EIC code displayed by LXR master/​include/​xsimd/​arch/​common/​xsimd_common_math.hpp	Source navigation Diff markup Identifier search General search
	Version: master test Revert   Change

File indexing completed on 2026-11-07 10:36:59

 /***************************************************************************
  * 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_COMMON_MATH_HPP
 #define XSIMD_COMMON_MATH_HPP
 
 #include "../xsimd_scalar.hpp"
 #include "./xsimd_common_details.hpp"
 #include "./xsimd_common_trigo.hpp"
 
 #include <type_traits>
 
 namespace xsimd
 {
 
     namespace kernel
     {
 
         using namespace types;
         // abs
         template <class A, class T, class>
         XSIMD_INLINE batch<T, A> abs(batch<T, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<T, A> abs(batch<std::complex<T>, A> const& z, requires_arch<common>) noexcept
         {
             return hypot(z.real(), z.imag());
         }
 
         // avg
         namespace detail
         {
             template <class A, class T>
             XSIMD_INLINE batch<T, A> avg(batch<T, A> const& x, batch<T, A> const& y, std::true_type, std::false_type) noexcept
             {
                 return (x & y) + ((x ^ y) >> 1);
             }
 
             template <class A, class T>
             XSIMD_INLINE batch<T, A> avg(batch<T, A> const& x, batch<T, A> const& y, std::true_type, std::true_type) noexcept
             {
                 // Inspired by
                 // https://stackoverflow.com/questions/5697500/take-the-average-of-two-signed-numbers-in-c
                 auto t = (x & y) + ((x ^ y) >> 1);
                 auto t_u = bitwise_cast<std::make_unsigned_t<T>>(t);
                 auto avg = t + (bitwise_cast<T>(t_u >> (8 * sizeof(T) - 1)) & (x ^ y));
                 return avg;
             }
 
             template <class A, class T>
             XSIMD_INLINE batch<T, A> avg(batch<T, A> const& x, batch<T, A> const& y, std::false_type, std::true_type) noexcept
             {
                 return (x + y) / 2;
             }
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<T, A> avg(batch<T, A> const& x, batch<T, A> const& y, requires_arch<common>) noexcept
         {
             return detail::avg(x, y, typename std::is_integral<T>::type {}, typename std::is_signed<T>::type {});
         }
 
         // avgr
         namespace detail
         {
             template <class A, class T>
             XSIMD_INLINE batch<T, A> avgr(batch<T, A> const& x, batch<T, A> const& y, std::true_type) noexcept
             {
                 constexpr unsigned shift = 8 * sizeof(T) - 1;
                 auto adj = std::is_signed<T>::value ? ((x ^ y) & 0x1) : (((x ^ y) << shift) >> shift);
                 return ::xsimd::kernel::avg(x, y, A {}) + adj;
             }
 
             template <class A, class T>
             XSIMD_INLINE batch<T, A> avgr(batch<T, A> const& x, batch<T, A> const& y, std::false_type) noexcept
             {
                 return ::xsimd::kernel::avg(x, y, A {});
             }
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<T, A> avgr(batch<T, A> const& x, batch<T, A> const& y, requires_arch<common>) noexcept
         {
             return detail::avgr(x, y, typename std::is_integral<T>::type {});
         }
 
         // batch_cast
         template <class A, class T>
         XSIMD_INLINE batch<T, A> batch_cast(batch<T, A> const& self, batch<T, A> const&, requires_arch<common>) noexcept
         {
             return self;
         }
 
         namespace detail
         {
             template <class A, class T_out, class T_in>
             XSIMD_INLINE batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const& out, requires_arch<common>, with_fast_conversion) noexcept
             {
                 return fast_cast(self, out, A {});
             }
 #if defined(__clang__) || __GNUC__
             template <class A, class T_out, class T_in>
             XSIMD_INLINE batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const&, requires_arch<common>, with_slow_conversion) noexcept
                 __attribute__((no_sanitize("undefined")));
 #endif
             template <class A, class T_out, class T_in>
             XSIMD_INLINE batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const&, requires_arch<common>, 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]);
                 for (size_t i = 0; i < batch_type_in::size; ++i)
                     buffer_out[i] = static_cast<T_out>(buffer_in[i]);
                 return batch_type_out::load_aligned(buffer_out);
             }
 
         }
 
         template <class A, class T_out, class T_in>
         XSIMD_INLINE batch<T_out, A> batch_cast(batch<T_in, A> const& self, batch<T_out, A> const& out, requires_arch<common>) noexcept
         {
             return detail::batch_cast(self, out, A {}, detail::conversion_type<A, T_in, T_out> {});
         }
 
         // bitofsign
         template <class A, class T>
         XSIMD_INLINE batch<T, A> bitofsign(batch<T, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<float, A> bitofsign(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             return self & constants::signmask<batch<float, A>>();
         }
         template <class A>
         XSIMD_INLINE batch<double, A> bitofsign(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             return self & constants::signmask<batch<double, A>>();
         }
 
         // bitwise_cast
         template <class A, class T>
         XSIMD_INLINE batch<T, A> bitwise_cast(batch<T, A> const& self, batch<T, A> const&, requires_arch<common>) noexcept
         {
             return self;
         }
 
         // 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>
         XSIMD_INLINE batch<float, A> cbrt(batch<float, A> const& self, requires_arch<common>) 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(batch_bool_cast<float>(flag), CBRT2, CBRT2I);
             const batch_type cbrt4 = select(batch_bool_cast<float>(flag), CBRT4, CBRT4I);
             batch_type fact = select(batch_bool_cast<float>(rem == i_type(1)), cbrt2, batch_type(1.));
             fact = select(batch_bool_cast<float>(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>
         XSIMD_INLINE batch<double, A> cbrt(batch<double, A> const& self, requires_arch<common>) 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(batch_bool_cast<double>(flag), CBRT2, CBRT2I);
             const batch_type cbrt4 = select(batch_bool_cast<double>(flag), CBRT4, CBRT4I);
             batch_type fact = select(batch_bool_cast<double>(rem == i_type(1)), cbrt2, batch_type(1.));
             fact = select(batch_bool_cast<double>(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>
         XSIMD_INLINE batch<T, A> clip(batch<T, A> const& self, batch<T, A> const& lo, batch<T, A> const& hi, requires_arch<common>) noexcept
         {
             return min(hi, max(self, lo));
         }
 
         // copysign
         template <class A, class T, class = std::enable_if_t<std::is_floating_point<T>::value>>
         XSIMD_INLINE batch<T, A> copysign(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             return abs(self) | bitofsign(other);
         }
 
         // erf
 
         namespace detail
         {
             /* origin: boost/simd/arch/common/detail/common/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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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>
         XSIMD_INLINE batch<float, A> erf(batch<float, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<double, A> erf(batch<double, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<float, A> erfc(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<float, A>;
             batch_type x = abs(self);
             auto test0 = self < batch_type(0.);
             batch_type r1(0.);
             auto test1 = 3.f * x < 2.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>
         XSIMD_INLINE batch<double, A> erfc(batch<double, A> const& self, requires_arch<common>) 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>
                 XSIMD_INLINE B operator()(const Ts&... coefs) noexcept
                 {
                     return eval(coefs...);
                 }
 
             private:
                 XSIMD_INLINE B eval(const B& c0) noexcept
                 {
                     return c0;
                 }
 
                 XSIMD_INLINE B eval(const B& c0, const B& c1) noexcept
                 {
                     return fma(x, c1, c0);
                 }
 
                 template <size_t... Is, class Tuple>
                 XSIMD_INLINE B eval(std::index_sequence<Is...>, const Tuple& tuple)
                 {
                     return estrin { x * x }(std::get<Is>(tuple)...);
                 }
 
                 template <class... Args>
                 XSIMD_INLINE B eval(const std::tuple<Args...>& tuple) noexcept
                 {
                     return eval(std::make_index_sequence<sizeof...(Args)>(), tuple);
                 }
 
                 template <class... Args>
                 XSIMD_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>
                 XSIMD_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>
                 XSIMD_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>
                 XSIMD_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>
         XSIMD_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 XSIMD_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 XSIMD_INLINE batch_type reduce(const batch_type& a, batch_type& x) noexcept
                 {
                     batch_type k = nearbyint(constants::invlog_2<batch_type>() * a);
                     detail::reassociation_barrier(k, "compensated exp range reduction");
                     x = fnma(k, constants::log_2hi<batch_type>(), a);
                     detail::reassociation_barrier(x, "compensated exp range reduction");
                     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 XSIMD_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 XSIMD_INLINE batch_type reduce(const batch_type& a, batch_type& x) noexcept
                 {
                     batch_type k = nearbyint(constants::invlog10_2<batch_type>() * a);
                     detail::reassociation_barrier(k, "compensated exp10 range reduction");
                     x = fnma(k, constants::log10_2hi<batch_type>(), a);
                     detail::reassociation_barrier(x, "compensated exp10 range reduction");
                     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 XSIMD_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 XSIMD_INLINE batch_type reduce(const batch_type& a, batch_type& x) noexcept
                 {
                     batch_type k = nearbyint(a);
                     detail::reassociation_barrier(k, "compensated exp2 range reduction");
                     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 XSIMD_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 XSIMD_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);
                     detail::reassociation_barrier(k, "compensated exp range reduction");
                     hi = fnma(k, constants::log_2hi<batch_type>(), a);
                     detail::reassociation_barrier(hi, "compensated exp range reduction");
                     lo = k * constants::log_2lo<batch_type>();
                     x = hi - lo;
                     return k;
                 }
 
                 static XSIMD_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 XSIMD_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 XSIMD_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);
                     detail::reassociation_barrier(k, "compensated exp10 range reduction");
                     x = fnma(k, constants::log10_2hi<batch_type>(), a);
                     detail::reassociation_barrier(x, "compensated exp10 range reduction");
                     x = fnma(k, constants::log10_2lo<batch_type>(), x);
                     return k;
                 }
 
                 static XSIMD_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 XSIMD_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 XSIMD_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>();
                     detail::reassociation_barrier(x, "keep reduced exponent ordered before finalize");
                     return k;
                 }
 
                 static XSIMD_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>
             XSIMD_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)));
 #ifndef __FAST_MATH__
                 x = select(self >= reducer_t::maxlog(), constants::infinity<batch_type>(), x);
 #endif
                 return x;
             }
 
             template <exp_reduction_tag Tag, class A>
             XSIMD_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)));
 #ifndef __FAST_MATH__
                 c = select(self >= reducer_t::maxlog(), constants::infinity<batch_type>(), c);
 #endif
                 return c;
             }
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<T, A> exp(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return detail::exp<detail::exp_tag>(self);
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> exp(batch<std::complex<T>, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<T, A> exp10(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             batch_type out = detail::exp<detail::exp10_tag>(self);
             detail::reassociation_barrier(out, "prevent folding exp10 for literal inputs");
             return out;
         }
 
         // exp2
         template <class A, class T>
         XSIMD_INLINE batch<T, A> exp2(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return detail::exp<detail::exp2_tag>(self);
         }
 
         // expm1
         namespace detail
         {
             /* origin: boost/simd/arch/common/detail/common/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 XSIMD_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<float>((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 XSIMD_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<double>((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>
         XSIMD_INLINE batch<T, A> expm1(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             auto x = detail::expm1(self);
 #ifndef __FAST_MATH__
             x = select(self > constants::maxlog<batch_type>(), constants::infinity<batch_type>(), x);
 #endif
             return select(self < constants::logeps<batch_type>(), batch_type(-1.), x);
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> expm1(const batch<std::complex<T>, A>& z, requires_arch<common>) 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 };
         }
 
         // polar
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> polar(const batch<T, A>& r, const batch<T, A>& theta, requires_arch<common>) noexcept
         {
             auto sincosTheta = sincos(theta);
             return { r * sincosTheta.second, r * sincosTheta.first };
         }
 
         // fdim
         template <class A, class T>
         XSIMD_INLINE batch<T, A> fdim(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             return fmax(batch<T, A>(0), self - other);
         }
 
         // fmod
         template <class A, class T>
         XSIMD_INLINE batch<T, A> fmod(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) 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>
         XSIMD_INLINE batch<T, A> frexp(const batch<T, A>& self, batch<as_integer_t<T>, A>& exp, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             using int_type = as_integer_t<T>;
             using i_type = batch<int_type, A>;
             i_type m1f = constants::mask1frexp<batch_type>();
             i_type r1 = m1f & ::xsimd::bitwise_cast<int_type>(self);
             batch_type x = self & ::xsimd::bitwise_cast<T>(~m1f);
             exp = (r1 >> constants::nmb<batch_type>()) - constants::maxexponentm1<batch_type>();
             exp = select(batch_bool_cast<typename i_type::value_type>(self != batch_type(0.)), exp, i_type(typename i_type::value_type(0)));
             return select((self != batch_type(0.)), x | ::xsimd::bitwise_cast<T>(constants::mask2frexp<batch_type>()), batch_type(0.));
         }
 
         // from bool
         template <class A, class T>
         XSIMD_INLINE batch<T, A> from_bool(batch_bool<T, A> const& self, requires_arch<common>) noexcept
         {
             return batch<T, A>((typename batch<T, A>::register_type)self.data) & batch<T, A>(1);
         }
 
         // horner
         template <class T, class A, uint64_t... Coefs>
         XSIMD_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>
         XSIMD_INLINE batch<T, A> hypot(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             return sqrt(fma(self, self, other * other));
         }
 
         // ipow
         template <class A, class T, class ITy>
         XSIMD_INLINE batch<T, A> ipow(batch<T, A> const& self, ITy other, requires_arch<common>) 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>
         XSIMD_INLINE batch<T, A> ldexp(const batch<T, A>& self, const batch<as_integer_t<T>, A>& other, requires_arch<common>) 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<T>(ik);
         }
 
         // lgamma
         template <class A, class T>
         XSIMD_INLINE batch<T, A> lgamma(batch<T, A> const& self, requires_arch<common>) noexcept;
 
         namespace detail
         {
             /* origin: boost/simd/arch/common/detail/common/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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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 XSIMD_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(0);
                     batch_type r1 = other(q);
                     if (any(ltza))
                     {
 #ifdef __FAST_MATH__
                         r = negative(q, r1);
 #else
                         r = select(inf_result, constants::infinity<batch_type>(), negative(q, r1));
 #endif
                         if (all(ltza))
                             return r;
                     }
                     batch_type r2 = select(ltza, r, r1);
 #ifdef __FAST_MATH__
                     return r2;
 #else
                     return select(a == constants::minusinfinity<batch_type>(), constants::nan<batch_type>(), select(inf_result, constants::infinity<batch_type>(), r2));
 #endif
                 }
 
             private:
                 static XSIMD_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 XSIMD_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 XSIMD_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);
 #ifdef __FAST_MATH__
                     return r2;
 #else
                     return select(a == constants::minusinfinity<batch_type>(), constants::nan<batch_type>(), select(inf_result, constants::infinity<batch_type>(), r2));
 #endif
                 }
 
             private:
                 // FIXME: cannot mark this one as XSIMD_INLINE because there's a
                 // recursive loop on `lgamma'.
                 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 XSIMD_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>
         XSIMD_INLINE batch<T, A> lgamma(batch<T, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<float, A> log(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<float, A>;
             using int_type = as_integer_t<float>;
             using i_type = batch<int_type, A>;
             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(batch_bool_cast<int_type>(test), k - i_type(23), k);
                 x = select(test, x * batch_type(8388608ul), x);
             }
 #endif
             i_type ix = ::xsimd::bitwise_cast<int_type>(x);
             ix += 0x3f800000 - 0x3f3504f3;
             k += (ix >> 23) - 0x7f;
             ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
             x = ::xsimd::bitwise_cast<float>(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);
             detail::reassociation_barrier(dk, "keep compensated k conversion before split log(2) scaling");
             batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, (hfsq + R), dk * constants::log_2lo<batch_type>()) - hfsq + f);
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A>
         XSIMD_INLINE batch<double, A> log(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<double, A>;
             using int_type = as_integer_t<double>;
             using i_type = batch<int_type, A>;
 
             batch_type x = self;
             i_type hx = ::xsimd::bitwise_cast<int_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(batch_bool_cast<int_type>(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);
             detail::reassociation_barrier(dk, "keep compensated k conversion before split log(2) scaling");
             hx = (hx & i_type(0x000fffff)) + 0x3fe6a09e;
             x = ::xsimd::bitwise_cast<double>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<int_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);
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> log(const batch<std::complex<T>, A>& z, requires_arch<common>) noexcept
         {
             return batch<std::complex<T>, A>(log(abs(z)), atan2(z.imag(), z.real()));
         }
 
         // log2
         template <class A>
         XSIMD_INLINE batch<float, A> log2(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<float, A>;
             using int_type = as_integer_t<float>;
             using i_type = batch<int_type, A>;
             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(batch_bool_cast<int_type>(test), k - i_type(25), k);
                 x = select(test, x * batch_type(33554432ul), x);
             }
 #endif
             i_type ix = ::xsimd::bitwise_cast<int_type>(x);
             ix += 0x3f800000 - 0x3f3504f3;
             k += (ix >> 23) - 0x7f;
             ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
             x = ::xsimd::bitwise_cast<float>(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);
             detail::reassociation_barrier(dk, "prevent distributing multiplies through compensated exponent conversion");
             batch_type r = fma(fms(s, hfsq + R, hfsq) + f, constants::invlog_2<batch_type>(), dk);
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A>
         XSIMD_INLINE batch<double, A> log2(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<double, A>;
             using int_type = as_integer_t<double>;
             using i_type = batch<int_type, A>;
             batch_type x = self;
             i_type hx = ::xsimd::bitwise_cast<int_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(batch_bool_cast<typename i_type::value_type>(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<double>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<int_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<double>((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);
             detail::reassociation_barrier(dk, "Kahan compensated log2 summation");
             batch_type w1 = dk + val_hi;
             detail::reassociation_barrier(w1, "Kahan compensated log2 summation");
             val_lo += (dk - w1) + val_hi;
             val_hi = w1;
             batch_type r = val_lo + val_hi;
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         namespace detail
         {
             template <class T, class A>
             XSIMD_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>
         XSIMD_INLINE batch<std::complex<T>, A> log2(batch<std::complex<T>, A> const& self, requires_arch<common>) 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>
         XSIMD_INLINE batch<float, A> log10(batch<float, A> const& self, requires_arch<common>) 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 int_type = as_integer_t<float>;
             using i_type = batch<int_type, A>;
             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(batch_bool_cast<int_type>(test), k - i_type(25), k);
                 x = select(test, x * batch_type(33554432ul), x);
             }
 #endif
             i_type ix = ::xsimd::bitwise_cast<int_type>(x);
             ix += 0x3f800000 - 0x3f3504f3;
             k += (ix >> 23) - 0x7f;
             ix = (ix & i_type(0x007fffff)) + 0x3f3504f3;
             x = ::xsimd::bitwise_cast<float>(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);
             detail::reassociation_barrier(dk, "prevent distributing multiplies through compensated exponent conversion");
             batch_type hfsq = batch_type(0.5) * f * f;
             batch_type hibits = f - hfsq;
             hibits &= ::xsimd::bitwise_cast<float>(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
             assert(all(isnez) && "Calling log10 on a batch with zero value while XSIMD_NO_INFINITIES is active");
             batch_type zz = r;
 #endif
 #ifndef XSIMD_NO_NANS
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #else
             assert(all(self >= batch_type(0.)) && "Calling log10 on a batch with negative value while XSIMD_NO_NANS is active");
             return zz;
 #endif
         }
 
         template <class A>
         XSIMD_INLINE batch<double, A> log10(batch<double, A> const& self, requires_arch<common>) 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 int_type = as_integer_t<double>;
             using i_type = batch<int_type, A>;
             batch_type x = self;
             i_type hx = ::xsimd::bitwise_cast<int_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(batch_bool_cast<int_type>(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);
             detail::reassociation_barrier(dk, "prevent distributing multiplies through compensated exponent conversion");
             hx = (hx & i_type(0x000fffff)) + 0x3fe6a09e;
             x = ::xsimd::bitwise_cast<double>(hx << 32 | (i_type(0xffffffff) & ::xsimd::bitwise_cast<int_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<double>(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;
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(self >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> log10(const batch<std::complex<T>, A>& z, requires_arch<common>) 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>
         XSIMD_INLINE batch<float, A> log1p(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<float, A>;
             using int_type = as_integer_t<float>;
             using i_type = batch<int_type, A>;
             const batch_type uf = self + batch_type(1.);
             auto isnez = (uf != batch_type(0.));
             i_type iu = ::xsimd::bitwise_cast<int_type>(uf);
             iu += 0x3f800000 - 0x3f3504f3;
             i_type k = (iu >> 23) - 0x7f;
             iu = (iu & i_type(0x007fffff)) + 0x3f3504f3;
             batch_type f = --(::xsimd::bitwise_cast<float>(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);
             detail::reassociation_barrier(dk, "prevent distributing multiplies through compensated exponent conversion");
             /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
             batch_type c = select(batch_bool_cast<float>(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);
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(uf >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A>
         XSIMD_INLINE batch<double, A> log1p(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<double, A>;
             using int_type = as_integer_t<double>;
             using i_type = batch<int_type, A>;
             const batch_type uf = self + batch_type(1.);
             auto isnez = (uf != batch_type(0.));
             i_type hu = ::xsimd::bitwise_cast<int_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(batch_bool_cast<double>(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<double>((hu << 32) | (i_type(0xffffffff) & ::xsimd::bitwise_cast<int_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);
             detail::reassociation_barrier(dk, "prevent distributing multiplies through compensated exponent conversion");
             batch_type r = fma(dk, constants::log_2hi<batch_type>(), fma(s, hfsq + R, dk * constants::log_2lo<batch_type>() + c) - hfsq + f);
 #ifdef __FAST_MATH__
             return r;
 #else
             batch_type zz = select(isnez, select(self == constants::infinity<batch_type>(), constants::infinity<batch_type>(), r), constants::minusinfinity<batch_type>());
             return select(!(uf >= batch_type(0.)), constants::nan<batch_type>(), zz);
 #endif
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> log1p(batch<std::complex<T>, A> const& self, requires_arch<common>) 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 = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> mod(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             return detail::apply([](T x, T y) noexcept -> T
                                  { return x % y; },
                                  self, other);
         }
 
         // nearbyint
         template <class A, class T, class = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> nearbyint(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return self;
         }
         namespace detail
         {
             template <class A, class T>
             XSIMD_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>();
                 batch_type d0 = v + t2n;
                 detail::reassociation_barrier(d0, "prevent collapsing (v + 2^n) - 2^n back to v");
                 batch_type d = d0 - t2n;
                 return s ^ select(v < t2n, d, v);
             }
         }
         template <class A>
         XSIMD_INLINE batch<float, A> nearbyint(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             return detail::nearbyintf(self);
         }
         template <class A>
         XSIMD_INLINE batch<double, A> nearbyint(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             return detail::nearbyintf(self);
         }
 
         // nearbyint_as_int
         template <class T, class A, class = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> nearbyint_as_int(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return self;
         }
 
         // nearbyint_as_int
         template <class A>
         XSIMD_INLINE batch<as_integer_t<float>, A>
         nearbyint_as_int(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             using U = as_integer_t<float>;
             return kernel::detail::apply_transform<U>([](float x) noexcept -> U
                                                       { return std::nearbyintf(x); },
                                                       self);
         }
 
         template <class A>
         XSIMD_INLINE batch<as_integer_t<double>, A>
         nearbyint_as_int(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             using U = as_integer_t<double>;
             return kernel::detail::apply_transform<U>([](double x) noexcept -> U
                                                       { return std::nearbyint(x); },
                                                       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 XSIMD_INLINE batch_type next(batch_type const& b) noexcept
                 {
                     return b;
                 }
 
                 static XSIMD_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 XSIMD_INLINE batch_type next(const batch_type& b) noexcept
                 {
                     batch_type n = ::xsimd::bitwise_cast<T>(::xsimd::bitwise_cast<int_type>(b) + int_type(1));
 #ifdef __FAST_MATH__
                     return n;
 #else
                     return select(b == constants::infinity<batch_type>(), b, n);
 #endif
                 }
 
                 static XSIMD_INLINE batch_type prev(const batch_type& b) noexcept
                 {
                     batch_type p = ::xsimd::bitwise_cast<T>(::xsimd::bitwise_cast<int_type>(b) - int_type(1));
 #ifdef __FAST_MATH__
                     return p;
 #else
                     return select(b == constants::minusinfinity<batch_type>(), b, p);
 #endif
                 }
             };
         }
         template <class A, class T>
         XSIMD_INLINE batch<T, A> nextafter(batch<T, A> const& from, batch<T, A> const& to, requires_arch<common>) 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>
         XSIMD_INLINE batch<T, A> pow(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             const auto zero = batch_type(0.);
             auto negself = self < zero;
             auto iszeropowpos = self == zero && other >= zero;
             auto adj_self = select(iszeropowpos, batch_type(1), abs(self));
             batch_type z = exp(other * log(adj_self));
             z = select(iszeropowpos, zero, z);
             z = select(is_odd(other) && negself, -z, z);
             auto invalid = negself && !(is_flint(other) || isinf(other));
             return select(invalid, constants::nan<batch_type>(), z);
         }
 
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> pow(const batch<std::complex<T>, A>& a, const batch<std::complex<T>, A>& z, requires_arch<common>) 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));
             auto sincosTheta = xsimd::sincos(theta);
             return select(absa == ze, cplx_batch(ze), cplx_batch(r * sincosTheta.second, r * sincosTheta.first));
         }
 
         template <class A, class T>
         inline batch<std::complex<T>, A> pow(const batch<std::complex<T>, A>& a, const batch<T, A>& z, requires_arch<common>) noexcept
         {
             using cplx_batch = batch<std::complex<T>, A>;
 
             auto absa = abs(a);
             auto arga = arg(a);
             auto r = pow(absa, z);
 
             auto theta = z * arga;
             auto sincosTheta = xsimd::sincos(theta);
             return select(absa == 0, cplx_batch(0), cplx_batch(r * sincosTheta.second, r * sincosTheta.first));
         }
 
         template <class A, class T>
         inline batch<std::complex<T>, A> pow(const batch<T, A>& a, const batch<std::complex<T>, A>& z, requires_arch<common>) noexcept
         {
             return pow(batch<std::complex<T>, A> { a, batch<T, A> {} }, z);
         }
 
         // reciprocal
         template <class T, class A, class = std::enable_if_t<std::is_floating_point<T>::value>>
         XSIMD_INLINE batch<T, A> reciprocal(batch<T, A> const& self,
                                             requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             return div(batch_type(1), self);
         }
 
         // reduce_add
         template <class A, class T>
         XSIMD_INLINE std::complex<T> reduce_add(batch<std::complex<T>, A> const& self, requires_arch<common>) noexcept
         {
             return { reduce_add(self.real()), reduce_add(self.imag()) };
         }
 
         template <class A, class T, class /*=std::enable_if_t<std::is_scalar<T>::value>*/>
         XSIMD_INLINE T reduce_add(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             alignas(A::alignment()) T buffer[batch<T, A>::size];
             self.store_aligned(buffer);
             T res = 0;
             for (T val : buffer)
             {
                 res += val;
             }
             return res;
         }
 
         namespace detail
         {
             template <class T, T N>
             struct split_high
             {
                 static constexpr T get(T i, T)
                 {
                     return i < N ? (i + N) : ((i % N) + N);
                 }
             };
 
             template <class Op, class A, class T>
             XSIMD_INLINE T reduce(Op, batch<T, A> const& self, std::integral_constant<unsigned, 1>) noexcept
             {
                 return ::xsimd::kernel::first(self, A {});
             }
 
             template <class Op, class A, class T, unsigned Lvl>
             XSIMD_INLINE T reduce(Op op, batch<T, A> const& self, std::integral_constant<unsigned, Lvl>) noexcept
             {
                 using index_type = as_unsigned_integer_t<T>;
                 batch<T, A> split = swizzle(self, make_batch_constant<index_type, split_high<index_type, Lvl / 2>, A>());
                 return reduce(op, op(split, self), std::integral_constant<unsigned, Lvl / 2>());
             }
         }
 
         // reduce_max
         template <class A, class T>
         XSIMD_INLINE T reduce_max(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return detail::reduce([](batch<T, A> const& x, batch<T, A> const& y)
                                   { return max(x, y); },
                                   self, std::integral_constant<unsigned, batch<T, A>::size>());
         }
 
         // reduce_min
         template <class A, class T>
         XSIMD_INLINE T reduce_min(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             return detail::reduce([](batch<T, A> const& x, batch<T, A> const& y)
                                   { return min(x, y); },
                                   self, std::integral_constant<unsigned, batch<T, A>::size>());
         }
 
         // reduce_mul
         template <class A, class T>
         XSIMD_INLINE std::complex<T> reduce_mul(batch<std::complex<T>, A> const& self, requires_arch<common>) noexcept
         {
             // FIXME: could do better
             alignas(A::alignment()) std::complex<T> buffer[batch<std::complex<T>, A>::size];
             self.store_aligned(buffer);
             std::complex<T> res = 1;
             for (auto val : buffer)
             {
                 res *= val;
             }
             return res;
         }
 
         template <class A, class T, class /*=std::enable_if_t<std::is_scalar<T>::value>*/>
         XSIMD_INLINE T reduce_mul(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             alignas(A::alignment()) T buffer[batch<T, A>::size];
             self.store_aligned(buffer);
             T res = 1;
             for (T val : buffer)
             {
                 res *= val;
             }
             return res;
         }
 
         // remainder
         template <class A>
         XSIMD_INLINE batch<float, A> remainder(batch<float, A> const& self, batch<float, A> const& other, requires_arch<common>) noexcept
         {
             batch<float, A> q = nearbyint(self / other);
             detail::reassociation_barrier(q, "prevent pulling multiply back through rounded quotient");
             return fnma(q, other, self);
         }
         template <class A>
         XSIMD_INLINE batch<double, A> remainder(batch<double, A> const& self, batch<double, A> const& other, requires_arch<common>) noexcept
         {
             batch<double, A> q = nearbyint(self / other);
             detail::reassociation_barrier(q, "prevent pulling multiply back through rounded quotient");
             return fnma(q, other, self);
         }
         template <class A, class T, class = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> remainder(batch<T, A> const& self, batch<T, A> const& other, requires_arch<common>) noexcept
         {
             auto mod = self % other;
             return select(mod <= other / 2, mod, mod - other);
         }
 
         // select
         template <class A, class T>
         XSIMD_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<common>) 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 = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> sign(batch<T, A> const& self, requires_arch<common>) 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>
             XSIMD_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>
         XSIMD_INLINE batch<float, A> sign(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             return detail::signf(self);
         }
         template <class A>
         XSIMD_INLINE batch<double, A> sign(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             return detail::signf(self);
         }
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> sign(const batch<std::complex<T>, A>& z, requires_arch<common>) 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 = std::enable_if_t<std::is_integral<T>::value>>
         XSIMD_INLINE batch<T, A> signnz(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             return (self >> (sizeof(T) * 8 - 1)) | batch_type(1.);
         }
 
         namespace detail
         {
             template <class T, class A>
             XSIMD_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>
         XSIMD_INLINE batch<float, A> signnz(batch<float, A> const& self, requires_arch<common>) noexcept
         {
             return detail::signnzf(self);
         }
         template <class A>
         XSIMD_INLINE batch<double, A> signnz(batch<double, A> const& self, requires_arch<common>) noexcept
         {
             return detail::signnzf(self);
         }
 
         // sqrt
         template <class A, class T>
         XSIMD_INLINE batch<std::complex<T>, A> sqrt(batch<std::complex<T>, A> const& z, requires_arch<common>) 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/common/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 XSIMD_INLINE batch_type compute(const batch_type& x) noexcept
                 {
                     return horner<batch_type,
                                   0x3daaaaab,
                                   0x3b638e39,
                                   0xbb2fb930,
                                   0xb970b359>(x);
                 }
 
                 static XSIMD_INLINE batch_type split_limit() noexcept
                 {
                     return batch_type(bit_cast<float>(uint32_t(0x41d628f6)));
                 }
 
                 static XSIMD_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 XSIMD_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 XSIMD_INLINE batch_type split_limit() noexcept
                 {
                     return batch_type(bit_cast<double>(uint64_t(0x4061e083ba3443d4)));
                 }
 
                 static XSIMD_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>
             XSIMD_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;
 #ifdef __FAST_MATH__
                 return y;
 #else
                 y = select(isinf(x), x, y);
                 return select(x > stirlinglargelim, constants::infinity<batch_type>(), y);
 #endif
             }
 
             /* origin: boost/simd/arch/common/detail/common/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 XSIMD_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 XSIMD_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>
             XSIMD_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>
             XSIMD_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>
         XSIMD_INLINE batch<T, A> tgamma(batch<T, A> const& self, requires_arch<common>) noexcept
         {
             using batch_type = batch<T, A>;
             auto nan_result = (self < batch_type(0.) && is_flint(self));
 #ifndef XSIMD_NO_NANS
             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);
 #ifdef __FAST_MATH__
             return r2;
 #else
             return select(self == batch_type(0.), copysign(constants::infinity<batch_type>(), self), select(nan_result, constants::nan<batch_type>(), r2));
 #endif
         }
 
     }
 
 }
 
 #endif