2172 lines · plain
1// Copyright John Maddock 2017.2// Copyright Paul A. Bristow 2016, 2017, 2018.3// Copyright Nicholas Thompson 20184 5// Distributed under the Boost Software License, Version 1.0.6// (See accompanying file LICENSE_1_0.txt or7// copy at http ://www.boost.org/LICENSE_1_0.txt).8 9#ifndef BOOST_MATH_SF_LAMBERT_W_HPP10#define BOOST_MATH_SF_LAMBERT_W_HPP11 12#ifdef _MSC_VER13#pragma warning(disable : 4127)14#endif15 16/*17Implementation of an algorithm for the Lambert W0 and W-1 real-only functions.18 19This code is based in part on the algorithm by20Toshio Fukushima,21"Precise and fast computation of Lambert W-functions without transcendental function evaluations",22J.Comp.Appl.Math. 244 (2013) 77-89,23and on a C/C++ version by Darko Veberic, darko.veberic@ijs.si24based on the Fukushima algorithm and Toshio Fukushima's FORTRAN version of his algorithm.25 26First derivative of Lambert_w is derived from27Princeton Companion to Applied Mathematics, 'The Lambert-W function', Section 1.3: Series and Generating Functions.28 29*/30 31/*32TODO revise this list of macros.33Some macros that will show some (or much) diagnostic values if #defined.34//[boost_math_instrument_lambert_w_macros35 36// #define-able macros37BOOST_MATH_INSTRUMENT_LAMBERT_W_HALLEY // Halley refinement diagnostics.38BOOST_MATH_INSTRUMENT_LAMBERT_W_PRECISION // Precision.39BOOST_MATH_INSTRUMENT_LAMBERT_WM1 // W1 branch diagnostics.40BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY // Halley refinement diagnostics only for W-1 branch.41BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY // K > 64, z > -1.0264389699511303e-2642BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP // Show results from W-1 lookup table.43BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER // Schroeder refinement diagnostics.44BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS // Number of terms used for near-singularity series.45BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES // Show evaluation of series near branch singularity.46BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES47BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS // Show evaluation of series for small z.48//] [/boost_math_instrument_lambert_w_macros]49*/50 51#include <boost/math/tools/config.hpp>52#include <boost/math/policies/error_handling.hpp>53#include <boost/math/policies/policy.hpp>54#include <boost/math/tools/promotion.hpp>55#include <boost/math/special_functions/fpclassify.hpp>56#include <boost/math/special_functions/log1p.hpp> // for log (1 + x)57#include <boost/math/constants/constants.hpp> // For exp_minus_one == 3.67879441171442321595523770161460867e-01.58#include <boost/math/special_functions/next.hpp> // for has_denorm_now59#include <boost/math/special_functions/pow.hpp> // powers with compile time exponent, used in arbitrary precision code.60#include <boost/math/tools/series.hpp> // series functor.61//#include <boost/math/tools/polynomial.hpp> // polynomial.62#include <boost/math/tools/rational.hpp> // evaluate_polynomial.63#include <boost/math/tools/precision.hpp> // boost::math::tools::max_value().64#include <boost/math/tools/big_constant.hpp>65#include <boost/math/tools/cxx03_warn.hpp>66 67#ifndef BOOST_MATH_STANDALONE68#include <boost/lexical_cast.hpp>69#endif70 71#include <limits>72#include <cmath>73#include <type_traits>74#include <cstdint>75 76// Needed for testing and diagnostics only.77//#include <iostream>78//#include <typeinfo>79#include <boost/math/special_functions/next.hpp> // For float_distance.80 81using lookup_t = double; // Type for lookup table (double or float, or even long double?)82 83//#include "J:\Cpp\Misc\lambert_w_lookup_table_generator\lambert_w_lookup_table.ipp"84// #include "lambert_w_lookup_table.ipp" // Boost.Math version.85#include <boost/math/special_functions/detail/lambert_w_lookup_table.ipp>86 87#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)88//89// This is the only way we can avoid90// warning: non-standard suffix on floating constant [-Wpedantic]91// when building with -Wall -pedantic. Neither __extension__92// nor #pragma diagnostic ignored work :(93//94#pragma GCC system_header95#endif96 97namespace boost {98namespace math {99namespace lambert_w_detail {100 101//! \brief Applies a single Halley step to make a better estimate of Lambert W.102//! \details Used the simplified formulae obtained from103//! http://www.wolframalpha.com/input/?i=%5B2(z+exp(z)-w)+d%2Fdx+(z+exp(z)-w)%5D+%2F+%5B2+(d%2Fdx+(z+exp(z)-w))%5E2+-+(z+exp(z)-w)+d%5E2%2Fdx%5E2+(z+exp(z)-w)%5D104//! [2(z exp(z)-w) d/dx (z exp(z)-w)] / [2 (d/dx (z exp(z)-w))^2 - (z exp(z)-w) d^2/dx^2 (z exp(z)-w)]105 106//! \tparam T floating-point (or fixed-point) type.107//! \param w_est Lambert W estimate.108//! \param z Argument z for Lambert_w function.109//! \returns New estimate of Lambert W, hopefully improved.110//!111template <typename T>112inline T lambert_w_halley_step(T w_est, const T z)113{114 BOOST_MATH_STD_USING115 T e = exp(w_est);116 w_est -= 2 * (w_est + 1) * (e * w_est - z) / (z * (w_est + 2) + e * (w_est * (w_est + 2) + 2));117 return w_est;118} // template <typename T> lambert_w_halley_step(T w_est, T z)119 120//! \brief Halley iterate to refine Lambert_w estimate,121//! taking at least one Halley_step.122//! Repeat Halley steps until the *last step* had fewer than half the digits wrong,123//! the step we've just taken should have been sufficient to have completed the iteration.124 125//! \tparam T floating-point (or fixed-point) type.126//! \param z Argument z for Lambert_w function.127//! \param w_est Lambert w estimate.128template <typename T>129inline T lambert_w_halley_iterate(T w_est, const T z)130{131 BOOST_MATH_STD_USING132 static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est);133 134 T w_new = lambert_w_halley_step(w_est, z);135 T diff = fabs(w_est - w_new);136 while (diff > max_diff)137 {138 w_est = w_new;139 w_new = lambert_w_halley_step(w_est, z);140 diff = fabs(w_est - w_new);141 }142 return w_new;143} // template <typename T> lambert_w_halley_iterate(T w_est, T z)144 145// Two Halley function versions that either146// single step (if std::false_type) or iterate (if std::true_type).147// Selected at compile-time using parameter 3.148template <typename T>149inline T lambert_w_maybe_halley_iterate(T z, T w, std::false_type const&)150{151 return lambert_w_halley_step(z, w); // Single step.152}153 154template <typename T>155inline T lambert_w_maybe_halley_iterate(T z, T w, std::true_type const&)156{157 return lambert_w_halley_iterate(z, w); // Iterate steps.158}159 160//! maybe_reduce_to_double function,161//! Two versions that have a compile-time option to162//! reduce argument z to double precision (if true_type).163//! Version is selected at compile-time using parameter 2.164 165template <typename T>166inline double maybe_reduce_to_double(const T& z, const std::true_type&)167{168 return static_cast<double>(z); // Reduce to double precision.169}170 171template <typename T>172inline T maybe_reduce_to_double(const T& z, const std::false_type&)173{ // Don't reduce to double.174 return z;175}176 177template <typename T>178inline double must_reduce_to_double(const T& z, const std::true_type&)179{180 return static_cast<double>(z); // Reduce to double precision.181}182 183template <typename T>184inline double must_reduce_to_double(const T& z, const std::false_type&)185{ // try a lexical_cast and hope for the best:186#ifndef BOOST_MATH_STANDALONE187 188 #ifdef BOOST_MATH_USE_CHARCONV_FOR_CONVERSION189 190 // Catches the C++23 floating point types191 if constexpr (std::is_arithmetic_v<T>)192 {193 return static_cast<double>(z);194 }195 else196 {197 return boost::lexical_cast<double>(z);198 }199 200 #else201 202 return boost::lexical_cast<double>(z);203 204 #endif205 206#else207 static_assert(sizeof(T) == 0, "Unsupported in standalone mode: don't know how to cast your number type to a double.");208 return 0.0;209#endif210}211 212//! \brief Schroeder method, fifth-order update formula,213//! \details See T. Fukushima page 80-81, and214//! A. Householder, The Numerical Treatment of a Single Nonlinear Equation,215//! McGraw-Hill, New York, 1970, section 4.4.216//! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections,217//! chosen to ensure that the result will be achieve the +/- 10 epsilon target.218//! \param w Lambert w estimate from bisection or series.219//! \param y bracketing value from bisection.220//! \returns Refined estimate of Lambert w.221 222// Schroeder refinement, called unless NOT required by precision policy.223template<typename T>224inline T schroeder_update(const T w, const T y)225{226 // Compute derivatives using 5th order Schroeder refinement.227 // Since this is the final step, it will always use the highest precision type T.228 // Example of Call:229 // result = schroeder_update(w, y);230 //where231 // w is estimate of Lambert W (from bisection or series).232 // y is z * e^-w.233 234 BOOST_MATH_STD_USING // Aid argument dependent lookup of abs.235#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER236 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);237 using boost::math::float_distance;238 T fd = float_distance<T>(w, y);239 std::cout << "Schroder ";240 if (abs(fd) < 214748000.)241 {242 std::cout << " Distance = "<< static_cast<int>(fd);243 }244 else245 {246 std::cout << "Difference w - y = " << (w - y) << ".";247 }248 std::cout << std::endl;249#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER250 // Fukushima equation 18, page 6.251 const T f0 = w - y; // f0 = w - y.252 const T f1 = 1 + y; // f1 = df/dW253 const T f00 = f0 * f0;254 const T f11 = f1 * f1;255 const T f0y = f0 * y;256 const T result =257 w - 4 * f0 * (6 * f1 * (f11 + f0y) + f00 * y) /258 (f11 * (24 * f11 + 36 * f0y) +259 f00 * (6 * y * y + 8 * f1 * y + f0y)); // Fukushima Page 81, equation 21 from equation 20.260 261#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER262 std::cout << "Schroeder refined " << w << " " << y << ", difference " << w-y << ", change " << w - result << ", to result " << result << std::endl;263 std::cout.precision(saved_precision); // Restore.264#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER265 266 return result;267} // template<typename T = double> T schroeder_update(const T w, const T y)268 269 //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944.270 //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]]271 //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was272 //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50]273 //! -1+p-p^2/3+(11 p^3)/72-(43 p^4)/540+(769 p^5)/17280-(221 p^6)/8505+(680863 p^7)/43545600 ...274 //! Decimal values of specifications for built-in floating-point types below275 //! are at least 21 digits precision == max_digits10 for long double.276 //! Longer decimal digits strings are rationals evaluated using Wolfram.277 278template<typename T>279T lambert_w_singularity_series(const T p)280{281#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES282 std::size_t saved_precision = std::cout.precision(3);283 std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl;284 std::cout285 //<< "Argument Type = " << typeid(T).name()286 //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10287 //<< ", epsilon = " << std::numeric_limits<T>::epsilon()288 << std::endl;289 std::cout.precision(saved_precision);290#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES291 292 static const T q[] =293 {294 -static_cast<T>(1), // j0295 +T(1), // j1296 -T(1) / 3, // 1/3 j2297 +T(11) / 72, // 0.152777777777777778, // 11/72 j3298 -T(43) / 540, // 0.0796296296296296296, // 43/540 j4299 +T(769) / 17280, // 0.0445023148148148148, j5300 -T(221) / 8505, // 0.0259847148736037625, j6301 //+T(0.0156356325323339212L), // j7302 //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50]303 +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7304 //-T(0.00961689202429943171L), // j8305 -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8306 //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50]307 +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9308 -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10309 //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550310 +T(169709463197uLL) / 69528040243200uLL, // j11311 // -T(0.00157693034468678425L), // j12 -0.0015769303446867842539234095399314115973161850314723312 -T(1118511313uLL) / 709296588000uLL, // j12313 +T(667874164916771uLL) / 650782456676352000uLL, // j13314 //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973315 -T(500525573uLL) / 744761417400uLL, // j14316 // -T(0.000672061631156136204L), j14317 //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big318 //+T(0.000442473061814620910L, // j15319 BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15320 // -T(0.000292677224729627445L), // j16321 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16322 //+T(0.000194387276054539318L), // j17323 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17324 //-T(0.000129574266852748819L), // j18325 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18326 //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288327 BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19328 //-T(0.0000581136075044138168L) // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913329 // -T(2853534237182741069uLL) / 49102686267859224000000uLL // j20 // error C2177: constant too big,330 // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others.331 //-T(0.000058113607504413816772205464778828177256611844221913L), // j20 N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913332 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20 - last used by Fukushima333 // More terms don't seem to give any improvement (worse in fact) and are not use for many z values.334 //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21335 //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22336 //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23337 //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24338 //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25339 //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26340 // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26341 // 21 to 26 Added for long double.342 }; // static const T q[]343 344 /*345 // Temporary copy of original double values for comparison; these are reproduced well.346 static const T q[] =347 {348 -1L, // j0349 +1L, // j1350 -0.333333333333333333L, // 1/3 j2351 +0.152777777777777778L, // 11/72 j3352 -0.0796296296296296296L, // 43/540353 +0.0445023148148148148L,354 -0.0259847148736037625L,355 +0.0156356325323339212L,356 -0.00961689202429943171L,357 +0.00601454325295611786L,358 -0.00381129803489199923L,359 +0.00244087799114398267L,360 -0.00157693034468678425L,361 +0.00102626332050760715L,362 -0.000672061631156136204L,363 +0.000442473061814620910L,364 -0.000292677224729627445L,365 +0.000194387276054539318L,366 -0.000129574266852748819L,367 +0.0000866503580520812717L,368 -0.0000581136075044138168L // j20369 };370 */371 372 // Decide how many series terms to use, increasing as z approaches the singularity,373 // balancing run-time versus computational noise from round-off.374 // In practice, we truncate the series expansion at a certain order.375 // If the order is too large, not only does the amount of computation increase,376 // but also the round-off errors accumulate.377 // See Fukushima equation 35, page 85 for logic of choice of number of series terms.378 379 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.380 381 const T absp = abs(p);382 383#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS384 {385 int terms = 20; // Default to using all terms.386 if (absp < 0.01159)387 { // Very near singularity.388 terms = 6;389 }390 else if (absp < 0.0766)391 { // Near singularity.392 terms = 10;393 }394 std::streamsize saved_precision = std::cout.precision(3);395 std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl;396 std::cout.precision(saved_precision);397 }398#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS399 400 if (absp < T(0.01159))401 { // Only 6 near-singularity series terms are useful.402 return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * q[6])))));403 }404 else if (absp < T(0.0766)) // Use 10 near-singularity series terms.405 { // Use 10 near-singularity series terms.406 return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * (q[6] + p * (q[7] + p * (q[8] + p * (q[9] + p * q[10])))))))));407 }408 // Use all 20 near-singularity series terms.409 return -1 + p * (1 + p * (q[2] + p * (q[3] + p * (q[4] + p * (q[5] + p * (q[6] + p * (q[7] + p * (q[8] + p * (q[9] + p * (q[10] + p * (q[11] + p * (q[12] + p * (q[13] + p * (q[14] + p * (q[15] + p * (q[16] + p * (q[17] + p * (q[18] + p * (q[19] + p * q[20] /* Last Fukushima term.*/)))))))))))))))))));410 // + // more terms for more precise T: long double ...411 //// but makes almost no difference, so don't use more terms?412 // p*q[21] +413 // p*q[22] +414 // p*q[23] +415 // p*q[24] +416 // p*q[25]417 // )))))))))))))))))));418 419} // template<typename T = double> T lambert_w_singularity_series(const T p)420 421 422 /////////////////////////////////////////////////////////////////////////////////////////////423 424 //! \brief Series expansion used near zero (abs(z) < 0.05).425 //! \details426 //! Coefficients of the inverted series expansion of the Lambert W function around z = 0.427 //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with428 //! InverseSeries[Series[z Exp[z],{z,0,17}]]429 //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86.430 431 //! Decimal values of specifications for built-in floating-point types below432 //! are 21 digits precision == max_digits10 for long double.433 //! Care! Some coefficients might overflow some fixed_point types.434 435 //! This version is intended to allow use by user-defined types436 //! like Boost.Multiprecision quad and cpp_dec_float types.437 //! The three specializations below for built-in float, double438 //! (and perhaps long double) will be chosen in preference for these types.439 440 //! This version uses rationals computed by Wolfram as far as possible,441 //! limited by maximum size of uLL integers.442 //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals,443 //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term444 //! until the precision required by the policy is achieved.445 //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed.446 447 // Series evaluation for LambertW(z) as z -> 0.448 // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/449 // http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif450 451 //! \brief lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type.452 //! The Lambert W is computed by lambert_w0_small_z for small z.453 //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05),454 //! but the optimum might be a function of the size of the type of z.455 456 //! \details457 //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10.458 //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits,459 //! and also compilers that have a float type using 64 bits and/or long double using 128-bits.460 //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection.461 //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose.462 //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit.463 //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10()464 //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent.465 //! So must rely on @c std::numeric_limits<long double>::max_digits10.466 467 //! Specialization of float zero series expansion used for small z (abs(z) < 0.05).468 //! Specializations of lambert_w0_small_z for built-in types.469 //! These specializations should be chosen in preference to T version.470 //! For example: lambert_w0_small_z(0.001F) should use the float version.471 //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation,472 //! but for the tag_type selection to work, they all must include Policy in their signature.473 474 // Forward declaration of variants of lambert_w0_small_z.475template <typename T, typename Policy>476T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 0> const&); // for float (32-bit) type.477 478template <typename T, typename Policy>479T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 1> const&); // for double (64-bit) type.480 481template <typename T, typename Policy>482T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 2> const&); // for long double (double extended 80-bit) type.483 484template <typename T, typename Policy>485T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 3> const&); // for long double (128-bit) type.486 487template <typename T, typename Policy>488T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 4> const&); // for float128 quadmath Q type.489 490template <typename T, typename Policy>491T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 5> const&); // Generic multiprecision T.492 // Set tag_type depending on max_digits10.493template <typename T, typename Policy>494T lambert_w0_small_z(T x, const Policy& pol)495{ //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double.496 using tag_type = std::integral_constant<int,497 std::numeric_limits<T>::is_specialized == 0 ? 5 :498#ifndef BOOST_NO_CXX11_NUMERIC_LIMITS499 std::numeric_limits<T>::max_digits10 <= 9 ? 0 : // for float 32-bit.500 std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit.501 std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended.502 std::numeric_limits<T>::max_digits10 < 37 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).503#else504 std::numeric_limits<T>::radix != 2 ? 5 :505 std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit.506 std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit.507 std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended.508 std::numeric_limits<T>::digits <= 113 ? 4 // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).509#endif510 : 5>; // All Generic multiprecision types.511 // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression.512 return lambert_w0_small_z(x, pol, tag_type());513} // template <typename T> T lambert_w0_small_z(T x)514 515 //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05).516 // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms.517 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction518 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],519 // as proposed by Tosio Fukushima and implemented by Darko Veberic.520 521template <typename T, typename Policy>522T lambert_w0_small_z(T z, const Policy&, std::integral_constant<int, 0> const&)523{524#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES525 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.526 std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision "527 << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl;528#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES529 T result =530 z * (1 - // j1 z^1 term = 1531 z * (1 - // j2 z^2 term = -1532 z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5.533 z * (2.6666666666666666667F - // 8/3 // j4534 z * (5.2083333333333333333F - // -125/24 // j5535 z * (10.8F - // j6536 z * (23.343055555555555556F - // j7537 z * (52.012698412698412698F - // j8538 z * 118.62522321428571429F)))))))); // j9539 540#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES541 std::cout << "return w = " << result << std::endl;542 std::cout.precision(prec); // Restore.543#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES544 545 return result;546} // template <typename T> T lambert_w0_small_z(T x, std::integral_constant<int, 0> const&)547 548 //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05).549 // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms.550 // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction551 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic.552 553template <typename T, typename Policy>554T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 1> const&)555{556#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES557 std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.558 std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "559 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;560#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES561 T result =562 z * (1. - // j1 z^1563 z * (1. - // j2 z^2564 z * (1.5 - // 3/2 // j3 z^3565 z * (2.6666666666666666667 - // 8/3 // j4566 z * (5.2083333333333333333 - // -125/24 // j5567 z * (10.8 - // j6568 z * (23.343055555555555556 - // j7569 z * (52.012698412698412698 - // j8570 z * (118.62522321428571429 - // j9571 z * (275.57319223985890653 - // j10572 z * (649.78717234347442681 - // j11573 z * (1551.1605194805194805 - // j12574 z * (3741.4497029592385495 - // j13575 z * (9104.5002411580189358 - // j14576 z * (22324.308512706601434 - // j15577 z * (55103.621972903835338 - // j16578 z * 136808.86090394293563)))))))))))))))); // j17 z^17579 580#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES581 std::cout << "return w = " << result << std::endl;582 std::cout.precision(prec); // Restore.583#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES584 585 return result;586} // T lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)587 588 //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05).589 // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some590 // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default).591 // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type.592 // Nor used for 128-bit float128.)593template <typename T, typename Policy>594T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 2> const&)595{596#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES597 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.598 std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, "599 << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl;600#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES601// T result =602// z * (1.L - // j1 z^1603// z * (1.L - // j2 z^2604// z * (1.5L - // 3/2 // j3605// z * (2.6666666666666666667L - // 8/3 // j4606// z * (5.2083333333333333333L - // -125/24 // j5607// z * (10.800000000000000000L - // j6608// z * (23.343055555555555556L - // j7609// z * (52.012698412698412698L - // j8610// z * (118.62522321428571429L - // j9611// z * (275.57319223985890653L - // j10612// z * (649.78717234347442681L - // j11613// z * (1551.1605194805194805L - // j12614// z * (3741.4497029592385495L - // j13615// z * (9104.5002411580189358L - // j14616// z * (22324.308512706601434L - // j15617// z * (55103.621972903835338L - // j16618// z * (136808.86090394293563L - // j17 z^17 last term used by Fukushima double.619// z * (341422.050665838363317L - // z^18620// z * (855992.9659966075514633L - // z^19621// z * (2.154990206091088289321e6L - // z^20622// z * 5.4455529223144624316423e6L // z^21623// ))))))))))))))))))));624//625 626 T result =627z * (1.L - // z j1628z * (1.L - // z^2629z * (1.500000000000000000000000000000000L - // z^3630z * (2.666666666666666666666666666666666L - // z ^ 4631z * (5.208333333333333333333333333333333L - // z ^ 5632z * (10.80000000000000000000000000000000L - // z ^ 6633z * (23.34305555555555555555555555555555L - // z ^ 7634z * (52.01269841269841269841269841269841L - // z ^ 8635z * (118.6252232142857142857142857142857L - // z ^ 9636z * (275.5731922398589065255731922398589L - // z ^ 10637z * (649.7871723434744268077601410934744L - // z ^ 11638z * (1551.160519480519480519480519480519L - // z ^ 12639z * (3741.449702959238549516327294105071L - //z ^ 13640z * (9104.500241158018935796713574491352L - // z ^ 14641z * (22324.308512706601434280005708577137L - // z ^ 15642z * (55103.621972903835337697771560205422L - // z ^ 16643z * (136808.86090394293563342215789305736L - // z ^ 17644z * (341422.05066583836331735491399356945L - // z^18645z * (855992.9659966075514633630250633224L - // z^19646z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20647))))))))))))))))))));648 649#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES650 std::cout << "return w = " << result << std::endl;651 std::cout.precision(precision); // Restore.652#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES653 return result;654} // long double lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)655 656//! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05).657// 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction658// N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],659// and are suffixed by L as they are assumed of type long double.660// (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q661// nor multiprecision type cpp_bin_float_quad that can only be initialized at full precision of the type662// constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".)663 664template <typename T, typename Policy>665T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 3> const&)666{667#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES668 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.669 std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "670 << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;671#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES672 T result =673 z * (1.L - // j1674 z * (1.L - // j2675 z * (1.5L - // 3/2 // j3676 z * (2.6666666666666666666666666666666666L - // 8/3 // j4677 z * (5.2052083333333333333333333333333333L - // -125/24 // j5678 z * (10.800000000000000000000000000000000L - // j6679 z * (23.343055555555555555555555555555555L - // j7680 z * (52.0126984126984126984126984126984126L - // j8681 z * (118.625223214285714285714285714285714L - // j9682 z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10683 z * (649.78717234347442680776014109347442680776014109347L - // j11684 z * (1551.1605194805194805194805194805194805194805194805L - // j12685 z * (3741.4497029592385495163272941050718828496606274384L - // j13686 z * (9104.5002411580189357967135744913522691300469078247L - // j14687 z * (22324.308512706601434280005708577137148565719994291L - // j15688 z * (55103.621972903835337697771560205422639285073147507L - // j16689 z * 136808.86090394293563342215789305736395683485630576L // j17690 ))))))))))))))));691 692#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES693 std::cout << "return w = " << result << std::endl;694 std::cout.precision(precision); // Restore.695#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES696 return result;697} // T lambert_w0_small_z(const T z, std::integral_constant<int, 3> const&)698 699//! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05).700// 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction701// N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],702// and are suffixed by Q as they are assumed of type quad.703// This could be used for 128-bit quad (which requires a suffix Q for full precision).704// But experiments with GCC 7.2.0 show that while this gives full 128-bit precision705// when the -f-ext-numeric-literals option is in force and the libquadmath library available,706// over the range -0.049 to +0.049,707// it is slightly slower than getting a double approximation followed by a single Halley step.708 709#ifdef BOOST_HAS_FLOAT128710template <typename T, typename Policy>711T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 4> const&)712{713#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES714 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.715 std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, "716 << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl;717#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES718 T result =719 z * (1.Q - // z j1720 z * (1.Q - // z^2721 z * (1.500000000000000000000000000000000Q - // z^3722 z * (2.666666666666666666666666666666666Q - // z ^ 4723 z * (5.208333333333333333333333333333333Q - // z ^ 5724 z * (10.80000000000000000000000000000000Q - // z ^ 6725 z * (23.34305555555555555555555555555555Q - // z ^ 7726 z * (52.01269841269841269841269841269841Q - // z ^ 8727 z * (118.6252232142857142857142857142857Q - // z ^ 9728 z * (275.5731922398589065255731922398589Q - // z ^ 10729 z * (649.7871723434744268077601410934744Q - // z ^ 11730 z * (1551.160519480519480519480519480519Q - // z ^ 12731 z * (3741.449702959238549516327294105071Q - //z ^ 13732 z * (9104.500241158018935796713574491352Q - // z ^ 14733 z * (22324.308512706601434280005708577137Q - // z ^ 15734 z * (55103.621972903835337697771560205422Q - // z ^ 16735 z * (136808.86090394293563342215789305736Q - // z ^ 17736 z * (341422.05066583836331735491399356945Q - // z^18737 z * (855992.9659966075514633630250633224Q - // z^19738 z * (2.154990206091088289321708745358647e6Q - // 20739 z * (5.445552922314462431642316420035073e6Q - // 21740 z * (1.380733000216662949061923813184508e7Q - // 22741 z * (3.511704498513923292853869855945334e7Q - // 23742 z * (8.956800256102797693072819557780090e7Q - // 24743 z * (2.290416846187949813964782641734774e8Q - // 25744 z * (5.871035041171798492020292225245235e8Q - // 26745 z * (1.508256053857792919641317138812957e9Q - // 27746 z * (3.882630161293188940385873468413841e9Q - // 28747 z * (1.001394313665482968013913601565723e10Q - // 29748 z * (2.587356736265760638992878359024929e10Q - // 30749 z * (6.696209709358073856946120522333454e10Q - // 31750 z * (1.735711659599198077777078238043644e11Q - // 32751 z * (4.505680465642353886756098108484670e11Q - // 33752 z * (1.171223178256487391904047636564823e12Q //z^34753 ))))))))))))))))))))))))))))))))));754 755 756 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES757 std::cout << "return w = " << result << std::endl;758 std::cout.precision(precision); // Restore.759#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES760 761 return result;762} // T lambert_w0_small_z(const T z, std::integral_constant<int, 4> const&) float128763 764#else765 766template <typename T, typename Policy>767inline T lambert_w0_small_z(const T z, const Policy& pol, std::integral_constant<int, 4> const&) // LCOV_EXCL_LINE body is covered, strangley this line is not.768{769 return lambert_w0_small_z(z, pol, std::integral_constant<int, 5>());770}771 772#endif // BOOST_HAS_FLOAT128773 774//! Series functor to compute series term using pow and factorial.775//! \details Functor is called after evaluating polynomial with the coefficients as rationals below.776template <typename T>777struct lambert_w0_small_z_series_term778{779 using result_type = T;780 //! \param _z Lambert W argument z.781 //! \param -term -pow<18>(z) / 6402373705728000uLL782 //! \param _k number of terms == initially 18783 784 // Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N.785 786 lambert_w0_small_z_series_term(T _z, T _term, int _k)787 : k(_k), z(_z), term(_term) { }788 789 T operator()()790 { // Called by sum_series until needs precision set by factor (policy::get_epsilon).791 using std::pow;792 ++k;793 term *= -z / k;794 //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k!795 T result = term * pow(T(k), T(-1 + k)); // term * k^(k-1)796 // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl;797 return result; //798 }799private:800 int k;801 T z;802 T term;803}; // template <typename T> struct lambert_w0_small_z_series_term804 805 //! Generic variant for T a User-defined types like Boost.Multiprecision.806template <typename T, typename Policy>807inline T lambert_w0_small_z(T z, const Policy& pol, std::integral_constant<int, 5> const&)808{809#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES810 std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.811 std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl;812 std::cout << "Argument z is of type " << typeid(T).name() << std::endl;813#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES814 815 // First several terms of the series are tabulated and evaluated as a polynomial:816 // this will save us a bunch of expensive calls to pow.817 // Then our series functor is initialized "as if" it had already reached term 18,818 // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types.819 820 // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i].821 static const T coeff[] =822 {823 0, // z^0 Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different!824 1, // z^1 term.825 -1, // z^2 term826 static_cast<T>(3uLL) / 2uLL, // z^3 term.827 -static_cast<T>(8uLL) / 3uLL, // z^4828 static_cast<T>(125uLL) / 24uLL, // z^5829 -static_cast<T>(54uLL) / 5uLL, // z^6830 static_cast<T>(16807uLL) / 720uLL, // z^7831 -static_cast<T>(16384uLL) / 315uLL, // z^8832 static_cast<T>(531441uLL) / 4480uLL, // z^9833 -static_cast<T>(156250uLL) / 567uLL, // z^10834 static_cast<T>(2357947691uLL) / 3628800uLL, // z^11835 -static_cast<T>(2985984uLL) / 1925uLL, // z^12836 static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13837 -static_cast<T>(7909306972uLL) / 868725uLL, // z^14838 static_cast<T>(320361328125uLL) / 14350336uLL, // z^15839 -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16840 static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term841 -static_cast<T>(5083731656658uLL) / 14889875uLL,842 // z^18 term. = 136808.86090394293563342215789305735851647769682393843 844 // z^18 is biggest that can be computed as rational using the largest possible uLL integers,845 // so higher terms cannot be potentially compiler-computed as uLL rationals.846 // Wolfram (5083731656658 z ^ 18) / 14889875 or847 // -341422.05066583836331735491399356945575432970390954 z^18848 849 // See note below calling the functor to compute another term,850 // sufficient for 80-bit long double precision.851 // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term.852 // (5480386857784802185939 z^19)/6402373705728000853 // But now this variant is not used to compute long double854 // as specializations are provided above.855 }; // static const T coeff[]856 857 /*858 Table of 19 computed coefficients:859 860 #0 0861 #1 1862 #2 -1863 #3 1.5864 #4 -2.6666666666666666666666666666666665382713370408509865 #5 5.2083333333333333333333333333333330765426740817019866 #6 -10.800000000000000000000000000000000616297582203915867 #7 23.343055555555555555555555555555555076212991619177868 #8 -52.012698412698412698412698412698412659282693193402869 #9 118.62522321428571428571428571428571146835390992496870 #10 -275.57319223985890652557319223985891400375196748314871 #11 649.7871723434744268077601410934743969785223845882872 #12 -1551.1605194805194805194805194805194947599566007429873 #13 3741.4497029592385495163272941050719510009019331763874 #14 -9104.5002411580189357967135744913524243896052869184875 #15 22324.308512706601434280005708577137322392070452582876 #16 -55103.621972903835337697771560205423203318720697224877 #17 136808.86090394293563342215789305735851647769682393878 136808.86090394293563342215789305735851647769682393 == Exactly same as Wolfram computed value.879 #18 -341422.05066583836331735491399356947486381600607416880 341422.05066583836331735491399356945575432970390954 z^19 Wolfram value differs at 36 decimal digit, as expected.881 */882 883 using boost::math::policies::get_epsilon; // for type T.884 using boost::math::tools::sum_series;885 using boost::math::tools::evaluate_polynomial;886 // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html887 888 // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10);889 890 T result = evaluate_polynomial(coeff, z); // LCOV_EXCL_LINE next line covered but not this one strangely - GCOV SNAFU?891 // template <std::size_t N, typename T, typename V>892 // V evaluate_polynomial(const T(&poly)[N], const V& val);893 // Size of coeff found from N894 //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl;895 //std::cout << "result = " << result << std::endl;896 // It's an artefact of the way I wrote the functor: *after* evaluating N897 // terms, its internal state has k = N and term = (-1)^N z^N. So after898 // evaluating 18 terms, we initialize the functor to the term we've just899 // evaluated, and then when it's called, it increments itself to the next term.900 // So 18!is 6402373705728000, which is where that comes from.901 902 // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!=903 // 104127350297911241532841 / 121645100408832000 which after removing GCDs904 // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000.905 // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000906 // +855992.96599660755146336302506332246623424823099755 z^19907 908 //! Evaluate Functor.909 lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18);910 911 // Temporary to list the coefficients.912 //std::cout << " Table of coefficients" << std::endl;913 //std::streamsize saved_precision = std::cout.precision(50);914 //for (size_t i = 0; i != 19; i++)915 //{916 // std::cout << "#" << i << " " << coeff[i] << std::endl;917 //}918 //std::cout.precision(saved_precision);919 920 std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy.921#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES922 std::cout << "max iter from policy = " << max_iter << std::endl;923 // // max iter from policy = 1000000 is default.924#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES925 926 result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result);927 // result == evaluate_polynomial.928 //sum_series(Functor& func, int bits, std::uintmax_t& max_terms, const U& init_value)929 // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl;930 931 //T epsilon = get_epsilon<T, Policy>();932 //std::cout << "epsilon from policy = " << epsilon << std::endl;933 // epsilon from policy = 1.93e-34 for T == quad934 // 5.35e-51 for t = cpp_bin_float_50935 936 // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51937 policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol);938#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS939 std::cout << "z = " << z << " needed " << max_iter << " iterations." << std::endl;940 std::cout.precision(prec); // Restore.941#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS942 return result;943} // template <typename T, typename Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol)944 945// Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions)946// Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.947template <typename T>948inline T lambert_w0_approx(T z)949{950 BOOST_MATH_STD_USING951 T lz = log(z);952 T llz = log(lz);953 T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.954 return w;955 // std::cout << "w max " << max_w << std::endl; // double 703.227956}957 958 //////////////////////////////////////////////////////////////////////////////////////////959 960//! \brief Lambert_w0 implementations for float, double and higher precisions.961//! 3rd parameter used to select which version is used.962 963//! /details Rational polynomials are provided for several range of argument z.964//! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879),965//! two other series functions are used.966 967//! float precision polynomials are used for 32-bit (usually float) precision (for speed)968//! double precision polynomials are used for 64-bit (usually double) precision.969//! For higher precisions, a 64-bit double approximation is computed first,970//! and then refined using Halley iterations.971 972template <typename T>973inline T do_get_near_singularity_param(T z)974{975 BOOST_MATH_STD_USING976 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);977 const T p = sqrt(p2);978 return p;979}980template <typename T, typename Policy>981inline T get_near_singularity_param(T z, const Policy)982{983 using value_type = typename policies::evaluation<T, Policy>::type;984 return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z)));985}986 987// Forward declarations:988 989//template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy& pol);990//template <typename T, typename Policy>991//T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 0>&); // 32 bit usually float.992//template <typename T, typename Policy>993//T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 1>&); // 64 bit usually double.994//template <typename T, typename Policy>995//T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 2>&); // 80-bit long double.996 997template <typename T>998T lambert_w_positive_rational_float(T z)999{1000 BOOST_MATH_STD_USING1001 if (z < 2)1002 {1003 if (z < T(0.5))1004 { // 0.05 < z < 0.51005 // Maximum Deviation Found: 2.993e-081006 // Expected Error Term : 2.993e-081007 // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-011008 // LCOV_EXCL_START1009 static const T Y = 8.196592331e-01f;1010 static const T P[] = {1011 1.803388345e-01f,1012 -4.820256838e-01f,1013 -1.068349741e+00f,1014 -3.506624319e-02f,1015 };1016 static const T Q[] = {1017 1.000000000e+00f,1018 2.871703469e+00f,1019 1.690949264e+00f,1020 };1021 // LCOV_EXCL_STOP1022 return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));1023 }1024 else1025 { // 0.5 < z < 21026 // Max error in interpolated form: 1.018e-081027 // LCOV_EXCL_START1028 static const T Y = 5.503368378e-01f;1029 static const T P[] = {1030 4.493332766e-01f,1031 2.543432707e-01f,1032 -4.808788799e-01f,1033 -1.244425316e-01f,1034 };1035 static const T Q[] = {1036 1.000000000e+00f,1037 2.780661241e+00f,1038 1.830840318e+00f,1039 2.407221031e-01f,1040 };1041 // LCOV_EXCL_STOP1042 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));1043 }1044 }1045 else if (z < 6)1046 {1047 // 2 < z < 61048 // Max error in interpolated form: 2.944e-081049 // LCOV_EXCL_START1050 static const T Y = 1.162393570e+00f;1051 static const T P[] = {1052 -1.144183394e+00f,1053 -4.712732855e-01f,1054 1.563162512e-01f,1055 1.434010911e-02f,1056 };1057 static const T Q[] = {1058 1.000000000e+00f,1059 1.192626340e+00f,1060 2.295580708e-01f,1061 5.477869455e-03f,1062 };1063 // LCOV_EXCL_STOP1064 return Y + boost::math::tools::evaluate_rational(P, Q, z);1065 }1066 else if (z < 18)1067 {1068 // 6 < z < 181069 // Max error in interpolated form: 5.893e-081070 // LCOV_EXCL_START1071 static const T Y = 1.809371948e+00f;1072 static const T P[] = {1073 -1.689291769e+00f,1074 -3.337812742e-01f,1075 3.151434873e-02f,1076 1.134178734e-03f,1077 };1078 static const T Q[] = {1079 1.000000000e+00f,1080 5.716915685e-01f,1081 4.489521292e-02f,1082 4.076716763e-04f,1083 };1084 // LCOV_EXCL_STOP1085 return Y + boost::math::tools::evaluate_rational(P, Q, z);1086 }1087 else if (z < T(9897.12905874)) // 2.8 < log(z) < 9.21088 {1089 // Max error in interpolated form: 1.771e-081090 // LCOV_EXCL_START1091 static const T Y = -1.402973175e+00f;1092 static const T P[] = {1093 1.966174312e+00f,1094 2.350864728e-01f,1095 -5.098074353e-02f,1096 -1.054818339e-02f,1097 };1098 static const T Q[] = {1099 1.000000000e+00f,1100 4.388208264e-01f,1101 8.316639634e-02f,1102 3.397187918e-03f,1103 -1.321489743e-05f,1104 };1105 // LCOV_EXCL_STOP1106 T log_w = log(z);1107 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);1108 }1109 else if (z < T(7.896296e+13)) // 9.2 < log(z) <= 321110 {1111 // Max error in interpolated form: 5.821e-081112 // LCOV_EXCL_START1113 static const T Y = -2.735729218e+00f;1114 static const T P[] = {1115 3.424903470e+00f,1116 7.525631787e-02f,1117 -1.427309584e-02f,1118 -1.435974178e-05f,1119 };1120 static const T Q[] = {1121 1.000000000e+00f,1122 2.514005579e-01f,1123 6.118994652e-03f,1124 -1.357889535e-05f,1125 7.312865624e-08f,1126 };1127 // LCOV_EXCL_STOP1128 T log_w = log(z);1129 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);1130 }1131 1132 // Max error in interpolated form: 1.491e-081133 // LCOV_EXCL_START1134 static const T Y = -4.012863159e+00f;1135 static const T P[] = {1136 4.431629226e+00f,1137 2.756690487e-01f,1138 -2.992956930e-03f,1139 -4.912259384e-05f,1140 };1141 static const T Q[] = {1142 1.000000000e+00f,1143 2.015434591e-01f,1144 4.949426142e-03f,1145 1.609659944e-05f,1146 -5.111523436e-09f,1147 };1148 // LCOV_EXCL_STOP1149 T log_w = log(z);1150 return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);1151 1152}1153 1154template <typename T, typename Policy>1155T lambert_w_negative_rational_float(T z, const Policy& pol)1156{1157 BOOST_MATH_STD_USING1158 if (z > T(-0.27))1159 {1160 if (z < T(-0.051))1161 {1162 // -0.27 < z < -0.0511163 // Max error in interpolated form: 5.080e-081164 // LCOV_EXCL_START1165 static const T Y = 1.255809784e+00f;1166 static const T P[] = {1167 -2.558083412e-01f,1168 -2.306524098e+00f,1169 -5.630887033e+00f,1170 -3.803974556e+00f,1171 };1172 static const T Q[] = {1173 1.000000000e+00f,1174 5.107680783e+00f,1175 7.914062868e+00f,1176 3.501498501e+00f,1177 };1178 // LCOV_EXCL_STOP1179 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));1180 }1181 else1182 {1183 // Very small z so use a series function.1184 return lambert_w0_small_z(z, pol);1185 }1186 }1187 else if (z > T(-0.3578794411714423215955237701))1188 { // Very close to branch singularity.1189 // Max error in interpolated form: 5.269e-081190 // LCOV_EXCL_START1191 static const T Y = 1.220928431e-01f;1192 static const T P[] = {1193 -1.221787446e-01f,1194 -6.816155875e+00f,1195 7.144582035e+01f,1196 1.128444390e+03f,1197 };1198 static const T Q[] = {1199 1.000000000e+00f,1200 6.480326790e+01f,1201 1.869145243e+02f,1202 -1.361804274e+03f,1203 1.117826726e+03f,1204 };1205 // LCOV_EXCL_STOP1206 T d = z + 0.367879441171442321595523770161460867445811f;1207 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));1208 }1209 1210 return lambert_w_singularity_series(get_near_singularity_param(z, pol));1211}1212 1213//! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision.1214template <typename T, typename Policy>1215inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&)1216{1217 static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages.1218 BOOST_MATH_STD_USING // Aid ADL of std functions.1219 1220 if ((boost::math::isnan)(z))1221 {1222 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);1223 }1224 if ((boost::math::isinf)(z))1225 {1226 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);1227 }1228 1229 if (z >= T(0.05)) // Fukushima switch point.1230 // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045.1231 { // Normal ranges using several rational polynomials.1232 return lambert_w_positive_rational_float(z);1233 }1234 else if (z <= -0.3678794411714423215955237701614608674458111310f)1235 {1236 if (z < -0.3678794411714423215955237701614608674458111310f)1237 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);1238 return -1;1239 }1240 1241 return lambert_w_negative_rational_float(z, pol);1242} // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) for 32-bit usually float.1243 1244template <typename T>1245T lambert_w_positive_rational_double(T z)1246{1247 BOOST_MATH_STD_USING1248 if (z < 2)1249 {1250 if (z < 0.5)1251 {1252 // Max error in interpolated form: 2.255e-171253 // LCOV_EXCL_START1254 static const T offset = 8.19659233093261719e-01;1255 static const T P[] = {1256 1.80340766906685177e-01,1257 3.28178241493119307e-01,1258 -2.19153620687139706e+00,1259 -7.24750929074563990e+00,1260 -7.28395876262524204e+00,1261 -2.57417169492512916e+00,1262 -2.31606948888704503e-011263 };1264 static const T Q[] = {1265 1.00000000000000000e+00,1266 7.36482529307436604e+00,1267 2.03686007856430677e+01,1268 2.62864592096657307e+01,1269 1.59742041380858333e+01,1270 4.03760534788374589e+00,1271 2.91327346750475362e-011272 };1273 // LCOV_EXCL_STOP1274 return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));1275 }1276 else1277 {1278 // Max error in interpolated form: 3.806e-181279 // LCOV_EXCL_START1280 static const T offset = 5.50335884094238281e-01;1281 static const T P[] = {1282 4.49664083944098322e-01,1283 1.90417666196776909e+00,1284 1.99951368798255994e+00,1285 -6.91217310299270265e-01,1286 -1.88533935998617058e+00,1287 -7.96743968047750836e-01,1288 -1.02891726031055254e-01,1289 -3.09156013592636568e-031290 };1291 static const T Q[] = {1292 1.00000000000000000e+00,1293 6.45854489419584014e+00,1294 1.54739232422116048e+01,1295 1.72606164253337843e+01,1296 9.29427055609544096e+00,1297 2.29040824649748117e+00,1298 2.21610620995418981e-01,1299 5.70597669908194213e-031300 };// LCOV_EXCL_STOP1301 return z * (offset + boost::math::tools::evaluate_rational(P, Q, z));1302 }1303 }1304 else if (z < 6)1305 {1306 // 2 < z < 61307 // Max error in interpolated form: 1.216e-171308 // LCOV_EXCL_START1309 static const T Y = 1.16239356994628906e+00;1310 static const T P[] = {1311 -1.16230494982099475e+00,1312 -3.38528144432561136e+00,1313 -2.55653717293161565e+00,1314 -3.06755172989214189e-01,1315 1.73149743765268289e-01,1316 3.76906042860014206e-02,1317 1.84552217624706666e-03,1318 1.69434126904822116e-05,1319 };1320 static const T Q[] = {1321 1.00000000000000000e+00,1322 3.77187616711220819e+00,1323 4.58799960260143701e+00,1324 2.24101228462292447e+00,1325 4.54794195426212385e-01,1326 3.60761772095963982e-02,1327 9.25176499518388571e-04,1328 4.43611344705509378e-06,1329 };1330 // LCOV_EXCL_STOP1331 return Y + boost::math::tools::evaluate_rational(P, Q, z);1332 }1333 else if (z < 18)1334 {1335 // 6 < z < 181336 // Max error in interpolated form: 1.985e-191337 // LCOV_EXCL_START1338 static const T offset = 1.80937194824218750e+00;1339 static const T P[] =1340 {1341 -1.80690935424793635e+00,1342 -3.66995929380314602e+00,1343 -1.93842957940149781e+00,1344 -2.94269984375794040e-01,1345 1.81224710627677778e-03,1346 2.48166798603547447e-03,1347 1.15806592415397245e-04,1348 1.43105573216815533e-06,1349 3.47281483428369604e-091350 };1351 static const T Q[] = {1352 1.00000000000000000e+00,1353 2.57319080723908597e+00,1354 1.96724528442680658e+00,1355 5.84501352882650722e-01,1356 7.37152837939206240e-02,1357 3.97368430940416778e-03,1358 8.54941838187085088e-05,1359 6.05713225608426678e-07,1360 8.17517283816615732e-101361 };1362 // LCOV_EXCL_STOP1363 return offset + boost::math::tools::evaluate_rational(P, Q, z);1364 }1365 else if (z < 9897.12905874) // 2.8 < log(z) < 9.21366 {1367 // Max error in interpolated form: 1.195e-181368 // LCOV_EXCL_START1369 static const T Y = -1.40297317504882812e+00;1370 static const T P[] = {1371 1.97011826279311924e+00,1372 1.05639945701546704e+00,1373 3.33434529073196304e-01,1374 3.34619153200386816e-02,1375 -5.36238353781326675e-03,1376 -2.43901294871308604e-03,1377 -2.13762095619085404e-04,1378 -4.85531936495542274e-06,1379 -2.02473518491905386e-08,1380 };1381 static const T Q[] = {1382 1.00000000000000000e+00,1383 8.60107275833921618e-01,1384 4.10420467985504373e-01,1385 1.18444884081994841e-01,1386 2.16966505556021046e-02,1387 2.24529766630769097e-03,1388 9.82045090226437614e-05,1389 1.36363515125489502e-06,1390 3.44200749053237945e-09,1391 };1392 // LCOV_EXCL_STOP1393 T log_w = log(z);1394 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);1395 }1396 else if (z < 7.896296e+13) // 9.2 < log(z) <= 321397 {1398 // Max error in interpolated form: 6.529e-181399 // LCOV_EXCL_START1400 static const T Y = -2.73572921752929688e+00;1401 static const T P[] = {1402 3.30547638424076217e+00,1403 1.64050071277550167e+00,1404 4.57149576470736039e-01,1405 4.03821227745424840e-02,1406 -4.99664976882514362e-04,1407 -1.28527893803052956e-04,1408 -2.95470325373338738e-06,1409 -1.76662025550202762e-08,1410 -1.98721972463709290e-11,1411 };1412 static const T Q[] = {1413 1.00000000000000000e+00,1414 6.91472559412458759e-01,1415 2.48154578891676774e-01,1416 4.60893578284335263e-02,1417 3.60207838982301946e-03,1418 1.13001153242430471e-04,1419 1.33690948263488455e-06,1420 4.97253225968548872e-09,1421 3.39460723731970550e-12,1422 };1423 // LCOV_EXCL_STOP1424 T log_w = log(z);1425 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);1426 }1427 else if (z < 2.6881171e+43) // 32 < log(z) < 1001428 {1429 // Max error in interpolated form: 2.015e-181430 // LCOV_EXCL_START1431 static const T Y = -4.01286315917968750e+00;1432 static const T P[] = {1433 5.07714858354309672e+00,1434 -3.32994414518701458e+00,1435 -8.61170416909864451e-01,1436 -4.01139705309486142e-02,1437 -1.85374201771834585e-04,1438 1.08824145844270666e-05,1439 1.17216905810452396e-07,1440 2.97998248101385990e-10,1441 1.42294856434176682e-13,1442 };1443 static const T Q[] = {1444 1.00000000000000000e+00,1445 -4.85840770639861485e-01,1446 -3.18714850604827580e-01,1447 -3.20966129264610534e-02,1448 -1.06276178044267895e-03,1449 -1.33597828642644955e-05,1450 -6.27900905346219472e-08,1451 -9.35271498075378319e-11,1452 -2.60648331090076845e-14,1453 };1454 // LCOV_EXCL_STOP1455 T log_w = log(z);1456 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);1457 }1458 else // 100 < log(z) < 7101459 {1460 // Max error in interpolated form: 5.277e-181461 // LCOV_EXCL_START1462 static const T Y = -5.70115661621093750e+00;1463 static const T P[] = {1464 6.42275660145116698e+00,1465 1.33047964073367945e+00,1466 6.72008923401652816e-02,1467 1.16444069958125895e-03,1468 7.06966760237470501e-06,1469 5.48974896149039165e-09,1470 -7.00379652018853621e-11,1471 -1.89247635913659556e-13,1472 -1.55898770790170598e-16,1473 -4.06109208815303157e-20,1474 -2.21552699006496737e-24,1475 };1476 static const T Q[] = {1477 1.00000000000000000e+00,1478 3.34498588416632854e-01,1479 2.51519862456384983e-02,1480 6.81223810622416254e-04,1481 7.94450897106903537e-06,1482 4.30675039872881342e-08,1483 1.10667669458467617e-10,1484 1.31012240694192289e-13,1485 6.53282047177727125e-17,1486 1.11775518708172009e-20,1487 3.78250395617836059e-25,1488 };1489 // LCOV_EXCL_STOP1490 T log_w = log(z);1491 return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);1492 }1493}1494 1495template <typename T, typename Policy>1496T lambert_w_negative_rational_double(T z, const Policy& pol)1497{1498 BOOST_MATH_STD_USING1499 if (z > -0.1)1500 {1501 if (z < -0.051)1502 {1503 // -0.1 < z < -0.0511504 // Maximum Deviation Found: 4.402e-221505 // Expected Error Term : 4.240e-221506 // Maximum Relative Change in Control Points : 4.115e-031507 // LCOV_EXCL_START1508 static const T Y = 1.08633995056152344e+00;1509 static const T P[] = {1510 -8.63399505615014331e-02,1511 -1.64303871814816464e+00,1512 -7.71247913918273738e+00,1513 -1.41014495545382454e+01,1514 -1.02269079949257616e+01,1515 -2.17236002836306691e+00,1516 };1517 static const T Q[] = {1518 1.00000000000000000e+00,1519 7.44775406945739243e+00,1520 2.04392643087266541e+01,1521 2.51001961077774193e+01,1522 1.31256080849023319e+01,1523 2.11640324843601588e+00,1524 };1525 // LCOV_EXCL_STOP1526 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));1527 }1528 else1529 {1530 // Very small z > 0.051:1531 return lambert_w0_small_z(z, pol);1532 }1533 }1534 else if (z > -0.2)1535 {1536 // -0.2 < z < -0.11537 // Maximum Deviation Found: 2.898e-201538 // Expected Error Term : 2.873e-201539 // Maximum Relative Change in Control Points : 3.779e-041540 // LCOV_EXCL_START1541 static const T Y = 1.20359611511230469e+00;1542 static const T P[] = {1543 -2.03596115108465635e-01,1544 -2.95029082937201859e+00,1545 -1.54287922188671648e+01,1546 -3.81185809571116965e+01,1547 -4.66384358235575985e+01,1548 -2.59282069989642468e+01,1549 -4.70140451266553279e+00,1550 };1551 static const T Q[] = {1552 1.00000000000000000e+00,1553 9.57921436074599929e+00,1554 3.60988119290234377e+01,1555 6.73977699505546007e+01,1556 6.41104992068148823e+01,1557 2.82060127225153607e+01,1558 4.10677610657724330e+00,1559 };1560 // LCOV_EXCL_STOP1561 return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));1562 }1563 else if (z > -0.3178794411714423215955237)1564 {1565 // Max error in interpolated form: 6.996e-181566 // LCOV_EXCL_START1567 static const T Y = 3.49680423736572266e-01;1568 static const T P[] = {1569 -3.49729841718749014e-01,1570 -6.28207407760709028e+01,1571 -2.57226178029669171e+03,1572 -2.50271008623093747e+04,1573 1.11949239154711388e+05,1574 1.85684566607844318e+06,1575 4.80802490427638643e+06,1576 2.76624752134636406e+06,1577 };1578 static const T Q[] = {1579 1.00000000000000000e+00,1580 1.82717661215113000e+02,1581 8.00121119810280100e+03,1582 1.06073266717010129e+05,1583 3.22848993926057721e+05,1584 -8.05684814514171256e+05,1585 -2.59223192927265737e+06,1586 -5.61719645211570871e+05,1587 6.27765369292636844e+04,1588 };1589 // LCOV_EXCL_STOP1590 T d = z + 0.367879441171442321595523770161460867445811;1591 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));1592 }1593 else if (z > -0.3578794411714423215955237701)1594 {1595 // Max error in interpolated form: 1.404e-171596 // LCOV_EXCL_START1597 static const T Y = 5.00126481056213379e-02;1598 static const T P[] = {1599 -5.00173570682372162e-02,1600 -4.44242461870072044e+01,1601 -9.51185533619946042e+03,1602 -5.88605699015429386e+05,1603 -1.90760843597427751e+06,1604 5.79797663818311404e+08,1605 1.11383352508459134e+10,1606 5.67791253678716467e+10,1607 6.32694500716584572e+10,1608 };1609 static const T Q[] = {1610 1.00000000000000000e+00,1611 9.08910517489981551e+02,1612 2.10170163753340133e+05,1613 1.67858612416470327e+07,1614 4.90435561733227953e+08,1615 4.54978142622939917e+09,1616 2.87716585708739168e+09,1617 -4.59414247951143131e+10,1618 -1.72845216404874299e+10,1619 };1620 // LCOV_EXCL_STOP1621 T d = z + 0.36787944117144232159552377016146086744581113103176804;1622 return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));1623 }1624 else1625 { // z is very close (within 0.01) of the singularity at -e^-1,1626 // so use a series expansion from R. M. Corless et al.1627 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);1628 const T p = sqrt(p2);1629 return lambert_w_detail::lambert_w_singularity_series(p);1630 }1631}1632 1633//! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision.1634template <typename T, typename Policy>1635inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&)1636{1637 static const char* function = "boost::math::lambert_w0<%1%>";1638 BOOST_MATH_STD_USING // Aid ADL of std functions.1639 1640 // Detect unusual case of 32-bit double with a wider/64-bit long double1641 static_assert(std::numeric_limits<double>::digits >= 53,1642 "Our double precision coefficients will be truncated, "1643 "please file a bug report with details of your platform's floating point types "1644 "- or possibly edit the coefficients to have "1645 "an appropriate size-suffix for 64-bit floats on your platform - L?");1646 1647 if ((boost::math::isnan)(z))1648 {1649 return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);1650 }1651 if ((boost::math::isinf)(z))1652 {1653 return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);1654 }1655 1656 if (z >= 0.05)1657 {1658 return lambert_w_positive_rational_double(z);1659 }1660 else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50).1661 {1662 if (z < -0.36787944117144232159552377016146086744581113103176804)1663 {1664 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);1665 }1666 return -1;1667 }1668 else1669 {1670 return lambert_w_negative_rational_double(z, pol);1671 }1672} // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) 64-bit precision, usually double.1673 1674//! lambert_W0 implementation for extended precision types including1675//! long double (80-bit and 128-bit), ???1676//! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50...1677 1678template <typename T, typename Policy>1679inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&)1680{1681 static const char* function = "boost::math::lambert_w0<%1%>";1682 BOOST_MATH_STD_USING // Aid ADL of std functions.1683 1684 // Filter out special cases first:1685 if ((boost::math::isnan)(z))1686 {1687 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);1688 }1689 if (fabs(z) <= 0.05f)1690 {1691 // Very small z:1692 return lambert_w0_small_z(z, pol);1693 }1694 if (z > (std::numeric_limits<double>::max)())1695 {1696 if ((boost::math::isinf)(z))1697 {1698 return policies::raise_overflow_error<T>(function, nullptr, pol);1699 // Or might return infinity if available else max_value,1700 // but other Boost.Math special functions raise overflow.1701 }1702 // z is larger than the largest double, so cannot use the polynomial to get an approximation,1703 // so use the asymptotic approximation and Halley iterate:1704 1705 T w = lambert_w0_approx(z); // Make an inline function as also used elsewhere.1706 //T lz = log(z);1707 //T llz = log(lz);1708 //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.1709 return lambert_w_halley_iterate(w, z);1710 }1711 if (z < -0.3578794411714423215955237701)1712 { // Very close to branch point so rational polynomials are not usable.1713 if (z <= -boost::math::constants::exp_minus_one<T>())1714 {1715 if (z == -boost::math::constants::exp_minus_one<T>())1716 { // Exactly at the branch point singularity.1717 return -1;1718 }1719 return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);1720 }1721 // z is very close (within 0.01) of the branch singularity at -e^-11722 // so use a series approximation proposed by Corless et al.1723 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);1724 const T p = sqrt(p2);1725 T w = lambert_w_detail::lambert_w_singularity_series(p);1726 return lambert_w_halley_iterate(w, z);1727 }1728 1729 // Phew! If we get here we are in the normal range of the function,1730 // so get a double precision approximation first, then iterate to full precision of T.1731 // We define a tag_type that is:1732 // true_type if there are so many digits precision wanted that iteration is necessary.1733 // false_type if a single Halley step is sufficient.1734 1735 using precision_type = typename policies::precision<T, Policy>::type;1736 using tag_type = std::integral_constant<bool,1737 (precision_type::value == 0) || (precision_type::value > 113) ?1738 true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision.1739 : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step.1740 >;1741 1742 // For speed, we also cast z to type double when that is possible1743 // if (std::is_constructible<double, T>() == true).1744 T w = lambert_w0_imp(maybe_reduce_to_double(z, std::is_constructible<double, T>()), pol, std::integral_constant<int, 2>());1745 1746 return lambert_w_maybe_halley_iterate(w, z, tag_type());1747 1748} // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&) all extended precision types.1749 1750 // Lambert w-1 implementation1751// ==============================================================================================1752 1753 //! Lambert W for W-1 branch, -max(z) < z <= -1/e.1754 // TODO is -max(z) allowed?1755template<typename T, typename Policy>1756T lambert_wm1_imp(const T z, const Policy& pol)1757{1758 // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1).1759 // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L),1760 // or static_casted integer, for example: static_cast<float>(1) or static_cast<cpp_dec_float_50>(1).1761 // Want to allow fixed_point types too, so do not just test for floating-point.1762 // Integral types should be promoted to double by user Lambert w functions.1763 // If integral type provided to user function lambert_w0 or lambert_wm1,1764 // then should already have been promoted to double.1765 static_assert(!std::is_integral<T>::value,1766 "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!");1767 1768 BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.1769 1770 const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages.1771 1772 // Check for edge and corner cases first:1773 if ((boost::math::isnan)(z))1774 {1775 return policies::raise_domain_error(function, "Argument z is NaN!", z, pol);1776 } // isnan1777 1778 if ((boost::math::isinf)(z))1779 {1780 return policies::raise_domain_error(function, "Argument z is infinite!", z, pol);1781 } // isinf1782 1783 if (z == static_cast<T>(0))1784 { // z is exactly zero so return -std::numeric_limits<T>::infinity();1785 return -policies::raise_overflow_error(function, nullptr, z, pol);1786 }1787 if (boost::math::detail::has_denorm_now<T>())1788 { // All real types except arbitrary precision.1789 if (!(boost::math::isnormal)(z))1790 { // Almost zero - might also just return infinity like z == 0 or max_value?1791 return -policies::raise_overflow_error(function, "Argument z = %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)", z, pol);1792 }1793 }1794 1795 if (z > static_cast<T>(0))1796 { //1797 return policies::raise_domain_error(function, "Argument z = %1% is out of range (z <= 0) for Lambert W-1 branch! (Try Lambert W0 branch?)", z, pol);1798 }1799 if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -0.36787944.1800 { // At singularity, so return exactly -1.1801 return -static_cast<T>(1);1802 }1803 // z is too negative for the W-1 (or W0) branch.1804 if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -0.36787944.1805 {1806 return policies::raise_domain_error(function, "Argument z = %1% is out of range (require -exp(-1) = -0.36787944... < z <= 0) for Lambert W-1 (or W0) branch!", z, pol);1807 }1808 if (z < static_cast<T>(-0.35))1809 { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch.1810 const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);1811 // Commented out, requires z = -1 / 2e which is greater than -0.35 so this whole branch is not taken.1812 //if (p2 == 0)1813 //{ // At the singularity at branch point.1814 // return -1;1815 // }1816 BOOST_MATH_ASSERT(p2 > 0);1817 T w_series = lambert_w_singularity_series(T(-sqrt(p2)));1818 if (boost::math::tools::digits<T>() > 53)1819 { // Multiprecision, so try a Halley refinement.1820 w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z);1821#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN1822 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1823 std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl;1824 std::cout.precision(saved_precision);1825#endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN1826 }1827 return w_series;1828 } // if (z < -0.35)1829 1830 using lambert_w_lookup::wm1es;1831 using lambert_w_lookup::wm1zs;1832 using lambert_w_lookup::noof_wm1zs; // size == 641833 1834 // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) = -1.0264389699511283e-261835 // Check that z argument value is not smaller than lookup_table G[64]1836 // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl;1837 1838 if (z >= T(wm1zs[63])) // wm1zs[63] = -1.0264389699511282259046957018510946438e-26L W = 64.000000000000000001839 { // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized).1840 1841 // Some info on Lambert W-1 values for extreme values of z.1842 // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1843 // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl;1844 // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl;1845 // -std::numeric_limits<float>::min() = -1.1754943508222875e-381846 // -std::numeric_limits<double>::min() = -2.2250738585072014e-3081847 // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.8567753245954795095677567300938239938341550278581848 // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.85445212305538535581321805184043636179680429421849 // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.999999999999999999999999999999999551850 1851 // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth,1852 // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996.1853 // Francois Chapeau-Blondeau and Abdelilah Monir1854 // Numerical Evaluation of the Lambert W Function1855 // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 20021856 // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf1857 // Estimate Lambert W using ln(-z) ...1858 // This is roughly the power of ten * ln(10) ~= 2.3. n ~= 10^n1859 // and improve by adding a second term -ln(ln(-z))1860 T guess; // bisect lowest possible Gk[=64] (for lookup_t type)1861 T lz = log(-z);1862 T llz = log(-lz);1863 guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162.1864#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY1865 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1866 std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl;1867 // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.0613112314473041941868 // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.0257384244237643111869 // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.00926459208218466221870 int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits101871 int d2 = policies::digits<T, Policy>(); // digits base 2 from policy.1872 std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 51873 << std::endl;1874 std::cout.precision(saved_precision);1875#endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY1876 if (policies::digits<T, Policy>() < 12)1877 { // For the worst case near w = 64, the error in the 'guess' is ~0.008, ratio ~ 0.0001 or 1 in 10,000 digits 10 ~= 4, or digits2 ~= 12.1878 return guess; // LCOV_EXCL_LINE We don't have a test type with few enough digits to trigger this.1879 }1880 T result = lambert_w_detail::lambert_w_halley_iterate(guess, z);1881 return result;1882 1883 // Was Fukushima1884 // G[k=64] == g[63] == -1.02643897e-261885 //return policies::raise_domain_error(function,1886 // "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.",1887 // z, pol);1888 } // Z too small so use approximation and Halley.1889 // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection.1890 1891 if (boost::math::tools::digits<T>() > 53)1892 { // T is more precise than 64-bit double (or long double, or ?),1893 // so compute an approximate value using only one Schroeder refinement,1894 // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 501895 // because are next going to use Halley refinement at full/high precision using this as an approximation).1896 using boost::math::policies::precision;1897 using boost::math::policies::digits10;1898 using boost::math::policies::digits2;1899 using boost::math::policies::policy;1900 // Compute a 50-bit precision approximate W0 in a double (no Halley refinement).1901 T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, std::is_constructible<double, T>()), policy<digits2<50>>())));1902#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN1903 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1904 std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl;1905 std::cout.precision(saved_precision);1906#endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM11907 // Perform additional Halley refinement(s) to ensure that1908 // get a near as possible to correct result (usually +/- one epsilon).1909 T result = lambert_w_halley_iterate(double_approx, z);1910#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM11911 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1912 std::cout << "Result " << typeid(T).name() << " precision Halley refinement = " << result << std::endl;1913 std::cout.precision(saved_precision);1914#endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM11915 return result;1916 } // digits > 53 - higher precision than double.1917 else // T is double or less precision.1918 { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection.1919 using namespace boost::math::lambert_w_detail::lambert_w_lookup;1920 // Bracketing sequence n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity)1921 // Since z is probably quite small, start with lowest n (=2).1922 int n = 2;1923 if (T(wm1zs[n - 1]) > z)1924 {1925 goto bisect;1926 }1927 for (int j = 1; j <= 5; ++j)1928 {1929 n *= 2;1930 if (T(wm1zs[n - 1]) > z)1931 {1932 goto overshot;1933 }1934 }1935 // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64.1936 // This should not now occur (should be caught by test and code above) so should be a logic_error?1937 return policies::raise_evaluation_error(function, "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)", z, pol); // LCOV_EXCL_LINE1938 overshot:1939 {1940 int nh = n / 2;1941 for (int j = 1; j <= 5; ++j)1942 {1943 nh /= 2; // halve step size.1944 if (nh <= 0)1945 {1946 break; // goto bisect;1947 }1948 if (T(wm1zs[n - nh - 1]) > z)1949 {1950 n -= nh;1951 }1952 }1953 }1954 bisect:1955 --n;1956 // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part;1957 // these are used as initial values for bisection.1958#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP1959 std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);1960 std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n]1961 << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl;1962 std::cout.precision(saved_precision);1963#endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP1964 1965 // Compute bisections is the number of bisections computed from n,1966 // such that a single application of the fifth-order Schroeder update formula1967 // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy.1968 // Fukushima established these by trial and error?1969 int bisections = 11; // Assume maximum number of bisections will be needed (most common case).1970 if (n >= 8)1971 {1972 bisections = 8;1973 }1974 else if (n >= 3)1975 {1976 bisections = 9;1977 }1978 else if (n >= 2)1979 {1980 bisections = 10;1981 }1982 // Bracketing, Fukushima section 2.3, page 82:1983 // (Avoiding using exponential function for speed).1984 // Only use @c lookup_t precision, default double, for bisection (again for speed),1985 // and use later Halley refinement for higher precisions.1986 using lambert_w_lookup::halves;1987 using lambert_w_lookup::sqrtwm1s;1988 1989 using calc_type = typename std::conditional<std::is_constructible<lookup_t, T>::value, lookup_t, T>::type;1990 1991 calc_type w = -static_cast<calc_type>(n); // Equation 25,1992 calc_type y = static_cast<calc_type>(z * T(wm1es[n - 1])); // Equation 26,1993 // Perform the bisections fractional bisections for necessary precision.1994 for (int j = 0; j < bisections; ++j)1995 { // Equation 27.1996 calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ...1997 calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ...1998 if (wj < yj)1999 {2000 w = wj;2001 y = yj;2002 }2003 } // for j2004 return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement.2005 2006// else // Perform additional Halley refinement(s) to ensure that2007// // get a near as possible to correct result (usually +/- epsilon).2008// {2009// // result = lambert_w_halley_iterate(result, z);2010// result = lambert_w_halley_step(result, z); // Just one Halley step should be enough.2011//#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY2012// std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);2013// std::cout << "Halley refinement estimate = " << result << std::endl;2014// std::cout.precision(saved_precision);2015//#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY2016// return result; // Halley2017// } // Schroeder or Schroeder and Halley.2018 }2019 } // template<typename T = double> T lambert_wm1_imp(const T z)2020} // namespace lambert_w_detail2021 2022///////////////////////////// User Lambert w functions. //////////////////////////////2023 2024//! Lambert W0 using User-defined policy.2025 template <typename T, typename Policy>2026 inline2027 typename boost::math::tools::promote_args<T>::type2028 lambert_w0(T z, const Policy& pol)2029 {2030 // Promote integer or expression template arguments to double,2031 // without doing any other internal promotion like float to double.2032 using result_type = typename tools::promote_args<T>::type;2033 2034 // Work out what precision has been selected,2035 // based on the Policy and the number type.2036 using precision_type = typename policies::precision<result_type, Policy>::type;2037 // and then select the correct implementation based on that precision (not the type T):2038 using tag_type = std::integral_constant<int,2039 (precision_type::value == 0) || (precision_type::value > 53) ?2040 0 // either variable precision (0), or greater than 64-bit precision.2041 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.2042 : 2 // 64-bit (probably double) precision.2043 >;2044 2045 return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); //2046 } // lambert_w0(T z, const Policy& pol)2047 2048 //! Lambert W0 using default policy.2049 template <typename T>2050 inline2051 typename tools::promote_args<T>::type2052 lambert_w0(T z)2053 {2054 // Promote integer or expression template arguments to double,2055 // without doing any other internal promotion like float to double.2056 using result_type = typename tools::promote_args<T>::type;2057 2058 // Work out what precision has been selected, based on the Policy and the number type.2059 // For the default policy version, we want the *default policy* precision for T.2060 using precision_type = typename policies::precision<result_type, policies::policy<>>::type;2061 // and then select the correct implementation based on that (not the type T):2062 using tag_type = std::integral_constant<int,2063 (precision_type::value == 0) || (precision_type::value > 53) ?2064 0 // either variable precision (0), or greater than 64-bit precision.2065 : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.2066 : 2 // 64-bit (probably double) precision.2067 >;2068 return lambert_w_detail::lambert_w0_imp(result_type(z), policies::policy<>(), tag_type());2069 } // lambert_w0(T z) using default policy.2070 2071 //! W-1 branch (-max(z) < z <= -1/e).2072 2073 //! Lambert W-1 using User-defined policy.2074 template <typename T, typename Policy>2075 inline2076 typename tools::promote_args<T>::type2077 lambert_wm1(T z, const Policy& pol)2078 {2079 // Promote integer or expression template arguments to double,2080 // without doing any other internal promotion like float to double.2081 using result_type = typename tools::promote_args<T>::type;2082 return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); //2083 }2084 2085 //! Lambert W-1 using default policy.2086 template <typename T>2087 inline2088 typename tools::promote_args<T>::type2089 lambert_wm1(T z)2090 {2091 using result_type = typename tools::promote_args<T>::type;2092 return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>());2093 } // lambert_wm1(T z)2094 2095 // First derivative of Lambert W0 and W-1.2096 namespace lambert_w_detail {2097 template <typename T, typename Policy>2098 inline typename tools::promote_args<T>::type2099 lambert_w0_prime(T z, const Policy& pol)2100 {2101 using result_type = typename tools::promote_args<T>::type;2102 using std::numeric_limits;2103 if (z == 0)2104 {2105 return static_cast<result_type>(1);2106 }2107 // This is the sensible choice if we regard the Lambert-W function as complex analytic.2108 // Of course on the real line, it's just undefined.2109 if (z == -boost::math::constants::exp_minus_one<result_type>())2110 {2111 return boost::math::policies::raise_overflow_error("lambert_w0_prime", nullptr, z, pol);2112 }2113 // if z < -1/e, we'll let lambert_w0 do the error handling:2114 result_type w = lambert_w0(result_type(z), pol);2115 // If w ~ -1, then presumably this can get inaccurate.2116 // Is there an accurate way to evaluate 1 + W(-1/e + eps)?2117 // Yes: This is discussed in the Princeton Companion to Applied Mathematics,2118 // 'The Lambert-W function', Section 1.3: Series and Generating Functions.2119 // 1 + W(-1/e + x) ~ sqrt(2ex).2120 // Nick is not convinced this formula is more accurate than the naive one.2121 // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100).2122 return w / (z * (1 + w));2123 } // lambert_w0_prime(T z)2124 }2125 // First derivative of Lambert W0 and W-1.2126 template <typename T, typename Policy>2127 inline typename tools::promote_args<T>::type2128 lambert_w0_prime(T z, const Policy& pol)2129 {2130 using result_type = typename tools::promote_args<T>::type;2131 return lambert_w_detail::lambert_w0_prime(static_cast<result_type>(z), pol);2132 }2133 2134 template <typename T>2135 inline typename tools::promote_args<T>::type2136 lambert_w0_prime(T z)2137 {2138 return lambert_w0_prime(z, policies::policy<>());2139 }2140 2141 template <typename T, typename Policy>2142 inline typename tools::promote_args<T>::type2143 lambert_wm1_prime(T z, const Policy& pol)2144 {2145 using std::numeric_limits;2146 using result_type = typename tools::promote_args<T>::type;2147 //if (z == 0)2148 //{2149 // return static_cast<result_type>(1);2150 //}2151 //if (z == - boost::math::constants::exp_minus_one<result_type>())2152 if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>())2153 {2154 return -boost::math::policies::raise_overflow_error("lambert_wm1_prime", nullptr, z, pol);2155 }2156 2157 result_type w = lambert_wm1(z, pol);2158 return w/(z*(1+w));2159 } // lambert_wm1_prime(T z)2160 2161 template <typename T>2162 inline typename tools::promote_args<T>::type2163 lambert_wm1_prime(T z)2164 {2165 return lambert_wm1_prime(z, policies::policy<>());2166 }2167 2168}} //boost::math namespaces2169 2170#endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP2171 2172