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Ruta de la carpeta: \\game3dprogramming\materials\GameFactory\GameFactoryDemo\references\boost_1_35_0\boost\math\special_functions\detail\lgamma_small.hpp
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// (C) Copyright John Maddock 2006. // 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_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL #define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL namespace boost{ namespace math{ namespace detail{ // // lgamma for small arguments: // template
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<64>&, const Policy& /* l */, const L&) { // This version uses rational approximations for small // values of z accurate enough for 64-bit mantissas // (80-bit long doubles), works well for 53-bit doubles as well. // L is only used to select the Lanczos function. BOOST_MATH_STD_USING // for ADL of std names T result = 0; if(z < tools::epsilon
()) { result = -log(z); } else if((zm1 == 0) || (zm2 == 0)) { // nothing to do, result is zero.... } else if(z > 2) { // // Begin by performing argument reduction until // z is in [2,3): // if(z >= 3) { do { z -= 1; zm2 -= 1; result += log(z); }while(z >= 3); // Update zm2, we need it below: zm2 = z - 2; } // // Use the following form: // // lgamma(z) = (z-2)(z+1)(Y + R(z-2)) // // where R(z-2) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // R(z-2) has the following properties: // // At double: Max error found: 4.231e-18 // At long double: Max error found: 1.987e-21 // Maximum Deviation Found (approximation error): 5.900e-24 // static const T P[] = { static_cast
(-0.180355685678449379109e-1L), static_cast
(0.25126649619989678683e-1L), static_cast
(0.494103151567532234274e-1L), static_cast
(0.172491608709613993966e-1L), static_cast
(-0.259453563205438108893e-3L), static_cast
(-0.541009869215204396339e-3L), static_cast
(-0.324588649825948492091e-4L) }; static const T Q[] = { static_cast
(0.1e1), static_cast
(0.196202987197795200688e1L), static_cast
(0.148019669424231326694e1L), static_cast
(0.541391432071720958364e0L), static_cast
(0.988504251128010129477e-1L), static_cast
(0.82130967464889339326e-2L), static_cast
(0.224936291922115757597e-3L), static_cast
(-0.223352763208617092964e-6L) }; static const float Y = 0.158963680267333984375e0f; T r = zm2 * (z + 1); T R = tools::evaluate_polynomial(P, zm2); R /= tools::evaluate_polynomial(Q, zm2); result += r * Y + r * R; } else { // // If z is less than 1 use recurrance to shift to // z in the interval [1,2]: // if(z < 1) { result += -log(z); zm2 = zm1; zm1 = z; z += 1; } // // Two approximations, on for z in [1,1.5] and // one for z in [1.5,2]: // if(z <= 1.5) { // // Use the following form: // // lgamma(z) = (z-1)(z-2)(Y + R(z-1)) // // where R(z-1) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // R(z-1) has the following properties: // // At double precision: Max error found: 1.230011e-17 // At 80-bit long double precision: Max error found: 5.631355e-21 // Maximum Deviation Found: 3.139e-021 // Expected Error Term: 3.139e-021 // static const float Y = 0.52815341949462890625f; static const T P[] = { static_cast
(0.490622454069039543534e-1L), static_cast
(-0.969117530159521214579e-1L), static_cast
(-0.414983358359495381969e0L), static_cast
(-0.406567124211938417342e0L), static_cast
(-0.158413586390692192217e0L), static_cast
(-0.240149820648571559892e-1L), static_cast
(-0.100346687696279557415e-2L) }; static const T Q[] = { static_cast
(0.1e1L), static_cast
(0.302349829846463038743e1L), static_cast
(0.348739585360723852576e1L), static_cast
(0.191415588274426679201e1L), static_cast
(0.507137738614363510846e0L), static_cast
(0.577039722690451849648e-1L), static_cast
(0.195768102601107189171e-2L) }; T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1); T prefix = zm1 * zm2; result += prefix * Y + prefix * r; } else { // // Use the following form: // // lgamma(z) = (2-z)(1-z)(Y + R(2-z)) // // where R(2-z) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // R(2-z) has the following properties: // // At double precision, max error found: 1.797565e-17 // At 80-bit long double precision, max error found: 9.306419e-21 // Maximum Deviation Found: 2.151e-021 // Expected Error Term: 2.150e-021 // static const float Y = 0.452017307281494140625f; static const T P[] = { static_cast
(-0.292329721830270012337e-1L), static_cast
(0.144216267757192309184e0L), static_cast
(-0.142440390738631274135e0L), static_cast
(0.542809694055053558157e-1L), static_cast
(-0.850535976868336437746e-2L), static_cast
(0.431171342679297331241e-3L) }; static const T Q[] = { static_cast
(0.1e1), static_cast
(-0.150169356054485044494e1L), static_cast
(0.846973248876495016101e0L), static_cast
(-0.220095151814995745555e0L), static_cast
(0.25582797155975869989e-1L), static_cast
(-0.100666795539143372762e-2L), static_cast
(-0.827193521891290553639e-6L) }; T r = zm2 * zm1; T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2); result += r * Y + r * R; } } return result; } template
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<113>&, const Policy& /* l */, const L&) { // // This version uses rational approximations for small // values of z accurate enough for 113-bit mantissas // (128-bit long doubles). // BOOST_MATH_STD_USING // for ADL of std names T result = 0; if(z < tools::epsilon
()) { result = -log(z); BOOST_MATH_INSTRUMENT_CODE(result); } else if((zm1 == 0) || (zm2 == 0)) { // nothing to do, result is zero.... } else if(z > 2) { // // Begin by performing argument reduction until // z is in [2,3): // if(z >= 3) { do { z -= 1; result += log(z); }while(z >= 3); zm2 = z - 2; } BOOST_MATH_INSTRUMENT_CODE(zm2); BOOST_MATH_INSTRUMENT_CODE(z); BOOST_MATH_INSTRUMENT_CODE(result); // // Use the following form: // // lgamma(z) = (z-2)(z+1)(Y + R(z-2)) // // where R(z-2) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // Maximum Deviation Found (approximation error) 3.73e-37 static const T P[] = { -0.018035568567844937910504030027467476655L, 0.013841458273109517271750705401202404195L, 0.062031842739486600078866923383017722399L, 0.052518418329052161202007865149435256093L, 0.01881718142472784129191838493267755758L, 0.0025104830367021839316463675028524702846L, -0.00021043176101831873281848891452678568311L, -0.00010249622350908722793327719494037981166L, -0.11381479670982006841716879074288176994e-4L, -0.49999811718089980992888533630523892389e-6L, -0.70529798686542184668416911331718963364e-8L }; static const T Q[] = { 1L, 2.5877485070422317542808137697939233685L, 2.8797959228352591788629602533153837126L, 1.8030885955284082026405495275461180977L, 0.69774331297747390169238306148355428436L, 0.17261566063277623942044077039756583802L, 0.02729301254544230229429621192443000121L, 0.0026776425891195270663133581960016620433L, 0.00015244249160486584591370355730402168106L, 0.43997034032479866020546814475414346627e-5L, 0.46295080708455613044541885534408170934e-7L, -0.93326638207459533682980757982834180952e-11L, 0.42316456553164995177177407325292867513e-13L }; T R = tools::evaluate_polynomial(P, zm2); R /= tools::evaluate_polynomial(Q, zm2); static const float Y = 0.158963680267333984375F; T r = zm2 * (z + 1); result += r * Y + r * R; BOOST_MATH_INSTRUMENT_CODE(result); } else { // // If z is less than 1 use recurrance to shift to // z in the interval [1,2]: // if(z < 1) { result += -log(z); zm2 = zm1; zm1 = z; z += 1; } BOOST_MATH_INSTRUMENT_CODE(result); BOOST_MATH_INSTRUMENT_CODE(z); BOOST_MATH_INSTRUMENT_CODE(zm2); // // Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1] // if(z <= 1.35) { // // Use the following form: // // lgamma(z) = (z-1)(z-2)(Y + R(z-1)) // // where R(z-1) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // R(z-1) has the following properties: // // Maximum Deviation Found (approximation error) 1.659e-36 // Expected Error Term (theoretical error) 1.343e-36 // Max error found at 128-bit long double precision 1.007e-35 // static const float Y = 0.54076099395751953125f; static const T P[] = { 0.036454670944013329356512090082402429697L, -0.066235835556476033710068679907798799959L, -0.67492399795577182387312206593595565371L, -1.4345555263962411429855341651960000166L, -1.4894319559821365820516771951249649563L, -0.87210277668067964629483299712322411566L, -0.29602090537771744401524080430529369136L, -0.0561832587517836908929331992218879676L, -0.0053236785487328044334381502530383140443L, -0.00018629360291358130461736386077971890789L, -0.10164985672213178500790406939467614498e-6L, 0.13680157145361387405588201461036338274e-8L }; static const T Q[] = { 1, 4.9106336261005990534095838574132225599L, 10.258804800866438510889341082793078432L, 11.88588976846826108836629960537466889L, 8.3455000546999704314454891036700998428L, 3.6428823682421746343233362007194282703L, 0.97465989807254572142266753052776132252L, 0.15121052897097822172763084966793352524L, 0.012017363555383555123769849654484594893L, 0.0003583032812720649835431669893011257277L }; T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1); T prefix = zm1 * zm2; result += prefix * Y + prefix * r; BOOST_MATH_INSTRUMENT_CODE(result); } else if(z <= 1.625) { // // Use the following form: // // lgamma(z) = (2-z)(1-z)(Y + R(2-z)) // // where R(2-z) is a rational approximation optimised for // low absolute error - as long as it's absolute error // is small compared to the constant Y - then any rounding // error in it's computation will get wiped out. // // R(2-z) has the following properties: // // Max error found at 128-bit long double precision 9.634e-36 // Maximum Deviation Found (approximation error) 1.538e-37 // Expected Error Term (theoretical error) 2.350e-38 // static const float Y = 0.483787059783935546875f; static const T P[] = { -0.017977422421608624353488126610933005432L, 0.18484528905298309555089509029244135703L, -0.40401251514859546989565001431430884082L, 0.40277179799147356461954182877921388182L, -0.21993421441282936476709677700477598816L, 0.069595742223850248095697771331107571011L, -0.012681481427699686635516772923547347328L, 0.0012489322866834830413292771335113136034L, -0.57058739515423112045108068834668269608e-4L, 0.8207548771933585614380644961342925976e-6L }; static const T Q[] = { 1, -2.9629552288944259229543137757200262073L, 3.7118380799042118987185957298964772755L, -2.5569815272165399297600586376727357187L, 1.0546764918220835097855665680632153367L, -0.26574021300894401276478730940980810831L, 0.03996289731752081380552901986471233462L, -0.0033398680924544836817826046380586480873L, 0.00013288854760548251757651556792598235735L, -0.17194794958274081373243161848194745111e-5L }; T r = zm2 * zm1; T R = tools::evaluate_polynomial(P, 0.625 - zm1) / tools::evaluate_polynomial(Q, 0.625 - zm1); result += r * Y + r * R; BOOST_MATH_INSTRUMENT_CODE(result); } else { // // Same form as above. // // Max error found (at 128-bit long double precision) 1.831e-35 // Maximum Deviation Found (approximation error) 8.588e-36 // Expected Error Term (theoretical error) 1.458e-36 // static const float Y = 0.443811893463134765625f; static const T P[] = { -0.021027558364667626231512090082402429494L, 0.15128811104498736604523586803722368377L, -0.26249631480066246699388544451126410278L, 0.21148748610533489823742352180628489742L, -0.093964130697489071999873506148104370633L, 0.024292059227009051652542804957550866827L, -0.0036284453226534839926304745756906117066L, 0.0002939230129315195346843036254392485984L, -0.11088589183158123733132268042570710338e-4L, 0.13240510580220763969511741896361984162e-6L }; static const T Q[] = { 1, -2.4240003754444040525462170802796471996L, 2.4868383476933178722203278602342786002L, -1.4047068395206343375520721509193698547L, 0.47583809087867443858344765659065773369L, -0.09865724264554556400463655444270700132L, 0.012238223514176587501074150988445109735L, -0.00084625068418239194670614419707491797097L, 0.2796574430456237061420839429225710602e-4L, -0.30202973883316730694433702165188835331e-6L }; // (2 - x) * (1 - x) * (c + R(2 - x)) T r = zm2 * zm1; T R = tools::evaluate_polynomial(P, -zm2) / tools::evaluate_polynomial(Q, -zm2); result += r * Y + r * R; BOOST_MATH_INSTRUMENT_CODE(result); } } BOOST_MATH_INSTRUMENT_CODE(result); return result; } template
T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<0>&, const Policy& pol, const L&) { // // No rational approximations are available because either // T has no numeric_limits support (so we can't tell how // many digits it has), or T has more digits than we know // what to do with.... we do have a Lanczos approximation // though, and that can be used to keep errors under control. // BOOST_MATH_STD_USING // for ADL of std names T result = 0; if(z < tools::epsilon
()) { result = -log(z); } else if(z < 0.5) { // taking the log of tgamma reduces the error, no danger of overflow here: result = log(gamma_imp(z, pol, L())); } else if(z >= 3) { // taking the log of tgamma reduces the error, no danger of overflow here: result = log(gamma_imp(z, pol, L())); } else if(z >= 1.5) { // special case near 2: T dz = zm2; result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e
()); result += boost::math::log1p(dz / (L::g() + T(1.5)), pol) * T(1.5); result += boost::math::log1p(L::lanczos_sum_near_2(dz), pol); } else { // special case near 1: T dz = zm1; result = dz * log((z + L::g() - T(0.5)) / boost::math::constants::e
()); result += boost::math::log1p(dz / (L::g() + T(0.5)), pol) / 2; result += boost::math::log1p(L::lanczos_sum_near_1(dz), pol); } return result; } }}} // namespaces #endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
lgamma_small.hpp
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