ÿØÿà JFIF    ÿÛ „ !.%+&8&+/1555$;@;4?.451 4,$,44444444444414444444444444444444444444444444444444ÿÀ  á á" ÿÄ     ÿÄ ?    !1AQaq"2‘¡±ÁðBRbrÑá#‚’¢²3S CñÿÄ   ÿÄ !    !1QAa‘2ÿÚ   ? 5˜Z¯V¦cø)›t/? z¨±>Õ5€¶‹Á¤·¼z¼Ü¬+ñ®v¤¨_ˆR­BFn©—˜ý®ç̝P8gýt·ÉSTŦˆìät?þé¼íìN/Þa)ì–í6ô… Ï¿øÃj´¿KÇü]ÿ ªô¹-eKànëÕHTx}ýSÜ›ÿ ”7Ø×&µ<¦  ¥ÑO¶[Ù¯ä¨ÞÃÿ PZ-¬;#õ|•oaÿ ©CìÞz3˜öː/¤­ñTûIØ}š^ mÓ%ªxˆ¥ÉŸu=Z+ISe¿45™¼u;ú&WØ÷€æßQ™®{|íx*TC“#ZŠìZ§²‹ 6pv…³¿¡äª*áZÐ%ÒOáˆo"x«OHk w±æ+¬V(kMúŸ5Vö«$ ÁrÏbàb57/luR ¸ÑÛj Òµì`Ðœq­û žICÀÊ•©4€Âcà¨Ï€O´<èÐ:›ù(Ë^L8þ‘ÍÌ#¸Ð_Ì©ÙK(Öz 4¬û+¸;ü’V’84‘¬ÃŽ:[â‡ÔÌáõp¢~§ªlæ£ö{®G>J¼"°‡7¯ÆÉèßû ‹É‹§ÁòÃýâßî ^ƾÙõ‹×óH#«LP½ïX=xÑÍ$|W?•~• îëÔ©ª‹ {ÝT…Kÿ ”hûâá)J*ö˜–ÔU;iÇ€/ ÆþjóZ\ýwØ=Ìm ºèËL9 ýèÆð/¨’¥öo=nË.%Îì ŽÕ¯È|{Oj²ƒE6e/ßdÄõ²Ìâ1O®ò×TsəԸhOMýíMˆ¿¼H˜l²,7Â¥#MF/Úf°Ö½± ¸–dr‹NýÊ íjqx{œÉ ä-È ¦ øÄër¨q°ð †nцýÑÄÆ’mä…n<0È™;ÁÝá¯ÁZƒ7FÀmì­ É&9ˆîéi¶ùN§Y• ÃZãAâ?•‡©‰ , ó¾IŸŠc1 4â&y­&pŠ­6;M À 0¹qç»p.á …ŸÅáK@%6·y6ƒ‰3?”úºŽ‰éX5ªPT §µ!=Mž«Ú½‹ÅgÂSâÉaþÓoö–¯ÁÔìR>5éÿ üs¶ÆUcÌ kÇR ]ÿ ù¬¼«VŽ;Â|‡~¢¦”ÏÅ°æ {L™Õ°Óv¹ò¸írŽ¡×¢CÃ!íVÕ {¶»sŒNPg/ "uÕbkm²“$ďå¿é¹§°½æz¯6 †s¿!s–wÚÝ“™Œ °.ûj>·+™Òa…©Œ&rÝÎtÛë긪Ît’LAVp%c Úý[ÄzJ¾ÇàXXç@˜ó<êL]·T˜¾¥1Ó©V‡g´æ½¦Ý@¹óø!_@´ÞâSÁ —S3™•& ]@JHÚý©ZŽ €×æÔr»Áf!‡yÞ4Mv*èÓã_{‘åóUuљØ«Oïé*®EvÑ Œ÷‡U \"㪒ÍK+À 4“M¡ï:0¥5í!'<@î´”>Ç»&Z–ïCCV˜Ì5Šo&îhè.žû |ÓK©h$s6KìŒëã)¹hI¦GïOåóI;ììü#É$Š0…Ææ¥TØ.5­¾gn´ “ÂÖ\:hœ89G)J@„}œ:’Ò{/Š"¦_Æ×7Æ3VÇŠÊa]ÚŒÙ€Ä–=®uÁßâACZƒ§§£ Qnâ:«,×{tyø¬iÛcœÜÄ€H½ÄÍCk´÷šß .W'b¤Íåh]÷€=,Žv×cÚEÚHXJX¶îo¨FÒtèöŸ>ªª6[J®Fµ£sGÁeqõfe\íjÒÐïÄÐGˆe1Ø‹.Ø”‘Ëuø Y­ˆÜ ŽG|zùªüMpDnQWÄ”%JŠ™)â*p@Örš«ÕT2Ð%ˆG#ª„ ·¤!°ŸOTÂT¸aÚ%4&h™LµšØüÐ.F¿²ÐÞ_Ç‚¾ÅÃaÜ÷09Æ q€öy˜v‡85õN÷]¬äѼóS{°_MŽ¬úÔ#°Ç¸0åÞè2ëôPcvÆw9®ií1Ä8F™˜à‰´+‰Ik1òÝ7“Ñ×ÒsÝ\x‚h`ÞÑ`ó"|µEcý£n˜h`}GÞ !±ù²Ápü²ß6 0ïi󜵩SÈÇ7˜-ÕURO˜¦´f$ªž-Í6(œ}<„ éc øs]ŽŽ„*—¾ ìdŽ„)méª\¿êÎIg¾ØÞ~I#C/¼¼´EÁÈŽi8“©õådô·>euä ƒ'Ê×लR1ÉJE1ÐAát`t;ÇР%Ý<‡¥„ÍÆ`×Oyó)õiI€ñQaŸ4Ûù\áàaÃÔ¹HÃu¹*k€¦<„e S‡&õÏ B!ŽhüÞ`yj}mªf×\¿ Ç~æ­9‡û\՞Ǖg²1Žû5V7 !àöšm° c`ܬøÇìµÒ'P"?…´Ö,"§^•õŽ¨sÔ)6˜sæéÍR¼ ò|Sl”‹7 nPW Gòú÷½§O¯‡„l¡kSÞŒr½PÊ@æ¢pŽ-mÿ #Ÿ˜Àº¶Áä¦;ïÔæ$1££`“Õ>„—·ž)ßð³ñ#Ï Ô$¶œ‰ÊE‹À;÷º ¯«P:Ñ”8–IÊtpÞ3ª“>ê“þës4ò2OÏÕ­±zô†Õ§‰.÷ä¸;¿˜“'œ›žª}«Œ{ª±Ì 9ÔóÞÕ‡0 $íWV3Üì¬ —@kÝ4@¿r¼±½¬™›?øØæ´'Áé®CË3-g$˜ö‡×auÚi´Žp/êÛ æF›Ú2v‹ã¿¿,nB1̨ƃqÞa5͝@&Æû“él÷ \C²½UÍc ¯k×¢U ÖéQå™—-r wô ÞÏ<Ò=&=ÿ Ôê Òêˈt,i—;LîÜ á¸*ÚÃ1$êL•LÍ <É)ýÐà’ ;F™{ƒ™˜€&'}‚ãÄK`¡ÞT@I;®žZóè‚s’7®°›+§O­Åq©é»²9<Ô J ¼9O’HL»Ùïì¸rk¼Ž_ý‘TŸu[²ßÚŒ·ü÷B%¯E ŸÔX5êO´ Ç•€’I0 ÉJX` ñ¹õ%;µŸD‘«´€àwÒ™U ûئžÖö\×®×´8 ½‡ºÐÆÓ§?Àkmœ=;d5*@-ì0F Rªýš[Ü6âö̃ڸr*KA9· u*µæ£?U¸Âêí†8@¦X4 e-ò„0s{ HâUpU?¼mñRa°®a%Ð'tÉ×’\¾ÊÉ]t›h>·(Ë@R¼¡Ãt h}’O÷au<+nT…Ö…MӐ??Óe95 q>í/;&JSû °¯ÊéÞ øƒ*Ã2½Ài&:nôUl=¾¿5eˆ3”ñc|Ú2V”>„»&eE;«ÚäC p¢Û úy 9š[ŒÌx¼擼A&DåÒ¯ˆ¤ÀÌ;"˜ ÏQä¸åhÊ}Ûq«Û0WžÒ|»€ø®öCm5•\ÇÀ§Pe3£]0ÃàLDÉ‰1øªxjgwT‚÷¿LΨK‹›ùs—xˆÜ±µ kæ¸f‰‰ÜGk/LÛØ6d9ò¶ùA{ƒA3š/¬D¬khÓk‰`˜"㯒r¿±Óã jx‡°e}<Ñø\3y:'À•/h½Í€Ç4~g ?Û(¼]v‘ªlKÎâ~?O‚W%{Ì:“'©úNq¾›úo(X’¥¯ˆ nFê{Ç€ü?º'ë ø‹ì Þ09ŒÌç9Æ —ËC`j@ÓÄ(+a‹un¸#ÂꟋ{K`‘ÑÍÍ'à´»/Û,KW;Þ4²þð ï Nm|~fGÏ(…³Ã)«1ö­Õ ¥‡¨©ƒÃ™ü-s=à=U66Ï«Ýc蓦W¹íž®›nÔ%êÇìŒ<#Ü×84ån®Ð ÒåOC` ñânÑs‡¢ç 1õ%Îhì½Ã½® e:ݼUZo™`  ÅZŸŒÊ«ê1ÏÄo$q¹Þ€©ˆhÐÉä¯ñ[!…Ú˜àJ:x2$Íß&PåT£6ç— ‡Í*4Ýšçjÿ ‰É nófÐ ó(L5C•åÆ\rMÒ@ò }y-W}™üýVù—ú¢=Ù”c®‘< M ž ´Phr ¦©TD ‘ù.$´÷O‡‘V2Æò.=IUŒ=ž‡â¬i™aþÓåÙ?òUø'ØÖ•.~* šTŒ!•-×áºTâ®ä#õü'´ eýlYÅÓeÕKÂrT"CÚ@u!Óxƒ{š3€}1¿(r}%«nËamjÑ%ÑNEò v ˜à  σöK³,*º.àzù¨™Ó ÚçâU¦*¿ 9{%Ö¹ njûdaXöb) kÛƱûÓ\°M7ˆÂ=û›ç¿Ã‚­V»Cg–8ÙêE- j)k$º`Ã-ùEýeBÆÇ]c¡°ñty&Òd0nõ'¡W+ƒ*|–øµFa\GQªEAÔp5\Ǽ·¼Ç8·õ -â§Ú[ ‡ uZeÖ 3}×d'+¹:ð+K†Û®s!Ï$úe€<Û”x)1»a­¡LC]¸µík…ÚàA»AYº{†ªS[¦5HÒ7ù --,ísòDØ€èk ÞÀîÜ ò@â( ËNˆë›4ô½•/¦o‡€Û7 ê•ÆêòðÜy'Án½µ á˜ݦ ndeo…[ì¶Ê,¥R³Ä=À±—–ß;£™´ñSâ*g§”ïaið‘Jå~™ÓÞ ß³Õ¢»8x埒²52>AÊb&-÷\7´éÄù€T˜,w;3{ï˜k…à¹ÄqÀ«œ{€\ ˆ¾[´¨јr &Úé„Ívˆ±8†¿]|¬ņ4I×pÞS1ÈÖz‰#Ìv‡G!YNògñ:màTz¢Ý1ô©^O=~ë|5Bã™ç•¼µõ•bÆ@úÕS¬ÈŒ#¬zünrŸ û” Z²•ï›èðV"ÁHÚý©wÝ €7¼Ìu1hÑa3Éä û f$o¿É ™Ú›ÝçnpÒ3äÌ3†Í§,Äï]$‰/pê †«À¼¸e9­Æê_C]žƒ·ý·frÁN«, E=›Çq -‰öŒ:aÏ¿±í&£Í:-} 84‘ÿ eƒQÑeëSsuiA ³g㟥ú£?ÿ ʼn*”“÷aühe:ÊWa@ÒÞk±eØ] F Ô—r.åä˜ @ö¥ªZoÐýYL·¥S²G/‡ñ <~*ZÆ´è>JlòàÛƽÿ 窘ìGN¢:I®KšJp/`íIÁÀõ#Ä-€ö­šµŒoF4|ÆQØÆ@Ì|£Ô…¢À{9˜è½Üó›€ôYÒÎYsið;ís¤€à²ˆ‚4qÉVŒI$ ‰"° æµ8cXGjœˏ¡Aâý•ËÜ¢ûï e·çLx']á"oÅÎê3¯Ç—¹”ó0nå‚âg{Œñ> S´˜îè°g238‚ãköÝfÚd´6Ò€;ò÷±¢™¼›º ¢Æ'¥Ðx'e¬ç ]bÈÆV¢ó‹kýBO ðÊâ$Ÿ!×T 3Mýמ žìٍàÌü‘8÷€àæØ8æ©6‰©L´«…oãpð„~Çk‰!ñ;‹”ÛžÍ àž±z Ÿôû øŸÝužÏ;ÿ #|u6™Þ¬ÚˆÐõA4¶â|ôl|Ê2ŽÇ¤ÝÅÇY.<#Aí.k§hóF‚”Y; M½Ö4hŸ4&›­¿tès´%FìL¥£Ãk‰ÇT¤haÁ¤ÚxfÉ`ÑìË›>i 3t‚:,–+^÷´–{Û–Nxi"x‘Ûg î¨>¥Õ܁ùZH,2Û“:8xÊ¢Çí9.É-Ìâã-=çjwµS˜dütžçwýGòú®®ûº_ˆýx$–¡ãøO EÚÛÏ÷R„×w+3£Á£öUMyR²¹âŒ°š›¸Ñãò9§Ó_Dl+Ùßc›úšGÅÌc†Ž!Ko=¶.‘Îÿ c²(2®V mª.ÿ ¹B›¹å ù„öŸSV>™ü¯$y:G¢Z×àøúdî¹û­·ýÇ´:•c LÍõi_‹ö+ÎæGÊè>OŠ•äž´§Þ{X}¨1ÚTc›»Qþ•êô°t¿OP?eæ~É{5]•ÙR£r5†nZ\ã@ &îJõ ¾àC°þV>fé¥/ü5ñÊIº_é5 ;e­h<@ Ä&æÃëE%;X,ÒãÆÞ`Oò¦kŸm#˜!ÀyÄ¢| óLšò¥Ä` ¶R=|ÈCâh5ò3DˆïF†ðÒ#ÅìÛœ?¸yhBãœí ZxßÎÄhºRK„`Þödvײ™ÀÈÑÒgŒuY w³%†ƒÓzõ ÖÏp‚dH®¦A´ù§»ÓÇMæ~)ˆð‡û:ù&Ä •vGD´À n ݇¼Ö8Fö óáà£~Ë¥x`oK|Ä?fxiØü%pìR>éò+Û±éÎ>núlFŤ'tq8LZÏvÃ?„¡ß±È⽆¯³íü@x|PöUäèØã¡ð‚ŒAìÏ"vÍwóŸÍ{ ý0.z È•Ö{,N¡£¡ŸKÕÙž>Ýœþ ÍÀ°<×EA!Å‚D™IúOÍ¡>ôG}Â` ÍßkÜL™Ž Þð™ {IøF²¹òQ3&!ÃÂÞz.d&Ï-sH¸,Ôõ˜ŽP€ 77ˆÝ¼ÊëÜw =cÕ Ú,ØÐ5ÎYÐ)ì´öœgŒ[¤ßv㙑8心>h]§µháYš£²ºÑ.{Ï7Sð•?´~×SÃKýJÛ˜ ™Íäiúu<µX¶1õ^kâçIÑ£sZ4h>j*ÔšD:4­¿_ ÷¸ Õxæÿ ¸?Mù _•­ÊÐ ä ÷ý ÑwL œ­ïnTkÛUÍN©ë:¦fV ¶ÜÔÜMªÅâA½–¿R×TXš-%iTÊT•‡Ù‚JôϐZxWÑè‰f‰òG º ×Õû2aZ7OU3[“×AT–ÞŒ…-‘¤”Ì ì&(ˆ¿­•ƒkï’:ðY¦W‘ Å)“†‘˜³Åtcø˜ñTÂwÚÇ4|üLÇªí–v- qˆèU qPE.†â‘˜µ Æ,ÐÅs]8¾„oúÑ i>ÜxxÈó)ƒ ´æÁâØ$À‰vžŸf$Ž |ãw;ÀÁIJ»b` {¦Ó¤Ú$©YÀ‘n@Óïž«9J¼êG m¤ ܯ¹ÌW4€ÐÒÅÛ‡#褕Ÿn-?í|с¥÷Ú¹¬'´ÞÜ9ÓK `hê£SÄSà?7—Wí_´…óB›»:=Ãïq`<8ñÓŒÑlú2d¬ê³£hÖ[l|$vÝro~'R®‰§°ñmY ͧäP |PUª¹·:3Œ[Û{Xÿ ºâ@‚W–Äé u‚ ¯´*=íή.pûÒdt @G‰¬ s¸ ëÉücr ÞæѨÊ@>¤¢Ö±. Þ'¯°ÌME[YéïĵÂCå½ Ué©Áû'Ê9%eÔðNU”ë‘ÌsD3/®+UI˜9h.WC”빓$#:pz:YÓ ¿xž* ³$Í +$kñAŠ‹†¢ Uê>¸)_š¬÷©ßAÂÔb9ÇU ¯¾á•9¯ÏÏ÷O÷¼¼Fähal1‰3Ì[Ïr•´UCksNÐ] R‘¸¥H+§Šé†c©vÖÞ0iÓ76s†î!§=ß ¼~Ô'°Ãmäoäš³ªøi1úÉ)³yV8 CLÄØÁ‘WYïi€H6ÖÑiámø^ÈY´°Ñ7¥Û*—Ñ©L«Qƒï—Ùrÿ ›£Ð*š¸ˆL©ˆ$ˆ ÷¾D§9È®«qbqC)–ˆïv´çñsÑVT­Ø, <àïºÀO«Jý·õ àfPìð .wFšir´þ’2_Y *Æ€x\« ì€9š@ Ž|F⇥ˆkZ@hÖÄ0t¿-<“‹qµ¾*ZL¤Ú)&BJpÓF5=$„at*Zš$’ÑtdûÝRI1 2Ž‰$€$I$#‰SÞ’Hë¬ï;Á$¡t$’`<(ñÇt)$‡Ð.Êf¢X’Kt=Éé$‚ˆªè¢oÝëòI%Rgcª÷ŠyI%¡‰ÿ !ñ)´õ $¤ Ô’IIGÿÙfrom test.support import requires_IEEE_754, cpython_only from test.test_math import parse_testfile, test_file import test.test_math as test_math import unittest import cmath, math from cmath import phase, polar, rect, pi import platform import sys INF = float('inf') NAN = float('nan') complex_zeros = [complex(x, y) for x in [0.0, -0.0] for y in [0.0, -0.0]] complex_infinities = [complex(x, y) for x, y in [ (INF, 0.0), # 1st quadrant (INF, 2.3), (INF, INF), (2.3, INF), (0.0, INF), (-0.0, INF), # 2nd quadrant (-2.3, INF), (-INF, INF), (-INF, 2.3), (-INF, 0.0), (-INF, -0.0), # 3rd quadrant (-INF, -2.3), (-INF, -INF), (-2.3, -INF), (-0.0, -INF), (0.0, -INF), # 4th quadrant (2.3, -INF), (INF, -INF), (INF, -2.3), (INF, -0.0) ]] complex_nans = [complex(x, y) for x, y in [ (NAN, -INF), (NAN, -2.3), (NAN, -0.0), (NAN, 0.0), (NAN, 2.3), (NAN, INF), (-INF, NAN), (-2.3, NAN), (-0.0, NAN), (0.0, NAN), (2.3, NAN), (INF, NAN) ]] class CMathTests(unittest.TestCase): # list of all functions in cmath test_functions = [getattr(cmath, fname) for fname in [ 'acos', 'acosh', 'asin', 'asinh', 'atan', 'atanh', 'cos', 'cosh', 'exp', 'log', 'log10', 'sin', 'sinh', 'sqrt', 'tan', 'tanh']] # test first and second arguments independently for 2-argument log test_functions.append(lambda x : cmath.log(x, 1729. + 0j)) test_functions.append(lambda x : cmath.log(14.-27j, x)) def setUp(self): self.test_values = open(test_file) def tearDown(self): self.test_values.close() def assertFloatIdentical(self, x, y): """Fail unless floats x and y are identical, in the sense that: (1) both x and y are nans, or (2) both x and y are infinities, with the same sign, or (3) both x and y are zeros, with the same sign, or (4) x and y are both finite and nonzero, and x == y """ msg = 'floats {!r} and {!r} are not identical' if math.isnan(x) or math.isnan(y): if math.isnan(x) and math.isnan(y): return elif x == y: if x != 0.0: return # both zero; check that signs match elif math.copysign(1.0, x) == math.copysign(1.0, y): return else: msg += ': zeros have different signs' self.fail(msg.format(x, y)) def assertComplexIdentical(self, x, y): """Fail unless complex numbers x and y have equal values and signs. In particular, if x and y both have real (or imaginary) part zero, but the zeros have different signs, this test will fail. """ self.assertFloatIdentical(x.real, y.real) self.assertFloatIdentical(x.imag, y.imag) def rAssertAlmostEqual(self, a, b, rel_err = 2e-15, abs_err = 5e-323, msg=None): """Fail if the two floating-point numbers are not almost equal. Determine whether floating-point values a and b are equal to within a (small) rounding error. The default values for rel_err and abs_err are chosen to be suitable for platforms where a float is represented by an IEEE 754 double. They allow an error of between 9 and 19 ulps. """ # special values testing if math.isnan(a): if math.isnan(b): return self.fail(msg or '{!r} should be nan'.format(b)) if math.isinf(a): if a == b: return self.fail(msg or 'finite result where infinity expected: ' 'expected {!r}, got {!r}'.format(a, b)) # if both a and b are zero, check whether they have the same sign # (in theory there are examples where it would be legitimate for a # and b to have opposite signs; in practice these hardly ever # occur). if not a and not b: if math.copysign(1., a) != math.copysign(1., b): self.fail(msg or 'zero has wrong sign: expected {!r}, ' 'got {!r}'.format(a, b)) # if a-b overflows, or b is infinite, return False. Again, in # theory there are examples where a is within a few ulps of the # max representable float, and then b could legitimately be # infinite. In practice these examples are rare. try: absolute_error = abs(b-a) except OverflowError: pass else: # test passes if either the absolute error or the relative # error is sufficiently small. The defaults amount to an # error of between 9 ulps and 19 ulps on an IEEE-754 compliant # machine. if absolute_error <= max(abs_err, rel_err * abs(a)): return self.fail(msg or '{!r} and {!r} are not sufficiently close'.format(a, b)) def test_constants(self): e_expected = 2.71828182845904523536 pi_expected = 3.14159265358979323846 self.assertAlmostEqual(cmath.pi, pi_expected, places=9, msg="cmath.pi is {}; should be {}".format(cmath.pi, pi_expected)) self.assertAlmostEqual(cmath.e, e_expected, places=9, msg="cmath.e is {}; should be {}".format(cmath.e, e_expected)) def test_infinity_and_nan_constants(self): self.assertEqual(cmath.inf.real, math.inf) self.assertEqual(cmath.inf.imag, 0.0) self.assertEqual(cmath.infj.real, 0.0) self.assertEqual(cmath.infj.imag, math.inf) self.assertTrue(math.isnan(cmath.nan.real)) self.assertEqual(cmath.nan.imag, 0.0) self.assertEqual(cmath.nanj.real, 0.0) self.assertTrue(math.isnan(cmath.nanj.imag)) # Check consistency with reprs. self.assertEqual(repr(cmath.inf), "inf") self.assertEqual(repr(cmath.infj), "infj") self.assertEqual(repr(cmath.nan), "nan") self.assertEqual(repr(cmath.nanj), "nanj") def test_user_object(self): # Test automatic calling of __complex__ and __float__ by cmath # functions # some random values to use as test values; we avoid values # for which any of the functions in cmath is undefined # (i.e. 0., 1., -1., 1j, -1j) or would cause overflow cx_arg = 4.419414439 + 1.497100113j flt_arg = -6.131677725 # a variety of non-complex numbers, used to check that # non-complex return values from __complex__ give an error non_complexes = ["not complex", 1, 5, 2., None, object(), NotImplemented] # Now we introduce a variety of classes whose instances might # end up being passed to the cmath functions # usual case: new-style class implementing __complex__ class MyComplex(object): def __init__(self, value): self.value = value def __complex__(self): return self.value # old-style class implementing __complex__ class MyComplexOS: def __init__(self, value): self.value = value def __complex__(self): return self.value # classes for which __complex__ raises an exception class SomeException(Exception): pass class MyComplexException(object): def __complex__(self): raise SomeException class MyComplexExceptionOS: def __complex__(self): raise SomeException # some classes not providing __float__ or __complex__ class NeitherComplexNorFloat(object): pass class NeitherComplexNorFloatOS: pass class Index: def __int__(self): return 2 def __index__(self): return 2 class MyInt: def __int__(self): return 2 # other possible combinations of __float__ and __complex__ # that should work class FloatAndComplex(object): def __float__(self): return flt_arg def __complex__(self): return cx_arg class FloatAndComplexOS: def __float__(self): return flt_arg def __complex__(self): return cx_arg class JustFloat(object): def __float__(self): return flt_arg class JustFloatOS: def __float__(self): return flt_arg for f in self.test_functions: # usual usage self.assertEqual(f(MyComplex(cx_arg)), f(cx_arg)) self.assertEqual(f(MyComplexOS(cx_arg)), f(cx_arg)) # other combinations of __float__ and __complex__ self.assertEqual(f(FloatAndComplex()), f(cx_arg)) self.assertEqual(f(FloatAndComplexOS()), f(cx_arg)) self.assertEqual(f(JustFloat()), f(flt_arg)) self.assertEqual(f(JustFloatOS()), f(flt_arg)) self.assertEqual(f(Index()), f(int(Index()))) # TypeError should be raised for classes not providing # either __complex__ or __float__, even if they provide # __int__ or __index__. An old-style class # currently raises AttributeError instead of a TypeError; # this could be considered a bug. self.assertRaises(TypeError, f, NeitherComplexNorFloat()) self.assertRaises(TypeError, f, MyInt()) self.assertRaises(Exception, f, NeitherComplexNorFloatOS()) # non-complex return value from __complex__ -> TypeError for bad_complex in non_complexes: self.assertRaises(TypeError, f, MyComplex(bad_complex)) self.assertRaises(TypeError, f, MyComplexOS(bad_complex)) # exceptions in __complex__ should be propagated correctly self.assertRaises(SomeException, f, MyComplexException()) self.assertRaises(SomeException, f, MyComplexExceptionOS()) def test_input_type(self): # ints should be acceptable inputs to all cmath # functions, by virtue of providing a __float__ method for f in self.test_functions: for arg in [2, 2.]: self.assertEqual(f(arg), f(arg.__float__())) # but strings should give a TypeError for f in self.test_functions: for arg in ["a", "long_string", "0", "1j", ""]: self.assertRaises(TypeError, f, arg) def test_cmath_matches_math(self): # check that corresponding cmath and math functions are equal # for floats in the appropriate range # test_values in (0, 1) test_values = [0.01, 0.1, 0.2, 0.5, 0.9, 0.99] # test_values for functions defined on [-1., 1.] unit_interval = test_values + [-x for x in test_values] + \ [0., 1., -1.] # test_values for log, log10, sqrt positive = test_values + [1.] + [1./x for x in test_values] nonnegative = [0.] + positive # test_values for functions defined on the whole real line real_line = [0.] + positive + [-x for x in positive] test_functions = { 'acos' : unit_interval, 'asin' : unit_interval, 'atan' : real_line, 'cos' : real_line, 'cosh' : real_line, 'exp' : real_line, 'log' : positive, 'log10' : positive, 'sin' : real_line, 'sinh' : real_line, 'sqrt' : nonnegative, 'tan' : real_line, 'tanh' : real_line} for fn, values in test_functions.items(): float_fn = getattr(math, fn) complex_fn = getattr(cmath, fn) for v in values: z = complex_fn(v) self.rAssertAlmostEqual(float_fn(v), z.real) self.assertEqual(0., z.imag) # test two-argument version of log with various bases for base in [0.5, 2., 10.]: for v in positive: z = cmath.log(v, base) self.rAssertAlmostEqual(math.log(v, base), z.real) self.assertEqual(0., z.imag) @requires_IEEE_754 def test_specific_values(self): # Some tests need to be skipped on ancient OS X versions. # See issue #27953. SKIP_ON_TIGER = {'tan0064'} osx_version = None if sys.platform == 'darwin': version_txt = platform.mac_ver()[0] try: osx_version = tuple(map(int, version_txt.split('.'))) except ValueError: pass def rect_complex(z): """Wrapped version of rect that accepts a complex number instead of two float arguments.""" return cmath.rect(z.real, z.imag) def polar_complex(z): """Wrapped version of polar that returns a complex number instead of two floats.""" return complex(*polar(z)) for id, fn, ar, ai, er, ei, flags in parse_testfile(test_file): arg = complex(ar, ai) expected = complex(er, ei) # Skip certain tests on OS X 10.4. if osx_version is not None and osx_version < (10, 5): if id in SKIP_ON_TIGER: continue if fn == 'rect': function = rect_complex elif fn == 'polar': function = polar_complex else: function = getattr(cmath, fn) if 'divide-by-zero' in flags or 'invalid' in flags: try: actual = function(arg) except ValueError: continue else: self.fail('ValueError not raised in test ' '{}: {}(complex({!r}, {!r}))'.format(id, fn, ar, ai)) if 'overflow' in flags: try: actual = function(arg) except OverflowError: continue else: self.fail('OverflowError not raised in test ' '{}: {}(complex({!r}, {!r}))'.format(id, fn, ar, ai)) actual = function(arg) if 'ignore-real-sign' in flags: actual = complex(abs(actual.real), actual.imag) expected = complex(abs(expected.real), expected.imag) if 'ignore-imag-sign' in flags: actual = complex(actual.real, abs(actual.imag)) expected = complex(expected.real, abs(expected.imag)) # for the real part of the log function, we allow an # absolute error of up to 2e-15. if fn in ('log', 'log10'): real_abs_err = 2e-15 else: real_abs_err = 5e-323 error_message = ( '{}: {}(complex({!r}, {!r}))\n' 'Expected: complex({!r}, {!r})\n' 'Received: complex({!r}, {!r})\n' 'Received value insufficiently close to expected value.' ).format(id, fn, ar, ai, expected.real, expected.imag, actual.real, actual.imag) self.rAssertAlmostEqual(expected.real, actual.real, abs_err=real_abs_err, msg=error_message) self.rAssertAlmostEqual(expected.imag, actual.imag, msg=error_message) def check_polar(self, func): def check(arg, expected): got = func(arg) for e, g in zip(expected, got): self.rAssertAlmostEqual(e, g) check(0, (0., 0.)) check(1, (1., 0.)) check(-1, (1., pi)) check(1j, (1., pi / 2)) check(-3j, (3., -pi / 2)) inf = float('inf') check(complex(inf, 0), (inf, 0.)) check(complex(-inf, 0), (inf, pi)) check(complex(3, inf), (inf, pi / 2)) check(complex(5, -inf), (inf, -pi / 2)) check(complex(inf, inf), (inf, pi / 4)) check(complex(inf, -inf), (inf, -pi / 4)) check(complex(-inf, inf), (inf, 3 * pi / 4)) check(complex(-inf, -inf), (inf, -3 * pi / 4)) nan = float('nan') check(complex(nan, 0), (nan, nan)) check(complex(0, nan), (nan, nan)) check(complex(nan, nan), (nan, nan)) check(complex(inf, nan), (inf, nan)) check(complex(-inf, nan), (inf, nan)) check(complex(nan, inf), (inf, nan)) check(complex(nan, -inf), (inf, nan)) def test_polar(self): self.check_polar(polar) @cpython_only def test_polar_errno(self): # Issue #24489: check a previously set C errno doesn't disturb polar() from _testcapi import set_errno def polar_with_errno_set(z): set_errno(11) try: return polar(z) finally: set_errno(0) self.check_polar(polar_with_errno_set) def test_phase(self): self.assertAlmostEqual(phase(0), 0.) self.assertAlmostEqual(phase(1.), 0.) self.assertAlmostEqual(phase(-1.), pi) self.assertAlmostEqual(phase(-1.+1E-300j), pi) self.assertAlmostEqual(phase(-1.-1E-300j), -pi) self.assertAlmostEqual(phase(1j), pi/2) self.assertAlmostEqual(phase(-1j), -pi/2) # zeros self.assertEqual(phase(complex(0.0, 0.0)), 0.0) self.assertEqual(phase(complex(0.0, -0.0)), -0.0) self.assertEqual(phase(complex(-0.0, 0.0)), pi) self.assertEqual(phase(complex(-0.0, -0.0)), -pi) # infinities self.assertAlmostEqual(phase(complex(-INF, -0.0)), -pi) self.assertAlmostEqual(phase(complex(-INF, -2.3)), -pi) self.assertAlmostEqual(phase(complex(-INF, -INF)), -0.75*pi) self.assertAlmostEqual(phase(complex(-2.3, -INF)), -pi/2) self.assertAlmostEqual(phase(complex(-0.0, -INF)), -pi/2) self.assertAlmostEqual(phase(complex(0.0, -INF)), -pi/2) self.assertAlmostEqual(phase(complex(2.3, -INF)), -pi/2) self.assertAlmostEqual(phase(complex(INF, -INF)), -pi/4) self.assertEqual(phase(complex(INF, -2.3)), -0.0) self.assertEqual(phase(complex(INF, -0.0)), -0.0) self.assertEqual(phase(complex(INF, 0.0)), 0.0) self.assertEqual(phase(complex(INF, 2.3)), 0.0) self.assertAlmostEqual(phase(complex(INF, INF)), pi/4) self.assertAlmostEqual(phase(complex(2.3, INF)), pi/2) self.assertAlmostEqual(phase(complex(0.0, INF)), pi/2) self.assertAlmostEqual(phase(complex(-0.0, INF)), pi/2) self.assertAlmostEqual(phase(complex(-2.3, INF)), pi/2) self.assertAlmostEqual(phase(complex(-INF, INF)), 0.75*pi) self.assertAlmostEqual(phase(complex(-INF, 2.3)), pi) self.assertAlmostEqual(phase(complex(-INF, 0.0)), pi) # real or imaginary part NaN for z in complex_nans: self.assertTrue(math.isnan(phase(z))) def test_abs(self): # zeros for z in complex_zeros: self.assertEqual(abs(z), 0.0) # infinities for z in complex_infinities: self.assertEqual(abs(z), INF) # real or imaginary part NaN self.assertEqual(abs(complex(NAN, -INF)), INF) self.assertTrue(math.isnan(abs(complex(NAN, -2.3)))) self.assertTrue(math.isnan(abs(complex(NAN, -0.0)))) self.assertTrue(math.isnan(abs(complex(NAN, 0.0)))) self.assertTrue(math.isnan(abs(complex(NAN, 2.3)))) self.assertEqual(abs(complex(NAN, INF)), INF) self.assertEqual(abs(complex(-INF, NAN)), INF) self.assertTrue(math.isnan(abs(complex(-2.3, NAN)))) self.assertTrue(math.isnan(abs(complex(-0.0, NAN)))) self.assertTrue(math.isnan(abs(complex(0.0, NAN)))) self.assertTrue(math.isnan(abs(complex(2.3, NAN)))) self.assertEqual(abs(complex(INF, NAN)), INF) self.assertTrue(math.isnan(abs(complex(NAN, NAN)))) @requires_IEEE_754 def test_abs_overflows(self): # result overflows self.assertRaises(OverflowError, abs, complex(1.4e308, 1.4e308)) def assertCEqual(self, a, b): eps = 1E-7 if abs(a.real - b[0]) > eps or abs(a.imag - b[1]) > eps: self.fail((a ,b)) def test_rect(self): self.assertCEqual(rect(0, 0), (0, 0)) self.assertCEqual(rect(1, 0), (1., 0)) self.assertCEqual(rect(1, -pi), (-1., 0)) self.assertCEqual(rect(1, pi/2), (0, 1.)) self.assertCEqual(rect(1, -pi/2), (0, -1.)) def test_isfinite(self): real_vals = [float('-inf'), -2.3, -0.0, 0.0, 2.3, float('inf'), float('nan')] for x in real_vals: for y in real_vals: z = complex(x, y) self.assertEqual(cmath.isfinite(z), math.isfinite(x) and math.isfinite(y)) def test_isnan(self): self.assertFalse(cmath.isnan(1)) self.assertFalse(cmath.isnan(1j)) self.assertFalse(cmath.isnan(INF)) self.assertTrue(cmath.isnan(NAN)) self.assertTrue(cmath.isnan(complex(NAN, 0))) self.assertTrue(cmath.isnan(complex(0, NAN))) self.assertTrue(cmath.isnan(complex(NAN, NAN))) self.assertTrue(cmath.isnan(complex(NAN, INF))) self.assertTrue(cmath.isnan(complex(INF, NAN))) def test_isinf(self): self.assertFalse(cmath.isinf(1)) self.assertFalse(cmath.isinf(1j)) self.assertFalse(cmath.isinf(NAN)) self.assertTrue(cmath.isinf(INF)) self.assertTrue(cmath.isinf(complex(INF, 0))) self.assertTrue(cmath.isinf(complex(0, INF))) self.assertTrue(cmath.isinf(complex(INF, INF))) self.assertTrue(cmath.isinf(complex(NAN, INF))) self.assertTrue(cmath.isinf(complex(INF, NAN))) @requires_IEEE_754 def testTanhSign(self): for z in complex_zeros: self.assertComplexIdentical(cmath.tanh(z), z) # The algorithm used for atan and atanh makes use of the system # log1p function; If that system function doesn't respect the sign # of zero, then atan and atanh will also have difficulties with # the sign of complex zeros. @requires_IEEE_754 def testAtanSign(self): for z in complex_zeros: self.assertComplexIdentical(cmath.atan(z), z) @requires_IEEE_754 def testAtanhSign(self): for z in complex_zeros: self.assertComplexIdentical(cmath.atanh(z), z) class IsCloseTests(test_math.IsCloseTests): isclose = cmath.isclose def test_reject_complex_tolerances(self): with self.assertRaises(TypeError): self.isclose(1j, 1j, rel_tol=1j) with self.assertRaises(TypeError): self.isclose(1j, 1j, abs_tol=1j) with self.assertRaises(TypeError): self.isclose(1j, 1j, rel_tol=1j, abs_tol=1j) def test_complex_values(self): # test complex values that are close to within 12 decimal places complex_examples = [(1.0+1.0j, 1.000000000001+1.0j), (1.0+1.0j, 1.0+1.000000000001j), (-1.0+1.0j, -1.000000000001+1.0j), (1.0-1.0j, 1.0-0.999999999999j), ] self.assertAllClose(complex_examples, rel_tol=1e-12) self.assertAllNotClose(complex_examples, rel_tol=1e-13) def test_complex_near_zero(self): # test values near zero that are near to within three decimal places near_zero_examples = [(0.001j, 0), (0.001, 0), (0.001+0.001j, 0), (-0.001+0.001j, 0), (0.001-0.001j, 0), (-0.001-0.001j, 0), ] self.assertAllClose(near_zero_examples, abs_tol=1.5e-03) self.assertAllNotClose(near_zero_examples, abs_tol=0.5e-03) self.assertIsClose(0.001-0.001j, 0.001+0.001j, abs_tol=2e-03) self.assertIsNotClose(0.001-0.001j, 0.001+0.001j, abs_tol=1e-03) if __name__ == "__main__": unittest.main()