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boost/math/bindings/detail/big_digamma.hpp

//  (C) Copyright John Maddock 2006-8.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

#ifndef BOOST_MATH_NTL_DIGAMMA
#define BOOST_MATH_NTL_DIGAMMA

#include <boost/math/tools/rational.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/constants/constants.hpp>

namespace boost{ namespace math{ namespace detail{

template <class T>
T big_digamma_helper(T x)
{
      static const T P[61] = {
         boost::lexical_cast<T>("0.6660133691143982067148122682345055274952e81"),
         boost::lexical_cast<T>("0.6365271516829242456324234577164675383137e81"),
         boost::lexical_cast<T>("0.2991038873096202943405966144203628966976e81"),
         boost::lexical_cast<T>("0.9211116495503170498076013367421231351115e80"),
         boost::lexical_cast<T>("0.2090792764676090716286400360584443891749e80"),
         boost::lexical_cast<T>("0.3730037777359591428226035156377978092809e79"),
         boost::lexical_cast<T>("0.5446396536956682043376492370432031543834e78"),
         boost::lexical_cast<T>("0.6692523966335177847425047827449069256345e77"),
         boost::lexical_cast<T>("0.7062543624100864681625612653756619116848e76"),
         boost::lexical_cast<T>("0.6499914905966283735005256964443226879158e75"),
         boost::lexical_cast<T>("0.5280364564853225211197557708655426736091e74"),
         boost::lexical_cast<T>("0.3823205608981176913075543599005095206953e73"),
         boost::lexical_cast<T>("0.2486733714214237704739129972671154532415e72"),
         boost::lexical_cast<T>("0.1462562139602039577983434547171318011675e71"),
         boost::lexical_cast<T>("0.7821169065036815012381267259559910324285e69"),
         boost::lexical_cast<T>("0.3820552182348155468636157988764435365078e68"),
         boost::lexical_cast<T>("0.1711618296983598244658239925535632505062e67"),
         boost::lexical_cast<T>("0.7056661618357643731419080738521475204245e65"),
         boost::lexical_cast<T>("0.2685246896473614017356264531791459936036e64"),
         boost::lexical_cast<T>("0.9455168125599643085283071944864977592391e62"),
         boost::lexical_cast<T>("0.3087541626972538362237309145177486236219e61"),
         boost::lexical_cast<T>("0.9367928873352980208052601301625005737407e59"),
         boost::lexical_cast<T>("0.2645306130689794942883818547314327466007e58"),
         boost::lexical_cast<T>("0.6961815141171454309161007351079576190079e56"),
         boost::lexical_cast<T>("0.1709637824471794552313802669803885946843e55"),
         boost::lexical_cast<T>("0.3921553258481531526663112728778759311158e53"),
         boost::lexical_cast<T>("0.8409006354449988687714450897575728228696e51"),
         boost::lexical_cast<T>("0.1686755204461325935742097669030363344927e50"),
         boost::lexical_cast<T>("0.3166653542877070999007425197729038754254e48"),
         boost::lexical_cast<T>("0.5566029092358215049069560272835654229637e46"),
         boost::lexical_cast<T>("0.9161766287916328133080586672953875116242e44"),
         boost::lexical_cast<T>("1412317772330871298317974693514430627922000"),
         boost::lexical_cast<T>("20387991986727877473732570146112459874790"),
         boost::lexical_cast<T>("275557928713904105182512535678580359839.3"),
         boost::lexical_cast<T>("3485719851040516559072031256589598330.723"),
         boost::lexical_cast<T>("41247046743564028399938106707656877.40859"),
         boost::lexical_cast<T>("456274078125709314602601667471879.0147312"),
         boost::lexical_cast<T>("4714450683242899367025707077155.310613012"),
         boost::lexical_cast<T>("45453933537925041680009544258.75073849996"),
         boost::lexical_cast<T>("408437900487067278846361972.302331241052"),
         boost::lexical_cast<T>("3415719344386166273085838.705771571751035"),
         boost::lexical_cast<T>("26541502879185876562320.93134691487351145"),
         boost::lexical_cast<T>("191261415065918713661.1571433274648417668"),
         boost::lexical_cast<T>("1275349770108718421.645275944284937551702"),
         boost::lexical_cast<T>("7849171120971773.318910987434906905704272"),
         boost::lexical_cast<T>("44455946386549.80866460312682983576538056"),
         boost::lexical_cast<T>("230920362395.3198137186361608905136598046"),
         boost::lexical_cast<T>("1095700096.240863858624279930600654130254"),
         boost::lexical_cast<T>("4727085.467506050153744334085516289728134"),
         boost::lexical_cast<T>("18440.75118859447173303252421991479005424"),
         boost::lexical_cast<T>("64.62515887799460295677071749181651317052"),
         boost::lexical_cast<T>("0.201851568864688406206528472883512147547"),
         boost::lexical_cast<T>("0.0005565091674187978029138500039504078098143"),
         boost::lexical_cast<T>("0.1338097668312907986354698683493366559613e-5"),
         boost::lexical_cast<T>("0.276308225077464312820179030238305271638e-8"),
         boost::lexical_cast<T>("0.4801582970473168520375942100071070575043e-11"),
         boost::lexical_cast<T>("0.6829184144212920949740376186058541800175e-14"),
         boost::lexical_cast<T>("0.7634080076358511276617829524639455399182e-17"),
         boost::lexical_cast<T>("0.6290035083727140966418512608156646142409e-20"),
         boost::lexical_cast<T>("0.339652245667538733044036638506893821352e-23"),
         boost::lexical_cast<T>("0.9017518064256388530773585529891677854909e-27")
      };
      static const T Q[61] = {
         boost::lexical_cast<T>("0"),
         boost::lexical_cast<T>("0.1386831185456898357379390197203894063459e81"),
         boost::lexical_cast<T>("0.6467076379487574703291056110838151259438e81"),
         boost::lexical_cast<T>("0.1394967823848615838336194279565285465161e82"),
         boost::lexical_cast<T>("0.1872927317344192945218570366455046340458e82"),
         boost::lexical_cast<T>("0.1772461045338946243584650759986310355937e82"),
         boost::lexical_cast<T>("0.1267294892200258648315971144069595555118e82"),
         boost::lexical_cast<T>("0.7157764838362416821508872117623058626589e81"),
         boost::lexical_cast<T>("0.329447266909948668265277828268378274513e81"),
         boost::lexical_cast<T>("0.1264376077317689779509250183194342571207e81"),
         boost::lexical_cast<T>("0.4118230304191980787640446056583623228873e80"),
         boost::lexical_cast<T>("0.1154393529762694616405952270558316515261e80"),
         boost::lexical_cast<T>("0.281655612889423906125295485693696744275e79"),
         boost::lexical_cast<T>("0.6037483524928743102724159846414025482077e78"),
         boost::lexical_cast<T>("0.1145927995397835468123576831800276999614e78"),
         boost::lexical_cast<T>("0.1938624296151985600348534009382865995154e77"),
         boost::lexical_cast<T>("0.293980925856227626211879961219188406675e76"),
         boost::lexical_cast<T>("0.4015574518216966910319562902099567437832e75"),
         boost::lexical_cast<T>("0.4961475457509727343545565970423431880907e74"),
         boost::lexical_cast<T>("0.5565482348278933960215521991000378896338e73"),
         boost::lexical_cast<T>("0.5686112924615820754631098622770303094938e72"),
         boost::lexical_cast<T>("0.5305988545844796293285410303747469932856e71"),
         boost::lexical_cast<T>("0.4533363413802585060568537458067343491358e70"),
         boost::lexical_cast<T>("0.3553932059473516064068322757331575565718e69"),
         boost::lexical_cast<T>("0.2561198565218704414618802902533972354203e68"),
         boost::lexical_cast<T>("0.1699519313292900324098102065697454295572e67"),
         boost::lexical_cast<T>("0.1039830160862334505389615281373574959236e66"),
         boost::lexical_cast<T>("0.5873082967977428281000961954715372504986e64"),
         boost::lexical_cast<T>("0.3065255179030575882202133042549783442446e63"),
         boost::lexical_cast<T>("0.1479494813481364701208655943688307245459e62"),
         boost::lexical_cast<T>("0.6608150467921598615495180659808895663164e60"),
         boost::lexical_cast<T>("0.2732535313770902021791888953487787496976e59"),
         boost::lexical_cast<T>("0.1046402297662493314531194338414508049069e58"),
         boost::lexical_cast<T>("0.3711375077192882936085049147920021549622e56"),
         boost::lexical_cast<T>("0.1219154482883895482637944309702972234576e55"),
         boost::lexical_cast<T>("0.3708359374149458741391374452286837880162e53"),
         boost::lexical_cast<T>("0.1044095509971707189716913168889769471468e52"),
         boost::lexical_cast<T>("0.271951506225063286130946773813524945052e50"),
         boost::lexical_cast<T>("0.6548016291215163843464133978454065823866e48"),
         boost::lexical_cast<T>("0.1456062447610542135403751730809295219344e47"),
         boost::lexical_cast<T>("0.2986690175077969760978388356833006028929e45"),
         boost::lexical_cast<T>("5643149706574013350061247429006443326844000"),
         boost::lexical_cast<T>("98047545414467090421964387960743688053480"),
         boost::lexical_cast<T>("1563378767746846395507385099301468978550"),
         boost::lexical_cast<T>("22823360528584500077862274918382796495"),
         boost::lexical_cast<T>("304215527004115213046601295970388750"),
         boost::lexical_cast<T>("3690289075895685793844344966820325"),
         boost::lexical_cast<T>("40584512015702371433911456606050"),
         boost::lexical_cast<T>("402834190897282802772754873905"),
         boost::lexical_cast<T>("3589522158493606918146495750"),
         boost::lexical_cast<T>("28530557707503483723634725"),
         boost::lexical_cast<T>("200714561335055753000730"),
         boost::lexical_cast<T>("1237953783437761888641"),
         boost::lexical_cast<T>("6614698701445762950"),
         boost::lexical_cast<T>("30155495647727505"),
         boost::lexical_cast<T>("114953256021450"),
         boost::lexical_cast<T>("356398020013"),
         boost::lexical_cast<T>("863113950"),
         boost::lexical_cast<T>("1531345"),
         boost::lexical_cast<T>("1770"),
         boost::lexical_cast<T>("1")
      };
      static const T PD[60] = {
         boost::lexical_cast<T>("0.6365271516829242456324234577164675383137e81"),
         2*boost::lexical_cast<T>("0.2991038873096202943405966144203628966976e81"),
         3*boost::lexical_cast<T>("0.9211116495503170498076013367421231351115e80"),
         4*boost::lexical_cast<T>("0.2090792764676090716286400360584443891749e80"),
         5*boost::lexical_cast<T>("0.3730037777359591428226035156377978092809e79"),
         6*boost::lexical_cast<T>("0.5446396536956682043376492370432031543834e78"),
         7*boost::lexical_cast<T>("0.6692523966335177847425047827449069256345e77"),
         8*boost::lexical_cast<T>("0.7062543624100864681625612653756619116848e76"),
         9*boost::lexical_cast<T>("0.6499914905966283735005256964443226879158e75"),
         10*boost::lexical_cast<T>("0.5280364564853225211197557708655426736091e74"),
         11*boost::lexical_cast<T>("0.3823205608981176913075543599005095206953e73"),
         12*boost::lexical_cast<T>("0.2486733714214237704739129972671154532415e72"),
         13*boost::lexical_cast<T>("0.1462562139602039577983434547171318011675e71"),
         14*boost::lexical_cast<T>("0.7821169065036815012381267259559910324285e69"),
         15*boost::lexical_cast<T>("0.3820552182348155468636157988764435365078e68"),
         16*boost::lexical_cast<T>("0.1711618296983598244658239925535632505062e67"),
         17*boost::lexical_cast<T>("0.7056661618357643731419080738521475204245e65"),
         18*boost::lexical_cast<T>("0.2685246896473614017356264531791459936036e64"),
         19*boost::lexical_cast<T>("0.9455168125599643085283071944864977592391e62"),
         20*boost::lexical_cast<T>("0.3087541626972538362237309145177486236219e61"),
         21*boost::lexical_cast<T>("0.9367928873352980208052601301625005737407e59"),
         22*boost::lexical_cast<T>("0.2645306130689794942883818547314327466007e58"),
         23*boost::lexical_cast<T>("0.6961815141171454309161007351079576190079e56"),
         24*boost::lexical_cast<T>("0.1709637824471794552313802669803885946843e55"),
         25*boost::lexical_cast<T>("0.3921553258481531526663112728778759311158e53"),
         26*boost::lexical_cast<T>("0.8409006354449988687714450897575728228696e51"),
         27*boost::lexical_cast<T>("0.1686755204461325935742097669030363344927e50"),
         28*boost::lexical_cast<T>("0.3166653542877070999007425197729038754254e48"),
         29*boost::lexical_cast<T>("0.5566029092358215049069560272835654229637e46"),
         30*boost::lexical_cast<T>("0.9161766287916328133080586672953875116242e44"),
         31*boost::lexical_cast<T>("1412317772330871298317974693514430627922000"),
         32*boost::lexical_cast<T>("20387991986727877473732570146112459874790"),
         33*boost::lexical_cast<T>("275557928713904105182512535678580359839.3"),
         34*boost::lexical_cast<T>("3485719851040516559072031256589598330.723"),
         35*boost::lexical_cast<T>("41247046743564028399938106707656877.40859"),
         36*boost::lexical_cast<T>("456274078125709314602601667471879.0147312"),
         37*boost::lexical_cast<T>("4714450683242899367025707077155.310613012"),
         38*boost::lexical_cast<T>("45453933537925041680009544258.75073849996"),
         39*boost::lexical_cast<T>("408437900487067278846361972.302331241052"),
         40*boost::lexical_cast<T>("3415719344386166273085838.705771571751035"),
         41*boost::lexical_cast<T>("26541502879185876562320.93134691487351145"),
         42*boost::lexical_cast<T>("191261415065918713661.1571433274648417668"),
         43*boost::lexical_cast<T>("1275349770108718421.645275944284937551702"),
         44*boost::lexical_cast<T>("7849171120971773.318910987434906905704272"),
         45*boost::lexical_cast<T>("44455946386549.80866460312682983576538056"),
         46*boost::lexical_cast<T>("230920362395.3198137186361608905136598046"),
         47*boost::lexical_cast<T>("1095700096.240863858624279930600654130254"),
         48*boost::lexical_cast<T>("4727085.467506050153744334085516289728134"),
         49*boost::lexical_cast<T>("18440.75118859447173303252421991479005424"),
         50*boost::lexical_cast<T>("64.62515887799460295677071749181651317052"),
         51*boost::lexical_cast<T>("0.201851568864688406206528472883512147547"),
         52*boost::lexical_cast<T>("0.0005565091674187978029138500039504078098143"),
         53*boost::lexical_cast<T>("0.1338097668312907986354698683493366559613e-5"),
         54*boost::lexical_cast<T>("0.276308225077464312820179030238305271638e-8"),
         55*boost::lexical_cast<T>("0.4801582970473168520375942100071070575043e-11"),
         56*boost::lexical_cast<T>("0.6829184144212920949740376186058541800175e-14"),
         57*boost::lexical_cast<T>("0.7634080076358511276617829524639455399182e-17"),
         58*boost::lexical_cast<T>("0.6290035083727140966418512608156646142409e-20"),
         59*boost::lexical_cast<T>("0.339652245667538733044036638506893821352e-23"),
         60*boost::lexical_cast<T>("0.9017518064256388530773585529891677854909e-27")
      };
      static const T QD[60] = {
         boost::lexical_cast<T>("0.1386831185456898357379390197203894063459e81"),
         2*boost::lexical_cast<T>("0.6467076379487574703291056110838151259438e81"),
         3*boost::lexical_cast<T>("0.1394967823848615838336194279565285465161e82"),
         4*boost::lexical_cast<T>("0.1872927317344192945218570366455046340458e82"),
         5*boost::lexical_cast<T>("0.1772461045338946243584650759986310355937e82"),
         6*boost::lexical_cast<T>("0.1267294892200258648315971144069595555118e82"),
         7*boost::lexical_cast<T>("0.7157764838362416821508872117623058626589e81"),
         8*boost::lexical_cast<T>("0.329447266909948668265277828268378274513e81"),
         9*boost::lexical_cast<T>("0.1264376077317689779509250183194342571207e81"),
         10*boost::lexical_cast<T>("0.4118230304191980787640446056583623228873e80"),
         11*boost::lexical_cast<T>("0.1154393529762694616405952270558316515261e80"),
         12*boost::lexical_cast<T>("0.281655612889423906125295485693696744275e79"),
         13*boost::lexical_cast<T>("0.6037483524928743102724159846414025482077e78"),
         14*boost::lexical_cast<T>("0.1145927995397835468123576831800276999614e78"),
         15*boost::lexical_cast<T>("0.1938624296151985600348534009382865995154e77"),
         16*boost::lexical_cast<T>("0.293980925856227626211879961219188406675e76"),
         17*boost::lexical_cast<T>("0.4015574518216966910319562902099567437832e75"),
         18*boost::lexical_cast<T>("0.4961475457509727343545565970423431880907e74"),
         19*boost::lexical_cast<T>("0.5565482348278933960215521991000378896338e73"),
         20*boost::lexical_cast<T>("0.5686112924615820754631098622770303094938e72"),
         21*boost::lexical_cast<T>("0.5305988545844796293285410303747469932856e71"),
         22*boost::lexical_cast<T>("0.4533363413802585060568537458067343491358e70"),
         23*boost::lexical_cast<T>("0.3553932059473516064068322757331575565718e69"),
         24*boost::lexical_cast<T>("0.2561198565218704414618802902533972354203e68"),
         25*boost::lexical_cast<T>("0.1699519313292900324098102065697454295572e67"),
         26*boost::lexical_cast<T>("0.1039830160862334505389615281373574959236e66"),
         27*boost::lexical_cast<T>("0.5873082967977428281000961954715372504986e64"),
         28*boost::lexical_cast<T>("0.3065255179030575882202133042549783442446e63"),
         29*boost::lexical_cast<T>("0.1479494813481364701208655943688307245459e62"),
         30*boost::lexical_cast<T>("0.6608150467921598615495180659808895663164e60"),
         31*boost::lexical_cast<T>("0.2732535313770902021791888953487787496976e59"),
         32*boost::lexical_cast<T>("0.1046402297662493314531194338414508049069e58"),
         33*boost::lexical_cast<T>("0.3711375077192882936085049147920021549622e56"),
         34*boost::lexical_cast<T>("0.1219154482883895482637944309702972234576e55"),
         35*boost::lexical_cast<T>("0.3708359374149458741391374452286837880162e53"),
         36*boost::lexical_cast<T>("0.1044095509971707189716913168889769471468e52"),
         37*boost::lexical_cast<T>("0.271951506225063286130946773813524945052e50"),
         38*boost::lexical_cast<T>("0.6548016291215163843464133978454065823866e48"),
         39*boost::lexical_cast<T>("0.1456062447610542135403751730809295219344e47"),
         40*boost::lexical_cast<T>("0.2986690175077969760978388356833006028929e45"),
         41*boost::lexical_cast<T>("5643149706574013350061247429006443326844000"),
         42*boost::lexical_cast<T>("98047545414467090421964387960743688053480"),
         43*boost::lexical_cast<T>("1563378767746846395507385099301468978550"),
         44*boost::lexical_cast<T>("22823360528584500077862274918382796495"),
         45*boost::lexical_cast<T>("304215527004115213046601295970388750"),
         46*boost::lexical_cast<T>("3690289075895685793844344966820325"),
         47*boost::lexical_cast<T>("40584512015702371433911456606050"),
         48*boost::lexical_cast<T>("402834190897282802772754873905"),
         49*boost::lexical_cast<T>("3589522158493606918146495750"),
         50*boost::lexical_cast<T>("28530557707503483723634725"),
         51*boost::lexical_cast<T>("200714561335055753000730"),
         52*boost::lexical_cast<T>("1237953783437761888641"),
         53*boost::lexical_cast<T>("6614698701445762950"),
         54*boost::lexical_cast<T>("30155495647727505"),
         55*boost::lexical_cast<T>("114953256021450"),
         56*boost::lexical_cast<T>("356398020013"),
         57*boost::lexical_cast<T>("863113950"),
         58*boost::lexical_cast<T>("1531345"),
         59*boost::lexical_cast<T>("1770"),
         60*boost::lexical_cast<T>("1")
      };
   static const double g = 63.192152;

   T zgh = x + g - 0.5;

   T result = (x - 0.5) / zgh;
   result += log(zgh);
   result += tools::evaluate_polynomial(PD, x) / tools::evaluate_polynomial(P, x);
   result -= tools::evaluate_polynomial(QD, x) / tools::evaluate_polynomial(Q, x);
   result -= 1;

   return result;
}

template <class T>
T big_digamma(T x)
{
   BOOST_MATH_STD_USING
   if(x < 0)
   {
      return big_digamma_helper(static_cast<T>(1-x)) + constants::pi<T>() / tan(constants::pi<T>() * (1-x));
   }
   return big_digamma_helper(x);
}

}}}

#endif // include guard