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author Shinji KONO Mon, 25 May 2020 18:13:55 +0900 1830386684a0
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/*							j0l.c
*
*	Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* long double x, y, j0l();
*
* y = j0l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of first kind, order zero of the argument.
*
* The domain is divided into two major intervals [0, 2] and
* (2, infinity). In the first interval the rational approximation
* is J0(x) = 1 - x^2 / 4 + x^4 R(x^2)
* The second interval is further partitioned into eight equal segments
* of 1/x.
*
* J0(x) = sqrt(2/(pi x)) (P0(x) cos(X) - Q0(x) sin(X)),
* X = x - pi/4,
*
* and the auxiliary functions are given by
*
* J0(x)cos(X) + Y0(x)sin(X) = sqrt( 2/(pi x)) P0(x),
* P0(x) = 1 + 1/x^2 R(1/x^2)
*
* Y0(x)cos(X) - J0(x)sin(X) = sqrt( 2/(pi x)) Q0(x),
* Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
*
*
*
* ACCURACY:
*
*                      Absolute error:
* arithmetic   domain      # trials      peak         rms
*    IEEE      0, 30       100000      1.7e-34      2.4e-35
*
*
*/

/*							y0l.c
*
*	Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, y0l();
*
* y = y0l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The approximation is the same as for J0(x), and
* Y0(x) = sqrt(2/(pi x)) (P0(x) sin(X) + Q0(x) cos(X)).
*
* ACCURACY:
*
*  Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic   domain     # trials      peak         rms
*    IEEE      0, 30       100000      3.0e-34     2.7e-35
*
*/

/* Copyright 2001 by Stephen L. Moshier (moshier@na-net.ornl.gov).

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
version 2.1 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see

/* 1 / sqrt(pi) */
static const __float128 ONEOSQPI = 5.6418958354775628694807945156077258584405E-1Q;
/* 2 / pi */
static const __float128 TWOOPI = 6.3661977236758134307553505349005744813784E-1Q;
static const __float128 zero = 0;

/* J0(x) = 1 - x^2/4 + x^2 x^2 R(x^2)
Peak relative error 3.4e-37
0 <= x <= 2  */
#define NJ0_2N 6
static const __float128 J0_2N[NJ0_2N + 1] = {
3.133239376997663645548490085151484674892E16Q,
-5.479944965767990821079467311839107722107E14Q,
6.290828903904724265980249871997551894090E12Q,
-3.633750176832769659849028554429106299915E10Q,
1.207743757532429576399485415069244807022E8Q,
-2.107485999925074577174305650549367415465E5Q,
1.562826808020631846245296572935547005859E2Q,
};
#define NJ0_2D 6
static const __float128 J0_2D[NJ0_2D + 1] = {
2.005273201278504733151033654496928968261E18Q,
2.063038558793221244373123294054149790864E16Q,
1.053350447931127971406896594022010524994E14Q,
3.496556557558702583143527876385508882310E11Q,
8.249114511878616075860654484367133976306E8Q,
1.402965782449571800199759247964242790589E6Q,
1.619910762853439600957801751815074787351E3Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2),
0 <= 1/x <= .0625
Peak relative error 3.3e-36  */
#define NP16_IN 9
static const __float128 P16_IN[NP16_IN + 1] = {
-1.901689868258117463979611259731176301065E-16Q,
-1.798743043824071514483008340803573980931E-13Q,
-6.481746687115262291873324132944647438959E-11Q,
-1.150651553745409037257197798528294248012E-8Q,
-1.088408467297401082271185599507222695995E-6Q,
-5.551996725183495852661022587879817546508E-5Q,
-1.477286941214245433866838787454880214736E-3Q,
-1.882877976157714592017345347609200402472E-2Q,
-9.620983176855405325086530374317855880515E-2Q,
-1.271468546258855781530458854476627766233E-1Q,
};
#define NP16_ID 9
static const __float128 P16_ID[NP16_ID + 1] = {
2.704625590411544837659891569420764475007E-15Q,
2.562526347676857624104306349421985403573E-12Q,
9.259137589952741054108665570122085036246E-10Q,
1.651044705794378365237454962653430805272E-7Q,
1.573561544138733044977714063100859136660E-5Q,
8.134482112334882274688298469629884804056E-4Q,
2.219259239404080863919375103673593571689E-2Q,
2.976990606226596289580242451096393862792E-1Q,
1.713895630454693931742734911930937246254E0Q,
3.231552290717904041465898249160757368855E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
0.0625 <= 1/x <= 0.125
Peak relative error 2.4e-35  */
#define NP8_16N 10
static const __float128 P8_16N[NP8_16N + 1] = {
-2.335166846111159458466553806683579003632E-15Q,
-1.382763674252402720401020004169367089975E-12Q,
-3.192160804534716696058987967592784857907E-10Q,
-3.744199606283752333686144670572632116899E-8Q,
-2.439161236879511162078619292571922772224E-6Q,
-9.068436986859420951664151060267045346549E-5Q,
-1.905407090637058116299757292660002697359E-3Q,
-2.164456143936718388053842376884252978872E-2Q,
-1.212178415116411222341491717748696499966E-1Q,
-2.782433626588541494473277445959593334494E-1Q,
-1.670703190068873186016102289227646035035E-1Q,
};
#define NP8_16D 10
static const __float128 P8_16D[NP8_16D + 1] = {
3.321126181135871232648331450082662856743E-14Q,
1.971894594837650840586859228510007703641E-11Q,
4.571144364787008285981633719513897281690E-9Q,
5.396419143536287457142904742849052402103E-7Q,
3.551548222385845912370226756036899901549E-5Q,
1.342353874566932014705609788054598013516E-3Q,
2.899133293006771317589357444614157734385E-2Q,
3.455374978185770197704507681491574261545E-1Q,
2.116616964297512311314454834712634820514E0Q,
5.850768316827915470087758636881584174432E0Q,
5.655273858938766830855753983631132928968E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
0.125 <= 1/x <= 0.1875
Peak relative error 2.7e-35  */
#define NP5_8N 10
static const __float128 P5_8N[NP5_8N + 1] = {
-1.270478335089770355749591358934012019596E-12Q,
-4.007588712145412921057254992155810347245E-10Q,
-4.815187822989597568124520080486652009281E-8Q,
-2.867070063972764880024598300408284868021E-6Q,
-9.218742195161302204046454768106063638006E-5Q,
-1.635746821447052827526320629828043529997E-3Q,
-1.570376886640308408247709616497261011707E-2Q,
-7.656484795303305596941813361786219477807E-2Q,
-1.659371030767513274944805479908858628053E-1Q,
-1.185340550030955660015841796219919804915E-1Q,
-8.920026499909994671248893388013790366712E-3Q,
};
#define NP5_8D 9
static const __float128 P5_8D[NP5_8D + 1] = {
1.806902521016705225778045904631543990314E-11Q,
5.728502760243502431663549179135868966031E-9Q,
6.938168504826004255287618819550667978450E-7Q,
4.183769964807453250763325026573037785902E-5Q,
1.372660678476925468014882230851637878587E-3Q,
2.516452105242920335873286419212708961771E-2Q,
2.550502712902647803796267951846557316182E-1Q,
1.365861559418983216913629123778747617072E0Q,
3.523825618308783966723472468855042541407E0Q,
3.656365803506136165615111349150536282434E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
Peak relative error 3.5e-35
0.1875 <= 1/x <= 0.25  */
#define NP4_5N 9
static const __float128 P4_5N[NP4_5N + 1] = {
-9.791405771694098960254468859195175708252E-10Q,
-1.917193059944531970421626610188102836352E-7Q,
-1.393597539508855262243816152893982002084E-5Q,
-4.881863490846771259880606911667479860077E-4Q,
-8.946571245022470127331892085881699269853E-3Q,
-8.707474232568097513415336886103899434251E-2Q,
-4.362042697474650737898551272505525973766E-1Q,
-1.032712171267523975431451359962375617386E0Q,
-9.630502683169895107062182070514713702346E-1Q,
-2.251804386252969656586810309252357233320E-1Q,
};
#define NP4_5D 9
static const __float128 P4_5D[NP4_5D + 1] = {
1.392555487577717669739688337895791213139E-8Q,
2.748886559120659027172816051276451376854E-6Q,
2.024717710644378047477189849678576659290E-4Q,
7.244868609350416002930624752604670292469E-3Q,
1.373631762292244371102989739300382152416E-1Q,
1.412298581400224267910294815260613240668E0Q,
7.742495637843445079276397723849017617210E0Q,
2.138429269198406512028307045259503811861E1Q,
2.651547684548423476506826951831712762610E1Q,
1.167499382465291931571685222882909166935E1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
Peak relative error 2.3e-36
0.25 <= 1/x <= 0.3125  */
#define NP3r2_4N 9
static const __float128 P3r2_4N[NP3r2_4N + 1] = {
-2.589155123706348361249809342508270121788E-8Q,
-3.746254369796115441118148490849195516593E-6Q,
-1.985595497390808544622893738135529701062E-4Q,
-5.008253705202932091290132760394976551426E-3Q,
-6.529469780539591572179155511840853077232E-2Q,
-4.468736064761814602927408833818990271514E-1Q,
-1.556391252586395038089729428444444823380E0Q,
-2.533135309840530224072920725976994981638E0Q,
-1.605509621731068453869408718565392869560E0Q,
-2.518966692256192789269859830255724429375E-1Q,
};
#define NP3r2_4D 9
static const __float128 P3r2_4D[NP3r2_4D + 1] = {
3.682353957237979993646169732962573930237E-7Q,
5.386741661883067824698973455566332102029E-5Q,
2.906881154171822780345134853794241037053E-3Q,
7.545832595801289519475806339863492074126E-2Q,
1.029405357245594877344360389469584526654E0Q,
7.565706120589873131187989560509757626725E0Q,
2.951172890699569545357692207898667665796E1Q,
5.785723537170311456298467310529815457536E1Q,
5.095621464598267889126015412522773474467E1Q,
1.602958484169953109437547474953308401442E1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
Peak relative error 1.0e-35
0.3125 <= 1/x <= 0.375  */
#define NP2r7_3r2N 9
static const __float128 P2r7_3r2N[NP2r7_3r2N + 1] = {
-1.917322340814391131073820537027234322550E-7Q,
-1.966595744473227183846019639723259011906E-5Q,
-7.177081163619679403212623526632690465290E-4Q,
-1.206467373860974695661544653741899755695E-2Q,
-1.008656452188539812154551482286328107316E-1Q,
-4.216016116408810856620947307438823892707E-1Q,
-8.378631013025721741744285026537009814161E-1Q,
-6.973895635309960850033762745957946272579E-1Q,
-1.797864718878320770670740413285763554812E-1Q,
-4.098025357743657347681137871388402849581E-3Q,
};
#define NP2r7_3r2D 8
static const __float128 P2r7_3r2D[NP2r7_3r2D + 1] = {
2.726858489303036441686496086962545034018E-6Q,
2.840430827557109238386808968234848081424E-4Q,
1.063826772041781947891481054529454088832E-2Q,
1.864775537138364773178044431045514405468E-1Q,
1.665660052857205170440952607701728254211E0Q,
7.723745889544331153080842168958348568395E0Q,
1.810726427571829798856428548102077799835E1Q,
1.986460672157794440666187503833545388527E1Q,
8.645503204552282306364296517220055815488E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
Peak relative error 1.3e-36
0.3125 <= 1/x <= 0.4375  */
#define NP2r3_2r7N 9
static const __float128 P2r3_2r7N[NP2r3_2r7N + 1] = {
-1.594642785584856746358609622003310312622E-6Q,
-1.323238196302221554194031733595194539794E-4Q,
-3.856087818696874802689922536987100372345E-3Q,
-5.113241710697777193011470733601522047399E-2Q,
-3.334229537209911914449990372942022350558E-1Q,
-1.075703518198127096179198549659283422832E0Q,
-1.634174803414062725476343124267110981807E0Q,
-1.030133247434119595616826842367268304880E0Q,
-1.989811539080358501229347481000707289391E-1Q,
-3.246859189246653459359775001466924610236E-3Q,
};
#define NP2r3_2r7D 8
static const __float128 P2r3_2r7D[NP2r3_2r7D + 1] = {
2.267936634217251403663034189684284173018E-5Q,
1.918112982168673386858072491437971732237E-3Q,
5.771704085468423159125856786653868219522E-2Q,
8.056124451167969333717642810661498890507E-1Q,
5.687897967531010276788680634413789328776E0Q,
2.072596760717695491085444438270778394421E1Q,
3.801722099819929988585197088613160496684E1Q,
3.254620235902912339534998592085115836829E1Q,
1.104847772130720331801884344645060675036E1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* J0(x)cosX + Y0(x)sinX = sqrt( 2/(pi x)) P0(x), P0(x) = 1 + 1/x^2 R(1/x^2)
Peak relative error 1.2e-35
0.4375 <= 1/x <= 0.5  */
#define NP2_2r3N 8
static const __float128 P2_2r3N[NP2_2r3N + 1] = {
-1.001042324337684297465071506097365389123E-4Q,
-6.289034524673365824853547252689991418981E-3Q,
-1.346527918018624234373664526930736205806E-1Q,
-1.268808313614288355444506172560463315102E0Q,
-5.654126123607146048354132115649177406163E0Q,
-1.186649511267312652171775803270911971693E1Q,
-1.094032424931998612551588246779200724257E1Q,
-3.728792136814520055025256353193674625267E0Q,
-3.000348318524471807839934764596331810608E-1Q,
};
#define NP2_2r3D 8
static const __float128 P2_2r3D[NP2_2r3D + 1] = {
1.423705538269770974803901422532055612980E-3Q,
9.171476630091439978533535167485230575894E-2Q,
2.049776318166637248868444600215942828537E0Q,
2.068970329743769804547326701946144899583E1Q,
1.025103500560831035592731539565060347709E2Q,
2.528088049697570728252145557167066708284E2Q,
2.992160327587558573740271294804830114205E2Q,
1.540193761146551025832707739468679973036E2Q,
2.779516701986912132637672140709452502650E1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 2.2e-35
0 <= 1/x <= .0625  */
#define NQ16_IN 10
static const __float128 Q16_IN[NQ16_IN + 1] = {
2.343640834407975740545326632205999437469E-18Q,
2.667978112927811452221176781536278257448E-15Q,
1.178415018484555397390098879501969116536E-12Q,
2.622049767502719728905924701288614016597E-10Q,
3.196908059607618864801313380896308968673E-8Q,
2.179466154171673958770030655199434798494E-6Q,
8.139959091628545225221976413795645177291E-5Q,
1.563900725721039825236927137885747138654E-3Q,
1.355172364265825167113562519307194840307E-2Q,
3.928058355906967977269780046844768588532E-2Q,
1.107891967702173292405380993183694932208E-2Q,
};
#define NQ16_ID 9
static const __float128 Q16_ID[NQ16_ID + 1] = {
3.199850952578356211091219295199301766718E-17Q,
3.652601488020654842194486058637953363918E-14Q,
1.620179741394865258354608590461839031281E-11Q,
3.629359209474609630056463248923684371426E-9Q,
4.473680923894354600193264347733477363305E-7Q,
3.106368086644715743265603656011050476736E-5Q,
1.198239259946770604954664925153424252622E-3Q,
2.446041004004283102372887804475767568272E-2Q,
2.403235525011860603014707768815113698768E-1Q,
9.491006790682158612266270665136910927149E-1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 5.1e-36
0.0625 <= 1/x <= 0.125  */
#define NQ8_16N 11
static const __float128 Q8_16N[NQ8_16N + 1] = {
1.001954266485599464105669390693597125904E-17Q,
7.545499865295034556206475956620160007849E-15Q,
2.267838684785673931024792538193202559922E-12Q,
3.561909705814420373609574999542459912419E-10Q,
3.216201422768092505214730633842924944671E-8Q,
1.731194793857907454569364622452058554314E-6Q,
5.576944613034537050396518509871004586039E-5Q,
1.051787760316848982655967052985391418146E-3Q,
1.102852974036687441600678598019883746959E-2Q,
5.834647019292460494254225988766702933571E-2Q,
1.290281921604364618912425380717127576529E-1Q,
7.598886310387075708640370806458926458301E-2Q,
};
#define NQ8_16D 11
static const __float128 Q8_16D[NQ8_16D + 1] = {
1.368001558508338469503329967729951830843E-16Q,
1.034454121857542147020549303317348297289E-13Q,
3.128109209247090744354764050629381674436E-11Q,
4.957795214328501986562102573522064468671E-9Q,
4.537872468606711261992676606899273588899E-7Q,
2.493639207101727713192687060517509774182E-5Q,
8.294957278145328349785532236663051405805E-4Q,
1.646471258966713577374948205279380115839E-2Q,
1.878910092770966718491814497982191447073E-1Q,
1.152641605706170353727903052525652504075E0Q,
3.383550240669773485412333679367792932235E0Q,
3.823875252882035706910024716609908473970E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 3.9e-35
0.125 <= 1/x <= 0.1875  */
#define NQ5_8N 10
static const __float128 Q5_8N[NQ5_8N + 1] = {
1.750399094021293722243426623211733898747E-13Q,
6.483426211748008735242909236490115050294E-11Q,
9.279430665656575457141747875716899958373E-9Q,
6.696634968526907231258534757736576340266E-7Q,
2.666560823798895649685231292142838188061E-5Q,
6.025087697259436271271562769707550594540E-4Q,
7.652807734168613251901945778921336353485E-3Q,
5.226269002589406461622551452343519078905E-2Q,
1.748390159751117658969324896330142895079E-1Q,
2.378188719097006494782174902213083589660E-1Q,
8.383984859679804095463699702165659216831E-2Q,
};
#define NQ5_8D 10
static const __float128 Q5_8D[NQ5_8D + 1] = {
2.389878229704327939008104855942987615715E-12Q,
8.926142817142546018703814194987786425099E-10Q,
1.294065862406745901206588525833274399038E-7Q,
9.524139899457666250828752185212769682191E-6Q,
3.908332488377770886091936221573123353489E-4Q,
9.250427033957236609624199884089916836748E-3Q,
1.263420066165922645975830877751588421451E-1Q,
9.692527053860420229711317379861733180654E-1Q,
3.937813834630430172221329298841520707954E0Q,
7.603126427436356534498908111445191312181E0Q,
5.670677653334105479259958485084550934305E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 3.2e-35
0.1875 <= 1/x <= 0.25  */
#define NQ4_5N 10
static const __float128 Q4_5N[NQ4_5N + 1] = {
2.233870042925895644234072357400122854086E-11Q,
5.146223225761993222808463878999151699792E-9Q,
4.459114531468296461688753521109797474523E-7Q,
1.891397692931537975547242165291668056276E-5Q,
4.279519145911541776938964806470674565504E-4Q,
5.275239415656560634702073291768904783989E-3Q,
3.468698403240744801278238473898432608887E-2Q,
1.138773146337708415188856882915457888274E-1Q,
1.622717518946443013587108598334636458955E-1Q,
7.249040006390586123760992346453034628227E-2Q,
1.941595365256460232175236758506411486667E-3Q,
};
#define NQ4_5D 9
static const __float128 Q4_5D[NQ4_5D + 1] = {
3.049977232266999249626430127217988047453E-10Q,
7.120883230531035857746096928889676144099E-8Q,
6.301786064753734446784637919554359588859E-6Q,
2.762010530095069598480766869426308077192E-4Q,
6.572163250572867859316828886203406361251E-3Q,
8.752566114841221958200215255461843397776E-2Q,
6.487654992874805093499285311075289932664E-1Q,
2.576550017826654579451615283022812801435E0Q,
5.056392229924022835364779562707348096036E0Q,
4.179770081068251464907531367859072157773E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 1.4e-36
0.25 <= 1/x <= 0.3125  */
#define NQ3r2_4N 10
static const __float128 Q3r2_4N[NQ3r2_4N + 1] = {
6.126167301024815034423262653066023684411E-10Q,
1.043969327113173261820028225053598975128E-7Q,
6.592927270288697027757438170153763220190E-6Q,
2.009103660938497963095652951912071336730E-4Q,
3.220543385492643525985862356352195896964E-3Q,
2.774405975730545157543417650436941650990E-2Q,
1.258114008023826384487378016636555041129E-1Q,
2.811724258266902502344701449984698323860E-1Q,
2.691837665193548059322831687432415014067E-1Q,
7.949087384900985370683770525312735605034E-2Q,
1.229509543620976530030153018986910810747E-3Q,
};
#define NQ3r2_4D 9
static const __float128 Q3r2_4D[NQ3r2_4D + 1] = {
8.364260446128475461539941389210166156568E-9Q,
1.451301850638956578622154585560759862764E-6Q,
9.431830010924603664244578867057141839463E-5Q,
3.004105101667433434196388593004526182741E-3Q,
5.148157397848271739710011717102773780221E-2Q,
4.901089301726939576055285374953887874895E-1Q,
2.581760991981709901216967665934142240346E0Q,
7.257105880775059281391729708630912791847E0Q,
1.006014717326362868007913423810737369312E1Q,
5.879416600465399514404064187445293212470E0Q,
/* 1.000000000000000000000000000000000000000E0*/
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 3.8e-36
0.3125 <= 1/x <= 0.375  */
#define NQ2r7_3r2N 9
static const __float128 Q2r7_3r2N[NQ2r7_3r2N + 1] = {
7.584861620402450302063691901886141875454E-8Q,
9.300939338814216296064659459966041794591E-6Q,
4.112108906197521696032158235392604947895E-4Q,
8.515168851578898791897038357239630654431E-3Q,
8.971286321017307400142720556749573229058E-2Q,
4.885856732902956303343015636331874194498E-1Q,
1.334506268733103291656253500506406045846E0Q,
1.681207956863028164179042145803851824654E0Q,
8.165042692571721959157677701625853772271E-1Q,
9.805848115375053300608712721986235900715E-2Q,
};
#define NQ2r7_3r2D 9
static const __float128 Q2r7_3r2D[NQ2r7_3r2D + 1] = {
1.035586492113036586458163971239438078160E-6Q,
1.301999337731768381683593636500979713689E-4Q,
5.993695702564527062553071126719088859654E-3Q,
1.321184892887881883489141186815457808785E-1Q,
1.528766555485015021144963194165165083312E0Q,
9.561463309176490874525827051566494939295E0Q,
3.203719484883967351729513662089163356911E1Q,
5.497294687660930446641539152123568668447E1Q,
4.391158169390578768508675452986948391118E1Q,
1.347836630730048077907818943625789418378E1Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 2.2e-35
0.375 <= 1/x <= 0.4375  */
#define NQ2r3_2r7N 9
static const __float128 Q2r3_2r7N[NQ2r3_2r7N + 1] = {
4.455027774980750211349941766420190722088E-7Q,
4.031998274578520170631601850866780366466E-5Q,
1.273987274325947007856695677491340636339E-3Q,
1.818754543377448509897226554179659122873E-2Q,
1.266748858326568264126353051352269875352E-1Q,
4.327578594728723821137731555139472880414E-1Q,
6.892532471436503074928194969154192615359E-1Q,
4.490775818438716873422163588640262036506E-1Q,
8.649615949297322440032000346117031581572E-2Q,
7.261345286655345047417257611469066147561E-4Q,
};
#define NQ2r3_2r7D 8
static const __float128 Q2r3_2r7D[NQ2r3_2r7D + 1] = {
6.082600739680555266312417978064954793142E-6Q,
5.693622538165494742945717226571441747567E-4Q,
1.901625907009092204458328768129666975975E-2Q,
2.958689532697857335456896889409923371570E-1Q,
2.343124711045660081603809437993368799568E0Q,
9.665894032187458293568704885528192804376E0Q,
2.035273104990617136065743426322454881353E1Q,
2.044102010478792896815088858740075165531E1Q,
8.445937177863155827844146643468706599304E0Q,
/* 1.000000000000000000000000000000000000000E0 */
};

/* Y0(x)cosX - J0(x)sinX = sqrt( 2/(pi x)) Q0(x),
Q0(x) = 1/x (-.125 + 1/x^2 R(1/x^2))
Peak relative error 3.1e-36
0.4375 <= 1/x <= 0.5  */
#define NQ2_2r3N 9
static const __float128 Q2_2r3N[NQ2_2r3N + 1] = {
2.817566786579768804844367382809101929314E-6Q,
2.122772176396691634147024348373539744935E-4Q,
5.501378031780457828919593905395747517585E-3Q,
6.355374424341762686099147452020466524659E-2Q,
3.539652320122661637429658698954748337223E-1Q,
9.571721066119617436343740541777014319695E-1Q,
1.196258777828426399432550698612171955305E0Q,
6.069388659458926158392384709893753793967E-1Q,
9.026746127269713176512359976978248763621E-2Q,
5.317668723070450235320878117210807236375E-4Q,
};
#define NQ2_2r3D 8
static const __float128 Q2_2r3D[NQ2_2r3D + 1] = {
3.846924354014260866793741072933159380158E-5Q,
3.017562820057704325510067178327449946763E-3Q,
8.356305620686867949798885808540444210935E-2Q,
1.068314930499906838814019619594424586273E0Q,
6.900279623894821067017966573640732685233E0Q,
2.307667390886377924509090271780839563141E1Q,
3.921043465412723970791036825401273528513E1Q,
3.167569478939719383241775717095729233436E1Q,
1.051023841699200920276198346301543665909E1Q,
/* 1.000000000000000000000000000000000000000E0*/
};

/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
neval (__float128 x, const __float128 *p, int n)
{
__float128 y;

p += n;
y = *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}

/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */

static __float128
deval (__float128 x, const __float128 *p, int n)
{
__float128 y;

p += n;
y = x + *p--;
do
{
y = y * x + *p--;
}
while (--n > 0);
return y;
}

/* Bessel function of the first kind, order zero.  */

__float128
j0q (__float128 x)
{
__float128 xx, xinv, z, p, q, c, s, cc, ss;

if (! finiteq (x))
{
if (x != x)
return x + x;
else
return 0;
}
if (x == 0)
return 1;

xx = fabsq (x);
if (xx <= 2)
{
if (xx < 0x1p-57Q)
return 1;
/* 0 <= x <= 2 */
z = xx * xx;
p = z * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D);
p -= 0.25Q * z;
p += 1;
return p;
}

/* X = x - pi/4
cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
= 1/sqrt(2) * (cos(x) + sin(x))
sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
= 1/sqrt(2) * (sin(x) - cos(x))
sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
cf. Fdlibm.  */
sincosq (xx, &s, &c);
ss = s - c;
cc = s + c;
if (xx <= FLT128_MAX / 2)
{
z = -cosq (xx + xx);
if ((s * c) < 0)
cc = z / ss;
else
ss = z / cc;
}

if (xx > 0x1p256Q)
return ONEOSQPI * cc / sqrtq (xx);

xinv = 1 / xx;
z = xinv * xinv;
if (xinv <= 0.25)
{
if (xinv <= 0.125)
{
if (xinv <= 0.0625)
{
p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
}
else
{
p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
}
}
else if (xinv <= 0.1875)
{
p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
}
else
{
p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
}
}				/* .25 */
else /* if (xinv <= 0.5) */
{
if (xinv <= 0.375)
{
if (xinv <= 0.3125)
{
p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
}
else
{
p = neval (z, P2r7_3r2N, NP2r7_3r2N)
/ deval (z, P2r7_3r2D, NP2r7_3r2D);
q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
/ deval (z, Q2r7_3r2D, NQ2r7_3r2D);
}
}
else if (xinv <= 0.4375)
{
p = neval (z, P2r3_2r7N, NP2r3_2r7N)
/ deval (z, P2r3_2r7D, NP2r3_2r7D);
q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
/ deval (z, Q2r3_2r7D, NQ2r3_2r7D);
}
else
{
p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
}
}
p = 1 + z * p;
q = z * xinv * q;
q = q - 0.125Q * xinv;
z = ONEOSQPI * (p * cc - q * ss) / sqrtq (xx);
return z;
}

/* Y0(x) = 2/pi * log(x) * J0(x) + R(x^2)
Peak absolute error 1.7e-36 (relative where Y0 > 1)
0 <= x <= 2   */
#define NY0_2N 7
static const __float128 Y0_2N[NY0_2N + 1] = {
-1.062023609591350692692296993537002558155E19Q,
2.542000883190248639104127452714966858866E19Q,
-1.984190771278515324281415820316054696545E18Q,
4.982586044371592942465373274440222033891E16Q,
-5.529326354780295177243773419090123407550E14Q,
3.013431465522152289279088265336861140391E12Q,
-7.959436160727126750732203098982718347785E9Q,
8.230845651379566339707130644134372793322E6Q,
};
#define NY0_2D 7
static const __float128 Y0_2D[NY0_2D + 1] = {
1.438972634353286978700329883122253752192E20Q,
1.856409101981569254247700169486907405500E18Q,
1.219693352678218589553725579802986255614E16Q,
5.389428943282838648918475915779958097958E13Q,
1.774125762108874864433872173544743051653E11Q,
4.522104832545149534808218252434693007036E8Q,
8.872187401232943927082914504125234454930E5Q,
1.251945613186787532055610876304669413955E3Q,
/* 1.000000000000000000000000000000000000000E0 */
};

static const __float128 U0 = -7.3804295108687225274343927948483016310862e-02Q;

/* Bessel function of the second kind, order zero.  */

__float128
y0q(__float128 x)
{
__float128 xx, xinv, z, p, q, c, s, cc, ss;

if (! finiteq (x))
return 1 / (x + x * x);
if (x <= 0)
{
if (x < 0)
return (zero / (zero * x));
return -1 / zero; /* -inf and divide by zero exception.  */
}
xx = fabsq (x);
if (xx <= 0x1p-57)
return U0 + TWOOPI * logq (x);
if (xx <= 2)
{
/* 0 <= x <= 2 */
z = xx * xx;
p = neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D);
p = TWOOPI * logq (x) * j0q (x) + p;
return p;
}

/* X = x - pi/4
cos(X) = cos(x) cos(pi/4) + sin(x) sin(pi/4)
= 1/sqrt(2) * (cos(x) + sin(x))
sin(X) = sin(x) cos(pi/4) - cos(x) sin(pi/4)
= 1/sqrt(2) * (sin(x) - cos(x))
sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
cf. Fdlibm.  */
sincosq (x, &s, &c);
ss = s - c;
cc = s + c;
if (xx <= FLT128_MAX / 2)
{
z = -cosq (x + x);
if ((s * c) < 0)
cc = z / ss;
else
ss = z / cc;
}

if (xx > 0x1p256Q)
return ONEOSQPI * ss / sqrtq (x);

xinv = 1 / xx;
z = xinv * xinv;
if (xinv <= 0.25)
{
if (xinv <= 0.125)
{
if (xinv <= 0.0625)
{
p = neval (z, P16_IN, NP16_IN) / deval (z, P16_ID, NP16_ID);
q = neval (z, Q16_IN, NQ16_IN) / deval (z, Q16_ID, NQ16_ID);
}
else
{
p = neval (z, P8_16N, NP8_16N) / deval (z, P8_16D, NP8_16D);
q = neval (z, Q8_16N, NQ8_16N) / deval (z, Q8_16D, NQ8_16D);
}
}
else if (xinv <= 0.1875)
{
p = neval (z, P5_8N, NP5_8N) / deval (z, P5_8D, NP5_8D);
q = neval (z, Q5_8N, NQ5_8N) / deval (z, Q5_8D, NQ5_8D);
}
else
{
p = neval (z, P4_5N, NP4_5N) / deval (z, P4_5D, NP4_5D);
q = neval (z, Q4_5N, NQ4_5N) / deval (z, Q4_5D, NQ4_5D);
}
}				/* .25 */
else /* if (xinv <= 0.5) */
{
if (xinv <= 0.375)
{
if (xinv <= 0.3125)
{
p = neval (z, P3r2_4N, NP3r2_4N) / deval (z, P3r2_4D, NP3r2_4D);
q = neval (z, Q3r2_4N, NQ3r2_4N) / deval (z, Q3r2_4D, NQ3r2_4D);
}
else
{
p = neval (z, P2r7_3r2N, NP2r7_3r2N)
/ deval (z, P2r7_3r2D, NP2r7_3r2D);
q = neval (z, Q2r7_3r2N, NQ2r7_3r2N)
/ deval (z, Q2r7_3r2D, NQ2r7_3r2D);
}
}
else if (xinv <= 0.4375)
{
p = neval (z, P2r3_2r7N, NP2r3_2r7N)
/ deval (z, P2r3_2r7D, NP2r3_2r7D);
q = neval (z, Q2r3_2r7N, NQ2r3_2r7N)
/ deval (z, Q2r3_2r7D, NQ2r3_2r7D);
}
else
{
p = neval (z, P2_2r3N, NP2_2r3N) / deval (z, P2_2r3D, NP2_2r3D);
q = neval (z, Q2_2r3N, NQ2_2r3N) / deval (z, Q2_2r3D, NQ2_2r3D);
}
}
p = 1 + z * p;
q = z * xinv * q;
q = q - 0.125Q * xinv;
z = ONEOSQPI * (p * ss + q * cc) / sqrtq (x);
return z;
}
```