = mp.dps | |
| #mp.dps += 3 | |
| m = 10 | |
| r = r.replace(s[:-m],'') | |
| r = r.replace("mpf('",'').replace("')",'') | |
| num0 = 0 | |
| for c in r: | |
| if c == '0': | |
| num0 += 1 | |
| else: | |
| break | |
| b = float(int(r))/10**(len(r) - m) | |
| if b >= 10**m - 0.5: # pragma: no cover | |
| raise NotImplementedError | |
| n = int(round(b)) | |
| sn = str(n) | |
| s = s[:-m] + '0'*num0 + sn | |
| return s[-20:] | |
| # values checked with Mathematica | |
| def test_log_hp(): | |
| mp.dps = 2000 | |
| a = mpf(10)**15000/3 | |
| r = log(a) | |
| res = last_digits(r) | |
| # Mathematica N[Log[10^15000/3], 2000] | |
| # ...7443804441768333470331 | |
| assert res == '43804441768333470331' | |
| # see issue 145 | |
| r = log(mpf(3)/2) | |
| # Mathematica N[Log[3/2], 2000] | |
| # ...69653749808140753263288 | |
| res = last_digits(r) | |
| assert res == '53749808140753263288' | |
| mp.dps = 10000 | |
| r = log(2) | |
| res = last_digits(r) | |
| # Mathematica N[Log[2], 10000] | |
| # ...695615913401856601359655561 | |
| assert res == '13401856601359655561' | |
| r = log(mpf(10)**10/3) | |
| res = last_digits(r) | |
| # Mathematica N[Log[10^10/3], 10000] | |
| # ...587087654020631943060007154 | |
| assert res == '54020631943060007154', res | |
| r = log(mpf(10)**100/3) | |
| res = last_digits(r) | |
| # Mathematica N[Log[10^100/3], 10000] | |
| # ,,,59246336539088351652334666 | |
| assert res == '36539088351652334666', res | |
| mp.dps += 10 | |
| a = 1 - mpf(1)/10**10 | |
| mp.dps -= 10 | |
| r = log(a) | |
| res = last_digits(r) | |
| # ...3310334360482956137216724048322957404 | |
| # 372167240483229574038733026370 | |
| # Mathematica N[Log[1 - 10^-10]*10^10, 10000] | |
| # ...60482956137216724048322957404 | |
| assert res == '37216724048322957404', res | |
| mp.dps = 10000 | |
| mp.dps += 100 | |
| a = 1 + mpf(1)/10**100 | |
| mp.dps -= 100 | |
| r = log(a) | |
| res = last_digits(+r) | |
| # Mathematica N[Log[1 + 10^-100]*10^10, 10030] | |
| # ...3994733877377412241546890854692521568292338268273 10^-91 | |
| assert res == '39947338773774122415', res | |
| mp.dps = 15 | |
| def test_exp_hp(): | |
| mp.dps = 4000 | |
| r = exp(mpf(1)/10) | |
| # IntegerPart[N[Exp[1/10] * 10^4000, 4000]] | |
| # ...92167105162069688129 | |
| assert int(r * 10**mp.dps) % 10**20 == 92167105162069688129 | |