text: /***************************************************************************
   * Copyright (c) Johan Mabille, Sylvain Corlay, Wolf Vollprecht and         *
   * Martin Renou                                                             *
   * Copyright (c) QuantStack                                                 *
   * Copyright (c) Serge Guelton                                              *
   *                                                                          *
   * Distributed under the terms of the BSD 3-Clause License.                 *
   *                                                                          *
   * The full license is in the file LICENSE, distributed with this software. *
   ****************************************************************************/
  
  #ifndef XSIMD_GENERIC_MATH_HPP
  #define XSIMD_GENERIC_MATH_HPP
  
  #include "../xsimd_scalar.hpp"
  #include "./xsimd_generic_details.hpp"
  #include "./xsimd_generic_trigo.hpp"
  
  #include <type_traits>
  
  namespace xsimd
  {
  
      namespace kernel
      {
  
          using namespace types;
          // abs
          template <class A, class T, class /*=typename std::enable_if<std::is_integral<T>::value, void>::type*/>
          inline batch<T, A> abs(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              if (std::is_unsigned<T>::value)
                  return self;
              else
              {
                  auto sign = bitofsign(self);
                  auto inv = self ^ sign;
                  return inv - sign;
              }
          }
  
          template <class A, class T>
          inline batch<T, A> abs(batch<std::complex<T>, A> const& z, requires_arch<generic>) noexcept
          {
              return hypot(z.real(), z.imag());
          }
  
          // batch_cast
          template <class A, class T>
          inline batch<T, A> batch_cast(batch<T, A> const& self, batch<T, A> const&, requires_arch<generic>) noexcept
          {
              return self;
          }
  
          namespace detail
          {
              template <class A, class T_out, class T_in>
              inline batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const& out, requires_arch<generic>, with_fast_conversion) noexcept
              {
                  return fast_cast(self, out, A {});
              }
              template <class A, class T_out, class T_in>
              inline batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const&, requires_arch<generic>, with_slow_conversion) noexcept
              {
                  static_assert(!std::is_same<T_in, T_out>::value, "there should be no conversion for this type combination");
                  using batch_type_in = batch<T_in, A>;
                  using batch_type_out = batch<T_out, A>;
                  static_assert(batch_type_in::size == batch_type_out::size, "compatible sizes");
                  alignas(A::alignment()) T_in buffer_in[batch_type_in::size];
                  alignas(A::alignment()) T_out buffer_out[batch_type_out::size];
                  self.store_aligned(&buffer_in[0]);
                  std::copy(std::begin(buffer_in), std::end(buffer_in), std::begin(buffer_out));
                  return batch_type_out::load_aligned(buffer_out);
              }
  
          }
  
          template <class A, class T_out, class T_in>
          inline batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const& out, requires_arch<generic>) noexcept
          {
              return detail::batch_cast(self, out, A {}, detail::conversion_type<A, T_in, T_out> {});
          }
  
          // bitofsign
          template <class A, class T>
          inline batch<T, A> bitofsign(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              static_assert(std::is_integral<T>::value, "int type implementation");
              if (std::is_unsigned<T>::value)
                  return batch<T, A>(0);
              else
                  return self >> (T)(8 * sizeof(T) - 1);
          }
  
          template <class A>
          inline batch<float, A> bitofsign(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              return self & constants::minuszero<batch<float, A>>();
          }
          template <class A>
          inline batch<double, A> bitofsign(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              return self & constants::minuszero<batch<double, A>>();
          }
  
          // cbrt
          /* origin: boost/simd/arch/common/simd/function/cbrt.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A>
          inline batch<float, A> cbrt(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              batch_type z = abs(self);
  #ifndef XSIMD_NO_DENORMALS
              auto denormal = z < constants::smallestposval<batch_type>();
              z = select(denormal, z * constants::twotonmb<batch_type>(), z);
              batch_type f = select(denormal, constants::twotonmbo3<batch_type>(), batch_type(1.));
  #endif
              const batch_type CBRT2(bit_cast<float>(0x3fa14518));
              const batch_type CBRT4(bit_cast<float>(0x3fcb2ff5));
              const batch_type CBRT2I(bit_cast<float>(0x3f4b2ff5));
              const batch_type CBRT4I(bit_cast<float>(0x3f214518));
              using i_type = as_integer_t<batch_type>;
              i_type e;
              batch_type x = frexp(z, e);
              x = detail::horner<batch_type,
                                 0x3ece0609,
                                 0x3f91eb77,
                                 0xbf745265,
                                 0x3f0bf0fe,
                                 0xbe09e49a>(x);
              auto flag = e >= i_type(0);
              i_type e1 = abs(e);
              i_type rem = e1;
              e1 /= i_type(3);
              rem -= e1 * i_type(3);
              e = e1 * sign(e);
              const batch_type cbrt2 = select(bool_cast(flag), CBRT2, CBRT2I);
              const batch_type cbrt4 = select(bool_cast(flag), CBRT4, CBRT4I);
              batch_type fact = select(bool_cast(rem == i_type(1)), cbrt2, batch_type(1.));
              fact = select(bool_cast(rem == i_type(2)), cbrt4, fact);
              x = ldexp(x * fact, e);
              x -= (x - z / (x * x)) * batch_type(1.f / 3.f);
  #ifndef XSIMD_NO_DENORMALS
              x = (x | bitofsign(self)) * f;
  #else
              x = x | bitofsign(self);
  #endif
  #ifndef XSIMD_NO_INFINITIES
              return select(self == batch_type(0.) || isinf(self), self, x);
  #else
              return select(self == batch_type(0.), self, x);
  #endif
          }
  
          template <class A>
          inline batch<double, A> cbrt(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              batch_type z = abs(self);
  #ifndef XSIMD_NO_DENORMALS
              auto denormal = z < constants::smallestposval<batch_type>();
              z = select(denormal, z * constants::twotonmb<batch_type>(), z);
              batch_type f = select(denormal, constants::twotonmbo3<batch_type>(), batch_type(1.));
  #endif
              const batch_type CBRT2(bit_cast<double>(int64_t(0x3ff428a2f98d728b)));
              const batch_type CBRT4(bit_cast<double>(int64_t(0x3ff965fea53d6e3d)));
              const batch_type CBRT2I(bit_cast<double>(int64_t(0x3fe965fea53d6e3d)));
              const batch_type CBRT4I(bit_cast<double>(int64_t(0x3fe428a2f98d728b)));
              using i_type = as_integer_t<batch_type>;
              i_type e;
              batch_type x = frexp(z, e);
              x = detail::horner<batch_type,
                                 0x3fd9c0c12122a4feull,
                                 0x3ff23d6ee505873aull,
                                 0xbfee8a4ca3ba37b8ull,
                                 0x3fe17e1fc7e59d58ull,
                                 0xbfc13c93386fdff6ull>(x);
              auto flag = e >= typename i_type::value_type(0);
              i_type e1 = abs(e);
              i_type rem = e1;
              e1 /= i_type(3);
              rem -= e1 * i_type(3);
              e = e1 * sign(e);
              const batch_type cbrt2 = select(bool_cast(flag), CBRT2, CBRT2I);
              const batch_type cbrt4 = select(bool_cast(flag), CBRT4, CBRT4I);
              batch_type fact = select(bool_cast(rem == i_type(1)), cbrt2, batch_type(1.));
              fact = select(bool_cast(rem == i_type(2)), cbrt4, fact);
              x = ldexp(x * fact, e);
              x -= (x - z / (x * x)) * batch_type(1. / 3.);
              x -= (x - z / (x * x)) * batch_type(1. / 3.);
  #ifndef XSIMD_NO_DENORMALS
              x = (x | bitofsign(self)) * f;
  #else
              x = x | bitofsign(self);
  #endif
  #ifndef XSIMD_NO_INFINITIES
              return select(self == batch_type(0.) || isinf(self), self, x);
  #else
              return select(self == batch_type(0.), self, x);
  #endif
          }
  
          // clip
          template <class A, class T>
          inline batch<T, A> clip(batch<T, A> const& self, batch<T, A> const& lo, batch<T, A> const& hi, requires_arch<generic>) noexcept
          {
              return min(hi, max(self, lo));
          }
  
          // copysign
          template <class A, class T>
          inline batch<T, A> copysign(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              return abs(self) | bitofsign(other);
          }
  
          // erf
  
          namespace detail
          {
              /* origin: boost/simd/arch/common/detail/generic/erf_kernel.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class B>
              struct erf_kernel;
  
              template <class A>
              struct erf_kernel<batch<float, A>>
              {
                  using batch_type = batch<float, A>;
                  // computes erf(a0)/a0
                  // x is sqr(a0) and 0 <= abs(a0) <= 2/3
                  static inline batch_type erf1(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0x3f906eba, //   1.128379154774254e+00
                                            0xbec0937e, //  -3.761252839094832e-01
                                            0x3de70f22, //   1.128218315189123e-01
                                            0xbcdb61f4, //  -2.678010670585737e-02
                                            0x3ba4468d, //   5.013293006147870e-03
                                            0xba1fc83b //  -6.095205117313012e-04
                                            >(x);
                  }
  
                  // computes erfc(x)*exp(sqr(x))
                  // x >=  2/3
                  static inline batch_type erfc2(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0x3f0a0e8b, //   5.392844046572836e-01
                                            0xbf918a62, //  -1.137035586823118e+00
                                            0x3e243828, //   1.603704761054187e-01
                                            0x3ec4ca6e, //   3.843569094305250e-01
                                            0x3e1175c7, //   1.420508523645926e-01
                                            0x3e2006f0, //   1.562764709849380e-01
                                            0xbfaea865, //  -1.364514006347145e+00
                                            0x4050b063, //   3.260765682222576e+00
                                            0xc0cd1a85, //  -6.409487379234005e+00
                                            0x40d67e3b, //   6.702908785399893e+00
                                            0xc0283611 //  -2.628299919293280e+00
                                            >(x);
                  }
  
                  static inline batch_type erfc3(const batch_type& x) noexcept
                  {
                      return (batch_type(1.) - x) * detail::horner<batch_type,
                                                                   0x3f7ffffe, //   9.9999988e-01
                                                                   0xbe036d7e, //  -1.2834737e-01
                                                                   0xbfa11698, //  -1.2585020e+00
                                                                   0xbffc9284, //  -1.9732213e+00
                                                                   0xc016c985, //  -2.3560498e+00
                                                                   0x3f2cff3b, //   6.7576951e-01
                                                                   0xc010d956, //  -2.2632651e+00
                                                                   0x401b5680, //   2.4271545e+00
                                                                   0x41aa8e55 //   2.1319498e+01
                                                                   >(x);
                  }
              };
  
              template <class A>
              struct erf_kernel<batch<double, A>>
              {
                  using batch_type = batch<double, A>;
                  // computes erf(a0)/a0
                  // x is sqr(a0) and 0 <= abs(a0) <= 0.65
                  static inline batch_type erf1(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0x3ff20dd750429b61ull, // 1.12837916709551
                                            0x3fc16500f106c0a5ull, // 0.135894887627278
                                            0x3fa4a59a4f02579cull, // 4.03259488531795E-02
                                            0x3f53b7664358865aull, // 1.20339380863079E-03
                                            0x3f110512d5b20332ull // 6.49254556481904E-05
                                            >(x)
                          / detail::horner<batch_type,
                                           0x3ff0000000000000ull, // 1
                                           0x3fdd0a84eb1ca867ull, // 0.453767041780003
                                           0x3fb64536ca92ea2full, // 8.69936222615386E-02
                                           0x3f8166f75999dbd1ull, // 8.49717371168693E-03
                                           0x3f37ea4332348252ull // 3.64915280629351E-04
                                           >(x);
                  }
  
                  // computes erfc(x)*exp(x*x)
                  // 0.65 <= abs(x) <= 2.2
                  static inline batch_type erfc2(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0x3feffffffbbb552bull, // 0.999999992049799
                                            0x3ff54dfe9b258a60ull, // 1.33154163936765
                                            0x3fec1986509e687bull, // 0.878115804155882
                                            0x3fd53dd7a67c7e9full, // 0.331899559578213
                                            0x3fb2488a6b5cb5e5ull, // 7.14193832506776E-02
                                            0x3f7cf4cfe0aacbb4ull, // 7.06940843763253E-03
                                            0x0ull // 0
                                            >(x)
                          / detail::horner<batch_type,
                                           0x3ff0000000000000ull, // 1
                                           0x4003adeae79b9708ull, // 2.45992070144246
                                           0x40053b1052dca8bdull, // 2.65383972869776
                                           0x3ff9e677c2777c3cull, // 1.61876655543871
                                           0x3fe307622fcff772ull, // 0.594651311286482
                                           0x3fc033c113a7deeeull, // 0.126579413030178
                                           0x3f89a996639b0d00ull // 1.25304936549413E-02
                                           >(x);
                  }
  
                  // computes erfc(x)*exp(x*x)
                  // 2.2 <= abs(x) <= 6
                  static inline batch_type erfc3(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0x3fefff5a9e697ae2ull, // 0.99992114009714
                                            0x3ff9fa202deb88e5ull, // 1.62356584489367
                                            0x3ff44744306832aeull, // 1.26739901455873
                                            0x3fe29be1cff90d94ull, // 0.581528574177741
                                            0x3fc42210f88b9d43ull, // 0.157289620742839
                                            0x3f971d0907ea7a92ull, // 2.25716982919218E-02
                                            0x0ll // 0
                                            >(x)
                          / detail::horner<batch_type,
                                           0x3ff0000000000000ull, // 1
                                           0x400602f24bf3fdb6ull, // 2.75143870676376
                                           0x400afd487397568full, // 3.37367334657285
                                           0x400315ffdfd5ce91ull, // 2.38574194785344
                                           0x3ff0cfd4cb6cde9full, // 1.05074004614827
                                           0x3fd1d7ab774bb837ull, // 0.278788439273629
                                           0x3fa47bd61bbb3843ull // 4.00072964526861E-02
                                           >(x);
                  }
  
                  // computes erfc(rx)*exp(rx*rx)
                  // x >=  6 rx = 1/x
                  static inline batch_type erfc4(const batch_type& x) noexcept
                  {
                      return detail::horner<batch_type,
                                            0xbc7e4ad1ec7d0000ll, // -2.627435221016534e-17
                                            0x3fe20dd750429a16ll, // 5.641895835477182e-01
                                            0x3db60000e984b501ll, // 2.000889609806154e-11
                                            0xbfd20dd753ae5dfdll, // -2.820947949598745e-01
                                            0x3e907e71e046a820ll, // 2.457786367990903e-07
                                            0x3fdb1494cac06d39ll, // 4.231311779019112e-01
                                            0x3f34a451701654f1ll, // 3.149699042180451e-04
                                            0xbff105e6b8ef1a63ll, // -1.063940737150596e+00
                                            0x3fb505a857e9ccc8ll, // 8.211757799454056e-02
                                            0x40074fbabc514212ll, // 2.913930388669777e+00
                                            0x4015ac7631f7ac4fll, // 5.418419628850713e+00
                                            0xc0457e03041e9d8bll, // -4.298446704382794e+01
                                            0x4055803d26c4ec4fll, // 8.600373238783617e+01
                                            0xc0505fce04ec4ec5ll // -6.549694941594051e+01
                                            >(x);
                  }
              };
          }
          /* origin: boost/simd/arch/common/simd/function/erf.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
  
          template <class A>
          inline batch<float, A> erf(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              batch_type x = abs(self);
              batch_type r1(0.);
              auto test1 = x < batch_type(2.f / 3.f);
              if (any(test1))
              {
                  r1 = self * detail::erf_kernel<batch_type>::erf1(x * x);
                  if (all(test1))
                      return r1;
              }
              batch_type z = x / (batch_type(1.) + x);
              z -= batch_type(0.4f);
              batch_type r2 = batch_type(1.) - exp(-x * x) * detail::erf_kernel<batch_type>::erfc2(z);
              r2 = select(self < batch_type(0.), -r2, r2);
              r1 = select(test1, r1, r2);
  #ifndef XSIMD_NO_INFINITIES
              r1 = select(xsimd::isinf(self), sign(self), r1);
  #endif
              return r1;
          }
  
          template <class A>
          inline batch<double, A> erf(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              batch_type x = abs(self);
              batch_type xx = x * x;
              batch_type lim1(0.65);
              batch_type lim2(2.2);
              auto test1 = x < lim1;
              batch_type r1(0.);
              if (any(test1))
              {
                  r1 = self * detail::erf_kernel<batch_type>::erf1(xx);
                  if (all(test1))
                      return r1;
              }
              auto test2 = x < lim2;
              auto test3 = test2 && !test1;
              batch_type ex = exp(-xx);
              if (any(test3))
              {
                  batch_type z = batch_type(1.) - ex * detail::erf_kernel<batch_type>::erfc2(x);
                  batch_type r2 = select(self < batch_type(0.), -z, z);
                  r1 = select(test1, r1, r2);
                  if (all(test1 || test3))
                      return r1;
              }
              batch_type z = batch_type(1.) - ex * detail::erf_kernel<batch_type>::erfc3(x);
              z = select(self < batch_type(0.), -z, z);
  #ifndef XSIMD_NO_INFINITIES
              z = select(xsimd::isinf(self), sign(self), z);
  #endif
              return select(test2, r1, z);
          }
  
          // erfc
          template <class A>
          inline batch<float, A> erfc(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              batch_type x = abs(self);
              auto test0 = self < batch_type(0.);
              batch_type r1(0.);
              auto test1 = x < batch_type(2.f / 3.f);
              batch_type z = x / (batch_type(1.) + x);
              if (any(test1))
              {
                  r1 = detail::erf_kernel<batch_type>::erfc3(z);
                  if (all(test1))
                      return select(test0, batch_type(2.) - r1, r1);
              }
              z -= batch_type(0.4f);
              batch_type r2 = exp(-x * x) * detail::erf_kernel<batch_type>::erfc2(z);
              r1 = select(test1, r1, r2);
  #ifndef XSIMD_NO_INFINITIES
              r1 = select(x == constants::infinity<batch_type>(), batch_type(0.), r1);
  #endif
              return select(test0, batch_type(2.) - r1, r1);
          }
  
          template <class A>
          inline batch<double, A> erfc(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              batch_type x = abs(self);
              batch_type xx = x * x;
              batch_type lim1(0.65);
              batch_type lim2(2.2);
              auto test0 = self < batch_type(0.);
              auto test1 = x < lim1;
              batch_type r1(0.);
              if (any(test1))
              {
                  r1 = batch_type(1.) - x * detail::erf_kernel<batch_type>::erf1(xx);
                  if (all(test1))
                      return select(test0, batch_type(2.) - r1, r1);
              }
              auto test2 = x < lim2;
              auto test3 = test2 && !test1;
              batch_type ex = exp(-xx);
              if (any(test3))
              {
                  batch_type z = ex * detail::erf_kernel<batch_type>::erfc2(x);
                  r1 = select(test1, r1, z);
                  if (all(test1 || test3))
                      return select(test0, batch_type(2.) - r1, r1);
              }
              batch_type z = ex * detail::erf_kernel<batch_type>::erfc3(x);
              r1 = select(test2, r1, z);
  #ifndef XSIMD_NO_INFINITIES
              r1 = select(x == constants::infinity<batch_type>(), batch_type(0.), r1);
  #endif
              return select(test0, batch_type(2.) - r1, r1);
          }
  
          // estrin
          namespace detail
          {
  
              template <class B>
              struct estrin
              {
                  B x;
  
                  template <typename... Ts>
                  inline B operator()(const Ts&... coefs) noexcept
                  {
                      return eval(coefs...);
                  }
  
              private:
                  inline B eval(const B& c0) noexcept
                  {
                      return c0;
                  }
  
                  inline B eval(const B& c0, const B& c1) noexcept
                  {
                      return fma(x, c1, c0);
                  }
  
                  template <size_t... Is, class Tuple>
                  inline B eval(::xsimd::detail::index_sequence<Is...>, const Tuple& tuple)
                  {
                      return estrin { x * x }(std::get<Is>(tuple)...);
                  }
  
                  template <class... Args>
                  inline B eval(const std::tuple<Args...>& tuple) noexcept
                  {
                      return eval(::xsimd::detail::make_index_sequence<sizeof...(Args)>(), tuple);
                  }
  
                  template <class... Args>
                  inline B eval(const std::tuple<Args...>& tuple, const B& c0) noexcept
                  {
                      return eval(std::tuple_cat(tuple, std::make_tuple(eval(c0))));
                  }
  
                  template <class... Args>
                  inline B eval(const std::tuple<Args...>& tuple, const B& c0, const B& c1) noexcept
                  {
                      return eval(std::tuple_cat(tuple, std::make_tuple(eval(c0, c1))));
                  }
  
                  template <class... Args, class... Ts>
                  inline B eval(const std::tuple<Args...>& tuple, const B& c0, const B& c1, const Ts&... coefs) noexcept
                  {
                      return eval(std::tuple_cat(tuple, std::make_tuple(eval(c0, c1))), coefs...);
                  }
  
                  template <class... Ts>
                  inline B eval(const B& c0, const B& c1, const Ts&... coefs) noexcept
                  {
                      return eval(std::make_tuple(eval(c0, c1)), coefs...);
                  }
              };
          }
  
          template <class T, class A, uint64_t... Coefs>
          inline batch<T, A> estrin(const batch<T, A>& self) noexcept
          {
              using batch_type = batch<T, A>;
              return detail::estrin<batch_type> { self }(detail::coef<batch_type, Coefs>()...);
          }
  
          // exp
          /* origin: boost/simd/arch/common/detail/simd/expo_base.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          namespace detail
          {
              enum exp_reduction_tag
              {
                  exp_tag,
                  exp2_tag,
                  exp10_tag
              };
  
              template <class B, exp_reduction_tag Tag>
              struct exp_reduction_base;
  
              template <class B>
              struct exp_reduction_base<B, exp_tag>
              {
                  static constexpr B maxlog() noexcept
                  {
                      return constants::maxlog<B>();
                  }
  
                  static constexpr B minlog() noexcept
                  {
                      return constants::minlog<B>();
                  }
              };
  
              template <class B>
              struct exp_reduction_base<B, exp10_tag>
              {
                  static constexpr B maxlog() noexcept
                  {
                      return constants::maxlog10<B>();
                  }
  
                  static constexpr B minlog() noexcept
                  {
                      return constants::minlog10<B>();
                  }
              };
  
              template <class B>
              struct exp_reduction_base<B, exp2_tag>
              {
                  static constexpr B maxlog() noexcept
                  {
                      return constants::maxlog2<B>();
                  }
  
                  static constexpr B minlog() noexcept
                  {
                      return constants::minlog2<B>();
                  }
              };
  
              template <class T, class A, exp_reduction_tag Tag>
              struct exp_reduction;
  
              template <class A>
              struct exp_reduction<float, A, exp_tag> : exp_reduction_base<batch<float, A>, exp_tag>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      batch_type y = detail::horner<batch_type,
                                                    0x3f000000, //  5.0000000e-01
                                                    0x3e2aa9a5, //  1.6666277e-01
                                                    0x3d2aa957, //  4.1665401e-02
                                                    0x3c098d8b, //  8.3955629e-03
                                                    0x3ab778cf //  1.3997796e-03
                                                    >(x);
                      return ++fma(y, x * x, x);
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(constants::invlog_2<batch_type>() * a);
                      x = fnma(k, constants::log_2hi<batch_type>(), a);
                      x = fnma(k, constants::log_2lo<batch_type>(), x);
                      return k;
                  }
              };
  
              template <class A>
              struct exp_reduction<float, A, exp10_tag> : exp_reduction_base<batch<float, A>, exp10_tag>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      return ++(detail::horner<batch_type,
                                               0x40135d8e, //    2.3025851e+00
                                               0x4029a926, //    2.6509490e+00
                                               0x400237da, //    2.0346589e+00
                                               0x3f95eb4c, //    1.1712432e+00
                                               0x3f0aacef, //    5.4170126e-01
                                               0x3e54dff1 //    2.0788552e-01
                                               >(x)
                                * x);
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(constants::invlog10_2<batch_type>() * a);
                      x = fnma(k, constants::log10_2hi<batch_type>(), a);
                      x -= k * constants::log10_2lo<batch_type>();
                      return k;
                  }
              };
  
              template <class A>
              struct exp_reduction<float, A, exp2_tag> : exp_reduction_base<batch<float, A>, exp2_tag>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      batch_type y = detail::horner<batch_type,
                                                    0x3e75fdf1, //    2.4022652e-01
                                                    0x3d6356eb, //    5.5502813e-02
                                                    0x3c1d9422, //    9.6178371e-03
                                                    0x3ab01218, //    1.3433127e-03
                                                    0x3922c8c4 //    1.5524315e-04
                                                    >(x);
                      return ++fma(y, x * x, x * constants::log_2<batch_type>());
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(a);
                      x = (a - k);
                      return k;
                  }
              };
  
              template <class A>
              struct exp_reduction<double, A, exp_tag> : exp_reduction_base<batch<double, A>, exp_tag>
              {
                  using batch_type = batch<double, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      batch_type t = x * x;
                      return fnma(t,
                                  detail::horner<batch_type,
                                                 0x3fc555555555553eull,
                                                 0xbf66c16c16bebd93ull,
                                                 0x3f11566aaf25de2cull,
                                                 0xbebbbd41c5d26bf1ull,
                                                 0x3e66376972bea4d0ull>(t),
                                  x);
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type& hi, batch_type& lo, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(constants::invlog_2<batch_type>() * a);
                      hi = fnma(k, constants::log_2hi<batch_type>(), a);
                      lo = k * constants::log_2lo<batch_type>();
                      x = hi - lo;
                      return k;
                  }
  
                  static inline batch_type finalize(const batch_type& x, const batch_type& c, const batch_type& hi, const batch_type& lo) noexcept
                  {
                      return batch_type(1.) - (((lo - (x * c) / (batch_type(2.) - c)) - hi));
                  }
              };
  
              template <class A>
              struct exp_reduction<double, A, exp10_tag> : exp_reduction_base<batch<double, A>, exp10_tag>
              {
                  using batch_type = batch<double, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      batch_type xx = x * x;
                      batch_type px = x * detail::horner<batch_type, 0x40a2b4798e134a01ull, 0x40796b7a050349e4ull, 0x40277d9474c55934ull, 0x3fa4fd75f3062dd4ull>(xx);
                      batch_type x2 = px / (detail::horner1<batch_type, 0x40a03f37650df6e2ull, 0x4093e05eefd67782ull, 0x405545fdce51ca08ull>(xx) - px);
                      return ++(x2 + x2);
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type&, batch_type&, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(constants::invlog10_2<batch_type>() * a);
                      x = fnma(k, constants::log10_2hi<batch_type>(), a);
                      x = fnma(k, constants::log10_2lo<batch_type>(), x);
                      return k;
                  }
  
                  static inline batch_type finalize(const batch_type&, const batch_type& c, const batch_type&, const batch_type&) noexcept
                  {
                      return c;
                  }
              };
  
              template <class A>
              struct exp_reduction<double, A, exp2_tag> : exp_reduction_base<batch<double, A>, exp2_tag>
              {
                  using batch_type = batch<double, A>;
                  static inline batch_type approx(const batch_type& x) noexcept
                  {
                      batch_type t = x * x;
                      return fnma(t,
                                  detail::horner<batch_type,
                                                 0x3fc555555555553eull,
                                                 0xbf66c16c16bebd93ull,
                                                 0x3f11566aaf25de2cull,
                                                 0xbebbbd41c5d26bf1ull,
                                                 0x3e66376972bea4d0ull>(t),
                                  x);
                  }
  
                  static inline batch_type reduce(const batch_type& a, batch_type&, batch_type&, batch_type& x) noexcept
                  {
                      batch_type k = nearbyint(a);
                      x = (a - k) * constants::log_2<batch_type>();
                      return k;
                  }
  
                  static inline batch_type finalize(const batch_type& x, const batch_type& c, const batch_type&, const batch_type&) noexcept
                  {
                      return batch_type(1.) + x + x * c / (batch_type(2.) - c);
                  }
              };
  
              template <exp_reduction_tag Tag, class A>
              inline batch<float, A> exp(batch<float, A> const& self) noexcept
              {
                  using batch_type = batch<float, A>;
                  using reducer_t = exp_reduction<float, A, Tag>;
                  batch_type x;
                  batch_type k = reducer_t::reduce(self, x);
                  x = reducer_t::approx(x);
                  x = select(self <= reducer_t::minlog(), batch_type(0.), ldexp(x, to_int(k)));
                  x = select(self >= reducer_t::maxlog(), constants::infinity<batch_type>(), x);
                  return x;
              }
  
              template <exp_reduction_tag Tag, class A>
              inline batch<double, A> exp(batch<double, A> const& self) noexcept
              {
                  using batch_type = batch<double, A>;
                  using reducer_t = exp_reduction<double, A, Tag>;
                  batch_type hi, lo, x;
                  batch_type k = reducer_t::reduce(self, hi, lo, x);
                  batch_type c = reducer_t::approx(x);
                  c = reducer_t::finalize(x, c, hi, lo);
                  c = select(self <= reducer_t::minlog(), batch_type(0.), ldexp(c, to_int(k)));
                  c = select(self >= reducer_t::maxlog(), constants::infinity<batch_type>(), c);
                  return c;
              }
          }
  
          template <class A, class T>
          inline batch<T, A> exp(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::exp<detail::exp_tag>(self);
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> exp(batch<std::complex<T>, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<std::complex<T>, A>;
              auto isincos = sincos(self.imag());
              return exp(self.real()) * batch_type(std::get<1>(isincos), std::get<0>(isincos));
          }
  
          // exp10
          template <class A, class T>
          inline batch<T, A> exp10(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::exp<detail::exp10_tag>(self);
          }
  
          // exp2
          template <class A, class T>
          inline batch<T, A> exp2(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::exp<detail::exp2_tag>(self);
          }
  
          // expm1
          namespace detail
          {
              /* origin: boost/simd/arch/common/detail/generic/expm1_kernel.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class A>
              static inline batch<float, A> expm1(const batch<float, A>& a) noexcept
              {
                  using batch_type = batch<float, A>;
                  batch_type k = nearbyint(constants::invlog_2<batch_type>() * a);
                  batch_type x = fnma(k, constants::log_2hi<batch_type>(), a);
                  x = fnma(k, constants::log_2lo<batch_type>(), x);
                  batch_type hx = x * batch_type(0.5);
                  batch_type hxs = x * hx;
                  batch_type r = detail::horner<batch_type,
                                                0X3F800000UL, // 1
                                                0XBD08887FUL, // -3.3333298E-02
                                                0X3ACF6DB4UL // 1.582554
                                                >(hxs);
                  batch_type t = fnma(r, hx, batch_type(3.));
                  batch_type e = hxs * ((r - t) / (batch_type(6.) - x * t));
                  e = fms(x, e, hxs);
                  using i_type = as_integer_t<batch_type>;
                  i_type ik = to_int(k);
                  batch_type two2mk = ::xsimd::bitwise_cast<batch_type>((constants::maxexponent<batch_type>() - ik) << constants::nmb<batch_type>());
                  batch_type y = batch_type(1.) - two2mk - (e - x);
                  return ldexp(y, ik);
              }
  
              template <class A>
              static inline batch<double, A> expm1(const batch<double, A>& a) noexcept
              {
                  using batch_type = batch<double, A>;
                  batch_type k = nearbyint(constants::invlog_2<batch_type>() * a);
                  batch_type hi = fnma(k, constants::log_2hi<batch_type>(), a);
                  batch_type lo = k * constants::log_2lo<batch_type>();
                  batch_type x = hi - lo;
                  batch_type hxs = x * x * batch_type(0.5);
                  batch_type r = detail::horner<batch_type,
                                                0X3FF0000000000000ULL,
                                                0XBFA11111111110F4ULL,
                                                0X3F5A01A019FE5585ULL,
                                                0XBF14CE199EAADBB7ULL,
                                                0X3ED0CFCA86E65239ULL,
                                                0XBE8AFDB76E09C32DULL>(hxs);
                  batch_type t = batch_type(3.) - r * batch_type(0.5) * x;
                  batch_type e = hxs * ((r - t) / (batch_type(6) - x * t));
                  batch_type c = (hi - x) - lo;
                  e = (x * (e - c) - c) - hxs;
                  using i_type = as_integer_t<batch_type>;
                  i_type ik = to_int(k);
                  batch_type two2mk = ::xsimd::bitwise_cast<batch_type>((constants::maxexponent<batch_type>() - ik) << constants::nmb<batch_type>());
                  batch_type ct1 = batch_type(1.) - two2mk - (e - x);
                  batch_type ct2 = ++(x - (e + two2mk));
                  batch_type y = select(k < batch_type(20.), ct1, ct2);
                  return ldexp(y, ik);
              }
  
          }
  
          template <class A, class T>
          inline batch<T, A> expm1(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              return select(self < constants::logeps<batch_type>(),
                            batch_type(-1.),
                            select(self > constants::maxlog<batch_type>(),
                                   constants::infinity<batch_type>(),
                                   detail::expm1(self)));
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> expm1(const batch<std::complex<T>, A>& z, requires_arch<generic>) noexcept
          {
              using batch_type = batch<std::complex<T>, A>;
              using real_batch = typename batch_type::real_batch;
              real_batch isin = sin(z.imag());
              real_batch rem1 = expm1(z.real());
              real_batch re = rem1 + 1.;
              real_batch si = sin(z.imag() * 0.5);
              return { rem1 - 2. * re * si * si, re * isin };
          }
  
          // fdim
          template <class A, class T>
          inline batch<T, A> fdim(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              return fmax(batch<T, A>(0), self - other);
          }
  
          // fmod
          template <class A, class T>
          inline batch<T, A> fmod(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              return fnma(trunc(self / other), other, self);
          }
  
          // frexp
          /* origin: boost/simd/arch/common/simd/function/ifrexp.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A, class T>
          inline batch<T, A> frexp(const batch<T, A>& self, batch<as_integer_t<T>, A>& exp, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              using i_type = batch<as_integer_t<T>, A>;
              i_type m1f = constants::mask1frexp<batch_type>();
              i_type r1 = m1f & ::xsimd::bitwise_cast<i_type>(self);
              batch_type x = self & ::xsimd::bitwise_cast<batch_type>(~m1f);
              exp = (r1 >> constants::nmb<batch_type>()) - constants::maxexponentm1<batch_type>();
              exp = select(bool_cast(self != batch_type(0.)), exp, i_type(typename i_type::value_type(0)));
              return select((self != batch_type(0.)), x | ::xsimd::bitwise_cast<batch_type>(constants::mask2frexp<batch_type>()), batch_type(0.));
          }
  
          // from bool
          template <class A, class T>
          inline batch<T, A> from_bool(batch_bool<T, A> const& self, requires_arch<generic>) noexcept
          {
              return batch<T, A>(self.data) & batch<T, A>(1);
          }
  
          // hadd
          template <class A, class T>
          inline std::complex<T> hadd(batch<std::complex<T>, A> const& self, requires_arch<generic>) noexcept
          {
              return { hadd(self.real()), hadd(self.imag()) };
          }
  
          // horner
          template <class T, class A, uint64_t... Coefs>
          inline batch<T, A> horner(const batch<T, A>& self) noexcept
          {
              return detail::horner<batch<T, A>, Coefs...>(self);
          }
  
          // hypot
          template <class A, class T>
          inline batch<T, A> hypot(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              return sqrt(fma(self, self, other * other));
          }
  
          // ipow
          template <class A, class T, class ITy>
          inline batch<T, A> ipow(batch<T, A> const& self, ITy other, requires_arch<generic>) noexcept
          {
              return ::xsimd::detail::ipow(self, other);
          }
  
          // ldexp
          /* origin: boost/simd/arch/common/simd/function/ldexp.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A, class T>
          inline batch<T, A> ldexp(const batch<T, A>& self, const batch<as_integer_t<T>, A>& other, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              using itype = as_integer_t<batch_type>;
              itype ik = other + constants::maxexponent<T>();
              ik = ik << constants::nmb<T>();
              return self * ::xsimd::bitwise_cast<batch_type>(ik);
          }
  
          // lgamma
          template <class A, class T>
          inline batch<T, A> lgamma(batch<T, A> const& self, requires_arch<generic>) noexcept;
  
          namespace detail
          {
              /* origin: boost/simd/arch/common/detail/generic/gammaln_kernel.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class A>
              static inline batch<float, A> gammalnB(const batch<float, A>& x) noexcept
              {
                  return horner<batch<float, A>,
                                0x3ed87730, //    4.227843421859038E-001
                                0x3ea51a64, //    3.224669577325661E-001,
                                0xbd89f07e, //   -6.735323259371034E-002,
                                0x3ca89ed8, //    2.058355474821512E-002,
                                0xbbf164fd, //   -7.366775108654962E-003,
                                0x3b3ba883, //    2.863437556468661E-003,
                                0xbaabeab1, //   -1.311620815545743E-003,
                                0x3a1ebb94 //    6.055172732649237E-004
                                >(x);
              }
  
              template <class A>
              static inline batch<float, A> gammalnC(const batch<float, A>& x) noexcept
              {
                  return horner<batch<float, A>,
                                0xbf13c468, //   -5.772156501719101E-001
                                0x3f528d34, //    8.224670749082976E-001,
                                0xbecd27a8, //   -4.006931650563372E-001,
                                0x3e8a898b, //    2.705806208275915E-001,
                                0xbe53c04f, //   -2.067882815621965E-001,
                                0x3e2d4dab, //    1.692415923504637E-001,
                                0xbe22d329, //   -1.590086327657347E-001,
                                0x3e0c3c4f //    1.369488127325832E-001
                                >(x);
              }
  
              template <class A>
              static inline batch<float, A> gammaln2(const batch<float, A>& x) noexcept
              {
                  return horner<batch<float, A>,
                                0x3daaaa94, //   8.333316229807355E-002f
                                0xbb358701, //  -2.769887652139868E-003f,
                                0x3a31fd69 //   6.789774945028216E-004f
                                >(x);
              }
  
              template <class A>
              static inline batch<double, A> gammaln1(const batch<double, A>& x) noexcept
              {
                  return horner<batch<double, A>,
                                0xc12a0c675418055eull, //  -8.53555664245765465627E5
                                0xc13a45890219f20bull, //  -1.72173700820839662146E6,
                                0xc131bc82f994db51ull, //  -1.16237097492762307383E6,
                                0xc1143d73f89089e5ull, //  -3.31612992738871184744E5,
                                0xc0e2f234355bb93eull, //  -3.88016315134637840924E4,
                                0xc09589018ff36761ull //  -1.37825152569120859100E3
                                >(x)
                      / horner<batch<double, A>,
                               0xc13ece4b6a11e14aull, //  -2.01889141433532773231E6
                               0xc1435255892ff34cull, //  -2.53252307177582951285E6,
                               0xc131628671950043ull, //  -1.13933444367982507207E6,
                               0xc10aeb84b9744c9bull, //  -2.20528590553854454839E5,
                               0xc0d0aa0d7b89d757ull, //  -1.70642106651881159223E4,
                               0xc075fd0d1cf312b2ull, //  -3.51815701436523470549E2,
                               0x3ff0000000000000ull //   1.00000000000000000000E0
                               >(x);
              }
  
              template <class A>
              static inline batch<double, A> gammalnA(const batch<double, A>& x) noexcept
              {
                  return horner<batch<double, A>,
                                0x3fb555555555554bull, //    8.33333333333331927722E-2
                                0xbf66c16c16b0a5a1ull, //   -2.77777777730099687205E-3,
                                0x3f4a019f20dc5ebbull, //    7.93650340457716943945E-4,
                                0xbf437fbdb580e943ull, //   -5.95061904284301438324E-4,
                                0x3f4a985027336661ull //    8.11614167470508450300E-4
                                >(x);
              }
  
              /* origin: boost/simd/arch/common/simd/function/gammaln.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class B>
              struct lgamma_impl;
  
              template <class A>
              struct lgamma_impl<batch<float, A>>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type compute(const batch_type& a) noexcept
                  {
                      auto inf_result = (a <= batch_type(0.)) && is_flint(a);
                      batch_type x = select(inf_result, constants::nan<batch_type>(), a);
                      batch_type q = abs(x);
  #ifndef XSIMD_NO_INFINITIES
                      inf_result = (x == constants::infinity<batch_type>()) || inf_result;
  #endif
                      auto ltza = a < batch_type(0.);
                      batch_type r;
                      batch_type r1 = other(q);
                      if (any(ltza))
                      {
                          r = select(inf_result, constants::infinity<batch_type>(), negative(q, r1));
                          if (all(ltza))
                              return r;
                      }
                      batch_type r2 = select(ltza, r, r1);
                      return select(a == constants::minusinfinity<batch_type>(), constants::nan<batch_type>(), select(inf_result, constants::infinity<batch_type>(), r2));
                  }
  
              private:
                  static inline batch_type negative(const batch_type& q, const batch_type& w) noexcept
                  {
                      batch_type p = floor(q);
                      batch_type z = q - p;
                      auto test2 = z < batch_type(0.5);
                      z = select(test2, z - batch_type(1.), z);
                      z = q * sin(z, trigo_pi_tag());
                      return -log(constants::invpi<batch_type>() * abs(z)) - w;
                  }
  
                  static inline batch_type other(const batch_type& x) noexcept
                  {
                      auto xlt650 = (x < batch_type(6.5));
                      batch_type r0x = x;
                      batch_type r0z = x;
                      batch_type r0s = batch_type(1.);
                      batch_type r1 = batch_type(0.);
                      batch_type p = constants::nan<batch_type>();
                      if (any(xlt650))
                      {
                          batch_type z = batch_type(1.);
                          batch_type tx = select(xlt650, x, batch_type(0.));
                          batch_type nx = batch_type(0.);
                          const batch_type _075 = batch_type(0.75);
                          const batch_type _150 = batch_type(1.50);
                          const batch_type _125 = batch_type(1.25);
                          const batch_type _250 = batch_type(2.50);
                          auto xge150 = (x >= _150);
                          auto txgt250 = (tx > _250);
  
                          // x >= 1.5
                          while (any(xge150 && txgt250))
                          {
                              nx = select(txgt250, nx - batch_type(1.), nx);
                              tx = select(txgt250, x + nx, tx);
                              z = select(txgt250, z * tx, z);
                              txgt250 = (tx > _250);
                          }
                          r0x = select(xge150, x + nx - batch_type(2.), x);
                          r0z = select(xge150, z, r0z);
                          r0s = select(xge150, batch_type(1.), r0s);
  
                          // x >= 1.25 && x < 1.5
                          auto xge125 = (x >= _125);
                          auto xge125t = xge125 && !xge150;
                          if (any(xge125))
                          {
                              r0x = select(xge125t, x - batch_type(1.), r0x);
                              r0z = select(xge125t, z * x, r0z);
                              r0s = select(xge125t, batch_type(-1.), r0s);
                          }
  
                          // x >= 0.75 && x < 1.5
                          batch_bool<float, A> kernelC(false);
                          auto xge075 = (x >= _075);
                          auto xge075t = xge075 && !xge125;
                          if (any(xge075t))
                          {
                              kernelC = xge075t;
                              r0x = select(xge075t, x - batch_type(1.), x);
                              r0z = select(xge075t, batch_type(1.), r0z);
                              r0s = select(xge075t, batch_type(-1.), r0s);
                              p = gammalnC(r0x);
                          }
  
                          // tx < 1.5 && x < 0.75
                          auto txlt150 = (tx < _150) && !xge075;
                          if (any(txlt150))
                          {
                              auto orig = txlt150;
                              while (any(txlt150))
                              {
                                  z = select(txlt150, z * tx, z);
                                  nx = select(txlt150, nx + batch_type(1.), nx);
                                  tx = select(txlt150, x + nx, tx);
                                  txlt150 = (tx < _150) && !xge075;
                              }
                              r0x = select(orig, r0x + nx - batch_type(2.), r0x);
                              r0z = select(orig, z, r0z);
                              r0s = select(orig, batch_type(-1.), r0s);
                          }
                          p = select(kernelC, p, gammalnB(r0x));
                          if (all(xlt650))
                              return fma(r0x, p, r0s * log(abs(r0z)));
                      }
                      r0z = select(xlt650, abs(r0z), x);
                      batch_type m = log(r0z);
                      r1 = fma(r0x, p, r0s * m);
                      batch_type r2 = fma(x - batch_type(0.5), m, constants::logsqrt2pi<batch_type>() - x);
                      r2 += gammaln2(batch_type(1.) / (x * x)) / x;
                      return select(xlt650, r1, r2);
                  }
              };
  
              template <class A>
              struct lgamma_impl<batch<double, A>>
              {
                  using batch_type = batch<double, A>;
  
                  static inline batch_type compute(const batch_type& a) noexcept
                  {
                      auto inf_result = (a <= batch_type(0.)) && is_flint(a);
                      batch_type x = select(inf_result, constants::nan<batch_type>(), a);
                      batch_type q = abs(x);
  #ifndef XSIMD_NO_INFINITIES
                      inf_result = (q == constants::infinity<batch_type>());
  #endif
                      auto test = (a < batch_type(-34.));
                      batch_type r = constants::nan<batch_type>();
                      if (any(test))
                      {
                          r = large_negative(q);
                          if (all(test))
                              return select(inf_result, constants::nan<batch_type>(), r);
                      }
                      batch_type r1 = other(a);
                      batch_type r2 = select(test, r, r1);
                      return select(a == constants::minusinfinity<batch_type>(), constants::nan<batch_type>(), select(inf_result, constants::infinity<batch_type>(), r2));
                  }
  
              private:
                  static inline batch_type large_negative(const batch_type& q) noexcept
                  {
                      batch_type w = lgamma(q);
                      batch_type p = floor(q);
                      batch_type z = q - p;
                      auto test2 = (z < batch_type(0.5));
                      z = select(test2, z - batch_type(1.), z);
                      z = q * sin(z, trigo_pi_tag());
                      z = abs(z);
                      return constants::logpi<batch_type>() - log(z) - w;
                  }
  
                  static inline batch_type other(const batch_type& xx) noexcept
                  {
                      batch_type x = xx;
                      auto test = (x < batch_type(13.));
                      batch_type r1 = batch_type(0.);
                      if (any(test))
                      {
                          batch_type z = batch_type(1.);
                          batch_type p = batch_type(0.);
                          batch_type u = select(test, x, batch_type(0.));
                          auto test1 = (u >= batch_type(3.));
                          while (any(test1))
                          {
                              p = select(test1, p - batch_type(1.), p);
                              u = select(test1, x + p, u);
                              z = select(test1, z * u, z);
                              test1 = (u >= batch_type(3.));
                          }
  
                          auto test2 = (u < batch_type(2.));
                          while (any(test2))
                          {
                              z = select(test2, z / u, z);
                              p = select(test2, p + batch_type(1.), p);
                              u = select(test2, x + p, u);
                              test2 = (u < batch_type(2.));
                          }
  
                          z = abs(z);
                          x += p - batch_type(2.);
                          r1 = x * gammaln1(x) + log(z);
                          if (all(test))
                              return r1;
                      }
                      batch_type r2 = fma(xx - batch_type(0.5), log(xx), constants::logsqrt2pi<batch_type>() - xx);
                      batch_type p = batch_type(1.) / (xx * xx);
                      r2 += gammalnA(p) / xx;
                      return select(test, r1, r2);
                  }
              };
          }
  
          template <class A, class T>
          inline batch<T, A> lgamma(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::lgamma_impl<batch<T, A>>::compute(self);
          }
  
          // log
          /* origin: boost/simd/arch/common/simd/function/log.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A>
          inline batch<float, A> log(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              using i_type = as_integer_t<batch_type>;
              batch_type x = self;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(23), k);
                  x = select(test, x * batch_type(8388608ul), x);
              }
  #endif
              i_type ix = ::xsimd::bitwise_cast<i_type>(x);
              ix += 0x3f800000 - 0x3f3504f3;
              k += (ix >> 23) - 0x7f;
              ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
              x = ::xsimd::bitwise_cast<batch_type>(ix);
              batch_type f = --x;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3eccce13, 0x3e789e26>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3f2aaaaa, 0x3e91e9ee>(w);
              batch_type R = t2 + t1;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type dk = to_float(k);
              batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, (hfsq + R), dk * constants::log_2lo<batch_type>()) - hfsq + f);
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A>
          inline batch<double, A> log(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              using i_type = as_integer_t<batch_type>;
  
              batch_type x = self;
              i_type hx = ::xsimd::bitwise_cast<i_type>(x) >> 32;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(54), k);
                  x = select(test, x * batch_type(18014398509481984ull), x);
              }
  #endif
              hx += 0x3ff00000 - 0x3fe6a09e;
              k += (hx >> 20) - 0x3ff;
              batch_type dk = to_float(k);
              hx = (hx & i_type(0x000fffff)) + 0x3fe6a09e;
              x = ::xsimd::bitwise_cast<batch_type>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<i_type>(x)));
  
              batch_type f = --x;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
  
              batch_type t1 = w * detail::horner<batch_type, 0x3fd999999997fa04ll, 0x3fcc71c51d8e78afll, 0x3fc39a09d078c69fll>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3fe5555555555593ll, 0x3fd2492494229359ll, 0x3fc7466496cb03dell, 0x3fc2f112df3e5244ll>(w);
              batch_type R = t2 + t1;
              batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, (hfsq + R), dk * constants::log_2lo<batch_type>()) - hfsq + f);
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> log(const batch<std::complex<T>, A>& z, requires_arch<generic>) noexcept
          {
              return batch<std::complex<T>, A>(log(abs(z)), atan2(z.imag(), z.real()));
          }
  
          // log2
          template <class A>
          inline batch<float, A> log2(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              using i_type = as_integer_t<batch_type>;
              batch_type x = self;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(25), k);
                  x = select(test, x * batch_type(33554432ul), x);
              }
  #endif
              i_type ix = ::xsimd::bitwise_cast<i_type>(x);
              ix += 0x3f800000 - 0x3f3504f3;
              k += (ix >> 23) - 0x7f;
              ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
              x = ::xsimd::bitwise_cast<batch_type>(ix);
              batch_type f = --x;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3eccce13, 0x3e789e26>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3f2aaaaa, 0x3e91e9ee>(w);
              batch_type R = t1 + t2;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type dk = to_float(k);
              batch_type r = fma(fms(s, hfsq + R, hfsq) + f, constants::invlog_2<batch_type>(), dk);
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A>
          inline batch<double, A> log2(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              using i_type = as_integer_t<batch_type>;
              batch_type x = self;
              i_type hx = ::xsimd::bitwise_cast<i_type>(x) >> 32;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(54), k);
                  x = select(test, x * batch_type(18014398509481984ull), x);
              }
  #endif
              hx += 0x3ff00000 - 0x3fe6a09e;
              k += (hx >> 20) - 0x3ff;
              hx = (hx & i_type(0x000fffff)) + 0x3fe6a09e;
              x = ::xsimd::bitwise_cast<batch_type>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<i_type>(x)));
              batch_type f = --x;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3fd999999997fa04ll, 0x3fcc71c51d8e78afll, 0x3fc39a09d078c69fll>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3fe5555555555593ll, 0x3fd2492494229359ll, 0x3fc7466496cb03dell, 0x3fc2f112df3e5244ll>(w);
              batch_type R = t2 + t1;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type hi = f - hfsq;
              hi = hi & ::xsimd::bitwise_cast<batch_type>((constants::allbits<i_type>() << 32));
              batch_type lo = fma(s, hfsq + R, f - hi - hfsq);
              batch_type val_hi = hi * constants::invlog_2hi<batch_type>();
              batch_type val_lo = fma(lo + hi, constants::invlog_2lo<batch_type>(), lo * constants::invlog_2hi<batch_type>());
              batch_type dk = to_float(k);
              batch_type w1 = dk + val_hi;
              val_lo += (dk - w1) + val_hi;
              val_hi = w1;
              batch_type r = val_lo + val_hi;
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          namespace detail
          {
              template <class T, class A>
              inline batch<T, A> logN_complex_impl(const batch<T, A>& z, typename batch<T, A>::value_type base) noexcept
              {
                  using batch_type = batch<T, A>;
                  using rv_type = typename batch_type::value_type;
                  return log(z) / batch_type(rv_type(base));
              }
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> log2(batch<std::complex<T>, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::logN_complex_impl(self, std::log(2));
          }
  
          // log10
          /* origin: FreeBSD /usr/src/lib/msun/src/e_log10f.c */
          /*
           * ====================================================
           * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
           *
           * Developed at SunPro, a Sun Microsystems, Inc. business.
           * Permission to use, copy, modify, and distribute this
           * software is freely granted, provided that this notice
           * is preserved.
           * ====================================================
           */
          template <class A>
          inline batch<float, A> log10(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              const batch_type
                  ivln10hi(4.3432617188e-01f),
                  ivln10lo(-3.1689971365e-05f),
                  log10_2hi(3.0102920532e-01f),
                  log10_2lo(7.9034151668e-07f);
              using i_type = as_integer_t<batch_type>;
              batch_type x = self;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(25), k);
                  x = select(test, x * batch_type(33554432ul), x);
              }
  #endif
              i_type ix = ::xsimd::bitwise_cast<i_type>(x);
              ix += 0x3f800000 - 0x3f3504f3;
              k += (ix >> 23) - 0x7f;
              ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
              x = ::xsimd::bitwise_cast<batch_type>(ix);
              batch_type f = --x;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3eccce13, 0x3e789e26>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3f2aaaaa, 0x3e91e9ee>(w);
              batch_type R = t2 + t1;
              batch_type dk = to_float(k);
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type hibits = f - hfsq;
              hibits &= ::xsimd::bitwise_cast<batch_type>(i_type(0xfffff000));
              batch_type lobits = fma(s, hfsq + R, f - hibits - hfsq);
              batch_type r = fma(dk, log10_2hi,
                                 fma(hibits, ivln10hi,
                                     fma(lobits, ivln10hi,
                                         fma(lobits + hibits, ivln10lo, dk * log10_2lo))));
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A>
          inline batch<double, A> log10(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              const batch_type
                  ivln10hi(4.34294481878168880939e-01),
                  ivln10lo(2.50829467116452752298e-11),
                  log10_2hi(3.01029995663611771306e-01),
                  log10_2lo(3.69423907715893078616e-13);
              using i_type = as_integer_t<batch_type>;
              batch_type x = self;
              i_type hx = ::xsimd::bitwise_cast<i_type>(x) >> 32;
              i_type k(0);
              auto isnez = (self != batch_type(0.));
  #ifndef XSIMD_NO_DENORMALS
              auto test = (self < constants::smallestposval<batch_type>()) && isnez;
              if (any(test))
              {
                  k = select(bool_cast(test), k - i_type(54), k);
                  x = select(test, x * batch_type(18014398509481984ull), x);
              }
  #endif
              hx += 0x3ff00000 - 0x3fe6a09e;
              k += (hx >> 20) - 0x3ff;
              hx = (hx & i_type(0x000fffff)) + 0x3fe6a09e;
              x = ::xsimd::bitwise_cast<batch_type>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<i_type>(x)));
              batch_type f = --x;
              batch_type dk = to_float(k);
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3fd999999997fa04ll, 0x3fcc71c51d8e78afll, 0x3fc39a09d078c69fll>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3fe5555555555593ll, 0x3fd2492494229359ll, 0x3fc7466496cb03dell, 0x3fc2f112df3e5244ll>(w);
              batch_type R = t2 + t1;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type hi = f - hfsq;
              hi = hi & ::xsimd::bitwise_cast<batch_type>(constants::allbits<i_type>() << 32);
              batch_type lo = f - hi - hfsq + s * (hfsq + R);
              batch_type val_hi = hi * ivln10hi;
              batch_type y = dk * log10_2hi;
              batch_type val_lo = dk * log10_2lo + (lo + hi) * ivln10lo + lo * ivln10hi;
              batch_type w1 = y + val_hi;
              val_lo += (y - w1) + val_hi;
              val_hi = w1;
              batch_type r = val_lo + val_hi;
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> log10(const batch<std::complex<T>, A>& z, requires_arch<generic>) noexcept
          {
              return detail::logN_complex_impl(z, std::log(10));
          }
  
          // log1p
          /* origin: boost/simd/arch/common/simd/function/log1p.hpp */
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A>
          inline batch<float, A> log1p(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<float, A>;
              using i_type = as_integer_t<batch_type>;
              const batch_type uf = self + batch_type(1.);
              auto isnez = (uf != batch_type(0.));
              i_type iu = ::xsimd::bitwise_cast<i_type>(uf);
              iu += 0x3f800000 - 0x3f3504f3;
              i_type k = (iu >> 23) - 0x7f;
              iu = (iu & i_type(0x007fffff)) + 0x3f3504f3;
              batch_type f = --(::xsimd::bitwise_cast<batch_type>(iu));
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3eccce13, 0x3e789e26>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3f2aaaaa, 0x3e91e9ee>(w);
              batch_type R = t2 + t1;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type dk = to_float(k);
              /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
              batch_type c = select(bool_cast(k >= i_type(2)), batch_type(1.) - (uf - self), self - (uf - batch_type(1.))) / uf;
              batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, (hfsq + R), dk * constants::log_2lo<batch_type>() + c) - hfsq + f);
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(uf >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A>
          inline batch<double, A> log1p(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<double, A>;
              using i_type = as_integer_t<batch_type>;
              const batch_type uf = self + batch_type(1.);
              auto isnez = (uf != batch_type(0.));
              i_type hu = ::xsimd::bitwise_cast<i_type>(uf) >> 32;
              hu += 0x3ff00000 - 0x3fe6a09e;
              i_type k = (hu >> 20) - 0x3ff;
              /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
              batch_type c = select(bool_cast(k >= i_type(2)), batch_type(1.) - (uf - self), self - (uf - batch_type(1.))) / uf;
              hu = (hu & i_type(0x000fffff)) + 0x3fe6a09e;
              batch_type f = ::xsimd::bitwise_cast<batch_type>((hu << 32) | (i_type(0xffffffff) & ::xsimd::bitwise_cast<i_type>(uf)));
              f = --f;
              batch_type hfsq = batch_type(0.5) * f * f;
              batch_type s = f / (batch_type(2.) + f);
              batch_type z = s * s;
              batch_type w = z * z;
              batch_type t1 = w * detail::horner<batch_type, 0x3fd999999997fa04ll, 0x3fcc71c51d8e78afll, 0x3fc39a09d078c69fll>(w);
              batch_type t2 = z * detail::horner<batch_type, 0x3fe5555555555593ll, 0x3fd2492494229359ll, 0x3fc7466496cb03dell, 0x3fc2f112df3e5244ll>(w);
              batch_type R = t2 + t1;
              batch_type dk = to_float(k);
              batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, hfsq + R, dk * constants::log_2lo<batch_type>() + c) - hfsq + f);
  #ifndef XSIMD_NO_INFINITIES
              batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
  #else
              batch_type zz = select(isnez, r, constants::minusinfinity<batch_type>());
  #endif
              return select(!(uf >= batch_type(0.)), constants::nan<batch_type>(), zz);
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> log1p(batch<std::complex<T>, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<std::complex<T>, A>;
              using real_batch = typename batch_type::real_batch;
              batch_type u = 1 + self;
              batch_type logu = log(u);
              return select(u == batch_type(1.),
                            self,
                            select(u.real() <= real_batch(0.),
                                   logu,
                                   logu * self / (u - batch_type(1.))));
          }
  
          // mod
          template <class A, class T, class = typename std::enable_if<std::is_integral<T>::value, void>::type>
          inline batch<T, A> mod(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              return detail::apply([](T x, T y) noexcept -> T
                                   { return x % y; },
                                   self, other);
          }
  
          // nearbyint
          template <class A, class T, class = typename std::enable_if<std::is_integral<T>::value, void>::type>
          inline batch<T, A> nearbyint(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              return self;
          }
          namespace detail
          {
              template <class A, class T>
              inline batch<T, A> nearbyintf(batch<T, A> const& self) noexcept
              {
                  using batch_type = batch<T, A>;
                  batch_type s = bitofsign(self);
                  batch_type v = self ^ s;
                  batch_type t2n = constants::twotonmb<batch_type>();
                  // Under fast-math, reordering is possible and the compiler optimizes d
                  // to v. That's not what we want, so prevent compiler optimization here.
                  // FIXME: it may be better to emit a memory barrier here (?).
  #ifdef __FAST_MATH__
                  volatile batch_type d0 = v + t2n;
                  batch_type d = *(batch_type*)(void*)(&d0) - t2n;
  #else
                  batch_type d0 = v + t2n;
                  batch_type d = d0 - t2n;
  #endif
                  return s ^ select(v < t2n, d, v);
              }
          }
          template <class A>
          inline batch<float, A> nearbyint(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::nearbyintf(self);
          }
          template <class A>
          inline batch<double, A> nearbyint(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::nearbyintf(self);
          }
  
          // nextafter
          namespace detail
          {
              template <class T, class A, bool is_int = std::is_integral<T>::value>
              struct nextafter_kernel
              {
                  using batch_type = batch<T, A>;
  
                  static inline batch_type next(batch_type const& b) noexcept
                  {
                      return b;
                  }
  
                  static inline batch_type prev(batch_type const& b) noexcept
                  {
                      return b;
                  }
              };
  
              template <class T, class A>
              struct bitwise_cast_batch;
  
              template <class A>
              struct bitwise_cast_batch<float, A>
              {
                  using type = batch<int32_t, A>;
              };
  
              template <class A>
              struct bitwise_cast_batch<double, A>
              {
                  using type = batch<int64_t, A>;
              };
  
              template <class T, class A>
              struct nextafter_kernel<T, A, false>
              {
                  using batch_type = batch<T, A>;
                  using int_batch = typename bitwise_cast_batch<T, A>::type;
                  using int_type = typename int_batch::value_type;
  
                  static inline batch_type next(const batch_type& b) noexcept
                  {
                      batch_type n = ::xsimd::bitwise_cast<batch_type>(::xsimd::bitwise_cast<int_batch>(b) + int_type(1));
                      return select(b == constants::infinity<batch_type>(), b, n);
                  }
  
                  static inline batch_type prev(const batch_type& b) noexcept
                  {
                      batch_type p = ::xsimd::bitwise_cast<batch_type>(::xsimd::bitwise_cast<int_batch>(b) - int_type(1));
                      return select(b == constants::minusinfinity<batch_type>(), b, p);
                  }
              };
          }
          template <class A, class T>
          inline batch<T, A> nextafter(batch<T, A> const& from, batch<T, A> const& to, requires_arch<generic>) noexcept
          {
              using kernel = detail::nextafter_kernel<T, A>;
              return select(from == to, from,
                            select(to > from, kernel::next(from), kernel::prev(from)));
          }
  
          // pow
          /* origin: boost/simd/arch/common/simd/function/pow.hpp*/
          /*
           * ====================================================
           * copyright 2016 NumScale SAS
           *
           * Distributed under the Boost Software License, Version 1.0.
           * (See copy at http://boost.org/LICENSE_1_0.txt)
           * ====================================================
           */
          template <class A, class T>
          inline batch<T, A> pow(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              auto negx = self < batch_type(0.);
              batch_type z = exp(other * log(abs(self)));
              z = select(is_odd(other) && negx, -z, z);
              auto invalid = negx && !(is_flint(other) || isinf(other));
              return select(invalid, constants::nan<batch_type>(), z);
          }
  
          template <class A, class T>
          inline batch<std::complex<T>, A> pow(const batch<std::complex<T>, A>& a, const batch<std::complex<T>, A>& z, requires_arch<generic>) noexcept
          {
              using cplx_batch = batch<std::complex<T>, A>;
              using real_batch = typename cplx_batch::real_batch;
              real_batch absa = abs(a);
              real_batch arga = arg(a);
              real_batch x = z.real();
              real_batch y = z.imag();
              real_batch r = pow(absa, x);
              real_batch theta = x * arga;
              real_batch ze(0);
              auto cond = (y == ze);
              r = select(cond, r, r * exp(-y * arga));
              theta = select(cond, theta, theta + y * log(absa));
              return select(absa == ze, cplx_batch(ze), cplx_batch(r * cos(theta), r * sin(theta)));
          }
  
          // remainder
          template <class A>
          inline batch<float, A> remainder(batch<float, A> const& self, batch<float, A> const& other, requires_arch<generic>) noexcept
          {
              return fnma(nearbyint(self / other), other, self);
          }
          template <class A>
          inline batch<double, A> remainder(batch<double, A> const& self, batch<double, A> const& other, requires_arch<generic>) noexcept
          {
              return fnma(nearbyint(self / other), other, self);
          }
          template <class A, class T, class = typename std::enable_if<std::is_integral<T>::value, void>::type>
          inline batch<T, A> remainder(batch<T, A> const& self, batch<T, A> const& other, requires_arch<generic>) noexcept
          {
              auto mod = self % other;
              return select(mod <= other / 2, mod, mod - other);
          }
  
          // select
          template <class A, class T>
          inline batch<std::complex<T>, A> select(batch_bool<T, A> const& cond, batch<std::complex<T>, A> const& true_br, batch<std::complex<T>, A> const& false_br, requires_arch<generic>) noexcept
          {
              return { select(cond, true_br.real(), false_br.real()), select(cond, true_br.imag(), false_br.imag()) };
          }
  
          // sign
          template <class A, class T, class = typename std::enable_if<std::is_integral<T>::value, void>::type>
          inline batch<T, A> sign(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              batch_type res = select(self > batch_type(0), batch_type(1), batch_type(0)) - select(self < batch_type(0), batch_type(1), batch_type(0));
              return res;
          }
  
          namespace detail
          {
              template <class T, class A>
              inline batch<T, A> signf(batch<T, A> const& self) noexcept
              {
                  using batch_type = batch<T, A>;
                  batch_type res = select(self > batch_type(0.f), batch_type(1.f), batch_type(0.f)) - select(self < batch_type(0.f), batch_type(1.f), batch_type(0.f));
  #ifdef XSIMD_NO_NANS
                  return res;
  #else
                  return select(isnan(self), constants::nan<batch_type>(), res);
  #endif
              }
          }
  
          template <class A>
          inline batch<float, A> sign(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::signf(self);
          }
          template <class A>
          inline batch<double, A> sign(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::signf(self);
          }
          template <class A, class T>
          inline batch<std::complex<T>, A> sign(const batch<std::complex<T>, A>& z, requires_arch<generic>) noexcept
          {
              using batch_type = batch<std::complex<T>, A>;
              using real_batch = typename batch_type::real_batch;
              auto rz = z.real();
              auto iz = z.imag();
              return select(rz != real_batch(0.),
                            batch_type(sign(rz)),
                            batch_type(sign(iz)));
          }
  
          // signnz
          template <class A, class T, class = typename std::enable_if<std::is_integral<T>::value, void>::type>
          inline batch<T, A> signnz(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              return (self >> (sizeof(T) * 8 - 1)) | batch_type(1.);
          }
  
          namespace detail
          {
              template <class T, class A>
              inline batch<T, A> signnzf(batch<T, A> const& self) noexcept
              {
                  using batch_type = batch<T, A>;
  #ifndef XSIMD_NO_NANS
                  return select(isnan(self), constants::nan<batch_type>(), batch_type(1.) | (constants::signmask<batch_type>() & self));
  #else
                  return batch_type(1.) | (constants::signmask<batch_type>() & self);
  #endif
              }
          }
  
          template <class A>
          inline batch<float, A> signnz(batch<float, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::signnzf(self);
          }
          template <class A>
          inline batch<double, A> signnz(batch<double, A> const& self, requires_arch<generic>) noexcept
          {
              return detail::signnzf(self);
          }
  
          // sqrt
          template <class A, class T>
          inline batch<std::complex<T>, A> sqrt(batch<std::complex<T>, A> const& z, requires_arch<generic>) noexcept
          {
  
              constexpr T csqrt_scale_factor = std::is_same<T, float>::value ? 6.7108864e7f : 1.8014398509481984e16;
              constexpr T csqrt_scale = std::is_same<T, float>::value ? 1.220703125e-4f : 7.450580596923828125e-9;
              using batch_type = batch<std::complex<T>, A>;
              using real_batch = batch<T, A>;
              real_batch x = z.real();
              real_batch y = z.imag();
              real_batch sqrt_x = sqrt(fabs(x));
              real_batch sqrt_hy = sqrt(0.5 * fabs(y));
              auto cond = (fabs(x) > real_batch(4.) || fabs(y) > real_batch(4.));
              x = select(cond, x * 0.25, x * csqrt_scale_factor);
              y = select(cond, y * 0.25, y * csqrt_scale_factor);
              real_batch scale = select(cond, real_batch(2.), real_batch(csqrt_scale));
              real_batch r = abs(batch_type(x, y));
  
              auto condxp = x > real_batch(0.);
              real_batch t0 = select(condxp, xsimd::sqrt(0.5 * (r + x)), xsimd::sqrt(0.5 * (r - x)));
              real_batch r0 = scale * fabs((0.5 * y) / t0);
              t0 *= scale;
              real_batch t = select(condxp, t0, r0);
              r = select(condxp, r0, t0);
              batch_type resg = select(y < real_batch(0.), batch_type(t, -r), batch_type(t, r));
              real_batch ze(0.);
  
              return select(y == ze,
                            select(x == ze,
                                   batch_type(ze, ze),
                                   select(x < ze, batch_type(ze, sqrt_x), batch_type(sqrt_x, ze))),
                            select(x == ze,
                                   select(y > ze, batch_type(sqrt_hy, sqrt_hy), batch_type(sqrt_hy, -sqrt_hy)),
                                   resg));
          }
  
          // tgamma
  
          namespace detail
          {
              /* origin: boost/simd/arch/common/detail/generic/stirling_kernel.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class B>
              struct stirling_kernel;
  
              template <class A>
              struct stirling_kernel<batch<float, A>>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type compute(const batch_type& x) noexcept
                  {
                      return horner<batch_type,
                                    0x3daaaaab,
                                    0x3b638e39,
                                    0xbb2fb930,
                                    0xb970b359>(x);
                  }
  
                  static inline batch_type split_limit() noexcept
                  {
                      return batch_type(bit_cast<float>(uint32_t(0x41d628f6)));
                  }
  
                  static inline batch_type large_limit() noexcept
                  {
                      return batch_type(bit_cast<float>(uint32_t(0x420c28f3)));
                  }
              };
  
              template <class A>
              struct stirling_kernel<batch<double, A>>
              {
                  using batch_type = batch<double, A>;
                  static inline batch_type compute(const batch_type& x) noexcept
                  {
                      return horner<batch_type,
                                    0x3fb5555555555986ull, //   8.33333333333482257126E-2
                                    0x3f6c71c71b98c5fdull, //   3.47222221605458667310E-3
                                    0xbf65f72607d44fd7ull, //  -2.68132617805781232825E-3
                                    0xbf2e166b27e61d7cull, //  -2.29549961613378126380E-4
                                    0x3f49cc72592d7293ull //   7.87311395793093628397E-4
                                    >(x);
                  }
  
                  static inline batch_type split_limit() noexcept
                  {
                      return batch_type(bit_cast<double>(uint64_t(0x4061e083ba3443d4)));
                  }
  
                  static inline batch_type large_limit() noexcept
                  {
                      return batch_type(bit_cast<double>(uint64_t(0x4065800000000000)));
                  }
              };
  
              /* origin: boost/simd/arch/common/simd/function/stirling.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class T, class A>
              inline batch<T, A> stirling(const batch<T, A>& a) noexcept
              {
                  using batch_type = batch<T, A>;
                  const batch_type stirlingsplitlim = stirling_kernel<batch_type>::split_limit();
                  const batch_type stirlinglargelim = stirling_kernel<batch_type>::large_limit();
                  batch_type x = select(a >= batch_type(0.), a, constants::nan<batch_type>());
                  batch_type w = batch_type(1.) / x;
                  w = fma(w, stirling_kernel<batch_type>::compute(w), batch_type(1.));
                  batch_type y = exp(-x);
                  auto test = (x < stirlingsplitlim);
                  batch_type z = x - batch_type(0.5);
                  z = select(test, z, batch_type(0.5) * z);
                  batch_type v = exp(z * log(abs(x)));
                  y *= v;
                  y = select(test, y, y * v);
                  y *= constants::sqrt_2pi<batch_type>() * w;
  #ifndef XSIMD_NO_INFINITIES
                  y = select(isinf(x), x, y);
  #endif
                  return select(x > stirlinglargelim, constants::infinity<batch_type>(), y);
              }
  
              /* origin: boost/simd/arch/common/detail/generic/gamma_kernel.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class B>
              struct tgamma_kernel;
  
              template <class A>
              struct tgamma_kernel<batch<float, A>>
              {
                  using batch_type = batch<float, A>;
                  static inline batch_type compute(const batch_type& x) noexcept
                  {
                      return horner<batch_type,
                                    0x3f800000UL, //  9.999999757445841E-01
                                    0x3ed87799UL, //  4.227874605370421E-01
                                    0x3ed2d411UL, //  4.117741948434743E-01
                                    0x3da82a34UL, //  8.211174403261340E-02
                                    0x3d93ae7cUL, //  7.211014349068177E-02
                                    0x3b91db14UL, //  4.451165155708328E-03
                                    0x3ba90c99UL, //  5.158972571345137E-03
                                    0x3ad28b22UL //  1.606319369134976E-03
                                    >(x);
                  }
              };
  
              template <class A>
              struct tgamma_kernel<batch<double, A>>
              {
                  using batch_type = batch<double, A>;
                  static inline batch_type compute(const batch_type& x) noexcept
                  {
                      return horner<batch_type,
                                    0x3ff0000000000000ULL, // 9.99999999999999996796E-1
                                    0x3fdfa1373993e312ULL, // 4.94214826801497100753E-1
                                    0x3fca8da9dcae7d31ULL, // 2.07448227648435975150E-1
                                    0x3fa863d918c423d3ULL, // 4.76367800457137231464E-2
                                    0x3f8557cde9db14b0ULL, // 1.04213797561761569935E-2
                                    0x3f5384e3e686bfabULL, // 1.19135147006586384913E-3
                                    0x3f24fcb839982153ULL // 1.60119522476751861407E-4
                                    >(x)
                          / horner<batch_type,
                                   0x3ff0000000000000ULL, //  1.00000000000000000320E00
                                   0x3fb24944c9cd3c51ULL, //  7.14304917030273074085E-2
                                   0xbfce071a9d4287c2ULL, // -2.34591795718243348568E-1
                                   0x3fa25779e33fde67ULL, //  3.58236398605498653373E-2
                                   0x3f8831ed5b1bb117ULL, //  1.18139785222060435552E-2
                                   0xBf7240e4e750b44aULL, // -4.45641913851797240494E-3
                                   0x3f41ae8a29152573ULL, //  5.39605580493303397842E-4
                                   0xbef8487a8400d3aFULL // -2.31581873324120129819E-5
                                   >(x);
                  }
              };
  
              /* origin: boost/simd/arch/common/simd/function/gamma.hpp */
              /*
               * ====================================================
               * copyright 2016 NumScale SAS
               *
               * Distributed under the Boost Software License, Version 1.0.
               * (See copy at http://boost.org/LICENSE_1_0.txt)
               * ====================================================
               */
              template <class B>
              inline B tgamma_large_negative(const B& a) noexcept
              {
                  B st = stirling(a);
                  B p = floor(a);
                  B sgngam = select(is_even(p), -B(1.), B(1.));
                  B z = a - p;
                  auto test2 = z < B(0.5);
                  z = select(test2, z - B(1.), z);
                  z = a * sin(z, trigo_pi_tag());
                  z = abs(z);
                  return sgngam * constants::pi<B>() / (z * st);
              }
  
              template <class B, class BB>
              inline B tgamma_other(const B& a, const BB& test) noexcept
              {
                  B x = select(test, B(2.), a);
  #ifndef XSIMD_NO_INFINITIES
                  auto inf_result = (a == constants::infinity<B>());
                  x = select(inf_result, B(2.), x);
  #endif
                  B z = B(1.);
                  auto test1 = (x >= B(3.));
                  while (any(test1))
                  {
                      x = select(test1, x - B(1.), x);
                      z = select(test1, z * x, z);
                      test1 = (x >= B(3.));
                  }
                  test1 = (x < B(0.));
                  while (any(test1))
                  {
                      z = select(test1, z / x, z);
                      x = select(test1, x + B(1.), x);
                      test1 = (x < B(0.));
                  }
                  auto test2 = (x < B(2.));
                  while (any(test2))
                  {
                      z = select(test2, z / x, z);
                      x = select(test2, x + B(1.), x);
                      test2 = (x < B(2.));
                  }
                  x = z * tgamma_kernel<B>::compute(x - B(2.));
  #ifndef XSIMD_NO_INFINITIES
                  return select(inf_result, a, x);
  #else
                  return x;
  #endif
              }
          }
  
          template <class A, class T>
          inline batch<T, A> tgamma(batch<T, A> const& self, requires_arch<generic>) noexcept
          {
              using batch_type = batch<T, A>;
              auto nan_result = (self < batch_type(0.) && is_flint(self));
  #ifndef XSIMD_NO_INVALIDS
              nan_result = isnan(self) || nan_result;
  #endif
              batch_type q = abs(self);
              auto test = (self < batch_type(-33.));
              batch_type r = constants::nan<batch_type>();
              if (any(test))
              {
                  r = detail::tgamma_large_negative(q);
                  if (all(test))
                      return select(nan_result, constants::nan<batch_type>(), r);
              }
              batch_type r1 = detail::tgamma_other(self, test);
              batch_type r2 = select(test, r, r1);
              return select(self == batch_type(0.), copysign(constants::infinity<batch_type>(), self), select(nan_result, constants::nan<batch_type>(), r2));
          }
  
      }
  
  }
  
  #endif
  