ÿØÿà JFIF    ÿÛ „  ( %"1!%)+...383,7(-.+  -+++--++++---+-+-----+---------------+---+-++7-----ÿÀ  ß â" ÿÄ     ÿÄ H    !1AQaq"‘¡2B±ÁÑð#R“Ò Tbr‚²á3csƒ’ÂñDS¢³$CÿÄ   ÿÄ %  !1AQa"23‘ÿÚ   ? ôÿ ¨pŸªáÿ —åYõõ\?àÒü©ŠÄï¨pŸªáÿ —åYõõ\?àÓü©ŠÄá 0Ÿªáÿ Ÿå[úƒ ú®ði~TÁbqÐ8OÕpÿ ƒOò¤Oè`–RÂáœá™êi€ßÉ< FtŸI“öÌ8úDf´°å}“¾œ6  öFá°y¥jñÇh†ˆ¢ã/ÃÐ:ªcÈ "Y¡ðÑl>ÿ ”ÏËte:qž\oäŠe÷󲍷˜HT4&ÿ ÓÐü6ö®¿øþßèô Ÿ•7Ñi’•j|“ñì>b…þS?*Óôÿ ÓÐü*h¥£ír¶ü UãSç‚Ÿ[AÐaè[ûª•õ&õj?†Éö+EzP—WeÒírJFt ‘BŒ†Ï‡%#tE Øz ¥OÛ«!1›üä±Í™%ºÍãö]°î(–:@<‹ŒÊö×òÆt¦ãº+‡¦%ÌÁ²h´OƒJŒtMÜ>ÀÜÊw3Y´•牋4ǍýʏTì>œú=Íwhyë,¾Ôò×õ¿ßÊa»«þˆѪQ|%6ž™A õ%:øj<>É—ÿ Å_ˆCbõ¥š±ý¯Ýƒï…¶|RëócÍf溪“t.СøTÿ *Ä¿-{†çàczůŽ_–^XþŒ±miB[X±d 1,é”zEù»& î9gœf™9Ð'.;—™i}!ôšåîqêÛ٤ёý£½ÆA–àôe"A$˝Úsäÿ ÷Û #°xŸëí(l »ý3—¥5m! rt`†0~'j2(]S¦¦kv,ÚÇ l¦øJA£Šƒ J3E8ÙiŽ:cÉžúeZ°€¯\®kÖ(79«Ž:¯X”¾³Š&¡* ….‰Ž(ÜíŸ2¥ª‡×Hi²TF¤ò[¨íÈRëÉ䢍mgÑ.Ÿ<öäS0í„ǹÁU´f#Vß;Õ–…P@3ío<ä-±»Ž.L|kªÀê›fÂ6@»eu‚|ÓaÞÆŸ…¨ááå>åŠ?cKü6ùTÍÆ”†sĤÚ;H2RÚ†õ\Ö·Ÿn'¾ ñ#ºI¤Å´%çÁ­‚â7›‹qT3Iï¨ÖÚ5I7Ë!ÅOóŸ¶øÝñØôת¦$Tcö‘[«Ö³šÒ';Aþ ¸èíg A2Z"i¸vdÄ÷.iõ®§)¿]¤À†–‡É&ä{V¶iŽ”.Ó×Õÿ û?h¬Mt–íª[ÿ Ñÿ ÌV(í}=ibÔ¡›¥¢±b Lô¥‡piη_Z<‡z§èŒ)iÖwiÇ 2hÙ3·=’d÷8éŽ1¦¸c¤µ€7›7Ø ð\á)} ¹fËí›pAÃL%âc2 í§æQz¿;T8sæ°qø)QFMð‰XŒÂ±N¢aF¨…8¯!U  Z©RÊ ÖPVÄÀÍin™Ì-GˆªÅËŠ›•zË}º±ŽÍFò¹}Uw×#ä5B¤{î}Ð<ÙD é©¤&‡ïDbàÁôMÁ." ¤‡ú*õ'VŽ|¼´Úgllº¼klz[Æüï÷Aób‡Eÿ dÑ»Xx9ÃÜ£ÁT/`¼¸vI±Ýµ·Ë‚“G³þ*Ÿû´r|*}<¨îºœ @¦mÄ’M¹”.œ«Y–|6ÏU¤jç¥ÕÞqO ˜kDÆÁ¨5ÿ š;Њ¦¦€GÙk \ –Þ=â¼=SͧµªS°ÚÍpÜãQűÀõ¬?ÃÁ1Ñ•õZà?hóœ€ L¦l{Y*K˜Ù›zc˜–ˆâ ø+¾ ­-Ök¥%ùEÜA'}ˆ><ÊIè“bpÍ/qÞâvoX€w,\úªò6Z[XdÒæ­@Ö—€$òJí#é>'°Ú ôª˜<)4ryÙ£|óAÅn5žêŸyÒäMÝ2{"}‰–¤l÷ûWX\l¾Á¸góÉOÔ /óñB¤f¸çñ[.P˜ZsÊË*ßT܈§QN¢’¡¨§V¼(Üù*eÕ“”5T¨‹Âê¥FŒã½Dü[8'Ò¥a…Ú¶k7a *•›¼'Ò·\8¨ª\@\õ¢¦íq+DÙrmÎ…_ªæ»ŠÓœ¡¯’Ré9MÅ×D™lælffc+ŒÑ,ý™ÿ ¯þǤ=Å’Á7µ÷ÚÛ/“Ü€ñýã¼àí¾ÕÑ+ƒ,uµMâÀÄbm:ÒÎPæ{˜Gz[ƒ¯«® KHà`ßØŠéí¯P8Aq.C‰ à€kòpj´kN¶qô€…Õ,ÜNŠª-­{Zö’æû44‰sŽè‰îVíRœÕm" 6?³D9¡ÇTíÅꋇ`4«¸ÝÁô ï’ýorqКÇZ«x4Žâéþuïf¹µö[P ,Q£éaX±`PÉÍZ ¸äYúg üAx ’6Lê‚xÝÓ*äQ  Ï’¨hÍ =²,6ï#rÃ<¯–£»ƒ‹,–ê•€ aÛsñ'%Æ"®ÛüìBᝠHÚ3ß°©$“XnœÖ’î2ËTeûìxîß ¦å¿çÉ ðK§þ{‘t‚Ϋ¬jéîZ[ ”š7L¥4VÚCE×]m¤Øy”ä4-dz£œ§¸x.*ãÊÊ b÷•h:©‡¦s`BTÁRû¾g⻩‹jø sF¢àJøFl‘È•Xᓁà~*j¯ +(ÚÕ6-£¯÷GŠØy‚<Ç’.F‹Hœw(+)ÜÜâÈzÄäT§FߘãÏ;DmVœ3Àu@mÚüXÝü•3B¨òÌÁÛ<·ÃÜ z,Ì@õÅ·d2]ü8s÷IôÞ¯^Ç9¢u„~ëAŸï4«M? K]­ÅàPl@s_ p:°¬ZR”´›JC[CS.h‹ƒïËœ«Æ]–÷ó‚wR×k7X‰k›‘´ù¦=¡«‰¨¨Â')—71ó’c‡Ðúµ `é.{§p¹ój\Ž{1h{o±Ý=áUÊïGÖŒõ–-BÄm+AZX¶¡ ïHðæ¥JmÙ;…ä¡Ÿˆ¦ ° äšiÉg«$üMk5¤L“’çÊvïâï ,=f“"íἊ5ô¬x6{ɏžID0e¸vçmi'︧ºð9$ò¹÷*£’9ÿ ²TÔ…×>JV¥}Œ}$p[bÔ®*[jzS*8 ”·T›Í–ñUîƒwo$áè=LT™ç—~ô·¤ÈÚ$榍q‰„+´kFm)ž‹©i–ËqÞŠ‰à¶ü( ‚•§ •°ò·‡#5ª•µÊ﯅¡X¨šÁ*F#TXJÊ ušJVÍ&=iÄs1‚3•'fý§5Ñ<=[íÞ­ PÚ;ѱÌ_~Ä££8rÞ ²w;’hDT°>ÈG¬8Á²ÚzŽ®ò®qZcqJêäÞ-ö[ܘbň±çb“Š¶31²n×iƒðÕ;1¶þÉ ªX‰,ßqÏ$>•î íZ¥Z 1{ç൵+ƒÕµ¥°T$§K]á»Ûï*·¤tMI’ÂZbŽÕiÒ˜}bÓ0£ª5›¨ [5Ž^ÝœWøÂÝh° ¢OWun£¤5 a2Z.G2³YL]jåtì”ä ÁÓ‘%"©<ÔúÊ°sº UZvä‡ÄiÆÒM .÷V·™ø#kèýiíÌ–ª)µT[)BˆõÑ xB¾B€ÖT¨.¥~ð@VĶr#¸ü*åZNDŽH;âi ],©£öØpù(šºãö¼T.uCê•4@ÿ GÕÛ)Cx›®0ø#:ÏðFÒbR\(€€Ä®fã4Þ‰Fä¯HXƒÅ,†öEÑÔÜ]Öv²?tLÃvBY£ú6Êu5ÅAQ³1‘’¬x–HŒÐ‡ ^ ¸KwJôÖŽ5×CÚ¨vÜ«/B0$×k°=ðbÇ(Ï)w±A†Á† 11Í=èQšµ626ŒÜ/`G«µ<}—-Ö7KEHÈÉðóȤmݱû±·ø«Snmá=“ä«šmݱŸ¡¶~ó·“äUóJæúòB|E LêŽy´jDÔ$G¢þÐñ7óR8ýÒ…Ç› WVe#·Ÿ p·Fx~•ݤF÷0Èÿ K¯æS<6’¡WШ; ´ÿ ¥Êø\Òuî†åÝ–VNœkÒ7oòX¨Á­Ø÷FÎÑä±g÷ÿ M~Çî=p,X´ ÝÌÚÅ‹’ÃjÖ.ØöÏñ qïQ¤ÓZE†° =6·]܈ s¸>v•Ž^Ý\wq9r‰Î\¸¡kURÒ$­*‹Nq?Þª*!sŠÆ:TU_u±T+øX¡ ®¹¡,ÄâÃBTsÜ$Ø›4m椴zÜK]’’›Pƒ @€#â˜`é¹=I‡fiV•Ôî“nRm+µFPOhÍ0B£ €+¬5c v•:P'ÒyÎ ‰V~‚Ó†ÖuókDoh$å\*ö%Ю=£«…aȼ½÷Û.-½VŒŠ¼'lyî±1¬3ó#ÞE¿ÔS¤gV£m›=§\û"—WU¤ÚǼÿ ÂnÁGŒÃ ‚õN D³õNÚíŒÕ;HôyÄÈ©P¹Ä{:?R‘Ô¨âF÷ø£bÅó® JS|‚R÷ivýáâ€Æé¡è³´IئÑT!§˜•Øª‚¬â@q€wnïCWÄ@JU€ê¯m6]Ï:£âx'+ÒðXvÓ¦Úm=–´7œ $ì“B£~p%ÕŸUþ« N@¼üï~w˜ñø5®—'Ôe»¤5ã//€ž~‰Tþ›Å7•#¤× Íö pÄ$ùeåì*«ÓŠEØWEÈsßg ¦ûvžSsLpºÊW–âµEWöˬH; ™!CYõZ ÃÄf æ#1W. \uWâ\,\Çf j’<qTbên›Î[vxx£ë 'ö¨1›˜ÀM¼Pÿ H)ƒêêŒA7s,|F“ 꺸k³9Ìö*ç®;Ö!Ö$Eiž•¹ÒÚ†ýóéÝû¾ÕS®ó$’NÝäŸz¤5r¦ãÄÃD÷Üø!°ø‡Ô&@m™Ì^Ãä­d q5Lnÿ N;.6½·N|#ä"1Nƒx“ã<3('&ñßt  ~ªu”1Tb㫨9ê–›–bìd$ߣ=#ÕãÒmU¯eí$EFù5ýYô櫨æ왍Ǘ±ssM]·á¿0ÕåJRÓªîiƒ+O58ÖñªŠÒx" \µâá¨i’¤i —Ö ” M+M¤ë9‚‰A¦°Qõ¾ßøK~¼Ã‘g…Ö´~÷Ï[3GUœÒ½#…kàÔ®Ò”‰³·dWV‰IP‰Ú8u¹”E ÖqLj¾êÕCBš{A^Âß;–¨`¯¬ìö ˼ ×tìø.tƐm*n¨y4o&Àx¥n¦×î‡aupáÛj8¿m›è¶ã!o½;ß0y^ý×^EÑ¿ÒjzŒ­)vÚÑnÄL …^ªô× ‡—‚3k Îý­hï]içå–îÏ*÷ñþ»Ô CÒjøjÍznˆ´ ¹#b'Fô‹ ‰v¥'’à'T´ƒHýÍ%M‰ ƒ&ÆÇŒï1 ‘ –Þ ‰i¬s žR-Ÿ kŠ¬á¬7:þ 0ŒÅÒÕ/aÙ¬ÃÝ#Úøœ ©aiVc‰. ¹¦ãµ” ›Yg¦›ÆÎýº°f³7ƒhá·¸­}&D9¡ÂsÉÙÞèŠõØàC™¨ñbFC|´Ü(ŸƒÚÒ-%»'a Ì¿)ËÇn¿úÿ ÞŽX…4ÊÅH^ôΑí@ù¹Eh¶“L8Çjù ¼ÎåVªóR©Ï5uà V4lZß®=€xÖŸ–ÑÈ ÷”¨°¾__yM1tÉ?uÆþIkÄgæ@þ[¢†°XÃJ£j·:nkÅ¢u ‘}âGzö­/IµèŠ¬¼48q¦F°ŽR¼=ûì{´¯RýicS ÕÛ íNtÍÙï£,w4rêì®»~x(©Uñ§#Ñ&œÕ¤>ÎåÍÓ9’Ö{9eV­[Öjâ²ãu]˜å2›qÑšÕJç0€sÄ|Êëè0튔bÁ>“{×_F`Ø©ºê:µä,v¤ðfc1±"«ÔÍän1#=· Âøv~H½ÐßA¾¿Ü€Óš]Õ; I¾÷ç‚Qi†î¹9ywÔKG˜áñ zQY—§ÃÕZ07§X‚ Áh;ÁM)iÌCH-¯T‘ë|A0{Ò½LÚ–TâÖkÜ’dÀ“rmm»”جPF³ÖcbE§T€ÒxKºû’Ó®7±²(\4ŽÃ¸Uu@j™yĵ;³µ!Á¢b.W¤=mõ´êµK k ¸K^ÜÛ#p*Ü14qkZç5ïë †°5Ï%ÍÛ<Õ¤×Ô¥ê†C Õ´¼ú$ƒÖ“”]Ù¬qÞÚ[4©ý!ûÏ—Áb쳐XµA¬â~`›Çr¸8ìùÝ䫦<>ä÷«?xs´ÇÑ /á;¹øüÊÈÙà{"@Žïzâ¬[âß‚ U_<ÇŸ½4èN˜ú61®qŠu ¦þF£»äJ_ˆÙÎ~ ÞAã–Ý„Ï—rŠD;xTž‘ô`É«…suãO`?³à™ô Lý#Íc5öoæØ‚y´´÷«ZR§<&JÇ+éâô´€i!Àˆ0æAoàðLèÖ-2ŸõW.’t^–(KÁmHµV@xÜÇy®Ñø­â^:Ú3w· 7½¹°ñ¸â¹®:',«Mœ—n­Á+Ãbš LÈ‘ÄnRÓÅœ%¦²‰¨ùQ:¤f‚ "PÕtô¸…cæl…&˜Ú˜Ôkv‹ž+vŠ,=¢v­6—Xy*¥t£«<™:“aîϲ=¦6rO]XI¿Œ÷¤zÚ­›¶ 6÷”w\d ü~v®ˆÌk«^m<ÿ ¢‰Õ\)ùºŽ;… lîÙÅEŠ®cѾ@vnMÏ,¼“ñ•ŽBxðÃzãÇç%3ˆ"}Ù•Åî> BÉú;Ò]V+P˜F_´ßé> Øše|ï‡ÄOmFæÇ ãqÞ$/xÐx­z`ï9"œÜij‚!7.\Td…9M‡•iŽ‹¾‘50ÞŽn¥ß4ÉôO ¹*í^QêËÜÇÌ8=Ž§s‰'ÂëÙ«á%Pú[O †ÅP¯VsŽ°.‰,kc¶ ¬A9n˜XÎ-ÞšN["¹QÕ‰ƒMýÁߺXJæÍaLj¾×Ãmã¾ãÚ uñÒþåQô¦¥ /ÄUx:‚ÍÜ’ Đ©ØÝ3V¨‰ÕnÐ6ó*óúK­«…c ¯U òhsý­jóÔj#,ímŒRµ«lbïUTŒÑ8†Ä0œÏr`ð¡¬É Ї ë"À² ™ 6¥ f¶ ¢ÚoܱԷ-<Àî)†a¶ž'Ú»¨TXqØæ¶÷YÄHy˜9ÈIW­YÀuMFë ºÏ’AqÌ4·/Ú †ô'i$øä­=Ä Ý|öK×40è|È6p‘0§)o¥ctî§H+CA-“ xØ|ÐXŠç l8íºð3Ø:³¤¬KX¯UÿÙ // Copyright 2010 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package math /* Bessel function of the first and second kinds of order zero. */ // The original C code and the long comment below are // from FreeBSD's /usr/src/lib/msun/src/e_j0.c and // came with this notice. The go code is a simplified // version of the original C. // // ==================================================== // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. // // Developed at SunPro, a Sun Microsystems, Inc. business. // Permission to use, copy, modify, and distribute this // software is freely granted, provided that this notice // is preserved. // ==================================================== // // __ieee754_j0(x), __ieee754_y0(x) // Bessel function of the first and second kinds of order zero. // Method -- j0(x): // 1. For tiny x, we use j0(x) = 1 - x**2/4 + x**4/64 - ... // 2. Reduce x to |x| since j0(x)=j0(-x), and // for x in (0,2) // j0(x) = 1-z/4+ z**2*R0/S0, where z = x*x; // (precision: |j0-1+z/4-z**2R0/S0 |<2**-63.67 ) // for x in (2,inf) // j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) // where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) // as follow: // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) // = 1/sqrt(2) * (cos(x) + sin(x)) // sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) // = 1/sqrt(2) * (sin(x) - cos(x)) // (To avoid cancellation, use // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) // to compute the worse one.) // // 3 Special cases // j0(nan)= nan // j0(0) = 1 // j0(inf) = 0 // // Method -- y0(x): // 1. For x<2. // Since // y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x**2/4 - ...) // therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. // We use the following function to approximate y0, // y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x**2 // where // U(z) = u00 + u01*z + ... + u06*z**6 // V(z) = 1 + v01*z + ... + v04*z**4 // with absolute approximation error bounded by 2**-72. // Note: For tiny x, U/V = u0 and j0(x)~1, hence // y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) // 2. For x>=2. // y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) // where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) // by the method mentioned above. // 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. // // J0 returns the order-zero Bessel function of the first kind. // // Special cases are: // // J0(±Inf) = 0 // J0(0) = 1 // J0(NaN) = NaN func J0(x float64) float64 { const ( Huge = 1e300 TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000 Two129 = 1 << 129 // 2**129 0x4800000000000000 // R0/S0 on [0, 2] R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9 R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919 R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4 S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4 S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9 S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F ) // special cases switch { case IsNaN(x): return x case IsInf(x, 0): return 0 case x == 0: return 1 } x = Abs(x) if x >= 2 { s, c := Sincos(x) ss := s - c cc := s + c // make sure x+x does not overflow if x < MaxFloat64/2 { z := -Cos(x + x) if s*c < 0 { cc = z / ss } else { ss = z / cc } } // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) var z float64 if x > Two129 { // |x| > ~6.8056e+38 z = (1 / SqrtPi) * cc / Sqrt(x) } else { u := pzero(x) v := qzero(x) z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x) } return z // |x| >= 2.0 } if x < TwoM13 { // |x| < ~1.2207e-4 if x < TwoM27 { return 1 // |x| < ~7.4506e-9 } return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4 } z := x * x r := z * (R02 + z*(R03+z*(R04+z*R05))) s := 1 + z*(S01+z*(S02+z*(S03+z*S04))) if x < 1 { return 1 + z*(-0.25+(r/s)) // |x| < 1.00 } u := 0.5 * x return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0 } // Y0 returns the order-zero Bessel function of the second kind. // // Special cases are: // // Y0(+Inf) = 0 // Y0(0) = -Inf // Y0(x < 0) = NaN // Y0(NaN) = NaN func Y0(x float64) float64 { const ( TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 Two129 = 1 << 129 // 2**129 0x4800000000000000 U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97 U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4 U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8 V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1 V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF ) // special cases switch { case x < 0 || IsNaN(x): return NaN() case IsInf(x, 1): return 0 case x == 0: return Inf(-1) } if x >= 2 { // |x| >= 2.0 // y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) // where x0 = x-pi/4 // Better formula: // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) // = 1/sqrt(2) * (sin(x) + cos(x)) // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) // = 1/sqrt(2) * (sin(x) - cos(x)) // To avoid cancellation, use // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) // to compute the worse one. s, c := Sincos(x) ss := s - c cc := s + c // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) // make sure x+x does not overflow if x < MaxFloat64/2 { z := -Cos(x + x) if s*c < 0 { cc = z / ss } else { ss = z / cc } } var z float64 if x > Two129 { // |x| > ~6.8056e+38 z = (1 / SqrtPi) * ss / Sqrt(x) } else { u := pzero(x) v := qzero(x) z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x) } return z // |x| >= 2.0 } if x <= TwoM27 { return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9 } z := x * x u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06))))) v := 1 + z*(V01+z*(V02+z*(V03+z*V04))) return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0 } // The asymptotic expansions of pzero is // 1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x. // For x >= 2, We approximate pzero by // pzero(x) = 1 + (R/S) // where R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10 // S = 1 + pS0*s**2 + ... + pS4*s**10 // and // | pzero(x)-1-R/S | <= 2 ** ( -60.26) // for x in [inf, 8]=1/[0,0.125] var p0R8 = [6]float64{ 0.00000000000000000000e+00, // 0x0000000000000000 -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32 -8.08167041275349795626e+00, // 0xC02029D0B44FA779 -2.57063105679704847262e+02, // 0xC07011027B19E863 -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC -5.25304380490729545272e+03, // 0xC0B4850B36CC643D } var p0S8 = [5]float64{ 1.16534364619668181717e+02, // 0x405D223307A96751 3.83374475364121826715e+03, // 0x40ADF37D50596938 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD 4.76277284146730962675e+04, // 0x40E741774F2C49DC } // for x in [8,4.5454]=1/[0.125,0.22001] var p0R5 = [6]float64{ -1.14125464691894502584e-11, // 0xBDA918B147E495CC -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6 -4.15961064470587782438e+00, // 0xC010A370F90C6BBF -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783 -3.31231299649172967747e+02, // 0xC074B3B36742CC63 -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7 } var p0S5 = [5]float64{ 6.07539382692300335975e+01, // 0x404E60810C98C5DE 1.05125230595704579173e+03, // 0x40906D025C7E2864 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864 } // for x in [4.547,2.8571]=1/[0.2199,0.35001] var p0R3 = [6]float64{ -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86 -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B -2.40903221549529611423e+00, // 0xC00345B2AEA48074 -2.19659774734883086467e+01, // 0xC035F74A4CB94E14 -5.80791704701737572236e+01, // 0xC04D0A22420A1A45 -3.14479470594888503854e+01, // 0xC03F72ACA892D80F } var p0S3 = [5]float64{ 3.58560338055209726349e+01, // 0x4041ED9284077DD3 3.61513983050303863820e+02, // 0x40769839464A7C0E 1.19360783792111533330e+03, // 0x4092A66E6D1061D6 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E 1.73580930813335754692e+02, // 0x4065B296FC379081 } // for x in [2.8570,2]=1/[0.3499,0.5] var p0R2 = [6]float64{ -8.87534333032526411254e-08, // 0xBE77D316E927026D -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42 -1.45073846780952986357e+00, // 0xBFF736398A24A843 -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3 -1.11931668860356747786e+01, // 0xC02662E6C5246303 -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F } var p0S2 = [5]float64{ 2.22202997532088808441e+01, // 0x40363865908B5959 1.36206794218215208048e+02, // 0x4061069E0EE8878F 2.70470278658083486789e+02, // 0x4070E78642EA079B 1.53875394208320329881e+02, // 0x40633C033AB6FAFF 1.46576176948256193810e+01, // 0x402D50B344391809 } func pzero(x float64) float64 { var p *[6]float64 var q *[5]float64 if x >= 8 { p = &p0R8 q = &p0S8 } else if x >= 4.5454 { p = &p0R5 q = &p0S5 } else if x >= 2.8571 { p = &p0R3 q = &p0S3 } else if x >= 2 { p = &p0R2 q = &p0S2 } z := 1 / (x * x) r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))) return 1 + r/s } // For x >= 8, the asymptotic expansions of qzero is // -1/8 s + 75/1024 s**3 - ..., where s = 1/x. // We approximate pzero by // qzero(x) = s*(-1.25 + (R/S)) // where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10 // S = 1 + qS0*s**2 + ... + qS5*s**12 // and // | qzero(x)/s +1.25-R/S | <= 2**(-61.22) // for x in [inf, 8]=1/[0,0.125] var q0R8 = [6]float64{ 0.00000000000000000000e+00, // 0x0000000000000000 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C 1.17682064682252693899e+01, // 0x402789525BB334D6 5.57673380256401856059e+02, // 0x40816D6315301825 8.85919720756468632317e+03, // 0x40C14D993E18F46D 3.70146267776887834771e+04, // 0x40E212D40E901566 } var q0S8 = [6]float64{ 1.63776026895689824414e+02, // 0x406478D5365B39BC 8.09834494656449805916e+03, // 0x40BFA2584E6B0563 1.42538291419120476348e+05, // 0x4101665254D38C3F 8.03309257119514397345e+05, // 0x412883DA83A52B43 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5 } // for x in [8,4.5454]=1/[0.125,0.22001] var q0R5 = [6]float64{ 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C 5.83563508962056953777e+00, // 0x401757B0B9953DD3 1.35111577286449829671e+02, // 0x4060E3920A8788E9 1.02724376596164097464e+03, // 0x40900CF99DC8C481 1.98997785864605384631e+03, // 0x409F17E953C6E3A6 } var q0S5 = [6]float64{ 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE 1.88472887785718085070e+04, // 0x40D267D27B591E6D 5.67511122894947329769e+04, // 0x40EBB5E397E02372 3.59767538425114471465e+04, // 0x40E191181F7A54A0 -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609 } // for x in [4.547,2.8571]=1/[0.2199,0.35001] var q0R3 = [6]float64{ 4.37741014089738620906e-09, // 0x3E32CD036ADECB82 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842 3.34423137516170720929e+00, // 0x400AC0FC61149CF5 4.26218440745412650017e+01, // 0x40454F98962DAEDD 1.70808091340565596283e+02, // 0x406559DBE25EFD1F 1.66733948696651168575e+02, // 0x4064D77C81FA21E0 } var q0S3 = [6]float64{ 4.87588729724587182091e+01, // 0x40486122BFE343A6 7.09689221056606015736e+02, // 0x40862D8386544EB3 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0 -1.49247451836156386662e+02, // 0xC062A7EB201CF40F } // for x in [2.8570,2]=1/[0.3499,0.5] var q0R2 = [6]float64{ 1.50444444886983272379e-07, // 0x3E84313B54F76BDB 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34 1.99819174093815998816e+00, // 0x3FFFF897E727779C 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A 1.62527075710929267416e+01, // 0x403040B171814BB4 } var q0S2 = [6]float64{ 3.03655848355219184498e+01, // 0x403E5D96F7C07AED 2.69348118608049844624e+02, // 0x4070D591E4D14B40 8.44783757595320139444e+02, // 0x408A664522B3BF22 8.82935845112488550512e+02, // 0x408B977C9C5CC214 2.12666388511798828631e+02, // 0x406A95530E001365 -5.31095493882666946917e+00, // 0xC0153E6AF8B32931 } func qzero(x float64) float64 { var p, q *[6]float64 if x >= 8 { p = &q0R8 q = &q0S8 } else if x >= 4.5454 { p = &q0R5 q = &q0S5 } else if x >= 2.8571 { p = &q0R3 q = &q0S3 } else if x >= 2 { p = &q0R2 q = &q0S2 } z := 1 / (x * x) r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))) return (-0.125 + r/s) / x }