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 // Copyright John Maddock 2017.
 // Copyright Paul A. Bristow 2016, 2017, 2018.
 // Copyright Nicholas Thompson 2018
 
 // Distributed under the Boost Software License, Version 1.0.
 // (See accompanying file LICENSE_1_0.txt or
 //  copy at http ://www.boost.org/LICENSE_1_0.txt).
 
 #ifndef BOOST_MATH_SF_LAMBERT_W_HPP
 #define BOOST_MATH_SF_LAMBERT_W_HPP
 
 #ifdef _MSC_VER
 #pragma warning(disable : 4127)
 #endif
 
 /*
 Implementation of an algorithm for the Lambert W0 and W-1 real-only functions.
 
 This code is based in part on the algorithm by
 Toshio Fukushima,
 "Precise and fast computation of Lambert W-functions without transcendental function evaluations",
 J.Comp.Appl.Math. 244 (2013) 77-89,
 and on a C/C++ version by Darko Veberic, darko.veberic@ijs.si
 based on the Fukushima algorithm and Toshio Fukushima's FORTRAN version of his algorithm.
 
 First derivative of Lambert_w is derived from
 Princeton Companion to Applied Mathematics, 'The Lambert-W function', Section 1.3: Series and Generating Functions.
 
 */
 
 /*
 TODO revise this list of macros.
 Some macros that will show some (or much) diagnostic values if #defined.
 //[boost_math_instrument_lambert_w_macros
 
 // #define-able macros
 BOOST_MATH_INSTRUMENT_LAMBERT_W_HALLEY                     // Halley refinement diagnostics.
 BOOST_MATH_INSTRUMENT_LAMBERT_W_PRECISION                  // Precision.
 BOOST_MATH_INSTRUMENT_LAMBERT_WM1                          // W1 branch diagnostics.
 BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY                   // Halley refinement diagnostics only for W-1 branch.
 BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY                     // K > 64, z > -1.0264389699511303e-26
 BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP                   // Show results from W-1 lookup table.
 BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER                  // Schroeder refinement diagnostics.
 BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS                      // Number of terms used for near-singularity series.
 BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES         // Show evaluation of series near branch singularity.
 BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS  // Show evaluation of series for small z.
 //] [/boost_math_instrument_lambert_w_macros]
 */
 
 #include <boost/math/tools/config.hpp>
 #include <boost/math/policies/error_handling.hpp>
 #include <boost/math/policies/policy.hpp>
 #include <boost/math/tools/promotion.hpp>
 #include <boost/math/special_functions/fpclassify.hpp>
 #include <boost/math/special_functions/log1p.hpp> // for log (1 + x)
 #include <boost/math/constants/constants.hpp> // For exp_minus_one == 3.67879441171442321595523770161460867e-01.
 #include <boost/math/special_functions/next.hpp>  // for has_denorm_now
 #include <boost/math/special_functions/pow.hpp> // powers with compile time exponent, used in arbitrary precision code.
 #include <boost/math/tools/series.hpp> // series functor.
 //#include <boost/math/tools/polynomial.hpp>  // polynomial.
 #include <boost/math/tools/rational.hpp>  // evaluate_polynomial.
 #include <boost/math/tools/precision.hpp> // boost::math::tools::max_value().
 #include <boost/math/tools/big_constant.hpp>
 #include <boost/math/tools/cxx03_warn.hpp>
 
 #ifndef BOOST_MATH_STANDALONE
 #include <boost/lexical_cast.hpp>
 #endif
 
 #include <limits>
 #include <cmath>
 #include <limits>
 #include <exception>
 #include <type_traits>
 #include <cstdint>
 
 // Needed for testing and diagnostics only.
 #include <iostream>
 #include <typeinfo>
 #include <boost/math/special_functions/next.hpp>  // For float_distance.
 
 using lookup_t = double; // Type for lookup table (double or float, or even long double?)
 
 //#include "J:\Cpp\Misc\lambert_w_lookup_table_generator\lambert_w_lookup_table.ipp"
 // #include "lambert_w_lookup_table.ipp" // Boost.Math version.
 #include <boost/math/special_functions/detail/lambert_w_lookup_table.ipp>
 
 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
 //
 // This is the only way we can avoid
 // warning: non-standard suffix on floating constant [-Wpedantic]
 // when building with -Wall -pedantic.  Neither __extension__
 // nor #pragma diagnostic ignored work :(
 //
 #pragma GCC system_header
 #endif
 
 namespace boost {
 namespace math {
 namespace lambert_w_detail {
 
 //! \brief Applies a single Halley step to make a better estimate of Lambert W.
 //! \details Used the simplified formulae obtained from
 //! 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)%5D
 //! [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)]
 
 //! \tparam T floating-point (or fixed-point) type.
 //! \param w_est Lambert W estimate.
 //! \param z Argument z for Lambert_w function.
 //! \returns New estimate of Lambert W, hopefully improved.
 //!
 template <typename T>
 inline T lambert_w_halley_step(T w_est, const T z)
 {
   BOOST_MATH_STD_USING
   T e = exp(w_est);
   w_est -= 2 * (w_est + 1) * (e * w_est - z) / (z * (w_est + 2) + e * (w_est * (w_est + 2) + 2));
   return w_est;
 } // template <typename T> lambert_w_halley_step(T w_est, T z)
 
 //! \brief Halley iterate to refine Lambert_w estimate,
 //! taking at least one Halley_step.
 //! Repeat Halley steps until the *last step* had fewer than half the digits wrong,
 //! the step we've just taken should have been sufficient to have completed the iteration.
 
 //! \tparam T floating-point (or fixed-point) type.
 //! \param z Argument z for Lambert_w function.
 //! \param w_est Lambert w estimate.
 template <typename T>
 inline T lambert_w_halley_iterate(T w_est, const T z)
 {
   BOOST_MATH_STD_USING
   static const T max_diff = boost::math::tools::root_epsilon<T>() * fabs(w_est);
 
   T w_new = lambert_w_halley_step(w_est, z);
   T diff = fabs(w_est - w_new);
   while (diff > max_diff)
   {
     w_est = w_new;
     w_new = lambert_w_halley_step(w_est, z);
     diff = fabs(w_est - w_new);
   }
   return w_new;
 } // template <typename T> lambert_w_halley_iterate(T w_est, T z)
 
 // Two Halley function versions that either
 // single step (if std::false_type) or iterate (if std::true_type).
 // Selected at compile-time using parameter 3.
 template <typename T>
 inline T lambert_w_maybe_halley_iterate(T z, T w, std::false_type const&)
 {
    return lambert_w_halley_step(z, w); // Single step.
 }
 
 template <typename T>
 inline T lambert_w_maybe_halley_iterate(T z, T w, std::true_type const&)
 {
    return lambert_w_halley_iterate(z, w); // Iterate steps.
 }
 
 //! maybe_reduce_to_double function,
 //! Two versions that have a compile-time option to
 //! reduce argument z to double precision (if true_type).
 //! Version is selected at compile-time using parameter 2.
 
 template <typename T>
 inline double maybe_reduce_to_double(const T& z, const std::true_type&)
 {
   return static_cast<double>(z); // Reduce to double precision.
 }
 
 template <typename T>
 inline T maybe_reduce_to_double(const T& z, const std::false_type&)
 { // Don't reduce to double.
   return z;
 }
 
 template <typename T>
 inline double must_reduce_to_double(const T& z, const std::true_type&)
 {
    return static_cast<double>(z); // Reduce to double precision.
 }
 
 template <typename T>
 inline double must_reduce_to_double(const T& z, const std::false_type&)
 { // try a lexical_cast and hope for the best:
 #ifndef BOOST_MATH_STANDALONE
 
    #ifdef BOOST_MATH_USE_CHARCONV_FOR_CONVERSION
 
    // Catches the C++23 floating point types
    if constexpr (std::is_arithmetic_v<T>)
    {
       return static_cast<double>(z);
    }
    else
    {
       return boost::lexical_cast<double>(z);
    }
 
    #else
    
    return boost::lexical_cast<double>(z);
    
    #endif
 
 #else
    static_assert(sizeof(T) == 0, "Unsupported in standalone mode: don't know how to cast your number type to a double.");
    return 0.0;
 #endif
 }
 
 //! \brief Schroeder method, fifth-order update formula,
 //! \details See T. Fukushima page 80-81, and
 //! A. Householder, The Numerical Treatment of a Single Nonlinear Equation,
 //! McGraw-Hill, New York, 1970, section 4.4.
 //! Fukushima algorithm switches to @c schroeder_update after pre-computed bisections,
 //! chosen to ensure that the result will be achieve the +/- 10 epsilon target.
 //! \param w Lambert w estimate from bisection or series.
 //! \param y bracketing value from bisection.
 //! \returns Refined estimate of Lambert w.
 
 // Schroeder refinement, called unless NOT required by precision policy.
 template<typename T>
 inline T schroeder_update(const T w, const T y)
 {
   // Compute derivatives using 5th order Schroeder refinement.
   // Since this is the final step, it will always use the highest precision type T.
   // Example of Call:
   //   result = schroeder_update(w, y);
   //where
   // w is estimate of Lambert W (from bisection or series).
   // y is z * e^-w.
 
   BOOST_MATH_STD_USING // Aid argument dependent lookup of abs.
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
     std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
   using boost::math::float_distance;
   T fd = float_distance<T>(w, y);
   std::cout << "Schroder ";
   if (abs(fd) < 214748000.)
   {
     std::cout << " Distance = "<< static_cast<int>(fd);
   }
   else
   {
     std::cout << "Difference w - y = " << (w - y) << ".";
   }
   std::cout << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
   //  Fukushima equation 18, page 6.
   const T f0 = w - y; // f0 = w - y.
   const T f1 = 1 + y; // f1 = df/dW
   const T f00 = f0 * f0;
   const T f11 = f1 * f1;
   const T f0y = f0 * y;
   const T result =
     w - 4 * f0 * (6 * f1 * (f11 + f0y)  +  f00 * y) /
     (f11 * (24 * f11 + 36 * f0y) +
       f00 * (6 * y * y  +  8 * f1 * y  +  f0y)); // Fukushima Page 81, equation 21 from equation 20.
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
   std::cout << "Schroeder refined " << w << "  " << y << ", difference  " << w-y  << ", change " << w - result << ", to result " << result << std::endl;
   std::cout.precision(saved_precision); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SCHROEDER
 
   return result;
 } // template<typename T = double> T schroeder_update(const T w, const T y)
 
   //! \brief Series expansion used near the singularity/branch point z = -exp(-1) = -3.6787944.
   //! Wolfram InverseSeries[Series[sqrt[2(p Exp[1 + p] + 1)], {p,-1, 20}]]
   //! Wolfram command used to obtain 40 series terms at 50 decimal digit precision was
   //! N[InverseSeries[Series[Sqrt[2(p Exp[1 + p] + 1)], { p,-1,40 }]], 50]
   //! -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 ...
   //! Decimal values of specifications for built-in floating-point types below
   //! are at least 21 digits precision == max_digits10 for long double.
   //! Longer decimal digits strings are rationals evaluated using Wolfram.
 
 template<typename T>
 T lambert_w_singularity_series(const T p)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
   std::size_t saved_precision = std::cout.precision(3);
   std::cout << "Singularity_series Lambert_w p argument = " << p << std::endl;
   std::cout
     //<< "Argument Type = " << typeid(T).name()
     //<< ", max_digits10 = " << std::numeric_limits<T>::max_digits10
     //<< ", epsilon = " << std::numeric_limits<T>::epsilon()
     << std::endl;
   std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SINGULARITY_SERIES
 
   static const T q[] =
   {
     -static_cast<T>(1), // j0
     +T(1), // j1
     -T(1) / 3, // 1/3  j2
     +T(11) / 72, // 0.152777777777777778, // 11/72 j3
     -T(43) / 540, // 0.0796296296296296296, // 43/540 j4
     +T(769) / 17280, // 0.0445023148148148148,  j5
     -T(221) / 8505, // 0.0259847148736037625,  j6
     //+T(0.0156356325323339212L), // j7
     //+T(0.015635632532333921222810111699000587889476778365667L), // j7 from Wolfram N[680863/43545600, 50]
     +T(680863uLL) / 43545600uLL, // +0.0156356325323339212, j7
     //-T(0.00961689202429943171L), // j8
     -T(1963uLL) / 204120uLL, // 0.00961689202429943171, j8
     //-T(0.0096168920242994317068391142465216539290613364687439L), // j8 from Wolfram N[1963/204120, 50]
     +T(226287557uLL) / 37623398400uLL, // 0.00601454325295611786, j9
     -T(5776369uLL) / 1515591000uLL, // 0.00381129803489199923, j10
     //+T(0.00244087799114398267L), j11 0.0024408779911439826658968585286437530215699919795550
     +T(169709463197uLL) / 69528040243200uLL, // j11
     // -T(0.00157693034468678425L), // j12  -0.0015769303446867842539234095399314115973161850314723
     -T(1118511313uLL) / 709296588000uLL, // j12
     +T(667874164916771uLL) / 650782456676352000uLL, // j13
     //+T(0.00102626332050760715L), // j13 0.0010262633205076071544375481533906861056468041465973
     -T(500525573uLL) / 744761417400uLL, // j14
     // -T(0.000672061631156136204L), j14
     //+T(1003663334225097487uLL) / 234281684403486720000uLL, // j15 0.00044247306181462090993020760858473726479232802068800 error C2177: constant too big
     //+T(0.000442473061814620910L, // j15
     BOOST_MATH_BIG_CONSTANT(T, 64, +0.000442473061814620910), // j15
     // -T(0.000292677224729627445L), // j16
     BOOST_MATH_BIG_CONSTANT(T, 64, -0.000292677224729627445), // j16
     //+T(0.000194387276054539318L), // j17
     BOOST_MATH_BIG_CONSTANT(T, 64, 0.000194387276054539318), // j17
     //-T(0.000129574266852748819L), // j18
     BOOST_MATH_BIG_CONSTANT(T, 64, -0.000129574266852748819), // j18
     //+T(0.0000866503580520812717L), // j19 N[+1150497127780071399782389/13277465363600276402995200000, 50] 0.000086650358052081271660451590462390293190597827783288
     BOOST_MATH_BIG_CONSTANT(T, 64, +0.0000866503580520812717), // j19
     //-T(0.0000581136075044138168L) // j20  N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
     // -T(2853534237182741069uLL) / 49102686267859224000000uLL  // j20 // error C2177: constant too big,
     // so must use BOOST_MATH_BIG_CONSTANT(T, ) format in hope of using suffix Q for quad or decimal digits string for others.
     //-T(0.000058113607504413816772205464778828177256611844221913L), // j20  N[2853534237182741069/49102686267859224000000, 50] 0.000058113607504413816772205464778828177256611844221913
     BOOST_MATH_BIG_CONSTANT(T, 113, -0.000058113607504413816772205464778828177256611844221913) // j20  - last used by Fukushima
     // More terms don't seem to give any improvement (worse in fact) and are not use for many z values.
     //BOOST_MATH_BIG_CONSTANT(T, +0.000039076684867439051635395583044527492132109160553593), // j21
     //BOOST_MATH_BIG_CONSTANT(T, -0.000026338064747231098738584082718649443078703982217219), // j22
     //BOOST_MATH_BIG_CONSTANT(T, +0.000017790345805079585400736282075184540383274460464169), // j23
     //BOOST_MATH_BIG_CONSTANT(T, -0.000012040352739559976942274116578992585158113153190354), // j24
     //BOOST_MATH_BIG_CONSTANT(T, +8.1635319824966121713827512573558687050675701559448E-6), // j25
     //BOOST_MATH_BIG_CONSTANT(T, -5.5442032085673591366657251660804575198155559225316E-6) // j26
     // -T(5.5442032085673591366657251660804575198155559225316E-6L) // j26
     // 21 to 26 Added for long double.
   }; // static const T q[]
 
      /*
      // Temporary copy of original double values for comparison; these are reproduced well.
      static const T q[] =
      {
      -1L,  // j0
      +1L,  // j1
      -0.333333333333333333L, // 1/3 j2
      +0.152777777777777778L, // 11/72 j3
      -0.0796296296296296296L, // 43/540
      +0.0445023148148148148L,
      -0.0259847148736037625L,
      +0.0156356325323339212L,
      -0.00961689202429943171L,
      +0.00601454325295611786L,
      -0.00381129803489199923L,
      +0.00244087799114398267L,
      -0.00157693034468678425L,
      +0.00102626332050760715L,
      -0.000672061631156136204L,
      +0.000442473061814620910L,
      -0.000292677224729627445L,
      +0.000194387276054539318L,
      -0.000129574266852748819L,
      +0.0000866503580520812717L,
      -0.0000581136075044138168L // j20
      };
      */
 
      // Decide how many series terms to use, increasing as z approaches the singularity,
      // balancing run-time versus computational noise from round-off.
      // In practice, we truncate the series expansion at a certain order.
      // If the order is too large, not only does the amount of computation increase,
      // but also the round-off errors accumulate.
      // See Fukushima equation 35, page 85 for logic of choice of number of series terms.
 
   BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
 
     const T absp = abs(p);
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
   {
     int terms = 20; // Default to using all terms.
     if (absp < 0.01159)
     { // Very near singularity.
       terms = 6;
     }
     else if (absp < 0.0766)
     { // Near singularity.
       terms = 10;
     }
     std::streamsize saved_precision = std::cout.precision(3);
     std::cout << "abs(p) = " << absp << ", terms = " << terms << std::endl;
     std::cout.precision(saved_precision);
   }
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_TERMS
 
   if (absp < T(0.01159))
   { // Only 6 near-singularity series terms are useful.
     return
       -1 +
       p * (1 +
         p * (q[2] +
           p * (q[3] +
             p * (q[4] +
               p * (q[5] +
                 p * q[6]
                 )))));
   }
   else if (absp < T(0.0766)) // Use 10 near-singularity series terms.
   { // Use 10 near-singularity series terms.
     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]
                         )))))))));
   }
    // Use all 20 near-singularity series terms.
     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.
                                             )))))))))))))))))));
     //                                                + // more terms for more precise T: long double ...
     //// but makes almost no difference, so don't use more terms?
     //                                          p*q[21] +
     //                                            p*q[22] +
     //                                              p*q[23] +
     //                                                p*q[24] +
     //                                                 p*q[25]
     //                                         )))))))))))))))))));
 
 } // template<typename T = double> T lambert_w_singularity_series(const T p)
 
 
  /////////////////////////////////////////////////////////////////////////////////////////////
 
   //! \brief Series expansion used near zero (abs(z) < 0.05).
   //! \details
   //! Coefficients of the inverted series expansion of the Lambert W function around z = 0.
   //! Tosio Fukushima always uses all 17 terms of a Taylor series computed using Wolfram with
   //!   InverseSeries[Series[z Exp[z],{z,0,17}]]
   //! Tosio Fukushima / Journal of Computational and Applied Mathematics 244 (2013) page 86.
 
   //! Decimal values of specifications for built-in floating-point types below
   //! are 21 digits precision == max_digits10 for long double.
   //! Care! Some coefficients might overflow some fixed_point types.
 
   //! This version is intended to allow use by user-defined types
   //! like Boost.Multiprecision quad and cpp_dec_float types.
   //! The three specializations below for built-in float, double
   //! (and perhaps long double) will be chosen in preference for these types.
 
   //! This version uses rationals computed by Wolfram as far as possible,
   //! limited by maximum size of uLL integers.
   //! For higher term, uses decimal digit strings computed by Wolfram up to the maximum possible using uLL rationals,
   //! and then higher coefficients are computed as necessary using function lambert_w0_small_z_series_term
   //! until the precision required by the policy is achieved.
   //! InverseSeries[Series[z Exp[z],{z,0,34}]] also computed.
 
   // Series evaluation for LambertW(z) as z -> 0.
   // See http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/
   //  http://functions.wolfram.com/ElementaryFunctions/ProductLog/06/01/01/0003/MainEq1.L.gif
 
   //! \brief  lambert_w0_small_z uses a tag_type to select a variant depending on the size of the type.
   //! The Lambert W is computed by lambert_w0_small_z for small z.
   //! The cutoff for z smallness determined by Tosio Fukushima by trial and error is (abs(z) < 0.05),
   //! but the optimum might be a function of the size of the type of z.
 
   //! \details
   //! The tag_type selection is based on the value @c std::numeric_limits<T>::max_digits10.
   //! This allows distinguishing between long double types that commonly vary between 64 and 80-bits,
   //! and also compilers that have a float type using 64 bits and/or long double using 128-bits.
   //! It assumes that max_digits10 is defined correctly or this might fail to make the correct selection.
   //! causing very small differences in computing lambert_w that would be very difficult to detect and diagnose.
   //! Cannot switch on @c std::numeric_limits<>::max() because comparison values may overflow the compiler limit.
   //! Cannot switch on @c std::numeric_limits<long double>::max_exponent10()
   //! because both 80 and 128 bit floating-point implementations use 11 bits for the exponent.
   //! So must rely on @c std::numeric_limits<long double>::max_digits10.
 
   //! Specialization of float zero series expansion used for small z (abs(z) < 0.05).
   //! Specializations of lambert_w0_small_z for built-in types.
   //! These specializations should be chosen in preference to T version.
   //! For example: lambert_w0_small_z(0.001F) should use the float version.
   //! (Parameter Policy is not used by built-in types when all terms are used during an inline computation,
   //! but for the tag_type selection to work, they all must include Policy in their signature.
 
   // Forward declaration of variants of lambert_w0_small_z.
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 0> const&);   //  for float (32-bit) type.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 1> const&);   //  for double (64-bit) type.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 2> const&);   //  for long double (double extended 80-bit) type.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 3> const&);   //  for long double (128-bit) type.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 4> const&);   //  for float128 quadmath Q type.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy&, std::integral_constant<int, 5> const&);   //  Generic multiprecision T.
                                                                         // Set tag_type depending on max_digits10.
 template <typename T, typename Policy>
 T lambert_w0_small_z(T x, const Policy& pol)
 { //std::numeric_limits<T>::max_digits10 == 36 ? 3 : // 128-bit long double.
   using tag_type = std::integral_constant<int,
      std::numeric_limits<T>::is_specialized == 0 ? 5 :
 #ifndef BOOST_NO_CXX11_NUMERIC_LIMITS
     std::numeric_limits<T>::max_digits10 <=  9 ? 0 : // for float 32-bit.
     std::numeric_limits<T>::max_digits10 <= 17 ? 1 : // for double 64-bit.
     std::numeric_limits<T>::max_digits10 <= 22 ? 2 : // for 80-bit double extended.
     std::numeric_limits<T>::max_digits10 <  37 ? 4  // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
 #else
      std::numeric_limits<T>::radix != 2 ? 5 :
      std::numeric_limits<T>::digits <= 24 ? 0 : // for float 32-bit.
      std::numeric_limits<T>::digits <= 53 ? 1 : // for double 64-bit.
      std::numeric_limits<T>::digits <= 64 ? 2 : // for 80-bit double extended.
      std::numeric_limits<T>::digits <= 113 ? 4  // for both 128-bit long double (3) and 128-bit quad suffix Q type (4).
 #endif
       :  5>;                                           // All Generic multiprecision types.
   // std::cout << "\ntag type = " << tag_type << std::endl; // error C2275: 'tag_type': illegal use of this type as an expression.
   return lambert_w0_small_z(x, pol, tag_type());
 } // template <typename T> T lambert_w0_small_z(T x)
 
   //! Specialization of float (32-bit) series expansion used for small z (abs(z) < 0.05).
   // Only 9 Coefficients are computed to 21 decimal digits precision, ample for 32-bit float used by most platforms.
   // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
   // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
   // as proposed by Tosio Fukushima and implemented by Darko Veberic.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(T z, const Policy&, std::integral_constant<int, 0> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "\ntag_type 0 float lambert_w0_small_z called with z = " << z << " using " << 9 << " terms of precision "
     << std::numeric_limits<float>::max_digits10 << " decimal digits. " << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   T result =
     z * (1 - // j1 z^1 term = 1
       z * (1 -  // j2 z^2 term = -1
         z * (static_cast<float>(3uLL) / 2uLL - // 3/2 // j3 z^3 term = 1.5.
           z * (2.6666666666666666667F -  // 8/3 // j4
             z * (5.2083333333333333333F - // -125/24 // j5
               z * (10.8F - // j6
                 z * (23.343055555555555556F - // j7
                   z * (52.012698412698412698F - // j8
                     z * 118.62522321428571429F)))))))); // j9
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "return w = " << result << std::endl;
   std::cout.precision(prec); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 
   return result;
 } // template <typename T>   T lambert_w0_small_z(T x, std::integral_constant<int, 0> const&)
 
   //! Specialization of double (64-bit double) series expansion used for small z (abs(z) < 0.05).
   // 17 Coefficients are computed to 21 decimal digits precision suitable for 64-bit double used by most platforms.
   // Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
   // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50], as proposed by Tosio Fukushima and implemented by Veberic.
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 1> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize prec = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "\ntag_type 1 double lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision, "
     << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   T result =
     z * (1. - // j1 z^1
       z * (1. -  // j2 z^2
         z * (1.5 - // 3/2 // j3 z^3
           z * (2.6666666666666666667 -  // 8/3 // j4
             z * (5.2083333333333333333 - // -125/24 // j5
               z * (10.8 - // j6
                 z * (23.343055555555555556 - // j7
                   z * (52.012698412698412698 - // j8
                     z * (118.62522321428571429 - // j9
                       z * (275.57319223985890653 - // j10
                         z * (649.78717234347442681 - // j11
                           z * (1551.1605194805194805 - // j12
                             z * (3741.4497029592385495 - // j13
                               z * (9104.5002411580189358 - // j14
                                 z * (22324.308512706601434 - // j15
                                   z * (55103.621972903835338 - // j16
                                     z * 136808.86090394293563)))))))))))))))); // j17 z^17
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "return w = " << result << std::endl;
   std::cout.precision(prec); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 
   return result;
 } // T lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)
 
   //! Specialization of long double (80-bit double extended) series expansion used for small z (abs(z) < 0.05).
   // 21 Coefficients are computed to 21 decimal digits precision suitable for 80-bit long double used by some
   // platforms including GCC and Clang when generating for Intel X86 floating-point processors with 80-bit operations enabled (the default).
   // (This is NOT used by Microsoft Visual Studio where double and long always both use only 64-bit type.
   // Nor used for 128-bit float128.)
 template <typename T, typename Policy>
 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 2> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "\ntag_type 2 long double (80-bit double extended) lambert_w0_small_z called with z = " << z << " using " << 21 << " terms of precision, "
     << std::numeric_limits<long double>::max_digits10 << " decimal digits. " << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 //  T  result =
 //    z * (1.L - // j1 z^1
 //      z * (1.L -  // j2 z^2
 //        z * (1.5L - // 3/2 // j3
 //          z * (2.6666666666666666667L -  // 8/3 // j4
 //            z * (5.2083333333333333333L - // -125/24 // j5
 //              z * (10.800000000000000000L - // j6
 //                z * (23.343055555555555556L - // j7
 //                  z * (52.012698412698412698L - // j8
 //                    z * (118.62522321428571429L - // j9
 //                      z * (275.57319223985890653L - // j10
 //                        z * (649.78717234347442681L - // j11
 //                          z * (1551.1605194805194805L - // j12
 //                            z * (3741.4497029592385495L - // j13
 //                              z * (9104.5002411580189358L - // j14
 //                                z * (22324.308512706601434L - // j15
 //                                  z * (55103.621972903835338L - // j16
 //                                    z * (136808.86090394293563L - // j17 z^17  last term used by Fukushima double.
 //                                      z * (341422.050665838363317L - // z^18
 //                                        z * (855992.9659966075514633L - // z^19
 //                                          z * (2.154990206091088289321e6L - // z^20
 //                                            z * 5.4455529223144624316423e6L   // z^21
 //                                              ))))))))))))))))))));
 //
 
   T result =
 z * (1.L - // z j1
 z * (1.L - // z^2
 z * (1.500000000000000000000000000000000L - // z^3
 z * (2.666666666666666666666666666666666L - // z ^ 4
 z * (5.208333333333333333333333333333333L - // z ^ 5
 z * (10.80000000000000000000000000000000L - // z ^ 6
 z * (23.34305555555555555555555555555555L - //  z ^ 7
 z * (52.01269841269841269841269841269841L - // z ^ 8
 z * (118.6252232142857142857142857142857L - // z ^ 9
 z * (275.5731922398589065255731922398589L - // z ^ 10
 z * (649.7871723434744268077601410934744L - // z ^ 11
 z * (1551.160519480519480519480519480519L - // z ^ 12
 z * (3741.449702959238549516327294105071L - //z ^ 13
 z * (9104.500241158018935796713574491352L - //  z ^ 14
 z * (22324.308512706601434280005708577137L - //  z ^ 15
 z * (55103.621972903835337697771560205422L - //  z ^ 16
 z * (136808.86090394293563342215789305736L - // z ^ 17
 z * (341422.05066583836331735491399356945L - //  z^18
 z * (855992.9659966075514633630250633224L - // z^19
 z * (2.154990206091088289321708745358647e6L // z^20 distance -5 without term 20
 ))))))))))))))))))));
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "return w = " << result << std::endl;
   std::cout.precision(precision); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   return result;
 }  // long double lambert_w0_small_z(const T z, std::integral_constant<int, 1> const&)
 
 //! Specialization of 128-bit long double series expansion used for small z (abs(z) < 0.05).
 // 34 Taylor series coefficients used are computed by Wolfram to 50 decimal digits using instruction
 // N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
 // and are suffixed by L as they are assumed of type long double.
 // (This is NOT used for 128-bit quad boost::multiprecision::float128 type which required a suffix Q
 // nor multiprecision type cpp_bin_float_quad that can only be initialized at full precision of the type
 // constructed with a decimal digit string like "2.6666666666666666666666666666666666666666666666667".)
 
 template <typename T, typename Policy>
 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 3> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "\ntag_type 3 long double (128-bit) lambert_w0_small_z called with z = " << z << " using " << 17 << " terms of precision,  "
     << std::numeric_limits<double>::max_digits10 << " decimal digits. " << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   T  result =
     z * (1.L - // j1
       z * (1.L -  // j2
         z * (1.5L - // 3/2 // j3
           z * (2.6666666666666666666666666666666666L -  // 8/3 // j4
             z * (5.2052083333333333333333333333333333L - // -125/24 // j5
               z * (10.800000000000000000000000000000000L - // j6
                 z * (23.343055555555555555555555555555555L - // j7
                   z * (52.0126984126984126984126984126984126L - // j8
                     z * (118.625223214285714285714285714285714L - // j9
                       z * (275.57319223985890652557319223985890L - // * z ^ 10 - // j10
                         z * (649.78717234347442680776014109347442680776014109347L - // j11
                           z * (1551.1605194805194805194805194805194805194805194805L - // j12
                             z * (3741.4497029592385495163272941050718828496606274384L - // j13
                               z * (9104.5002411580189357967135744913522691300469078247L - // j14
                                 z * (22324.308512706601434280005708577137148565719994291L - // j15
                                   z * (55103.621972903835337697771560205422639285073147507L - // j16
                                     z * 136808.86090394293563342215789305736395683485630576L    // j17
                                       ))))))))))))))));
 
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "return w = " << result << std::endl;
   std::cout.precision(precision); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   return result;
 }  // T lambert_w0_small_z(const T z, std::integral_constant<int, 3> const&)
 
 //! Specialization of 128-bit quad series expansion used for small z (abs(z) < 0.05).
 // 34 Taylor series coefficients used were computed by Wolfram to 50 decimal digits using instruction
 //   N[InverseSeries[Series[z Exp[z],{z,0,34}]],50],
 // and are suffixed by Q as they are assumed of type quad.
 // This could be used for 128-bit quad (which requires a suffix Q for full precision).
 // But experiments with GCC 7.2.0 show that while this gives full 128-bit precision
 // when the -f-ext-numeric-literals option is in force and the libquadmath library available,
 // over the range -0.049 to +0.049,
 // it is slightly slower than getting a double approximation followed by a single Halley step.
 
 #ifdef BOOST_HAS_FLOAT128
 template <typename T, typename Policy>
 T lambert_w0_small_z(const T z, const Policy&, std::integral_constant<int, 4> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "\ntag_type 4 128-bit quad float128 lambert_w0_small_z called with z = " << z << " using " << 34 << " terms of precision, "
     << std::numeric_limits<float128>::max_digits10 << " max decimal digits." << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   T  result =
     z * (1.Q - // z j1
       z * (1.Q - // z^2
         z * (1.500000000000000000000000000000000Q - // z^3
           z * (2.666666666666666666666666666666666Q - // z ^ 4
             z * (5.208333333333333333333333333333333Q - // z ^ 5
               z * (10.80000000000000000000000000000000Q - // z ^ 6
                 z * (23.34305555555555555555555555555555Q - //  z ^ 7
                   z * (52.01269841269841269841269841269841Q - // z ^ 8
                     z * (118.6252232142857142857142857142857Q - // z ^ 9
                       z * (275.5731922398589065255731922398589Q - // z ^ 10
                         z * (649.7871723434744268077601410934744Q - // z ^ 11
                           z * (1551.160519480519480519480519480519Q - // z ^ 12
                             z * (3741.449702959238549516327294105071Q - //z ^ 13
                               z * (9104.500241158018935796713574491352Q - //  z ^ 14
                                 z * (22324.308512706601434280005708577137Q - //  z ^ 15
                                   z * (55103.621972903835337697771560205422Q - //  z ^ 16
                                     z * (136808.86090394293563342215789305736Q - // z ^ 17
                                       z * (341422.05066583836331735491399356945Q - //  z^18
                                         z * (855992.9659966075514633630250633224Q - // z^19
                                           z * (2.154990206091088289321708745358647e6Q - //  20
                                             z * (5.445552922314462431642316420035073e6Q - // 21
                                               z * (1.380733000216662949061923813184508e7Q - // 22
                                                 z * (3.511704498513923292853869855945334e7Q - // 23
                                                   z * (8.956800256102797693072819557780090e7Q - // 24
                                                     z * (2.290416846187949813964782641734774e8Q - // 25
                                                       z * (5.871035041171798492020292225245235e8Q - // 26
                                                         z * (1.508256053857792919641317138812957e9Q - // 27
                                                           z * (3.882630161293188940385873468413841e9Q - // 28
                                                             z * (1.001394313665482968013913601565723e10Q - // 29
                                                               z * (2.587356736265760638992878359024929e10Q - // 30
                                                                 z * (6.696209709358073856946120522333454e10Q - // 31
                                                                   z * (1.735711659599198077777078238043644e11Q - // 32
                                                                     z * (4.505680465642353886756098108484670e11Q - // 33
                                                                       z * (1.171223178256487391904047636564823e12Q  //z^34
                                                                         ))))))))))))))))))))))))))))))))));
 
 
  #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "return w = " << result << std::endl;
   std::cout.precision(precision); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 
   return result;
 }  // T lambert_w0_small_z(const T z, std::integral_constant<int, 4> const&) float128
 
 #else
 
 template <typename T, typename Policy>
 inline T lambert_w0_small_z(const T z, const Policy& pol, std::integral_constant<int, 4> const&)
 {
    return lambert_w0_small_z(z, pol, std::integral_constant<int, 5>());
 }
 
 #endif // BOOST_HAS_FLOAT128
 
 //! Series functor to compute series term using pow and factorial.
 //! \details Functor is called after evaluating polynomial with the coefficients as rationals below.
 template <typename T>
 struct lambert_w0_small_z_series_term
 {
   using result_type = T;
   //! \param _z Lambert W argument z.
   //! \param -term  -pow<18>(z) / 6402373705728000uLL
   //! \param _k number of terms == initially 18
 
   //  Note *after* evaluating N terms, its internal state has k = N and term = (-1)^N z^N.
 
   lambert_w0_small_z_series_term(T _z, T _term, int _k)
     : k(_k), z(_z), term(_term) { }
 
   T operator()()
   { // Called by sum_series until needs precision set by factor (policy::get_epsilon).
     using std::pow;
     ++k;
     term *= -z / k;
     //T t = pow(z, k) * pow(T(k), -1 + k) / factorial<T>(k); // (z^k * k(k-1)^k) / k!
     T result = term * pow(T(k), T(-1 + k)); // term * k^(k-1)
                                          // std::cout << " k = " << k << ", term = " << term << ", result = " << result << std::endl;
     return result; //
   }
 private:
   int k;
   T z;
   T term;
 }; // template <typename T> struct lambert_w0_small_z_series_term
 
    //! Generic variant for T a User-defined types like Boost.Multiprecision.
 template <typename T, typename Policy>
 inline T lambert_w0_small_z(T z, const Policy& pol, std::integral_constant<int, 5> const&)
 {
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::streamsize precision = std::cout.precision(std::numeric_limits<T>::max_digits10); // Save.
   std::cout << "Generic lambert_w0_small_z called with z = " << z << " using as many terms needed for precision." << std::endl;
   std::cout << "Argument z is of type " << typeid(T).name() << std::endl;
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 
   // First several terms of the series are tabulated and evaluated as a polynomial:
   // this will save us a bunch of expensive calls to pow.
   // Then our series functor is initialized "as if" it had already reached term 18,
   // enough evaluation of built-in 64-bit double and float (and 80-bit long double?) types.
 
   // Coefficients should be stored such that the coefficients for the x^i terms are in poly[i].
   static const T coeff[] =
   {
     0, // z^0  Care: zeroth term needed by tools::evaluate_polynomial, but not in the Wolfram equation, so indexes are one different!
     1, // z^1 term.
     -1, // z^2 term
     static_cast<T>(3uLL) / 2uLL, // z^3 term.
     -static_cast<T>(8uLL) / 3uLL, // z^4
     static_cast<T>(125uLL) / 24uLL, // z^5
     -static_cast<T>(54uLL) / 5uLL, // z^6
     static_cast<T>(16807uLL) / 720uLL, // z^7
     -static_cast<T>(16384uLL) / 315uLL, // z^8
     static_cast<T>(531441uLL) / 4480uLL, // z^9
     -static_cast<T>(156250uLL) / 567uLL, // z^10
     static_cast<T>(2357947691uLL) / 3628800uLL, // z^11
     -static_cast<T>(2985984uLL) / 1925uLL, // z^12
     static_cast<T>(1792160394037uLL) / 479001600uLL, // z^13
     -static_cast<T>(7909306972uLL) / 868725uLL, // z^14
     static_cast<T>(320361328125uLL) / 14350336uLL, // z^15
     -static_cast<T>(35184372088832uLL) / 638512875uLL, // z^16
     static_cast<T>(2862423051509815793uLL) / 20922789888000uLL, // z^17 term
     -static_cast<T>(5083731656658uLL) / 14889875uLL,
     // z^18 term. = 136808.86090394293563342215789305735851647769682393
 
     // z^18 is biggest that can be computed as rational using the largest possible uLL integers,
     // so higher terms cannot be potentially compiler-computed as uLL rationals.
     // Wolfram (5083731656658 z ^ 18) / 14889875 or
     // -341422.05066583836331735491399356945575432970390954 z^18
 
     // See note below calling the functor to compute another term,
     // sufficient for 80-bit long double precision.
     // Wolfram -341422.05066583836331735491399356945575432970390954 z^19 term.
     // (5480386857784802185939 z^19)/6402373705728000
     // But now this variant is not used to compute long double
     // as specializations are provided above.
   }; // static const T coeff[]
 
      /*
      Table of 19 computed coefficients:
 
      #0 0
      #1 1
      #2 -1
      #3 1.5
      #4 -2.6666666666666666666666666666666665382713370408509
      #5 5.2083333333333333333333333333333330765426740817019
      #6 -10.800000000000000000000000000000000616297582203915
      #7 23.343055555555555555555555555555555076212991619177
      #8 -52.012698412698412698412698412698412659282693193402
      #9 118.62522321428571428571428571428571146835390992496
      #10 -275.57319223985890652557319223985891400375196748314
      #11 649.7871723434744268077601410934743969785223845882
      #12 -1551.1605194805194805194805194805194947599566007429
      #13 3741.4497029592385495163272941050719510009019331763
      #14 -9104.5002411580189357967135744913524243896052869184
      #15 22324.308512706601434280005708577137322392070452582
      #16 -55103.621972903835337697771560205423203318720697224
      #17 136808.86090394293563342215789305735851647769682393
          136808.86090394293563342215789305735851647769682393   == Exactly same as Wolfram computed value.
      #18 -341422.05066583836331735491399356947486381600607416
           341422.05066583836331735491399356945575432970390954  z^19  Wolfram value differs at 36 decimal digit, as expected.
      */
 
   using boost::math::policies::get_epsilon; // for type T.
   using boost::math::tools::sum_series;
   using boost::math::tools::evaluate_polynomial;
   // http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/roots/rational.html
 
   // std::streamsize prec = std::cout.precision(std::numeric_limits <T>::max_digits10);
 
   T result = evaluate_polynomial(coeff, z);
   //  template <std::size_t N, typename T, typename V>
   //  V evaluate_polynomial(const T(&poly)[N], const V& val);
   // Size of coeff found from N
   //std::cout << "evaluate_polynomial(coeff, z); == " << result << std::endl;
   //std::cout << "result = " << result << std::endl;
   // It's an artefact of the way I wrote the functor: *after* evaluating N
   // terms, its internal state has k = N and term = (-1)^N z^N.  So after
   // evaluating 18 terms, we initialize the functor to the term we've just
   // evaluated, and then when it's called, it increments itself to the next term.
   // So 18!is 6402373705728000, which is where that comes from.
 
   // The 19th coefficient of the polynomial is actually, 19 ^ 18 / 19!=
   // 104127350297911241532841 / 121645100408832000 which after removing GCDs
   // reduces down to Wolfram rational 5480386857784802185939 / 6402373705728000.
   // Wolfram z^19 term +(5480386857784802185939 z^19) /6402373705728000
   // +855992.96599660755146336302506332246623424823099755 z^19
 
   //! Evaluate Functor.
   lambert_w0_small_z_series_term<T> s(z, -pow<18>(z) / 6402373705728000uLL, 18);
 
   // Temporary to list the coefficients.
   //std::cout << " Table of coefficients" << std::endl;
   //std::streamsize saved_precision = std::cout.precision(50);
   //for (size_t i = 0; i != 19; i++)
   //{
   //  std::cout << "#" << i << " " << coeff[i] << std::endl;
   //}
   //std::cout.precision(saved_precision);
 
   std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); // Max iterations from policy.
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
   std::cout << "max iter from policy = " << max_iter << std::endl;
   // //   max iter from policy = 1000000 is default.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES
 
   result = sum_series(s, get_epsilon<T, Policy>(), max_iter, result);
   // result == evaluate_polynomial.
   //sum_series(Functor& func, int bits, std::uintmax_t& max_terms, const U& init_value)
   // std::cout << "sum_series(s, get_epsilon<T, Policy>(), max_iter, result); = " << result << std::endl;
 
   //T epsilon = get_epsilon<T, Policy>();
   //std::cout << "epsilon from policy = " << epsilon << std::endl;
   // epsilon from policy = 1.93e-34 for T == quad
   //  5.35e-51 for t = cpp_bin_float_50
 
   // std::cout << " get eps = " << get_epsilon<T, Policy>() << std::endl; // quad eps = 1.93e-34, bin_float_50 eps = 5.35e-51
   policies::check_series_iterations<T>("boost::math::lambert_w0_small_z<%1%>(%1%)", max_iter, pol);
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
   std::cout << "z = " << z << " needed  " << max_iter << " iterations." << std::endl;
   std::cout.precision(prec); // Restore.
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_W_SMALL_Z_SERIES_ITERATIONS
   return result;
 } // template <typename T, typename Policy> inline T lambert_w0_small_z_series(T z, const Policy& pol)
 
 // Approximate lambert_w0 (used for z values that are outside range of lookup table or rational functions)
 // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
 template <typename T>
 inline T lambert_w0_approx(T z)
 {
   BOOST_MATH_STD_USING
   T lz = log(z);
   T llz = log(lz);
   T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
   return w;
   // std::cout << "w max " << max_w << std::endl; // double 703.227
 }
 
   //////////////////////////////////////////////////////////////////////////////////////////
 
 //! \brief Lambert_w0 implementations for float, double and higher precisions.
 //! 3rd parameter used to select which version is used.
 
 //! /details Rational polynomials are provided for several range of argument z.
 //! For very small values of z, and for z very near the branch singularity at -e^-1 (~= -0.367879),
 //! two other series functions are used.
 
 //! float precision polynomials are used for 32-bit (usually float) precision (for speed)
 //! double precision polynomials are used for 64-bit (usually double) precision.
 //! For higher precisions, a 64-bit double approximation is computed first,
 //! and then refined using Halley iterations.
 
 template <typename T>
 inline T do_get_near_singularity_param(T z)
 {
    BOOST_MATH_STD_USING
    const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
    const T p = sqrt(p2);
    return p;
 }
 template <typename T, typename Policy>
 inline T get_near_singularity_param(T z, const Policy)
 {
    using value_type = typename policies::evaluation<T, Policy>::type;
    return static_cast<T>(do_get_near_singularity_param(static_cast<value_type>(z)));
 }
 
 // Forward declarations:
 
 //template <typename T, typename Policy> T lambert_w0_small_z(T z, const Policy& pol);
 //template <typename T, typename Policy>
 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 0>&); // 32 bit usually float.
 //template <typename T, typename Policy>
 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 1>&); //  64 bit usually double.
 //template <typename T, typename Policy>
 //T lambert_w0_imp(T w, const Policy& pol, const std::integral_constant<int, 2>&); // 80-bit long double.
 
 template <typename T>
 T lambert_w_positive_rational_float(T z)
 {
    BOOST_MATH_STD_USING
    if (z < 2)
    {
       if (z < T(0.5))
       { // 0.05 < z < 0.5
         // Maximum Deviation Found:                     2.993e-08
         // Expected Error Term : 2.993e-08
         // Maximum Relative Change in Control Points : 7.555e-04 Y offset : -8.196592331e-01
          static const T Y = 8.196592331e-01f;
          static const T P[] = {
             1.803388345e-01f,
             -4.820256838e-01f,
             -1.068349741e+00f,
             -3.506624319e-02f,
          };
          static const T Q[] = {
             1.000000000e+00f,
             2.871703469e+00f,
             1.690949264e+00f,
          };
          return z * (Y + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
       }
       else
       { // 0.5 < z < 2
         // Max error in interpolated form: 1.018e-08
          static const T Y = 5.503368378e-01f;
          static const T P[] = {
             4.493332766e-01f,
             2.543432707e-01f,
             -4.808788799e-01f,
             -1.244425316e-01f,
          };
          static const T Q[] = {
             1.000000000e+00f,
             2.780661241e+00f,
             1.830840318e+00f,
             2.407221031e-01f,
          };
          return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
       }
    }
    else if (z < 6)
    {
       // 2 < z < 6
       // Max error in interpolated form: 2.944e-08
       static const T Y = 1.162393570e+00f;
       static const T P[] = {
          -1.144183394e+00f,
          -4.712732855e-01f,
          1.563162512e-01f,
          1.434010911e-02f,
       };
       static const T Q[] = {
          1.000000000e+00f,
          1.192626340e+00f,
          2.295580708e-01f,
          5.477869455e-03f,
       };
       return Y + boost::math::tools::evaluate_rational(P, Q, z);
    }
    else if (z < 18)
    {
       // 6 < z < 18
       // Max error in interpolated form: 5.893e-08
       static const T Y = 1.809371948e+00f;
       static const T P[] = {
          -1.689291769e+00f,
          -3.337812742e-01f,
          3.151434873e-02f,
          1.134178734e-03f,
       };
       static const T Q[] = {
          1.000000000e+00f,
          5.716915685e-01f,
          4.489521292e-02f,
          4.076716763e-04f,
       };
       return Y + boost::math::tools::evaluate_rational(P, Q, z);
    }
    else if (z < T(9897.12905874))  // 2.8 < log(z) < 9.2
    {
       // Max error in interpolated form: 1.771e-08
       static const T Y = -1.402973175e+00f;
       static const T P[] = {
          1.966174312e+00f,
          2.350864728e-01f,
          -5.098074353e-02f,
          -1.054818339e-02f,
       };
       static const T Q[] = {
          1.000000000e+00f,
          4.388208264e-01f,
          8.316639634e-02f,
          3.397187918e-03f,
          -1.321489743e-05f,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
    }
    else if (z < T(7.896296e+13))  // 9.2 < log(z) <= 32
    {
       // Max error in interpolated form: 5.821e-08
       static const T Y = -2.735729218e+00f;
       static const T P[] = {
          3.424903470e+00f,
          7.525631787e-02f,
          -1.427309584e-02f,
          -1.435974178e-05f,
       };
       static const T Q[] = {
          1.000000000e+00f,
          2.514005579e-01f,
          6.118994652e-03f,
          -1.357889535e-05f,
          7.312865624e-08f,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
    }
 
     // Max error in interpolated form: 1.491e-08
     static const T Y = -4.012863159e+00f;
     static const T P[] = {
         4.431629226e+00f,
         2.756690487e-01f,
         -2.992956930e-03f,
         -4.912259384e-05f,
     };
     static const T Q[] = {
         1.000000000e+00f,
         2.015434591e-01f,
         4.949426142e-03f,
         1.609659944e-05f,
         -5.111523436e-09f,
     };
     T log_w = log(z);
     return log_w + Y + boost::math::tools::evaluate_polynomial(P, log_w) / boost::math::tools::evaluate_polynomial(Q, log_w);
 
 }
 
 template <typename T, typename Policy>
 T lambert_w_negative_rational_float(T z, const Policy& pol)
 {
    BOOST_MATH_STD_USING
    if (z > T(-0.27))
    {
       if (z < T(-0.051))
       {
          // -0.27 < z < -0.051
          // Max error in interpolated form: 5.080e-08
          static const T Y = 1.255809784e+00f;
          static const T P[] = {
             -2.558083412e-01f,
             -2.306524098e+00f,
             -5.630887033e+00f,
             -3.803974556e+00f,
          };
          static const T Q[] = {
             1.000000000e+00f,
             5.107680783e+00f,
             7.914062868e+00f,
             3.501498501e+00f,
          };
          return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
       }
       else
       {
          // Very small z so use a series function.
          return lambert_w0_small_z(z, pol);
       }
    }
    else if (z > T(-0.3578794411714423215955237701))
    { // Very close to branch singularity.
      // Max error in interpolated form: 5.269e-08
       static const T Y = 1.220928431e-01f;
       static const T P[] = {
          -1.221787446e-01f,
          -6.816155875e+00f,
          7.144582035e+01f,
          1.128444390e+03f,
       };
       static const T Q[] = {
          1.000000000e+00f,
          6.480326790e+01f,
          1.869145243e+02f,
          -1.361804274e+03f,
          1.117826726e+03f,
       };
       T d = z + 0.367879441171442321595523770161460867445811f;
       return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
    }
 
     return lambert_w_singularity_series(get_near_singularity_param(z, pol));
 }
 
 //! Lambert_w0 @b 'float' implementation, selected when T is 32-bit precision.
 template <typename T, typename Policy>
 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&)
 {
   static const char* function = "boost::math::lambert_w0<%1%>"; // For error messages.
   BOOST_MATH_STD_USING // Aid ADL of std functions.
 
   if ((boost::math::isnan)(z))
   {
     return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
   }
   if ((boost::math::isinf)(z))
   {
     return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
   }
 
    if (z >= T(0.05)) // Fukushima switch point.
    // if (z >= 0.045) // 34 terms makes 128-bit 'exact' below 0.045.
    { // Normal ranges using several rational polynomials.
       return lambert_w_positive_rational_float(z);
    }
    else if (z <= -0.3678794411714423215955237701614608674458111310f)
    {
       if (z < -0.3678794411714423215955237701614608674458111310f)
          return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
       return -1;
    }
 
    return lambert_w_negative_rational_float(z, pol);
 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 1>&) for 32-bit usually float.
 
 template <typename T>
 T lambert_w_positive_rational_double(T z)
 {
    BOOST_MATH_STD_USING
    if (z < 2)
    {
       if (z < 0.5)
       {
          // Max error in interpolated form: 2.255e-17
          static const T offset = 8.19659233093261719e-01;
          static const T P[] = {
             1.80340766906685177e-01,
             3.28178241493119307e-01,
             -2.19153620687139706e+00,
             -7.24750929074563990e+00,
             -7.28395876262524204e+00,
             -2.57417169492512916e+00,
             -2.31606948888704503e-01
          };
          static const T Q[] = {
             1.00000000000000000e+00,
             7.36482529307436604e+00,
             2.03686007856430677e+01,
             2.62864592096657307e+01,
             1.59742041380858333e+01,
             4.03760534788374589e+00,
             2.91327346750475362e-01
          };
          return z * (offset + boost::math::tools::evaluate_polynomial(P, z) / boost::math::tools::evaluate_polynomial(Q, z));
       }
       else
       {
          // Max error in interpolated form: 3.806e-18
          static const T offset = 5.50335884094238281e-01;
          static const T P[] = {
             4.49664083944098322e-01,
             1.90417666196776909e+00,
             1.99951368798255994e+00,
             -6.91217310299270265e-01,
             -1.88533935998617058e+00,
             -7.96743968047750836e-01,
             -1.02891726031055254e-01,
             -3.09156013592636568e-03
          };
          static const T Q[] = {
             1.00000000000000000e+00,
             6.45854489419584014e+00,
             1.54739232422116048e+01,
             1.72606164253337843e+01,
             9.29427055609544096e+00,
             2.29040824649748117e+00,
             2.21610620995418981e-01,
             5.70597669908194213e-03
          };
          return z * (offset + boost::math::tools::evaluate_rational(P, Q, z));
       }
    }
    else if (z < 6)
    {
       // 2 < z < 6
       // Max error in interpolated form: 1.216e-17
       static const T Y = 1.16239356994628906e+00;
       static const T P[] = {
          -1.16230494982099475e+00,
          -3.38528144432561136e+00,
          -2.55653717293161565e+00,
          -3.06755172989214189e-01,
          1.73149743765268289e-01,
          3.76906042860014206e-02,
          1.84552217624706666e-03,
          1.69434126904822116e-05,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          3.77187616711220819e+00,
          4.58799960260143701e+00,
          2.24101228462292447e+00,
          4.54794195426212385e-01,
          3.60761772095963982e-02,
          9.25176499518388571e-04,
          4.43611344705509378e-06,
       };
       return Y + boost::math::tools::evaluate_rational(P, Q, z);
    }
    else if (z < 18)
    {
       // 6 < z < 18
       // Max error in interpolated form: 1.985e-19
       static const T offset = 1.80937194824218750e+00;
       static const T P[] =
       {
          -1.80690935424793635e+00,
          -3.66995929380314602e+00,
          -1.93842957940149781e+00,
          -2.94269984375794040e-01,
          1.81224710627677778e-03,
          2.48166798603547447e-03,
          1.15806592415397245e-04,
          1.43105573216815533e-06,
          3.47281483428369604e-09
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          2.57319080723908597e+00,
          1.96724528442680658e+00,
          5.84501352882650722e-01,
          7.37152837939206240e-02,
          3.97368430940416778e-03,
          8.54941838187085088e-05,
          6.05713225608426678e-07,
          8.17517283816615732e-10
       };
       return offset + boost::math::tools::evaluate_rational(P, Q, z);
    }
    else if (z < 9897.12905874)  // 2.8 < log(z) < 9.2
    {
       // Max error in interpolated form: 1.195e-18
       static const T Y = -1.40297317504882812e+00;
       static const T P[] = {
          1.97011826279311924e+00,
          1.05639945701546704e+00,
          3.33434529073196304e-01,
          3.34619153200386816e-02,
          -5.36238353781326675e-03,
          -2.43901294871308604e-03,
          -2.13762095619085404e-04,
          -4.85531936495542274e-06,
          -2.02473518491905386e-08,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          8.60107275833921618e-01,
          4.10420467985504373e-01,
          1.18444884081994841e-01,
          2.16966505556021046e-02,
          2.24529766630769097e-03,
          9.82045090226437614e-05,
          1.36363515125489502e-06,
          3.44200749053237945e-09,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
    }
    else if (z < 7.896296e+13)  // 9.2 < log(z) <= 32
    {
       // Max error in interpolated form: 6.529e-18
       static const T Y = -2.73572921752929688e+00;
       static const T P[] = {
          3.30547638424076217e+00,
          1.64050071277550167e+00,
          4.57149576470736039e-01,
          4.03821227745424840e-02,
          -4.99664976882514362e-04,
          -1.28527893803052956e-04,
          -2.95470325373338738e-06,
          -1.76662025550202762e-08,
          -1.98721972463709290e-11,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          6.91472559412458759e-01,
          2.48154578891676774e-01,
          4.60893578284335263e-02,
          3.60207838982301946e-03,
          1.13001153242430471e-04,
          1.33690948263488455e-06,
          4.97253225968548872e-09,
          3.39460723731970550e-12,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
    }
    else if (z < 2.6881171e+43) // 32 < log(z) < 100
    {
       // Max error in interpolated form: 2.015e-18
       static const T Y = -4.01286315917968750e+00;
       static const T P[] = {
          5.07714858354309672e+00,
          -3.32994414518701458e+00,
          -8.61170416909864451e-01,
          -4.01139705309486142e-02,
          -1.85374201771834585e-04,
          1.08824145844270666e-05,
          1.17216905810452396e-07,
          2.97998248101385990e-10,
          1.42294856434176682e-13,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          -4.85840770639861485e-01,
          -3.18714850604827580e-01,
          -3.20966129264610534e-02,
          -1.06276178044267895e-03,
          -1.33597828642644955e-05,
          -6.27900905346219472e-08,
          -9.35271498075378319e-11,
          -2.60648331090076845e-14,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
    }
    else // 100 < log(z) < 710
    {
       // Max error in interpolated form: 5.277e-18
       static const T Y = -5.70115661621093750e+00;
       static const T P[] = {
          6.42275660145116698e+00,
          1.33047964073367945e+00,
          6.72008923401652816e-02,
          1.16444069958125895e-03,
          7.06966760237470501e-06,
          5.48974896149039165e-09,
          -7.00379652018853621e-11,
          -1.89247635913659556e-13,
          -1.55898770790170598e-16,
          -4.06109208815303157e-20,
          -2.21552699006496737e-24,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          3.34498588416632854e-01,
          2.51519862456384983e-02,
          6.81223810622416254e-04,
          7.94450897106903537e-06,
          4.30675039872881342e-08,
          1.10667669458467617e-10,
          1.31012240694192289e-13,
          6.53282047177727125e-17,
          1.11775518708172009e-20,
          3.78250395617836059e-25,
       };
       T log_w = log(z);
       return log_w + Y + boost::math::tools::evaluate_rational(P, Q, log_w);
    }
 }
 
 template <typename T, typename Policy>
 T lambert_w_negative_rational_double(T z, const Policy& pol)
 {
    BOOST_MATH_STD_USING
    if (z > -0.1)
    {
       if (z < -0.051)
       {
          // -0.1 < z < -0.051
          // Maximum Deviation Found:                     4.402e-22
          // Expected Error Term : 4.240e-22
          // Maximum Relative Change in Control Points : 4.115e-03
          static const T Y = 1.08633995056152344e+00;
          static const T P[] = {
             -8.63399505615014331e-02,
             -1.64303871814816464e+00,
             -7.71247913918273738e+00,
             -1.41014495545382454e+01,
             -1.02269079949257616e+01,
             -2.17236002836306691e+00,
          };
          static const T Q[] = {
             1.00000000000000000e+00,
             7.44775406945739243e+00,
             2.04392643087266541e+01,
             2.51001961077774193e+01,
             1.31256080849023319e+01,
             2.11640324843601588e+00,
          };
          return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
       }
       else
       {
          // Very small z > 0.051:
          return lambert_w0_small_z(z, pol);
       }
    }
    else if (z > -0.2)
    {
       // -0.2 < z < -0.1
       // Maximum Deviation Found:                     2.898e-20
       // Expected Error Term : 2.873e-20
       // Maximum Relative Change in Control Points : 3.779e-04
       static const T Y = 1.20359611511230469e+00;
       static const T P[] = {
          -2.03596115108465635e-01,
          -2.95029082937201859e+00,
          -1.54287922188671648e+01,
          -3.81185809571116965e+01,
          -4.66384358235575985e+01,
          -2.59282069989642468e+01,
          -4.70140451266553279e+00,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          9.57921436074599929e+00,
          3.60988119290234377e+01,
          6.73977699505546007e+01,
          6.41104992068148823e+01,
          2.82060127225153607e+01,
          4.10677610657724330e+00,
       };
       return z * (Y + boost::math::tools::evaluate_rational(P, Q, z));
    }
    else if (z > -0.3178794411714423215955237)
    {
       // Max error in interpolated form: 6.996e-18
       static const T Y = 3.49680423736572266e-01;
       static const T P[] = {
          -3.49729841718749014e-01,
          -6.28207407760709028e+01,
          -2.57226178029669171e+03,
          -2.50271008623093747e+04,
          1.11949239154711388e+05,
          1.85684566607844318e+06,
          4.80802490427638643e+06,
          2.76624752134636406e+06,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          1.82717661215113000e+02,
          8.00121119810280100e+03,
          1.06073266717010129e+05,
          3.22848993926057721e+05,
          -8.05684814514171256e+05,
          -2.59223192927265737e+06,
          -5.61719645211570871e+05,
          6.27765369292636844e+04,
       };
       T d = z + 0.367879441171442321595523770161460867445811;
       return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
    }
    else if (z > -0.3578794411714423215955237701)
    {
       // Max error in interpolated form: 1.404e-17
       static const T Y = 5.00126481056213379e-02;
       static const T  P[] = {
          -5.00173570682372162e-02,
          -4.44242461870072044e+01,
          -9.51185533619946042e+03,
          -5.88605699015429386e+05,
          -1.90760843597427751e+06,
          5.79797663818311404e+08,
          1.11383352508459134e+10,
          5.67791253678716467e+10,
          6.32694500716584572e+10,
       };
       static const T Q[] = {
          1.00000000000000000e+00,
          9.08910517489981551e+02,
          2.10170163753340133e+05,
          1.67858612416470327e+07,
          4.90435561733227953e+08,
          4.54978142622939917e+09,
          2.87716585708739168e+09,
          -4.59414247951143131e+10,
          -1.72845216404874299e+10,
       };
       T d = z + 0.36787944117144232159552377016146086744581113103176804;
       return -d / (Y + boost::math::tools::evaluate_polynomial(P, d) / boost::math::tools::evaluate_polynomial(Q, d));
    }
    else
    {  // z is very close (within 0.01) of the singularity at -e^-1,
       // so use a series expansion from R. M. Corless et al.
       const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
       const T p = sqrt(p2);
       return lambert_w_detail::lambert_w_singularity_series(p);
    }
 }
 
 //! Lambert_w0 @b 'double' implementation, selected when T is 64-bit precision.
 template <typename T, typename Policy>
 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&)
 {
    static const char* function = "boost::math::lambert_w0<%1%>";
    BOOST_MATH_STD_USING // Aid ADL of std functions.
 
    // Detect unusual case of 32-bit double with a wider/64-bit long double
    static_assert(std::numeric_limits<double>::digits >= 53,
    "Our double precision coefficients will be truncated, "
    "please file a bug report with details of your platform's floating point types "
    "- or possibly edit the coefficients to have "
    "an appropriate size-suffix for 64-bit floats on your platform - L?");
 
     if ((boost::math::isnan)(z))
     {
       return boost::math::policies::raise_domain_error<T>(function, "Expected a value > -e^-1 (-0.367879...) but got %1%.", z, pol);
     }
     if ((boost::math::isinf)(z))
     {
       return boost::math::policies::raise_overflow_error<T>(function, "Expected a finite value but got %1%.", z, pol);
     }
 
    if (z >= 0.05)
    {
       return lambert_w_positive_rational_double(z);
    }
    else if (z <= -0.36787944117144232159552377016146086744581113103176804) // Precision is max_digits10(cpp_bin_float_50).
    {
       if (z < -0.36787944117144232159552377016146086744581113103176804)
       {
          return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
       }
       return -1;
    }
    else
    {
       return lambert_w_negative_rational_double(z, pol);
    }
 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 2>&) 64-bit precision, usually double.
 
 //! lambert_W0 implementation for extended precision types including
 //! long double (80-bit and 128-bit), ???
 //! quad float128, Boost.Multiprecision types like cpp_bin_float_quad, cpp_bin_float_50...
 
 template <typename T, typename Policy>
 inline T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&)
 {
    static const char* function = "boost::math::lambert_w0<%1%>";
    BOOST_MATH_STD_USING // Aid ADL of std functions.
 
    // Filter out special cases first:
    if ((boost::math::isnan)(z))
    {
       return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
    }
    if (fabs(z) <= 0.05f)
    {
       // Very small z:
       return lambert_w0_small_z(z, pol);
    }
    if (z > (std::numeric_limits<double>::max)())
    {
       if ((boost::math::isinf)(z))
       {
          return policies::raise_overflow_error<T>(function, nullptr, pol);
          // Or might return infinity if available else max_value,
          // but other Boost.Math special functions raise overflow.
       }
       // z is larger than the largest double, so cannot use the polynomial to get an approximation,
       // so use the asymptotic approximation and Halley iterate:
 
      T w = lambert_w0_approx(z);  // Make an inline function as also used elsewhere.
       //T lz = log(z);
       //T llz = log(lz);
       //T w = lz - llz + (llz / lz); // Corless equation 4.19, page 349, and Chapeau-Blondeau equation 20, page 2162.
       return lambert_w_halley_iterate(w, z);
    }
    if (z < -0.3578794411714423215955237701)
    { // Very close to branch point so rational polynomials are not usable.
       if (z <= -boost::math::constants::exp_minus_one<T>())
       {
          if (z == -boost::math::constants::exp_minus_one<T>())
          { // Exactly at the branch point singularity.
             return -1;
          }
          return boost::math::policies::raise_domain_error<T>(function, "Expected z >= -e^-1 (-0.367879...) but got %1%.", z, pol);
       }
       // z is very close (within 0.01) of the branch singularity at -e^-1
       // so use a series approximation proposed by Corless et al.
       const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
       const T p = sqrt(p2);
       T w = lambert_w_detail::lambert_w_singularity_series(p);
       return lambert_w_halley_iterate(w, z);
    }
 
    // Phew!  If we get here we are in the normal range of the function,
    // so get a double precision approximation first, then iterate to full precision of T.
    // We define a tag_type that is:
    // true_type if there are so many digits precision wanted that iteration is necessary.
    // false_type if a single Halley step is sufficient.
 
    using precision_type = typename policies::precision<T, Policy>::type;
    using tag_type = std::integral_constant<bool,
       (precision_type::value == 0) || (precision_type::value > 113) ?
       true // Unknown at compile-time, variable/arbitrary, or more than float128 or cpp_bin_quad 128-bit precision.
       : false // float, double, float128, cpp_bin_quad 128-bit, so single Halley step.
    >;
 
    // For speed, we also cast z to type double when that is possible
    //   if (std::is_constructible<double, T>() == true).
    T w = lambert_w0_imp(maybe_reduce_to_double(z, std::is_constructible<double, T>()), pol, std::integral_constant<int, 2>());
 
    return lambert_w_maybe_halley_iterate(w, z, tag_type());
 
 } // T lambert_w0_imp(T z, const Policy& pol, const std::integral_constant<int, 0>&)  all extended precision types.
 
   // Lambert w-1 implementation
 // ==============================================================================================
 
   //! Lambert W for W-1 branch, -max(z) < z <= -1/e.
   // TODO is -max(z) allowed?
 template<typename T, typename Policy>
 T lambert_wm1_imp(const T z, const Policy&  pol)
 {
   // Catch providing an integer value as parameter x to lambert_w, for example, lambert_w(1).
   // Need to ensure it is a floating-point type (of the desired type, float 1.F, double 1., or long double 1.L),
   // or static_casted integer, for example:  static_cast<float>(1) or static_cast<cpp_dec_float_50>(1).
   // Want to allow fixed_point types too, so do not just test for floating-point.
   // Integral types should be promoted to double by user Lambert w functions.
   // If integral type provided to user function lambert_w0 or lambert_wm1,
   // then should already have been promoted to double.
   static_assert(!std::is_integral<T>::value,
     "Must be floating-point or fixed type (not integer type), for example: lambert_wm1(1.), not lambert_wm1(1)!");
 
   BOOST_MATH_STD_USING // Aid argument dependent lookup (ADL) of abs.
 
   const char* function = "boost::math::lambert_wm1<RealType>(<RealType>)"; // Used for error messages.
 
   // Check for edge and corner cases first:
   if ((boost::math::isnan)(z))
   {
     return policies::raise_domain_error(function,
       "Argument z is NaN!",
       z, pol);
   } // isnan
 
   if ((boost::math::isinf)(z))
   {
     return policies::raise_domain_error(function,
       "Argument z is infinite!",
       z, pol);
   } // isinf
 
   if (z == static_cast<T>(0))
   { // z is exactly zero so return -std::numeric_limits<T>::infinity();
     if (std::numeric_limits<T>::has_infinity)
     {
       return -std::numeric_limits<T>::infinity();
     }
     else
     {
       return -tools::max_value<T>();
     }
   }
   if (boost::math::detail::has_denorm_now<T>())
   { // All real types except arbitrary precision.
     if (!(boost::math::isnormal)(z))
     { // Almost zero - might also just return infinity like z == 0 or max_value?
       return policies::raise_overflow_error(function,
         "Argument z =  %1% is denormalized! (must be z > (std::numeric_limits<RealType>::min)() or z == 0)",
         z, pol);
     }
   }
 
   if (z > static_cast<T>(0))
   { //
     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);
   }
   if (z > -boost::math::tools::min_value<T>())
   { // z is denormalized, so cannot be computed.
     // -std::numeric_limits<T>::min() is smallest for type T,
     // for example, for double: lambert_wm1(-2.2250738585072014e-308) = -714.96865723796634
     return policies::raise_overflow_error(function,
       "Argument z = %1% is too small (z < -std::numeric_limits<T>::min so denormalized) for Lambert W-1 branch!",
       z, pol);
   }
   if (z == -boost::math::constants::exp_minus_one<T>()) // == singularity/branch point z = -exp(-1) = -3.6787944.
   { // At singularity, so return exactly -1.
     return -static_cast<T>(1);
   }
   // z is too negative for the W-1 (or W0) branch.
   if (z < -boost::math::constants::exp_minus_one<T>()) // > singularity/branch point z = -exp(-1) = -3.6787944.
   {
     return policies::raise_domain_error(function,
       "Argument z = %1% is out of range (z < -exp(-1) = -3.6787944... <= 0) for Lambert W-1 (or W0) branch!",
       z, pol);
   }
   if (z < static_cast<T>(-0.35))
   { // Close to singularity/branch point z = -0.3678794411714423215955237701614608727 but on W-1 branch.
     const T p2 = 2 * (boost::math::constants::e<T>() * z + 1);
     if (p2 == 0)
     { // At the singularity at branch point.
       return -1;
     }
     if (p2 > 0)
     {
       T w_series = lambert_w_singularity_series(T(-sqrt(p2)));
       if (boost::math::tools::digits<T>() > 53)
       { // Multiprecision, so try a Halley refinement.
         w_series = lambert_w_detail::lambert_w_halley_iterate(w_series, z);
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
         std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
         std::cout << "Lambert W-1 Halley updated to " << w_series << std::endl;
         std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
       }
       return w_series;
     }
     // Should not get here.
     return policies::raise_domain_error(function,
       "Argument z = %1% is out of range for Lambert W-1 branch. (Should not get here - please report!)",
       z, pol);
   } // if (z < -0.35)
 
   using lambert_w_lookup::wm1es;
   using lambert_w_lookup::wm1zs;
   using lambert_w_lookup::noof_wm1zs; // size == 64
 
   // std::cout <<" Wm1zs[63] (== G[64]) = " << " " << wm1zs[63] << std::endl; // Wm1zs[63] (== G[64]) =  -1.0264389699511283e-26
   // Check that z argument value is not smaller than lookup_table G[64]
   // std::cout << "(z > wm1zs[63]) = " << std::boolalpha << (z > wm1zs[63]) << std::endl;
 
   if (z >= T(wm1zs[63])) // wm1zs[63]  = -1.0264389699511282259046957018510946438e-26L  W = 64.00000000000000000
   {  // z >= -1.0264389699511303e-26 (but z != 0 and z >= std::numeric_limits<T>::min() and so NOT denormalized).
 
     // Some info on Lambert W-1 values for extreme values of z.
     // std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
     // std::cout << "-std::numeric_limits<float>::min() = " << -(std::numeric_limits<float>::min)() << std::endl;
     // std::cout << "-std::numeric_limits<double>::min() = " << -(std::numeric_limits<double>::min)() << std::endl;
     // -std::numeric_limits<float>::min() = -1.1754943508222875e-38
     // -std::numeric_limits<double>::min() = -2.2250738585072014e-308
     // N[productlog(-1, -1.1754943508222875 * 10^-38 ), 50] = -91.856775324595479509567756730093823993834155027858
     // N[productlog(-1, -2.2250738585072014e-308 * 10^-308 ), 50] = -1424.8544521230553853558132180518404363617968042942
     // N[productlog(-1, -1.4325445274604020119111357113179868158* 10^-27), 37] = -65.99999999999999999999999999999999955
 
     // R.M.Corless, G.H.Gonnet, D.E.G.Hare, D.J.Jeffrey, and D.E.Knuth,
     // On the Lambert W function, Adv.Comput.Math., vol. 5, pp. 329, 1996.
     // Francois Chapeau-Blondeau and Abdelilah Monir
     // Numerical Evaluation of the Lambert W Function
     // IEEE Transactions On Signal Processing, VOL. 50, NO. 9, Sep 2002
     // https://pdfs.semanticscholar.org/7a5a/76a9369586dd0dd34dda156d8f2779d1fd59.pdf
     // Estimate Lambert W using ln(-z)  ...
     // This is roughly the power of ten * ln(10) ~= 2.3.   n ~= 10^n
     //  and improve by adding a second term -ln(ln(-z))
     T guess; // bisect lowest possible Gk[=64] (for lookup_t type)
     T lz = log(-z);
     T llz = log(-lz);
     guess = lz - llz + (llz / lz); // Chapeau-Blondeau equation 20, page 2162.
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
     std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
     std::cout << "z = " << z << ", guess = " << guess << ", ln(-z) = " << lz << ", ln(-ln(-z) = " << llz << ", llz/lz = " << (llz / lz) << std::endl;
     // z = -1.0000000000000001e-30, guess = -73.312782616731482, ln(-z) = -69.077552789821368, ln(-ln(-z) = 4.2352298269101114, llz/lz = -0.061311231447304194
     // z = -9.9999999999999999e-91, guess = -212.56650048504233, ln(-z) = -207.23265836946410, ln(-ln(-z) = 5.3338421155782205, llz/lz = -0.025738424423764311
     // >z = -2.2250738585072014e-308, guess = -714.95942238244606, ln(-z) = -708.39641853226408, ln(-ln(-z) = 6.5630038501819854, llz/lz = -0.0092645920821846622
     int d10 = policies::digits_base10<T, Policy>(); // policy template parameter digits10
     int d2 = policies::digits<T, Policy>(); // digits base 2 from policy.
     std::cout << "digits10 = " << d10 << ", digits2 = " << d2 // For example: digits10 = 1, digits2 = 5
       << std::endl;
     std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_TINY
     if (policies::digits<T, Policy>() < 12)
     { // 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.
       return guess;
     }
     T result = lambert_w_detail::lambert_w_halley_iterate(guess, z);
     return result;
 
     // Was Fukushima
     // G[k=64] == g[63] == -1.02643897e-26
     //return policies::raise_domain_error(function,
     //  "Argument z = %1% is too small (< -1.02643897e-26) ! (Should not occur, please report.",
     //  z, pol);
   } // Z too small so use approximation and Halley.
     // Else Use a lookup table to find the nearest integer part of Lambert W-1 as starting point for Bisection.
 
   if (boost::math::tools::digits<T>() > 53)
   { // T is more precise than 64-bit double (or long double, or ?),
     // so compute an approximate value using only one Schroeder refinement,
     // (avoiding any double-precision Halley refinement from policy double_digits2<50> 53 - 3 = 50
     // because are next going to use Halley refinement at full/high precision using this as an approximation).
     using boost::math::policies::precision;
     using boost::math::policies::digits10;
     using boost::math::policies::digits2;
     using boost::math::policies::policy;
     // Compute a 50-bit precision approximate W0 in a double (no Halley refinement).
     T double_approx(static_cast<T>(lambert_wm1_imp(must_reduce_to_double(z, std::is_constructible<double, T>()), policy<digits2<50>>())));
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_NOT_BUILTIN
     std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
     std::cout << "Lambert_wm1 Argument Type " << typeid(T).name() << " approximation double = " << double_approx << std::endl;
     std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
     // Perform additional Halley refinement(s) to ensure that
     // get a near as possible to correct result (usually +/- one epsilon).
     T result = lambert_w_halley_iterate(double_approx, z);
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1
     std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
     std::cout << "Result " << typeid(T).name() << " precision Halley refinement =    " << result << std::endl;
     std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1
     return result;
   } // digits > 53  - higher precision than double.
   else // T is double or less precision.
   { // Use a lookup table to find the nearest integer part of Lambert W as starting point for Bisection.
     using namespace boost::math::lambert_w_detail::lambert_w_lookup;
     // Bracketing sequence  n = (2, 4, 8, 16, 32, 64) for W-1 branch. (0 is -infinity)
     // Since z is probably quite small, start with lowest n (=2).
     int n = 2;
     if (T(wm1zs[n - 1]) > z)
     {
       goto bisect;
     }
     for (int j = 1; j <= 5; ++j)
     {
       n *= 2;
       if (T(wm1zs[n - 1]) > z)
       {
         goto overshot;
       }
     }
     // else z < g[63] == -1.0264389699511303e-26, so Lambert W-1 integer part > 64.
     // This should not now occur (should be caught by test and code above) so should be a logic_error?
     return policies::raise_domain_error(function,
       "Argument z = %1% is too small (< -1.026439e-26) (logic error - please report!)",
       z, pol);
   overshot:
     {
       int nh = n / 2;
       for (int j = 1; j <= 5; ++j)
       {
         nh /= 2; // halve step size.
         if (nh <= 0)
         {
           break; // goto bisect;
         }
         if (T(wm1zs[n - nh - 1]) > z)
         {
           n -= nh;
         }
       }
     }
   bisect:
     --n;
     // g[n] now holds lambert W of floor integer n and g[n+1] the ceil part;
     // these are used as initial values for bisection.
 #ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
     std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
     std::cout << "Result lookup W-1(" << z << ") bisection between wm1zs[" << n - 1 << "] = " << wm1zs[n - 1] << " and wm1zs[" << n << "] = " << wm1zs[n]
       << ", bisect mean = " << (wm1zs[n - 1] + wm1zs[n]) / 2 << std::endl;
     std::cout.precision(saved_precision);
 #endif // BOOST_MATH_INSTRUMENT_LAMBERT_WM1_LOOKUP
 
     // Compute bisections is the number of bisections computed from n,
     // such that a single application of the fifth-order Schroeder update formula
     // after the bisections is enough to evaluate Lambert W-1 with (near?) 53-bit accuracy.
     // Fukushima established these by trial and error?
     int bisections = 11; //  Assume maximum number of bisections will be needed (most common case).
     if (n >= 8)
     {
       bisections = 8;
     }
     else if (n >= 3)
     {
       bisections = 9;
     }
     else if (n >= 2)
     {
       bisections = 10;
     }
     // Bracketing, Fukushima section 2.3, page 82:
     // (Avoiding using exponential function for speed).
     // Only use @c lookup_t precision, default double, for bisection (again for speed),
     // and use later Halley refinement for higher precisions.
     using lambert_w_lookup::halves;
     using lambert_w_lookup::sqrtwm1s;
 
     using calc_type = typename std::conditional<std::is_constructible<lookup_t, T>::value, lookup_t, T>::type;
 
     calc_type w = -static_cast<calc_type>(n); // Equation 25,
     calc_type y = static_cast<calc_type>(z * T(wm1es[n - 1])); // Equation 26,
                                                           // Perform the bisections fractional bisections for necessary precision.
     for (int j = 0; j < bisections; ++j)
     { // Equation 27.
       calc_type wj = w - halves[j]; // Subtract 1/2, 1/4, 1/8 ...
       calc_type yj = y * sqrtwm1s[j]; // Multiply by sqrt(1/e), ...
       if (wj < yj)
       {
         w = wj;
         y = yj;
       }
     } // for j
     return static_cast<T>(schroeder_update(w, y)); // Schroeder 5th order method refinement.
 
 //      else // Perform additional Halley refinement(s) to ensure that
 //           // get a near as possible to correct result (usually +/- epsilon).
 //      {
 //       // result = lambert_w_halley_iterate(result, z);
 //        result = lambert_w_halley_step(result, z);  // Just one Halley step should be enough.
 //#ifdef BOOST_MATH_INSTRUMENT_LAMBERT_WM1_HALLEY
 //        std::streamsize saved_precision = std::cout.precision(std::numeric_limits<T>::max_digits10);
 //        std::cout << "Halley refinement estimate =    " << result << std::endl;
 //        std::cout.precision(saved_precision);
 //#endif // BOOST_MATH_INSTRUMENT_LAMBERT_W1_HALLEY
 //        return result; // Halley
 //      } // Schroeder or Schroeder and Halley.
     }
   } // template<typename T = double> T lambert_wm1_imp(const T z)
 } // namespace lambert_w_detail
 
 /////////////////////////////  User Lambert w functions. //////////////////////////////
 
 //! Lambert W0 using User-defined policy.
   template <typename T, typename Policy>
   inline
     typename boost::math::tools::promote_args<T>::type
     lambert_w0(T z, const Policy& pol)
   {
      // Promote integer or expression template arguments to double,
      // without doing any other internal promotion like float to double.
     using result_type = typename tools::promote_args<T>::type;
 
     // Work out what precision has been selected,
     // based on the Policy and the number type.
     using precision_type = typename policies::precision<result_type, Policy>::type;
     // and then select the correct implementation based on that precision (not the type T):
     using tag_type = std::integral_constant<int,
       (precision_type::value == 0) || (precision_type::value > 53) ?
         0  // either variable precision (0), or greater than 64-bit precision.
       : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
       : 2  // 64-bit (probably double) precision.
       >;
 
     return lambert_w_detail::lambert_w0_imp(result_type(z), pol, tag_type()); //
   } // lambert_w0(T z, const Policy& pol)
 
   //! Lambert W0 using default policy.
   template <typename T>
   inline
     typename tools::promote_args<T>::type
     lambert_w0(T z)
   {
     // Promote integer or expression template arguments to double,
     // without doing any other internal promotion like float to double.
     using result_type = typename tools::promote_args<T>::type;
 
     // Work out what precision has been selected, based on the Policy and the number type.
     // For the default policy version, we want the *default policy* precision for T.
     using precision_type = typename policies::precision<result_type, policies::policy<>>::type;
     // and then select the correct implementation based on that (not the type T):
     using tag_type = std::integral_constant<int,
       (precision_type::value == 0) || (precision_type::value > 53) ?
       0  // either variable precision (0), or greater than 64-bit precision.
       : (precision_type::value <= 24) ? 1 // 32-bit (probably float) precision.
       : 2  // 64-bit (probably double) precision.
     >;
     return lambert_w_detail::lambert_w0_imp(result_type(z),  policies::policy<>(), tag_type());
   } // lambert_w0(T z) using default policy.
 
     //! W-1 branch (-max(z) < z <= -1/e).
 
     //! Lambert W-1 using User-defined policy.
   template <typename T, typename Policy>
   inline
     typename tools::promote_args<T>::type
     lambert_wm1(T z, const Policy& pol)
   {
     // Promote integer or expression template arguments to double,
     // without doing any other internal promotion like float to double.
     using result_type = typename tools::promote_args<T>::type;
     return lambert_w_detail::lambert_wm1_imp(result_type(z), pol); //
   }
 
   //! Lambert W-1 using default policy.
   template <typename T>
   inline
     typename tools::promote_args<T>::type
     lambert_wm1(T z)
   {
     using result_type = typename tools::promote_args<T>::type;
     return lambert_w_detail::lambert_wm1_imp(result_type(z), policies::policy<>());
   } // lambert_wm1(T z)
 
   // First derivative of Lambert W0 and W-1.
   template <typename T, typename Policy>
   inline typename tools::promote_args<T>::type
   lambert_w0_prime(T z, const Policy& pol)
   {
     using result_type = typename tools::promote_args<T>::type;
     using std::numeric_limits;
     if (z == 0)
     {
         return static_cast<result_type>(1);
     }
     // This is the sensible choice if we regard the Lambert-W function as complex analytic.
     // Of course on the real line, it's just undefined.
     if (z == - boost::math::constants::exp_minus_one<result_type>())
     {
         return numeric_limits<result_type>::has_infinity ? numeric_limits<result_type>::infinity() : boost::math::tools::max_value<result_type>();
     }
     // if z < -1/e, we'll let lambert_w0 do the error handling:
     result_type w = lambert_w0(result_type(z), pol);
     // If w ~ -1, then presumably this can get inaccurate.
     // Is there an accurate way to evaluate 1 + W(-1/e + eps)?
     //  Yes: This is discussed in the Princeton Companion to Applied Mathematics,
     // 'The Lambert-W function', Section 1.3: Series and Generating Functions.
     // 1 + W(-1/e + x) ~ sqrt(2ex).
     // Nick is not convinced this formula is more accurate than the naive one.
     // However, for z != -1/e, we never get rounded to w = -1 in any precision I've tested (up to cpp_bin_float_100).
     return w / (z * (1 + w));
   } // lambert_w0_prime(T z)
 
   template <typename T>
   inline typename tools::promote_args<T>::type
      lambert_w0_prime(T z)
   {
      return lambert_w0_prime(z, policies::policy<>());
   }
 
   template <typename T, typename Policy>
   inline typename tools::promote_args<T>::type
   lambert_wm1_prime(T z, const Policy& pol)
   {
     using std::numeric_limits;
     using result_type = typename tools::promote_args<T>::type;
     //if (z == 0)
     //{
     //      return static_cast<result_type>(1);
     //}
     //if (z == - boost::math::constants::exp_minus_one<result_type>())
     if (z == 0 || z == - boost::math::constants::exp_minus_one<result_type>())
     {
         return numeric_limits<result_type>::has_infinity ? -numeric_limits<result_type>::infinity() : -boost::math::tools::max_value<result_type>();
     }
 
     result_type w = lambert_wm1(z, pol);
     return w/(z*(1+w));
   } // lambert_wm1_prime(T z)
 
   template <typename T>
   inline typename tools::promote_args<T>::type
      lambert_wm1_prime(T z)
   {
      return lambert_wm1_prime(z, policies::policy<>());
   }
 
 }} //boost::math namespaces
 
 #endif // #ifdef BOOST_MATH_SF_LAMBERT_W_HPP
 

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