boost/math/cstdfloat/cstdfloat_cmath.hpp
/////////////////////////////////////////////////////////////////////////////// // Copyright Christopher Kormanyos 2014. // Copyright John Maddock 2014. // Copyright Paul Bristow 2014. // Distributed under 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) // // Implement quadruple-precision <cmath> support. #ifndef _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_ #define _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_ #include <boost/math/cstdfloat/cstdfloat_types.hpp> #include <boost/math/cstdfloat/cstdfloat_limits.hpp> #if defined(BOOST_CSTDFLOAT_HAS_INTERNAL_FLOAT128_T) && defined(BOOST_MATH_USE_FLOAT128) && !defined(BOOST_CSTDFLOAT_NO_LIBQUADMATH_SUPPORT) #include <cmath> #include <stdexcept> #include <boost/cstdint.hpp> #include <boost/static_assert.hpp> #include <boost/throw_exception.hpp> #if defined(_WIN32) && defined(__GNUC__) // Several versions of Mingw and probably cygwin too have broken // libquadmath implementations that segfault as soon as you call // expq or any function that depends on it. #define BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS #endif // Here is a helper function used for raising the value of a given // floating-point type to the power of n, where n has integral type. namespace boost { namespace math { namespace cstdfloat { namespace detail { template<class float_type, class integer_type> inline float_type pown(const float_type& x, const integer_type p) { const bool isneg = (x < 0); const bool isnan = (x != x); const bool isinf = ((!isneg) ? bool(+x > (std::numeric_limits<float_type>::max)()) : bool(-x > (std::numeric_limits<float_type>::max)())); if(isnan) { return x; } if(isinf) { return std::numeric_limits<float_type>::quiet_NaN(); } const bool x_is_neg = (x < 0); const float_type abs_x = (x_is_neg ? -x : x); if(p < static_cast<integer_type>(0)) { if(abs_x < (std::numeric_limits<float_type>::min)()) { return (x_is_neg ? -std::numeric_limits<float_type>::infinity() : +std::numeric_limits<float_type>::infinity()); } else { return float_type(1) / pown(x, static_cast<integer_type>(-p)); } } if(p == static_cast<integer_type>(0)) { return float_type(1); } else { if(p == static_cast<integer_type>(1)) { return x; } if(abs_x > (std::numeric_limits<float_type>::max)()) { return (x_is_neg ? -std::numeric_limits<float_type>::infinity() : +std::numeric_limits<float_type>::infinity()); } if (p == static_cast<integer_type>(2)) { return (x * x); } else if(p == static_cast<integer_type>(3)) { return ((x * x) * x); } else if(p == static_cast<integer_type>(4)) { const float_type x2 = (x * x); return (x2 * x2); } else { // The variable xn stores the binary powers of x. float_type result(((p % integer_type(2)) != integer_type(0)) ? x : float_type(1)); float_type xn (x); integer_type p2 = p; while(integer_type(p2 /= 2) != integer_type(0)) { // Square xn for each binary power. xn *= xn; const bool has_binary_power = (integer_type(p2 % integer_type(2)) != integer_type(0)); if(has_binary_power) { // Multiply the result with each binary power contained in the exponent. result *= xn; } } return result; } } } } } } } // boost::math::cstdfloat::detail // We will now define preprocessor symbols representing quadruple-precision <cmath> functions. #if defined(BOOST_INTEL) #define BOOST_CSTDFLOAT_FLOAT128_LDEXP __ldexpq #define BOOST_CSTDFLOAT_FLOAT128_FREXP __frexpq #define BOOST_CSTDFLOAT_FLOAT128_FABS __fabsq #define BOOST_CSTDFLOAT_FLOAT128_FLOOR __floorq #define BOOST_CSTDFLOAT_FLOAT128_CEIL __ceilq #if !defined(BOOST_CSTDFLOAT_FLOAT128_SQRT) #define BOOST_CSTDFLOAT_FLOAT128_SQRT __sqrtq #endif #define BOOST_CSTDFLOAT_FLOAT128_TRUNC __truncq #define BOOST_CSTDFLOAT_FLOAT128_EXP __expq #define BOOST_CSTDFLOAT_FLOAT128_EXPM1 __expm1q #define BOOST_CSTDFLOAT_FLOAT128_POW __powq #define BOOST_CSTDFLOAT_FLOAT128_LOG __logq #define BOOST_CSTDFLOAT_FLOAT128_LOG10 __log10q #define BOOST_CSTDFLOAT_FLOAT128_SIN __sinq #define BOOST_CSTDFLOAT_FLOAT128_COS __cosq #define BOOST_CSTDFLOAT_FLOAT128_TAN __tanq #define BOOST_CSTDFLOAT_FLOAT128_ASIN __asinq #define BOOST_CSTDFLOAT_FLOAT128_ACOS __acosq #define BOOST_CSTDFLOAT_FLOAT128_ATAN __atanq #define BOOST_CSTDFLOAT_FLOAT128_SINH __sinhq #define BOOST_CSTDFLOAT_FLOAT128_COSH __coshq #define BOOST_CSTDFLOAT_FLOAT128_TANH __tanhq #define BOOST_CSTDFLOAT_FLOAT128_ASINH __asinhq #define BOOST_CSTDFLOAT_FLOAT128_ACOSH __acoshq #define BOOST_CSTDFLOAT_FLOAT128_ATANH __atanhq #define BOOST_CSTDFLOAT_FLOAT128_FMOD __fmodq #define BOOST_CSTDFLOAT_FLOAT128_ATAN2 __atan2q #define BOOST_CSTDFLOAT_FLOAT128_LGAMMA __lgammaq #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA __tgammaq #elif defined(__GNUC__) #define BOOST_CSTDFLOAT_FLOAT128_LDEXP ldexpq #define BOOST_CSTDFLOAT_FLOAT128_FREXP frexpq #define BOOST_CSTDFLOAT_FLOAT128_FABS fabsq #define BOOST_CSTDFLOAT_FLOAT128_FLOOR floorq #define BOOST_CSTDFLOAT_FLOAT128_CEIL ceilq #if !defined(BOOST_CSTDFLOAT_FLOAT128_SQRT) #define BOOST_CSTDFLOAT_FLOAT128_SQRT sqrtq #endif #define BOOST_CSTDFLOAT_FLOAT128_TRUNC truncq #define BOOST_CSTDFLOAT_FLOAT128_POW powq #define BOOST_CSTDFLOAT_FLOAT128_LOG logq #define BOOST_CSTDFLOAT_FLOAT128_LOG10 log10q #define BOOST_CSTDFLOAT_FLOAT128_SIN sinq #define BOOST_CSTDFLOAT_FLOAT128_COS cosq #define BOOST_CSTDFLOAT_FLOAT128_TAN tanq #define BOOST_CSTDFLOAT_FLOAT128_ASIN asinq #define BOOST_CSTDFLOAT_FLOAT128_ACOS acosq #define BOOST_CSTDFLOAT_FLOAT128_ATAN atanq #define BOOST_CSTDFLOAT_FLOAT128_FMOD fmodq #define BOOST_CSTDFLOAT_FLOAT128_ATAN2 atan2q #define BOOST_CSTDFLOAT_FLOAT128_LGAMMA lgammaq #if !defined(BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS) #define BOOST_CSTDFLOAT_FLOAT128_EXP expq #define BOOST_CSTDFLOAT_FLOAT128_EXPM1 expm1q_internal #define BOOST_CSTDFLOAT_FLOAT128_SINH sinhq #define BOOST_CSTDFLOAT_FLOAT128_COSH coshq #define BOOST_CSTDFLOAT_FLOAT128_TANH tanhq #define BOOST_CSTDFLOAT_FLOAT128_ASINH asinhq #define BOOST_CSTDFLOAT_FLOAT128_ACOSH acoshq #define BOOST_CSTDFLOAT_FLOAT128_ATANH atanhq #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA tgammaq #else // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS #define BOOST_CSTDFLOAT_FLOAT128_EXP expq_patch #define BOOST_CSTDFLOAT_FLOAT128_SINH sinhq_patch #define BOOST_CSTDFLOAT_FLOAT128_COSH coshq_patch #define BOOST_CSTDFLOAT_FLOAT128_TANH tanhq_patch #define BOOST_CSTDFLOAT_FLOAT128_ASINH asinhq_patch #define BOOST_CSTDFLOAT_FLOAT128_ACOSH acoshq_patch #define BOOST_CSTDFLOAT_FLOAT128_ATANH atanhq_patch #define BOOST_CSTDFLOAT_FLOAT128_TGAMMA tgammaq_patch #endif // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS #endif // Implement quadruple-precision <cmath> functions in the namespace // boost::math::cstdfloat::detail. Subsequently inject these into the // std namespace via *using* directive. // Begin with some forward function declarations. Also implement patches // for compilers that have broken float128 exponential functions. extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LDEXP (boost::math::cstdfloat::detail::float_internal128_t, int) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FREXP (boost::math::cstdfloat::detail::float_internal128_t, int*) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FABS (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FLOOR (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_CEIL (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SQRT (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TRUNC (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_POW (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LOG (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LOG10 (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SIN (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COS (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TAN (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASIN (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOS (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATAN (boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_FMOD (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATAN2 (boost::math::cstdfloat::detail::float_internal128_t, boost::math::cstdfloat::detail::float_internal128_t) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_LGAMMA(boost::math::cstdfloat::detail::float_internal128_t) throw(); #if !defined(BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS) extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SINH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COSH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TANH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASINH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOSH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATANH (boost::math::cstdfloat::detail::float_internal128_t x) throw(); extern "C" boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TGAMMA(boost::math::cstdfloat::detail::float_internal128_t x) throw(); #else // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS // Forward declaration of the patched exponent function, exp(x). inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP (boost::math::cstdfloat::detail::float_internal128_t x); inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXPM1 (boost::math::cstdfloat::detail::float_internal128_t x) { // Compute exp(x) - 1 for x small. // Use an order-36 polynomial approximation of the exponential function // in the range of (-ln2 < x < ln2). Scale the argument to this range // and subsequently multiply the result by 2^n accordingly. // Derive the polynomial coefficients with Mathematica(R) by generating // a table of high-precision values of exp(x) in the range (-ln2 < x < ln2) // and subsequently applying the built-in *Fit* function. // Table[{x, Exp[x] - 1}, {x, -Log[2], Log[2], 1/180}] // N[%, 120] // Fit[%, {x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^12, // x^13, x^14, x^15, x^16, x^17, x^18, x^19, x^20, x^21, x^22, // x^23, x^24, x^25, x^26, x^27, x^28, x^29, x^30, x^31, x^32, // x^33, x^34, x^35, x^36}, x] typedef boost::math::cstdfloat::detail::float_internal128_t float_type; float_type sum; if(x > BOOST_FLOAT128_C(0.693147180559945309417232121458176568075500134360255)) { sum = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x) - float_type(1); } else { // Compute the polynomial approximation of exp(alpha). sum = (((((((((((((((((((((((((((((((((((( float_type(BOOST_FLOAT128_C(2.69291698127774166063293705964720493864630783729857438187365E-42)) * x + float_type(BOOST_FLOAT128_C(9.70937085471487654794114679403710456028986572118859594614033E-41))) * x + float_type(BOOST_FLOAT128_C(3.38715585158055097155585505318085512156885389014410753080500E-39))) * x + float_type(BOOST_FLOAT128_C(1.15162718532861050809222658798662695267019717760563645440433E-37))) * x + float_type(BOOST_FLOAT128_C(3.80039074689434663295873584133017767349635602413675471702393E-36))) * x + float_type(BOOST_FLOAT128_C(1.21612504934087520075905434734158045947460467096773246215239E-34))) * x + float_type(BOOST_FLOAT128_C(3.76998762883139753126119821241037824830069851253295480396224E-33))) * x + float_type(BOOST_FLOAT128_C(1.13099628863830344684998293828608215735777107850991029729440E-31))) * x + float_type(BOOST_FLOAT128_C(3.27988923706982293204067897468714277771890104022419696770352E-30))) * x + float_type(BOOST_FLOAT128_C(9.18368986379558482800593745627556950089950023355628325088207E-29))) * x + float_type(BOOST_FLOAT128_C(2.47959626322479746949155352659617642905315302382639380521497E-27))) * x + float_type(BOOST_FLOAT128_C(6.44695028438447337900255966737803112935639344283098705091949E-26))) * x + float_type(BOOST_FLOAT128_C(1.61173757109611834904452725462599961406036904573072897122957E-24))) * x + float_type(BOOST_FLOAT128_C(3.86817017063068403772269360016918092488847584660382953555804E-23))) * x + float_type(BOOST_FLOAT128_C(8.89679139245057328674891109315654704307721758924206107351744E-22))) * x + float_type(BOOST_FLOAT128_C(1.95729410633912612308475595397946731738088422488032228717097E-20))) * x + float_type(BOOST_FLOAT128_C(4.11031762331216485847799061511674191805055663711439605760231E-19))) * x + float_type(BOOST_FLOAT128_C(8.22063524662432971695598123977873600603370758794431071426640E-18))) * x + float_type(BOOST_FLOAT128_C(1.56192069685862264622163643500633782667263448653185159383285E-16))) * x + float_type(BOOST_FLOAT128_C(2.81145725434552076319894558300988749849555291507956994126835E-15))) * x + float_type(BOOST_FLOAT128_C(4.77947733238738529743820749111754320727153728139716409114011E-14))) * x + float_type(BOOST_FLOAT128_C(7.64716373181981647590113198578807092707697416852226691068627E-13))) * x + float_type(BOOST_FLOAT128_C(1.14707455977297247138516979786821056670509688396295740818677E-11))) * x + float_type(BOOST_FLOAT128_C(1.60590438368216145993923771701549479323291461578567184216302E-10))) * x + float_type(BOOST_FLOAT128_C(2.08767569878680989792100903212014323125428376052986408239620E-09))) * x + float_type(BOOST_FLOAT128_C(2.50521083854417187750521083854417187750523408006206780016659E-08))) * x + float_type(BOOST_FLOAT128_C(2.75573192239858906525573192239858906525573195144226062684604E-07))) * x + float_type(BOOST_FLOAT128_C(2.75573192239858906525573192239858906525573191310049321957902E-06))) * x + float_type(BOOST_FLOAT128_C(0.00002480158730158730158730158730158730158730158730149317774))) * x + float_type(BOOST_FLOAT128_C(0.00019841269841269841269841269841269841269841269841293575920))) * x + float_type(BOOST_FLOAT128_C(0.00138888888888888888888888888888888888888888888888889071045))) * x + float_type(BOOST_FLOAT128_C(0.00833333333333333333333333333333333333333333333333332986595))) * x + float_type(BOOST_FLOAT128_C(0.04166666666666666666666666666666666666666666666666666664876))) * x + float_type(BOOST_FLOAT128_C(0.16666666666666666666666666666666666666666666666666666669048))) * x + float_type(BOOST_FLOAT128_C(0.50000000000000000000000000000000000000000000000000000000006))) * x + float_type(BOOST_FLOAT128_C(0.99999999999999999999999999999999999999999999999999999999995))) * x); } return sum; } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_EXP (boost::math::cstdfloat::detail::float_internal128_t x) { // Patch the expq() function for a subset of broken GCC compilers // like GCC 4.7, 4.8 on MinGW. // Use an order-36 polynomial approximation of the exponential function // in the range of (-ln2 < x < ln2). Scale the argument to this range // and subsequently multiply the result by 2^n accordingly. // Derive the polynomial coefficients with Mathematica(R) by generating // a table of high-precision values of exp(x) in the range (-ln2 < x < ln2) // and subsequently applying the built-in *Fit* function. // Table[{x, Exp[x] - 1}, {x, -Log[2], Log[2], 1/180}] // N[%, 120] // Fit[%, {x, x^2, x^3, x^4, x^5, x^6, x^7, x^8, x^9, x^10, x^11, x^12, // x^13, x^14, x^15, x^16, x^17, x^18, x^19, x^20, x^21, x^22, // x^23, x^24, x^25, x^26, x^27, x^28, x^29, x^30, x^31, x^32, // x^33, x^34, x^35, x^36}, x] typedef boost::math::cstdfloat::detail::float_internal128_t float_type; // Scale the argument x to the range (-ln2 < x < ln2). BOOST_CONSTEXPR_OR_CONST float_type one_over_ln2 = float_type(BOOST_FLOAT128_C(1.44269504088896340735992468100189213742664595415299)); const float_type x_over_ln2 = x * one_over_ln2; boost::int_fast32_t n; if(x != x) { // The argument is NaN. return std::numeric_limits<float_type>::quiet_NaN(); } else if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) > BOOST_FLOAT128_C(+0.693147180559945309417232121458176568075500134360255)) { // The absolute value of the argument exceeds ln2. n = static_cast<boost::int_fast32_t>(::BOOST_CSTDFLOAT_FLOAT128_FLOOR(x_over_ln2)); } else if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) < BOOST_FLOAT128_C(+0.693147180559945309417232121458176568075500134360255)) { // The absolute value of the argument is less than ln2. n = static_cast<boost::int_fast32_t>(0); } else { // The absolute value of the argument is exactly equal to ln2 (in the sense of floating-point equality). return float_type(2); } // Check if the argument is very near an integer. const float_type floor_of_x = ::BOOST_CSTDFLOAT_FLOAT128_FLOOR(x); if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x - floor_of_x) < float_type(BOOST_CSTDFLOAT_FLOAT128_EPS)) { // Return e^n for arguments very near an integer. return boost::math::cstdfloat::detail::pown(BOOST_FLOAT128_C(2.71828182845904523536028747135266249775724709369996), static_cast<boost::int_fast32_t>(floor_of_x)); } // Compute the scaled argument alpha. const float_type alpha = x - (n * BOOST_FLOAT128_C(0.693147180559945309417232121458176568075500134360255)); // Compute the polynomial approximation of expm1(alpha) and add to it // in order to obtain the scaled result. const float_type scaled_result = ::BOOST_CSTDFLOAT_FLOAT128_EXPM1(alpha) + float_type(1); // Rescale the result and return it. return scaled_result * boost::math::cstdfloat::detail::pown(float_type(2), n); } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_SINH (boost::math::cstdfloat::detail::float_internal128_t x) { // Patch the sinhq() function for a subset of broken GCC compilers // like GCC 4.7, 4.8 on MinGW. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; // Here, we use the following: // Set: ex = exp(x) // Set: em1 = expm1(x) // Then // sinh(x) = (ex - 1/ex) / 2 ; for |x| >= 1 // sinh(x) = (2em1 + em1^2) / (2ex) ; for |x| < 1 const float_type ex = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x); if(::BOOST_CSTDFLOAT_FLOAT128_FABS(x) < float_type(+1)) { const float_type em1 = ::BOOST_CSTDFLOAT_FLOAT128_EXPM1(x); return ((em1 * 2) + (em1 * em1)) / (ex * 2); } else { return (ex - (float_type(1) / ex)) / 2; } } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_COSH (boost::math::cstdfloat::detail::float_internal128_t x) { // Patch the coshq() function for a subset of broken GCC compilers // like GCC 4.7, 4.8 on MinGW. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; const float_type ex = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x); return (ex + (float_type(1) / ex)) / 2; } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TANH (boost::math::cstdfloat::detail::float_internal128_t x) { // Patch the tanhq() function for a subset of broken GCC compilers // like GCC 4.7, 4.8 on MinGW. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; const float_type ex_plus = ::BOOST_CSTDFLOAT_FLOAT128_EXP(x); const float_type ex_minus = (float_type(1) / ex_plus); return (ex_plus - ex_minus) / (ex_plus + ex_minus); } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ASINH(boost::math::cstdfloat::detail::float_internal128_t x) throw() { // Patch the asinh() function since quadmath does not have it. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; return ::BOOST_CSTDFLOAT_FLOAT128_LOG(x + ::BOOST_CSTDFLOAT_FLOAT128_SQRT((x * x) + float_type(1))); } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ACOSH(boost::math::cstdfloat::detail::float_internal128_t x) throw() { // Patch the acosh() function since quadmath does not have it. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; const float_type zp(x + float_type(1)); const float_type zm(x - float_type(1)); return ::BOOST_CSTDFLOAT_FLOAT128_LOG(x + (zp * ::BOOST_CSTDFLOAT_FLOAT128_SQRT(zm / zp))); } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_ATANH(boost::math::cstdfloat::detail::float_internal128_t x) throw() { // Patch the atanh() function since quadmath does not have it. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; return ( ::BOOST_CSTDFLOAT_FLOAT128_LOG(float_type(1) + x) - ::BOOST_CSTDFLOAT_FLOAT128_LOG(float_type(1) - x)) / 2; } inline boost::math::cstdfloat::detail::float_internal128_t BOOST_CSTDFLOAT_FLOAT128_TGAMMA(boost::math::cstdfloat::detail::float_internal128_t x) throw() { // Patch the tgammaq() function for a subset of broken GCC compilers // like GCC 4.7, 4.8 on MinGW. typedef boost::math::cstdfloat::detail::float_internal128_t float_type; if(x > float_type(0)) { return ::BOOST_CSTDFLOAT_FLOAT128_EXP(::BOOST_CSTDFLOAT_FLOAT128_LGAMMA(x)); } else if(x < float_type(0)) { // For x < 0, compute tgamma(-x) and use the reflection formula. const float_type positive_x = -x; float_type gamma_value = ::BOOST_CSTDFLOAT_FLOAT128_TGAMMA(positive_x); const float_type floor_of_positive_x = ::BOOST_CSTDFLOAT_FLOAT128_FLOOR (positive_x); // Take the reflection checks (slightly adapted) from <boost/math/gamma.hpp>. const bool floor_of_z_is_equal_to_z = (positive_x == ::BOOST_CSTDFLOAT_FLOAT128_FLOOR(positive_x)); BOOST_CONSTEXPR_OR_CONST float_type my_pi = BOOST_FLOAT128_C(3.14159265358979323846264338327950288419716939937511); if(floor_of_z_is_equal_to_z) { const bool is_odd = ((boost::int32_t(floor_of_positive_x) % boost::int32_t(2)) != boost::int32_t(0)); return (is_odd ? -std::numeric_limits<float_type>::infinity() : +std::numeric_limits<float_type>::infinity()); } const float_type sinpx_value = x * ::BOOST_CSTDFLOAT_FLOAT128_SIN(my_pi * x); gamma_value *= sinpx_value; const bool result_is_too_large_to_represent = ( (::BOOST_CSTDFLOAT_FLOAT128_FABS(gamma_value) < float_type(1)) && (((std::numeric_limits<float_type>::max)() * ::BOOST_CSTDFLOAT_FLOAT128_FABS(gamma_value)) < my_pi)); if(result_is_too_large_to_represent) { const bool is_odd = ((boost::int32_t(floor_of_positive_x) % boost::int32_t(2)) != boost::int32_t(0)); return (is_odd ? -std::numeric_limits<float_type>::infinity() : +std::numeric_limits<float_type>::infinity()); } gamma_value = -my_pi / gamma_value; if((gamma_value > float_type(0)) || (gamma_value < float_type(0))) { return gamma_value; } else { // The value of gamma is too small to represent. Return 0.0 here. return float_type(0); } } else { // Gamma of zero is complex infinity. Return NaN here. return std::numeric_limits<float_type>::quiet_NaN(); } } #endif // BOOST_CSTDFLOAT_BROKEN_FLOAT128_MATH_FUNCTIONS // Define the quadruple-precision <cmath> functions in the namespace boost::math::cstdfloat::detail. namespace boost { namespace math { namespace cstdfloat { namespace detail { inline boost::math::cstdfloat::detail::float_internal128_t ldexp (boost::math::cstdfloat::detail::float_internal128_t x, int n) { return ::BOOST_CSTDFLOAT_FLOAT128_LDEXP (x, n); } inline boost::math::cstdfloat::detail::float_internal128_t frexp (boost::math::cstdfloat::detail::float_internal128_t x, int* pn) { return ::BOOST_CSTDFLOAT_FLOAT128_FREXP (x, pn); } inline boost::math::cstdfloat::detail::float_internal128_t fabs (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_FABS (x); } inline boost::math::cstdfloat::detail::float_internal128_t abs (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_FABS (x); } inline boost::math::cstdfloat::detail::float_internal128_t floor (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_FLOOR (x); } inline boost::math::cstdfloat::detail::float_internal128_t ceil (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_CEIL (x); } inline boost::math::cstdfloat::detail::float_internal128_t sqrt (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_SQRT (x); } inline boost::math::cstdfloat::detail::float_internal128_t trunc (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_TRUNC (x); } inline boost::math::cstdfloat::detail::float_internal128_t exp (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_EXP (x); } inline boost::math::cstdfloat::detail::float_internal128_t pow (boost::math::cstdfloat::detail::float_internal128_t x, boost::math::cstdfloat::detail::float_internal128_t a) { return ::BOOST_CSTDFLOAT_FLOAT128_POW (x, a); } inline boost::math::cstdfloat::detail::float_internal128_t pow (boost::math::cstdfloat::detail::float_internal128_t x, int a) { return ::BOOST_CSTDFLOAT_FLOAT128_POW (x, boost::math::cstdfloat::detail::float_internal128_t(a)); } inline boost::math::cstdfloat::detail::float_internal128_t log (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_LOG (x); } inline boost::math::cstdfloat::detail::float_internal128_t log10 (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_LOG10 (x); } inline boost::math::cstdfloat::detail::float_internal128_t sin (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_SIN (x); } inline boost::math::cstdfloat::detail::float_internal128_t cos (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_COS (x); } inline boost::math::cstdfloat::detail::float_internal128_t tan (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_TAN (x); } inline boost::math::cstdfloat::detail::float_internal128_t asin (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ASIN (x); } inline boost::math::cstdfloat::detail::float_internal128_t acos (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ACOS (x); } inline boost::math::cstdfloat::detail::float_internal128_t atan (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ATAN (x); } inline boost::math::cstdfloat::detail::float_internal128_t sinh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_SINH (x); } inline boost::math::cstdfloat::detail::float_internal128_t cosh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_COSH (x); } inline boost::math::cstdfloat::detail::float_internal128_t tanh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_TANH (x); } inline boost::math::cstdfloat::detail::float_internal128_t asinh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ASINH (x); } inline boost::math::cstdfloat::detail::float_internal128_t acosh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ACOSH (x); } inline boost::math::cstdfloat::detail::float_internal128_t atanh (boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ATANH (x); } inline boost::math::cstdfloat::detail::float_internal128_t fmod (boost::math::cstdfloat::detail::float_internal128_t a, boost::math::cstdfloat::detail::float_internal128_t b) { return ::BOOST_CSTDFLOAT_FLOAT128_FMOD (a, b); } inline boost::math::cstdfloat::detail::float_internal128_t atan2 (boost::math::cstdfloat::detail::float_internal128_t y, boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_ATAN2 (y, x); } inline boost::math::cstdfloat::detail::float_internal128_t lgamma(boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_LGAMMA(x); } inline boost::math::cstdfloat::detail::float_internal128_t tgamma(boost::math::cstdfloat::detail::float_internal128_t x) { return ::BOOST_CSTDFLOAT_FLOAT128_TGAMMA(x); } } } } } // boost::math::cstdfloat::detail // We will now inject the quadruple-precision <cmath> functions // into the std namespace. This is done via *using* directive. namespace std { using boost::math::cstdfloat::detail::ldexp; using boost::math::cstdfloat::detail::frexp; using boost::math::cstdfloat::detail::fabs; using boost::math::cstdfloat::detail::abs; using boost::math::cstdfloat::detail::floor; using boost::math::cstdfloat::detail::ceil; using boost::math::cstdfloat::detail::sqrt; using boost::math::cstdfloat::detail::trunc; using boost::math::cstdfloat::detail::exp; using boost::math::cstdfloat::detail::pow; using boost::math::cstdfloat::detail::log; using boost::math::cstdfloat::detail::log10; using boost::math::cstdfloat::detail::sin; using boost::math::cstdfloat::detail::cos; using boost::math::cstdfloat::detail::tan; using boost::math::cstdfloat::detail::asin; using boost::math::cstdfloat::detail::acos; using boost::math::cstdfloat::detail::atan; using boost::math::cstdfloat::detail::sinh; using boost::math::cstdfloat::detail::cosh; using boost::math::cstdfloat::detail::tanh; using boost::math::cstdfloat::detail::asinh; using boost::math::cstdfloat::detail::acosh; using boost::math::cstdfloat::detail::atanh; using boost::math::cstdfloat::detail::fmod; using boost::math::cstdfloat::detail::atan2; using boost::math::cstdfloat::detail::lgamma; using boost::math::cstdfloat::detail::tgamma; } // namespace std // We will now remove the preprocessor symbols representing quadruple-precision <cmath> // functions from the preprocessor. #undef BOOST_CSTDFLOAT_FLOAT128_LDEXP #undef BOOST_CSTDFLOAT_FLOAT128_FREXP #undef BOOST_CSTDFLOAT_FLOAT128_FABS #undef BOOST_CSTDFLOAT_FLOAT128_FLOOR #undef BOOST_CSTDFLOAT_FLOAT128_CEIL #undef BOOST_CSTDFLOAT_FLOAT128_SQRT #undef BOOST_CSTDFLOAT_FLOAT128_TRUNC #undef BOOST_CSTDFLOAT_FLOAT128_EXP #undef BOOST_CSTDFLOAT_FLOAT128_EXPM1 #undef BOOST_CSTDFLOAT_FLOAT128_POW #undef BOOST_CSTDFLOAT_FLOAT128_LOG #undef BOOST_CSTDFLOAT_FLOAT128_LOG10 #undef BOOST_CSTDFLOAT_FLOAT128_SIN #undef BOOST_CSTDFLOAT_FLOAT128_COS #undef BOOST_CSTDFLOAT_FLOAT128_TAN #undef BOOST_CSTDFLOAT_FLOAT128_ASIN #undef BOOST_CSTDFLOAT_FLOAT128_ACOS #undef BOOST_CSTDFLOAT_FLOAT128_ATAN #undef BOOST_CSTDFLOAT_FLOAT128_SINH #undef BOOST_CSTDFLOAT_FLOAT128_COSH #undef BOOST_CSTDFLOAT_FLOAT128_TANH #undef BOOST_CSTDFLOAT_FLOAT128_ASINH #undef BOOST_CSTDFLOAT_FLOAT128_ACOSH #undef BOOST_CSTDFLOAT_FLOAT128_ATANH #undef BOOST_CSTDFLOAT_FLOAT128_FMOD #undef BOOST_CSTDFLOAT_FLOAT128_ATAN2 #undef BOOST_CSTDFLOAT_FLOAT128_LGAMMA #undef BOOST_CSTDFLOAT_FLOAT128_TGAMMA #endif // Not BOOST_CSTDFLOAT_NO_LIBQUADMATH_SUPPORT (i.e., the user would like to have libquadmath support) #endif // _BOOST_CSTDFLOAT_CMATH_2014_02_15_HPP_