diff options
Diffstat (limited to 'sysdeps/ieee754')
| -rw-r--r-- | sysdeps/ieee754/ldbl-128/e_j0l.c | 908 |
1 files changed, 908 insertions, 0 deletions
diff --git a/sysdeps/ieee754/ldbl-128/e_j0l.c b/sysdeps/ieee754/ldbl-128/e_j0l.c new file mode 100644 index 0000000000..22a43848b2 --- /dev/null +++ b/sysdeps/ieee754/ldbl-128/e_j0l.c @@ -0,0 +1,908 @@ +/* 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). */ + +#include "math.h" +#include "math_private.h" + +/* 1 / sqrt(pi) */ +static long double ONEOSQPI = 5.6418958354775628694807945156077258584405E-1L; +/* 2 / pi */ +static long double TWOOPI = 6.3661977236758134307553505349005744813784E-1L; +static long double zero = 0.0L; + +/* 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 long double J0_2N[NJ0_2N + 1] = { + 3.133239376997663645548490085151484674892E16L, + -5.479944965767990821079467311839107722107E14L, + 6.290828903904724265980249871997551894090E12L, + -3.633750176832769659849028554429106299915E10L, + 1.207743757532429576399485415069244807022E8L, + -2.107485999925074577174305650549367415465E5L, + 1.562826808020631846245296572935547005859E2L, +}; +#define NJ0_2D 6 +static long double J0_2D[NJ0_2D + 1] = { + 2.005273201278504733151033654496928968261E18L, + 2.063038558793221244373123294054149790864E16L, + 1.053350447931127971406896594022010524994E14L, + 3.496556557558702583143527876385508882310E11L, + 8.249114511878616075860654484367133976306E8L, + 1.402965782449571800199759247964242790589E6L, + 1.619910762853439600957801751815074787351E3L, + /* 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 long double P16_IN[NP16_IN + 1] = { + -1.901689868258117463979611259731176301065E-16L, + -1.798743043824071514483008340803573980931E-13L, + -6.481746687115262291873324132944647438959E-11L, + -1.150651553745409037257197798528294248012E-8L, + -1.088408467297401082271185599507222695995E-6L, + -5.551996725183495852661022587879817546508E-5L, + -1.477286941214245433866838787454880214736E-3L, + -1.882877976157714592017345347609200402472E-2L, + -9.620983176855405325086530374317855880515E-2L, + -1.271468546258855781530458854476627766233E-1L, +}; +#define NP16_ID 9 +static long double P16_ID[NP16_ID + 1] = { + 2.704625590411544837659891569420764475007E-15L, + 2.562526347676857624104306349421985403573E-12L, + 9.259137589952741054108665570122085036246E-10L, + 1.651044705794378365237454962653430805272E-7L, + 1.573561544138733044977714063100859136660E-5L, + 8.134482112334882274688298469629884804056E-4L, + 2.219259239404080863919375103673593571689E-2L, + 2.976990606226596289580242451096393862792E-1L, + 1.713895630454693931742734911930937246254E0L, + 3.231552290717904041465898249160757368855E0L, + /* 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 long double P8_16N[NP8_16N + 1] = { + -2.335166846111159458466553806683579003632E-15L, + -1.382763674252402720401020004169367089975E-12L, + -3.192160804534716696058987967592784857907E-10L, + -3.744199606283752333686144670572632116899E-8L, + -2.439161236879511162078619292571922772224E-6L, + -9.068436986859420951664151060267045346549E-5L, + -1.905407090637058116299757292660002697359E-3L, + -2.164456143936718388053842376884252978872E-2L, + -1.212178415116411222341491717748696499966E-1L, + -2.782433626588541494473277445959593334494E-1L, + -1.670703190068873186016102289227646035035E-1L, +}; +#define NP8_16D 10 +static long double P8_16D[NP8_16D + 1] = { + 3.321126181135871232648331450082662856743E-14L, + 1.971894594837650840586859228510007703641E-11L, + 4.571144364787008285981633719513897281690E-9L, + 5.396419143536287457142904742849052402103E-7L, + 3.551548222385845912370226756036899901549E-5L, + 1.342353874566932014705609788054598013516E-3L, + 2.899133293006771317589357444614157734385E-2L, + 3.455374978185770197704507681491574261545E-1L, + 2.116616964297512311314454834712634820514E0L, + 5.850768316827915470087758636881584174432E0L, + 5.655273858938766830855753983631132928968E0L, + /* 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 long double P5_8N[NP5_8N + 1] = { + -1.270478335089770355749591358934012019596E-12L, + -4.007588712145412921057254992155810347245E-10L, + -4.815187822989597568124520080486652009281E-8L, + -2.867070063972764880024598300408284868021E-6L, + -9.218742195161302204046454768106063638006E-5L, + -1.635746821447052827526320629828043529997E-3L, + -1.570376886640308408247709616497261011707E-2L, + -7.656484795303305596941813361786219477807E-2L, + -1.659371030767513274944805479908858628053E-1L, + -1.185340550030955660015841796219919804915E-1L, + -8.920026499909994671248893388013790366712E-3L, +}; +#define NP5_8D 9 +static long double P5_8D[NP5_8D + 1] = { + 1.806902521016705225778045904631543990314E-11L, + 5.728502760243502431663549179135868966031E-9L, + 6.938168504826004255287618819550667978450E-7L, + 4.183769964807453250763325026573037785902E-5L, + 1.372660678476925468014882230851637878587E-3L, + 2.516452105242920335873286419212708961771E-2L, + 2.550502712902647803796267951846557316182E-1L, + 1.365861559418983216913629123778747617072E0L, + 3.523825618308783966723472468855042541407E0L, + 3.656365803506136165615111349150536282434E0L, + /* 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 long double P4_5N[NP4_5N + 1] = { + -9.791405771694098960254468859195175708252E-10L, + -1.917193059944531970421626610188102836352E-7L, + -1.393597539508855262243816152893982002084E-5L, + -4.881863490846771259880606911667479860077E-4L, + -8.946571245022470127331892085881699269853E-3L, + -8.707474232568097513415336886103899434251E-2L, + -4.362042697474650737898551272505525973766E-1L, + -1.032712171267523975431451359962375617386E0L, + -9.630502683169895107062182070514713702346E-1L, + -2.251804386252969656586810309252357233320E-1L, +}; +#define NP4_5D 9 +static long double P4_5D[NP4_5D + 1] = { + 1.392555487577717669739688337895791213139E-8L, + 2.748886559120659027172816051276451376854E-6L, + 2.024717710644378047477189849678576659290E-4L, + 7.244868609350416002930624752604670292469E-3L, + 1.373631762292244371102989739300382152416E-1L, + 1.412298581400224267910294815260613240668E0L, + 7.742495637843445079276397723849017617210E0L, + 2.138429269198406512028307045259503811861E1L, + 2.651547684548423476506826951831712762610E1L, + 1.167499382465291931571685222882909166935E1L, + /* 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 long double P3r2_4N[NP3r2_4N + 1] = { + -2.589155123706348361249809342508270121788E-8L, + -3.746254369796115441118148490849195516593E-6L, + -1.985595497390808544622893738135529701062E-4L, + -5.008253705202932091290132760394976551426E-3L, + -6.529469780539591572179155511840853077232E-2L, + -4.468736064761814602927408833818990271514E-1L, + -1.556391252586395038089729428444444823380E0L, + -2.533135309840530224072920725976994981638E0L, + -1.605509621731068453869408718565392869560E0L, + -2.518966692256192789269859830255724429375E-1L, +}; +#define NP3r2_4D 9 +static long double P3r2_4D[NP3r2_4D + 1] = { + 3.682353957237979993646169732962573930237E-7L, + 5.386741661883067824698973455566332102029E-5L, + 2.906881154171822780345134853794241037053E-3L, + 7.545832595801289519475806339863492074126E-2L, + 1.029405357245594877344360389469584526654E0L, + 7.565706120589873131187989560509757626725E0L, + 2.951172890699569545357692207898667665796E1L, + 5.785723537170311456298467310529815457536E1L, + 5.095621464598267889126015412522773474467E1L, + 1.602958484169953109437547474953308401442E1L, + /* 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 long double P2r7_3r2N[NP2r7_3r2N + 1] = { + -1.917322340814391131073820537027234322550E-7L, + -1.966595744473227183846019639723259011906E-5L, + -7.177081163619679403212623526632690465290E-4L, + -1.206467373860974695661544653741899755695E-2L, + -1.008656452188539812154551482286328107316E-1L, + -4.216016116408810856620947307438823892707E-1L, + -8.378631013025721741744285026537009814161E-1L, + -6.973895635309960850033762745957946272579E-1L, + -1.797864718878320770670740413285763554812E-1L, + -4.098025357743657347681137871388402849581E-3L, +}; +#define NP2r7_3r2D 8 +static long double P2r7_3r2D[NP2r7_3r2D + 1] = { + 2.726858489303036441686496086962545034018E-6L, + 2.840430827557109238386808968234848081424E-4L, + 1.063826772041781947891481054529454088832E-2L, + 1.864775537138364773178044431045514405468E-1L, + 1.665660052857205170440952607701728254211E0L, + 7.723745889544331153080842168958348568395E0L, + 1.810726427571829798856428548102077799835E1L, + 1.986460672157794440666187503833545388527E1L, + 8.645503204552282306364296517220055815488E0L, + /* 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 long double P2r3_2r7N[NP2r3_2r7N + 1] = { + -1.594642785584856746358609622003310312622E-6L, + -1.323238196302221554194031733595194539794E-4L, + -3.856087818696874802689922536987100372345E-3L, + -5.113241710697777193011470733601522047399E-2L, + -3.334229537209911914449990372942022350558E-1L, + -1.075703518198127096179198549659283422832E0L, + -1.634174803414062725476343124267110981807E0L, + -1.030133247434119595616826842367268304880E0L, + -1.989811539080358501229347481000707289391E-1L, + -3.246859189246653459359775001466924610236E-3L, +}; +#define NP2r3_2r7D 8 +static long double P2r3_2r7D[NP2r3_2r7D + 1] = { + 2.267936634217251403663034189684284173018E-5L, + 1.918112982168673386858072491437971732237E-3L, + 5.771704085468423159125856786653868219522E-2L, + 8.056124451167969333717642810661498890507E-1L, + 5.687897967531010276788680634413789328776E0L, + 2.072596760717695491085444438270778394421E1L, + 3.801722099819929988585197088613160496684E1L, + 3.254620235902912339534998592085115836829E1L, + 1.104847772130720331801884344645060675036E1L, + /* 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 long double P2_2r3N[NP2_2r3N + 1] = { + -1.001042324337684297465071506097365389123E-4L, + -6.289034524673365824853547252689991418981E-3L, + -1.346527918018624234373664526930736205806E-1L, + -1.268808313614288355444506172560463315102E0L, + -5.654126123607146048354132115649177406163E0L, + -1.186649511267312652171775803270911971693E1L, + -1.094032424931998612551588246779200724257E1L, + -3.728792136814520055025256353193674625267E0L, + -3.000348318524471807839934764596331810608E-1L, +}; +#define NP2_2r3D 8 +static long double P2_2r3D[NP2_2r3D + 1] = { + 1.423705538269770974803901422532055612980E-3L, + 9.171476630091439978533535167485230575894E-2L, + 2.049776318166637248868444600215942828537E0L, + 2.068970329743769804547326701946144899583E1L, + 1.025103500560831035592731539565060347709E2L, + 2.528088049697570728252145557167066708284E2L, + 2.992160327587558573740271294804830114205E2L, + 1.540193761146551025832707739468679973036E2L, + 2.779516701986912132637672140709452502650E1L, + /* 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 long double Q16_IN[NQ16_IN + 1] = { + 2.343640834407975740545326632205999437469E-18L, + 2.667978112927811452221176781536278257448E-15L, + 1.178415018484555397390098879501969116536E-12L, + 2.622049767502719728905924701288614016597E-10L, + 3.196908059607618864801313380896308968673E-8L, + 2.179466154171673958770030655199434798494E-6L, + 8.139959091628545225221976413795645177291E-5L, + 1.563900725721039825236927137885747138654E-3L, + 1.355172364265825167113562519307194840307E-2L, + 3.928058355906967977269780046844768588532E-2L, + 1.107891967702173292405380993183694932208E-2L, +}; +#define NQ16_ID 9 +static long double Q16_ID[NQ16_ID + 1] = { + 3.199850952578356211091219295199301766718E-17L, + 3.652601488020654842194486058637953363918E-14L, + 1.620179741394865258354608590461839031281E-11L, + 3.629359209474609630056463248923684371426E-9L, + 4.473680923894354600193264347733477363305E-7L, + 3.106368086644715743265603656011050476736E-5L, + 1.198239259946770604954664925153424252622E-3L, + 2.446041004004283102372887804475767568272E-2L, + 2.403235525011860603014707768815113698768E-1L, + 9.491006790682158612266270665136910927149E-1L, + /* 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 long double Q8_16N[NQ8_16N + 1] = { + 1.001954266485599464105669390693597125904E-17L, + 7.545499865295034556206475956620160007849E-15L, + 2.267838684785673931024792538193202559922E-12L, + 3.561909705814420373609574999542459912419E-10L, + 3.216201422768092505214730633842924944671E-8L, + 1.731194793857907454569364622452058554314E-6L, + 5.576944613034537050396518509871004586039E-5L, + 1.051787760316848982655967052985391418146E-3L, + 1.102852974036687441600678598019883746959E-2L, + 5.834647019292460494254225988766702933571E-2L, + 1.290281921604364618912425380717127576529E-1L, + 7.598886310387075708640370806458926458301E-2L, +}; +#define NQ8_16D 11 +static long double Q8_16D[NQ8_16D + 1] = { + 1.368001558508338469503329967729951830843E-16L, + 1.034454121857542147020549303317348297289E-13L, + 3.128109209247090744354764050629381674436E-11L, + 4.957795214328501986562102573522064468671E-9L, + 4.537872468606711261992676606899273588899E-7L, + 2.493639207101727713192687060517509774182E-5L, + 8.294957278145328349785532236663051405805E-4L, + 1.646471258966713577374948205279380115839E-2L, + 1.878910092770966718491814497982191447073E-1L, + 1.152641605706170353727903052525652504075E0L, + 3.383550240669773485412333679367792932235E0L, + 3.823875252882035706910024716609908473970E0L, + /* 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 long double Q5_8N[NQ5_8N + 1] = { + 1.750399094021293722243426623211733898747E-13L, + 6.483426211748008735242909236490115050294E-11L, + 9.279430665656575457141747875716899958373E-9L, + 6.696634968526907231258534757736576340266E-7L, + 2.666560823798895649685231292142838188061E-5L, + 6.025087697259436271271562769707550594540E-4L, + 7.652807734168613251901945778921336353485E-3L, + 5.226269002589406461622551452343519078905E-2L, + 1.748390159751117658969324896330142895079E-1L, + 2.378188719097006494782174902213083589660E-1L, + 8.383984859679804095463699702165659216831E-2L, +}; +#define NQ5_8D 10 +static long double Q5_8D[NQ5_8D + 1] = { + 2.389878229704327939008104855942987615715E-12L, + 8.926142817142546018703814194987786425099E-10L, + 1.294065862406745901206588525833274399038E-7L, + 9.524139899457666250828752185212769682191E-6L, + 3.908332488377770886091936221573123353489E-4L, + 9.250427033957236609624199884089916836748E-3L, + 1.263420066165922645975830877751588421451E-1L, + 9.692527053860420229711317379861733180654E-1L, + 3.937813834630430172221329298841520707954E0L, + 7.603126427436356534498908111445191312181E0L, + 5.670677653334105479259958485084550934305E0L, + /* 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 long double Q4_5N[NQ4_5N + 1] = { + 2.233870042925895644234072357400122854086E-11L, + 5.146223225761993222808463878999151699792E-9L, + 4.459114531468296461688753521109797474523E-7L, + 1.891397692931537975547242165291668056276E-5L, + 4.279519145911541776938964806470674565504E-4L, + 5.275239415656560634702073291768904783989E-3L, + 3.468698403240744801278238473898432608887E-2L, + 1.138773146337708415188856882915457888274E-1L, + 1.622717518946443013587108598334636458955E-1L, + 7.249040006390586123760992346453034628227E-2L, + 1.941595365256460232175236758506411486667E-3L, +}; +#define NQ4_5D 9 +static long double Q4_5D[NQ4_5D + 1] = { + 3.049977232266999249626430127217988047453E-10L, + 7.120883230531035857746096928889676144099E-8L, + 6.301786064753734446784637919554359588859E-6L, + 2.762010530095069598480766869426308077192E-4L, + 6.572163250572867859316828886203406361251E-3L, + 8.752566114841221958200215255461843397776E-2L, + 6.487654992874805093499285311075289932664E-1L, + 2.576550017826654579451615283022812801435E0L, + 5.056392229924022835364779562707348096036E0L, + 4.179770081068251464907531367859072157773E0L, + /* 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 long double Q3r2_4N[NQ3r2_4N + 1] = { + 6.126167301024815034423262653066023684411E-10L, + 1.043969327113173261820028225053598975128E-7L, + 6.592927270288697027757438170153763220190E-6L, + 2.009103660938497963095652951912071336730E-4L, + 3.220543385492643525985862356352195896964E-3L, + 2.774405975730545157543417650436941650990E-2L, + 1.258114008023826384487378016636555041129E-1L, + 2.811724258266902502344701449984698323860E-1L, + 2.691837665193548059322831687432415014067E-1L, + 7.949087384900985370683770525312735605034E-2L, + 1.229509543620976530030153018986910810747E-3L, +}; +#define NQ3r2_4D 9 +static long double Q3r2_4D[NQ3r2_4D + 1] = { + 8.364260446128475461539941389210166156568E-9L, + 1.451301850638956578622154585560759862764E-6L, + 9.431830010924603664244578867057141839463E-5L, + 3.004105101667433434196388593004526182741E-3L, + 5.148157397848271739710011717102773780221E-2L, + 4.901089301726939576055285374953887874895E-1L, + 2.581760991981709901216967665934142240346E0L, + 7.257105880775059281391729708630912791847E0L, + 1.006014717326362868007913423810737369312E1L, + 5.879416600465399514404064187445293212470E0L, + /* 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 long double Q2r7_3r2N[NQ2r7_3r2N + 1] = { + 7.584861620402450302063691901886141875454E-8L, + 9.300939338814216296064659459966041794591E-6L, + 4.112108906197521696032158235392604947895E-4L, + 8.515168851578898791897038357239630654431E-3L, + 8.971286321017307400142720556749573229058E-2L, + 4.885856732902956303343015636331874194498E-1L, + 1.334506268733103291656253500506406045846E0L, + 1.681207956863028164179042145803851824654E0L, + 8.165042692571721959157677701625853772271E-1L, + 9.805848115375053300608712721986235900715E-2L, +}; +#define NQ2r7_3r2D 9 +static long double Q2r7_3r2D[NQ2r7_3r2D + 1] = { + 1.035586492113036586458163971239438078160E-6L, + 1.301999337731768381683593636500979713689E-4L, + 5.993695702564527062553071126719088859654E-3L, + 1.321184892887881883489141186815457808785E-1L, + 1.528766555485015021144963194165165083312E0L, + 9.561463309176490874525827051566494939295E0L, + 3.203719484883967351729513662089163356911E1L, + 5.497294687660930446641539152123568668447E1L, + 4.391158169390578768508675452986948391118E1L, + 1.347836630730048077907818943625789418378E1L, + /* 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 long double Q2r3_2r7N[NQ2r3_2r7N + 1] = { + 4.455027774980750211349941766420190722088E-7L, + 4.031998274578520170631601850866780366466E-5L, + 1.273987274325947007856695677491340636339E-3L, + 1.818754543377448509897226554179659122873E-2L, + 1.266748858326568264126353051352269875352E-1L, + 4.327578594728723821137731555139472880414E-1L, + 6.892532471436503074928194969154192615359E-1L, + 4.490775818438716873422163588640262036506E-1L, + 8.649615949297322440032000346117031581572E-2L, + 7.261345286655345047417257611469066147561E-4L, +}; +#define NQ2r3_2r7D 8 +static long double Q2r3_2r7D[NQ2r3_2r7D + 1] = { + 6.082600739680555266312417978064954793142E-6L, + 5.693622538165494742945717226571441747567E-4L, + 1.901625907009092204458328768129666975975E-2L, + 2.958689532697857335456896889409923371570E-1L, + 2.343124711045660081603809437993368799568E0L, + 9.665894032187458293568704885528192804376E0L, + 2.035273104990617136065743426322454881353E1L, + 2.044102010478792896815088858740075165531E1L, + 8.445937177863155827844146643468706599304E0L, + /* 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 long double Q2_2r3N[NQ2_2r3N + 1] = { + 2.817566786579768804844367382809101929314E-6L, + 2.122772176396691634147024348373539744935E-4L, + 5.501378031780457828919593905395747517585E-3L, + 6.355374424341762686099147452020466524659E-2L, + 3.539652320122661637429658698954748337223E-1L, + 9.571721066119617436343740541777014319695E-1L, + 1.196258777828426399432550698612171955305E0L, + 6.069388659458926158392384709893753793967E-1L, + 9.026746127269713176512359976978248763621E-2L, + 5.317668723070450235320878117210807236375E-4L, +}; +#define NQ2_2r3D 8 +static long double Q2_2r3D[NQ2_2r3D + 1] = { + 3.846924354014260866793741072933159380158E-5L, + 3.017562820057704325510067178327449946763E-3L, + 8.356305620686867949798885808540444210935E-2L, + 1.068314930499906838814019619594424586273E0L, + 6.900279623894821067017966573640732685233E0L, + 2.307667390886377924509090271780839563141E1L, + 3.921043465412723970791036825401273528513E1L, + 3.167569478939719383241775717095729233436E1L, + 1.051023841699200920276198346301543665909E1L, + /* 1.000000000000000000000000000000000000000E0*/ +}; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static long double +neval (long double x, long double *p, int n) +{ + long double 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 long double +deval (long double x, long double *p, int n) +{ + long double y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + +/* Bessel function of the first kind, order zero. */ + +long double +__ieee754_j0l (long double x) +{ + long double xx, xinv, z, p, q, c, s, cc, ss; + + if (! finitel (x)) + { + if (x != x) + return x; + else + return 0.0L; + } + if (x == 0.0L) + return 1.0L; + + xx = fabsl (x); + if (xx <= 2.0L) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = z * z * neval (z, J0_2N, NJ0_2N) / deval (z, J0_2D, NJ0_2D); + p -= 0.25L * z; + p += 1.0L; + return p; + } + + xinv = 1.0L / 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 = 1.0L + z * p; + q = z * xinv * q; + q = q - 0.125L * xinv; + /* 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. */ + c = cosl (x); + s = sinl (x); + ss = s - c; + cc = s + c; + z = -cosl (x + x); + if ((s * c) < 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * cc - q * ss) / sqrtl (x); + 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 long double Y0_2N[NY0_2N + 1] = { + -1.062023609591350692692296993537002558155E19L, + 2.542000883190248639104127452714966858866E19L, + -1.984190771278515324281415820316054696545E18L, + 4.982586044371592942465373274440222033891E16L, + -5.529326354780295177243773419090123407550E14L, + 3.013431465522152289279088265336861140391E12L, + -7.959436160727126750732203098982718347785E9L, + 8.230845651379566339707130644134372793322E6L, +}; +#define NY0_2D 7 +static long double Y0_2D[NY0_2D + 1] = { + 1.438972634353286978700329883122253752192E20L, + 1.856409101981569254247700169486907405500E18L, + 1.219693352678218589553725579802986255614E16L, + 5.389428943282838648918475915779958097958E13L, + 1.774125762108874864433872173544743051653E11L, + 4.522104832545149534808218252434693007036E8L, + 8.872187401232943927082914504125234454930E5L, + 1.251945613186787532055610876304669413955E3L, + /* 1.000000000000000000000000000000000000000E0 */ +}; + + +/* Bessel function of the second kind, order zero. */ + +long double + __ieee754_y0l(long double x) +{ + long double xx, xinv, z, p, q, c, s, cc, ss; + + if (x <= 0.0L) + { + if (x < 0.0L) + return (zero / zero); + return 1.0L / zero; + } + if (! finitel (x)) + { + if (x != x) + return x; + else + return 0.0L; + } + xx = fabsl (x); + if (xx <= 2.0L) + { + /* 0 <= x <= 2 */ + z = xx * xx; + p = neval (z, Y0_2N, NY0_2N) / deval (z, Y0_2D, NY0_2D); + p = TWOOPI * logl(x) * __ieee754_j0l(x) + p; + return p; + } + + xinv = 1.0L / 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 = 1.0L + z * p; + q = z * xinv * q; + q = q - 0.125L * xinv; + /* 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. */ + c = cosl (x); + s = sinl (x); + ss = s - c; + cc = s + c; + z = -cosl (x + x); + if ((s * c) < 0) + cc = z / ss; + else + ss = z / cc; + z = ONEOSQPI * (p * ss + q * cc) / sqrtl (x); + return z; +} |