// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0

#include <limits.h>
#include "main.h"
#include "../Eigen/SpecialFunctions"

// Workaround to allow "implicit" conversions from double to Scalar via comma-initialization.
template <typename Derived>
Eigen::CommaInitializer<Derived> operator<<(Eigen::DenseBase<Derived>& dense, double v) {
  return (dense << static_cast<typename Derived::Scalar>(v));
}

template <typename XprType>
Eigen::CommaInitializer<XprType>& operator,(Eigen::CommaInitializer<XprType>& ci, double v) {
  return (ci, static_cast<typename XprType::Scalar>(v));
}

template <typename X, typename Y>
void verify_component_wise(const X& x, const Y& y) {
  for (Index i = 0; i < x.size(); ++i) {
    if ((numext::isfinite)(y(i)))
      VERIFY_IS_APPROX(x(i), y(i));
    else if ((numext::isnan)(y(i)))
      VERIFY((numext::isnan)(x(i)));
    else
      VERIFY_IS_EQUAL(x(i), y(i));
  }
}

template <typename ArrayType>
void array_special_functions() {
  using std::abs;
  using std::sqrt;
  typedef typename ArrayType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;

  Scalar plusinf = std::numeric_limits<Scalar>::infinity();
  Scalar nan = std::numeric_limits<Scalar>::quiet_NaN();

  Index rows = internal::random<Index>(1, 30);
  Index cols = 1;

  // API
  {
    ArrayType m1 = ArrayType::Random(rows, cols);
    VERIFY_IS_APPROX(m1.lgamma(), lgamma(m1));
    VERIFY_IS_APPROX(m1.digamma(), digamma(m1));
    VERIFY_IS_APPROX(m1.erf(), erf(m1));
    VERIFY_IS_APPROX(m1.erfc(), erfc(m1));
  }

  // check special functions (comparing against numpy implementation)
  if (!NumTraits<Scalar>::IsComplex) {
    {
      ArrayType m1 = ArrayType::Random(rows, cols);
      ArrayType m2 = ArrayType::Random(rows, cols);

      // Test various properties of igamma & igammac.  These are normalized
      // gamma integrals where
      //   igammac(a, x) = Gamma(a, x) / Gamma(a)
      //   igamma(a, x) = gamma(a, x) / Gamma(a)
      // where Gamma and gamma are considered the standard unnormalized
      // upper and lower incomplete gamma functions, respectively.
      ArrayType a = m1.abs() + Scalar(2);
      ArrayType x = m2.abs() + Scalar(2);
      ArrayType zero = ArrayType::Zero(rows, cols);
      ArrayType one = ArrayType::Constant(rows, cols, Scalar(1.0));
      ArrayType a_m1 = a - one;
      ArrayType Gamma_a_x = Eigen::igammac(a, x) * a.lgamma().exp();
      ArrayType Gamma_a_m1_x = Eigen::igammac(a_m1, x) * a_m1.lgamma().exp();
      ArrayType gamma_a_x = Eigen::igamma(a, x) * a.lgamma().exp();
      ArrayType gamma_a_m1_x = Eigen::igamma(a_m1, x) * a_m1.lgamma().exp();

      // Gamma(a, 0) == Gamma(a)
      VERIFY_IS_APPROX(Eigen::igammac(a, zero), one);

      // Gamma(a, x) + gamma(a, x) == Gamma(a)
      VERIFY_IS_APPROX(Gamma_a_x + gamma_a_x, a.lgamma().exp());

      // Gamma(a, x) == (a - 1) * Gamma(a-1, x) + x^(a-1) * exp(-x)
      VERIFY_IS_APPROX(Gamma_a_x, (a - Scalar(1)) * Gamma_a_m1_x + x.pow(a - Scalar(1)) * (-x).exp());

      // gamma(a, x) == (a - 1) * gamma(a-1, x) - x^(a-1) * exp(-x)
      VERIFY_IS_APPROX(gamma_a_x, (a - Scalar(1)) * gamma_a_m1_x - x.pow(a - Scalar(1)) * (-x).exp());
    }
    {
      // Verify for large a and x that values are between 0 and 1.
      ArrayType m1 = ArrayType::Random(rows, cols);
      ArrayType m2 = ArrayType::Random(rows, cols);
      int max_exponent = std::numeric_limits<Scalar>::max_exponent10;
      ArrayType a = m1.abs() * Scalar(pow(10., max_exponent - 1));
      ArrayType x = m2.abs() * Scalar(pow(10., max_exponent - 1));
      for (int i = 0; i < a.size(); ++i) {
        Scalar igam = numext::igamma(a(i), x(i));
        VERIFY(0 <= igam);
        VERIFY(igam <= 1);
      }
    }

    {
      // Check exact values of igamma and igammac against a third party calculation.
      Scalar a_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)};
      Scalar x_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)};

      // location i*6+j corresponds to a_s[i], x_s[j].
      Scalar igamma_s[][6] = {
          {Scalar(0.0), nan, nan, nan, nan, nan},
          {Scalar(0.0), Scalar(0.6321205588285578), Scalar(0.7768698398515702), Scalar(0.9816843611112658),
           Scalar(9.999500016666262e-05), Scalar(1.0)},
          {Scalar(0.0), Scalar(0.4275932955291202), Scalar(0.608374823728911), Scalar(0.9539882943107686),
           Scalar(7.522076445089201e-07), Scalar(1.0)},
          {Scalar(0.0), Scalar(0.01898815687615381), Scalar(0.06564245437845008), Scalar(0.5665298796332909),
           Scalar(4.166333347221828e-18), Scalar(1.0)},
          {Scalar(0.0), Scalar(0.9999780593618628), Scalar(0.9999899967080838), Scalar(0.9999996219837988),
           Scalar(0.9991370418689945), Scalar(1.0)},
          {Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.0), Scalar(0.5042041932513908)}};
      Scalar igammac_s[][6] = {
          {nan, nan, nan, nan, nan, nan},
          {Scalar(1.0), Scalar(0.36787944117144233), Scalar(0.22313016014842982), Scalar(0.018315638888734182),
           Scalar(0.9999000049998333), Scalar(0.0)},
          {Scalar(1.0), Scalar(0.5724067044708798), Scalar(0.3916251762710878), Scalar(0.04601170568923136),
           Scalar(0.9999992477923555), Scalar(0.0)},
          {Scalar(1.0), Scalar(0.9810118431238462), Scalar(0.9343575456215499), Scalar(0.4334701203667089), Scalar(1.0),
           Scalar(0.0)},
          {Scalar(1.0), Scalar(2.1940638138146658e-05), Scalar(1.0003291916285e-05), Scalar(3.7801620118431334e-07),
           Scalar(0.0008629581310054535), Scalar(0.0)},
          {Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(1.0), Scalar(0.49579580674813944)}};

      for (int i = 0; i < 6; ++i) {
        for (int j = 0; j < 6; ++j) {
          if ((std::isnan)(igamma_s[i][j])) {
            VERIFY((std::isnan)(numext::igamma(a_s[i], x_s[j])));
          } else {
            VERIFY_IS_APPROX(numext::igamma(a_s[i], x_s[j]), igamma_s[i][j]);
          }

          if ((std::isnan)(igammac_s[i][j])) {
            VERIFY((std::isnan)(numext::igammac(a_s[i], x_s[j])));
          } else {
            VERIFY_IS_APPROX(numext::igammac(a_s[i], x_s[j]), igammac_s[i][j]);
          }
        }
      }
    }
  }

  // Check the ndtri function against scipy.special.ndtri
  {
    ArrayType x(11), res(11), ref(11);
    x << 0.5, 0.2, 0.8, 0.9, 0.1, 0.99, 0.01, 0, 1, -0.01, 1.01;
    ref << 0., -0.8416212335729142, 0.8416212335729142, 1.2815515655446004, -1.2815515655446004, 2.3263478740408408,
        -2.3263478740408408, -plusinf, plusinf, nan, nan;
    CALL_SUBTEST(verify_component_wise(ref, ref););
    CALL_SUBTEST(res = x.ndtri(); verify_component_wise(res, ref););
    CALL_SUBTEST(res = ndtri(x); verify_component_wise(res, ref););

    // ndtri(normal_cdf(x)) ~= x. A rounding error of eps * p in p = normal_cdf(x) moves ndtri(p) by eps * p / pdf(x),
    // at most 3.5 eps on [-1, 1], so the error is bounded absolutely rather than relative to x, which may be near 0.
    // The rest of the 16 eps covers the rounding of erf and ndtri themselves.
    CALL_SUBTEST(ArrayType m1 = ArrayType::Random(32); using std::sqrt;

                 ArrayType cdf_val = (m1 / Scalar(sqrt(2.))).erf(); cdf_val = (cdf_val + Scalar(1)) / Scalar(2);
                 VERIFY(max_abs_coeff(cdf_val.ndtri() - m1) <= RealScalar(16) * NumTraits<RealScalar>::epsilon()););
  }

  // Check the zeta function against scipy.special.zeta
  {
    ArrayType x(11), q(11), res(11), ref(11);
    x << 1.5, 4, 10.5, 10000.5, 3, 1, 0.9, 2, 3, 4, 2000;
    q << 2, 1.5, 3, 1.0001, -2.5, 1.2345, 1.2345, -1, -2, -3, 2000;
    ref << 1.61237534869, 0.234848505667, 1.03086757337e-5, 0.367879440865, 0.054102025820864097, plusinf, nan, plusinf,
        nan, plusinf, 0;
    CALL_SUBTEST(verify_component_wise(ref, ref););
    CALL_SUBTEST(res = x.zeta(q); verify_component_wise(res, ref););
    CALL_SUBTEST(res = zeta(x, q); verify_component_wise(res, ref););
  }

  // digamma
  {
    ArrayType x(10), res(10), ref(10);
    x << 1, 1.5, 4, -10.5, 10000.5, 0.0, -0.0, -1, -2, -3;
    ref << -0.5772156649015329, 0.03648997397857645, 1.2561176684318, 2.398239129535781, 9.210340372392849, -plusinf,
        plusinf, nan, nan, nan;
    CALL_SUBTEST(verify_component_wise(ref, ref););

    CALL_SUBTEST(res = x.digamma(); verify_component_wise(res, ref););
    CALL_SUBTEST(res = digamma(x); verify_component_wise(res, ref););
  }

  {
    ArrayType n(16), x(16), res(16), ref(16);
    n << 1, 1, 1, 1.5, 17, 31, 28, 8, 42, 147, 170, -1, 0, 1, 2, 3;
    x << 2, 3, 25.5, 1.5, 4.7, 11.8, 17.7, 30.2, 15.8, 54.1, 64, -1, -2, -3, -4, -5;
    ref << 0.644934066848, 0.394934066848, 0.0399946696496, nan, 293.334565435, 0.445487887616, -2.47810300902e-07,
        -8.29668781082e-09, -0.434562276666, 0.567742190178, -0.0108615497927, nan, nan, plusinf, nan, plusinf;
    CALL_SUBTEST(verify_component_wise(ref, ref););

    if (sizeof(RealScalar) >= 8) {  // double
      // Reason for commented line: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232
      //       CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res, ref); );
      CALL_SUBTEST(res = polygamma(n, x); verify_component_wise(res, ref););
    } else {
      //       CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res.head(8), ref.head(8)); );
      CALL_SUBTEST(res = polygamma(n, x); verify_component_wise(res.head(8), ref.head(8)););
    }
  }

  {
    // Inputs and ground truth generated with scipy via:
    //   a = np.logspace(-3, 3, 5) - 1e-3
    //   b = np.logspace(-3, 3, 5) - 1e-3
    //   x = np.linspace(-0.1, 1.1, 5)
    //   (full_a, full_b, full_x) = np.vectorize(lambda a, b, x: (a, b, x))(*np.ix_(a, b, x))
    //   full_a = full_a.flatten().tolist()
    //   full_b = full_b.flatten().tolist()
    //   full_x = full_x.flatten().tolist()
    //   v = scipy.special.betainc(full_a, full_b, full_x).flatten().tolist()
    // Note in Eigen, we call betainc with arguments in the order (a, b, x);
    ArrayType a(125);
    ArrayType b(125);
    ArrayType x(125);
    ArrayType v(125);
    ArrayType res(125);

    a << 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
        0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
        0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999,
        999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999,
        999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999;

    b << 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999,
        0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379,
        31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999,
        0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999,
        999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999,
        999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379,
        0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999,
        31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999,
        999.999, 999.999, 999.999;

    x << -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1,
        0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2,
        0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5,
        0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8,
        1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1,
        -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1;

    v << nan, nan, nan, nan, nan, nan, 1.0, 1.0, 1.0, nan, nan, 1.0, 1.0, 1.0, nan, nan, 1.0, 1.0, 1.0, nan, nan, 1.0,
        1.0, 1.0, nan, nan, 0.0, 0.0, 0.0, nan, nan, 0.47972119876364683, 0.4999999999999997, 0.5202788012363533, nan,
        nan, 0.9518683957740043, 0.9789663010413745, 0.993172918807344, nan, nan, 0.999995949033062, 0.99999999999937,
        1.0, nan, nan, 1.0, 1.0, 1.0, nan, nan, 0.0, 0.0, 0.0, nan, nan, 0.0068270811926558735, 0.0210336989586256,
        0.048131604225995696, nan, nan, 0.20014344256217687, 0.5000000000000002, 0.7998565574378237, nan, nan,
        0.9991401428435834, 0.9999999996984031, 1.0, nan, nan, 1.0, 1.0, 1.0, nan, nan, 0.0, 0.0, 0.0, nan, nan,
        1.0646600232370862e-25, 6.30172287782625e-13, 4.050966937974916e-06, nan, nan, 7.864342668429712e-23,
        3.015969667594142e-10, 0.0008598571564165379, nan, nan, 6.031987710123845e-08, 0.5000000000000001,
        0.9999999396801229, nan, nan, 1.0, 1.0, 1.0, nan, nan, 0.0, 0.0, 0.0, nan, nan, 0.0, 7.029920380993552e-306,
        2.245072820861537e-101, nan, nan, 0.0, 9.275871147875753e-302, 1.223291302616039e-97, nan, nan, 0.0,
        3.089139308197642e-252, 2.9303043666230076e-60, nan, nan, 2.248913486881819e-196, 0.5000000000000036, 1.0, nan;

    CALL_SUBTEST(res = betainc(a, b, x); verify_component_wise(res, v););
  }

  // Test various properties of betainc
  {
    ArrayType m1 = ArrayType::Random(32);
    ArrayType m2 = ArrayType::Random(32);
    ArrayType m3 = ArrayType::Random(32);
    ArrayType one = ArrayType::Constant(32, Scalar(1.0));
    const Scalar eps = std::numeric_limits<Scalar>::epsilon();
    ArrayType a = (m1 * Scalar(4)).exp();
    ArrayType b = (m2 * Scalar(4)).exp();
    ArrayType x = m3.abs();

    // ARMv7 NEON flushes subnormal float intermediates, unlike the scalar betainc implementation. Form the
    // reference in at least double precision so x**a and the recurrence term retain subnormal float results.
    using ReferenceScalar = std::conditional_t<(sizeof(Scalar) < sizeof(double)), double, Scalar>;
    using ReferenceArray = Array<ReferenceScalar, Dynamic, 1>;
    const ReferenceArray& ar = a.template cast<ReferenceScalar>();
    const ReferenceArray& br = b.template cast<ReferenceScalar>();
    const ReferenceArray& xr = x.template cast<ReferenceScalar>();
    const Scalar tiny = (std::numeric_limits<Scalar>::min)();

    // betainc(a, 1, x) == x**a. The absolute floor covers rounding of subnormal results.
    CALL_SUBTEST(ArrayType test = betainc(a, one, x); ArrayType expected = xr.pow(ar).template cast<Scalar>();
                 for (Index i = 0; i < test.size(); ++i)
                     VERIFY_IS_APPROX_SCALED(test.segment(i, 1), expected.segment(i, 1), expected(i) + tiny););

    // betainc(1, b, x) == 1 - (1 - x)**b, evaluated as -expm1(b * log1p(-x)): for small b * x the direct form cancels
    // to a relative error near eps / (b * x).
    CALL_SUBTEST(ArrayType test = betainc(one, b, x); ArrayType expected = -(b * (-x).log1p()).expm1();
                 verify_component_wise(test, expected););

    // betainc(a, b, x) == 1 - betainc(b, a, 1-x)
    CALL_SUBTEST(ArrayType test = betainc(a, b, x) + betainc(b, a, one - x); ArrayType expected = one;
                 verify_component_wise(test, expected););

    // betainc(a+1, b, x) = betainc(a, b, x) - x**a * (1 - x)**b / (a * beta(a, b)). Bound cancellation relative
    // to the larger term. Log space avoids underflow of the numerator before division by a small beta(a, b).
    CALL_SUBTEST(ReferenceArray log_beta = ar.lgamma() + br.lgamma() - (ar + br).lgamma();
                 ReferenceArray term = (ar * xr.log() + br * (-xr).log1p() - ar.log() - log_beta).exp();
                 ArrayType lhs = betainc(a, b, x); ArrayType scale = (lhs.max)(term.template cast<Scalar>()) + tiny;
                 ArrayType expected = (lhs.template cast<ReferenceScalar>() - term).template cast<Scalar>();
                 ArrayType test = betainc(a + one, b, x); for (Index i = 0; i < test.size(); ++i)
                     VERIFY_IS_APPROX_SCALED(test.segment(i, 1), expected.segment(i, 1), scale(i)););

    // betainc(a, b+1, x) = betainc(a, b, x) + x**a * (1 - x)**b / (b * beta(a, b))
    CALL_SUBTEST(
        // Add eps to rhs and lhs so that component-wise test doesn't result in
        // nans when both outputs are zeros.
        ArrayType num = x.pow(a) * (one - x).pow(b);
        ArrayType denom = b * (a.lgamma() + b.lgamma() - (a + b).lgamma()).exp();
        ArrayType expected = betainc(a, b, x) + num / denom + eps; ArrayType test = betainc(a, b + one, x) + eps;
        verify_component_wise(test, expected););
  }

  /* Code to generate the data for the following two test cases.
  N = 5
  np.random.seed(3)

  a = np.logspace(-2, 3, 6)
  a = np.ravel(np.tile(np.reshape(a, [-1, 1]), [1, N]))
  x = np.random.gamma(a, 1.0)
  x = np.maximum(x, np.finfo(np.float32).tiny)

  def igamma(a, x):
    return mpmath.gammainc(a, 0, x, regularized=True)

  def igamma_der_a(a, x):
    res = mpmath.diff(lambda a_prime: igamma(a_prime, x), a)
    return np.float64(res)

  def gamma_sample_der_alpha(a, x):
    igamma_x = igamma(a, x)
    def igammainv_of_igamma(a_prime):
      return mpmath.findroot(lambda x_prime: igamma(a_prime, x_prime) -
          igamma_x, x, solver='newton')
    return np.float64(mpmath.diff(igammainv_of_igamma, a))

  v_igamma_der_a = np.vectorize(igamma_der_a)(a, x)
  v_gamma_sample_der_alpha = np.vectorize(gamma_sample_der_alpha)(a, x)
*/

  // Test igamma_der_a
  {
    ArrayType a(30);
    ArrayType x(30);
    ArrayType res(30);
    ArrayType v(30);

    a << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0,
        100.0, 100.0, 100.0, 100.0, 100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

    x << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05, 1.17549435082e-38, 1.17549435082e-38,
        5.66572070696e-16, 0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06, 0.333412038288,
        1.18135687766, 0.580629033777, 0.170631439426, 0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371, 86.3715926467, 96.0330217672, 82.6389930677,
        968.702906754, 969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

    v << -32.7256441441, -36.4394150514, -9.66467612263, -36.4394150514, -36.4394150514, -1.0891900302, -2.66351229645,
        -2.48666868596, -0.929700494428, -3.56327722764, -0.455320135314, -0.391437214323, -0.491352055991,
        -0.350454834292, -0.471773162921, -0.104084440522, -0.0723646747909, -0.0992828975532, -0.121638215446,
        -0.122619605294, -0.0317670267286, -0.0359974812869, -0.0154359225363, -0.0375775365921, -0.00794899153653,
        -0.00777303219211, -0.00796085782042, -0.0125850719397, -0.00455500206958, -0.00476436993148;

    CALL_SUBTEST(res = igamma_der_a(a, x); verify_component_wise(res, v););
  }

  // Test gamma_sample_der_alpha
  {
    ArrayType alpha(30);
    ArrayType sample(30);
    ArrayType res(30);
    ArrayType v(30);

    alpha << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0,
        10.0, 100.0, 100.0, 100.0, 100.0, 100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

    sample << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05, 1.17549435082e-38, 1.17549435082e-38,
        5.66572070696e-16, 0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06, 0.333412038288,
        1.18135687766, 0.580629033777, 0.170631439426, 0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371, 86.3715926467, 96.0330217672, 82.6389930677,
        968.702906754, 969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

    v << 7.42424742367e-23, 1.02004297287e-34, 0.0130155240738, 1.02004297287e-34, 1.02004297287e-34, 1.96505168277e-13,
        0.525575786243, 0.713903991771, 2.32077561808e-14, 0.000179348049886, 0.635500453302, 1.27561284917,
        0.878125852156, 0.41565819538, 1.03606488534, 0.885964824887, 1.16424049334, 1.10764479598, 1.04590810812,
        1.04193666963, 0.965193152414, 0.976217589464, 0.93008035061, 0.98153216096, 0.909196397698, 0.98434963993,
        0.984738050206, 1.00106492525, 0.97734200649, 1.02198794179;

    CALL_SUBTEST(res = gamma_sample_der_alpha(alpha, sample); verify_component_wise(res, v););
  }
}

template <typename Scalar>
void scalar_ndtri() {
  const Scalar sign_mask = internal::psignmask<Scalar>();
  VERIFY_IS_EQUAL(sign_mask, Scalar(0));
  VERIFY((std::signbit)(sign_mask));

  const Scalar packet_inf = internal::pinf<Scalar>();
  const Scalar packet_nan = internal::pnan<Scalar>();
  VERIFY((numext::isinf)(packet_inf));
  VERIFY(packet_inf > Scalar(0));
  VERIFY((numext::isnan)(packet_nan));

  const Scalar expected_quartile = Scalar(0.6744897501960817432L);
  const Scalar lower_quartile = numext::ndtri(Scalar(0.25L));
  const Scalar upper_quartile = numext::ndtri(Scalar(0.75L));
  VERIFY_IS_APPROX(lower_quartile, -expected_quartile);
  VERIFY_IS_APPROX(upper_quartile, expected_quartile);

  // These exactly complementary binary values exercise the scalar flipsign path in the tail approximation.
  const Scalar lower_tail = numext::ndtri(Scalar(0.125L));
  const Scalar upper_tail = numext::ndtri(Scalar(0.875L));
  VERIFY(lower_tail < Scalar(0));
  VERIFY(upper_tail > Scalar(0));
  VERIFY_IS_APPROX(lower_tail, -upper_tail);

  VERIFY_IS_EQUAL(numext::ndtri(Scalar(0.5L)), Scalar(0));
  const Scalar negative_inf = numext::ndtri(Scalar(0));
  const Scalar positive_inf = numext::ndtri(Scalar(1));
  VERIFY((numext::isinf)(negative_inf));
  VERIFY((numext::isinf)(positive_inf));
  VERIFY(negative_inf < Scalar(0));
  VERIFY(positive_inf > Scalar(0));
}

// A scalar type without a vectorized erf/erfc implementation must defer to
// std::erf/std::erfc instead of the float/double-tuned polynomials.
template <typename Scalar>
void scalar_erf_erfc() {
  EIGEN_USING_STD(erf);
  EIGEN_USING_STD(erfc);
  // Approximate comparison: the compiler may constant-fold one side to the
  // correctly rounded value while the other runs the libm implementation.
  for (Scalar x : {Scalar(-2.5L), Scalar(-0.5L), Scalar(0.0L), Scalar(0.25L), Scalar(3.0L)}) {
    VERIFY_IS_APPROX(numext::erf(x), Scalar(erf(x)));
    VERIFY_IS_APPROX(numext::erfc(x), Scalar(erfc(x)));
  }
}

namespace custom_scalar {
// Minimal custom real scalar providing erf/erfc via argument-dependent lookup,
// the way multiprecision types do. numext::erf/erfc must route to these instead
// of the float/double-tuned polynomials (issue #3023).
struct CustomReal {
  double value;
  CustomReal() : value(0) {}
  explicit CustomReal(double v) : value(v) {}
};
inline CustomReal erf(const CustomReal& x) { return CustomReal(std::erf(x.value)); }
inline CustomReal erfc(const CustomReal& x) { return CustomReal(std::erfc(x.value)); }

// The shape of boost::numeric::interval: neither erf nor erfc. Eigen's
// approximations are not valid for such a type, so it must not be routed into
// them; see failtest/erf_no_scalar_overload.cpp for the resulting error.
struct NoErfReal {
  double value;
};
}  // namespace custom_scalar

namespace Eigen {
template <>
struct NumTraits<custom_scalar::CustomReal> : NumTraits<double> {
  using Real = custom_scalar::CustomReal;
  using NonInteger = custom_scalar::CustomReal;
  using Nested = custom_scalar::CustomReal;
};
}  // namespace Eigen

void custom_scalar_erf_erfc() {
  using custom_scalar::CustomReal;
  using custom_scalar::NoErfReal;
  // Pin the routing decision in both directions.
  STATIC_CHECK((internal::has_erf<CustomReal>::value));
  STATIC_CHECK((internal::has_erfc<CustomReal>::value));
  STATIC_CHECK((internal::has_erf<long double>::value));
  STATIC_CHECK((internal::has_erfc<long double>::value));
  STATIC_CHECK((!internal::has_erf<NoErfReal>::value));
  STATIC_CHECK((!internal::has_erfc<NoErfReal>::value));

  Eigen::Array<CustomReal, 4, 1> x;
  x(0) = CustomReal(-2.5);
  x(1) = CustomReal(-0.5);
  x(2) = CustomReal(0.25);
  x(3) = CustomReal(3.0);
  Eigen::Array<CustomReal, 4, 1> e = x.erf();
  Eigen::Array<CustomReal, 4, 1> c = x.erfc();
  // The polynomial paths cannot even be instantiated for CustomReal, so
  // compiling proves the routing; the value check guards the plumbing.
  for (int i = 0; i < 4; ++i) {
    VERIFY_IS_APPROX(e(i).value, std::erf(x(i).value));
    VERIFY_IS_APPROX(c(i).value, std::erfc(x(i).value));
  }
}

template <typename Scalar>
void test_special_functions_edge_cases() {
  const Scalar plusinf = std::numeric_limits<Scalar>::infinity();
  const Scalar nan = std::numeric_limits<Scalar>::quiet_NaN();
  const Scalar max_val = (std::numeric_limits<Scalar>::max)();
  const Scalar denorm_min = std::numeric_limits<Scalar>::denorm_min();
  const Scalar min_val = (std::numeric_limits<Scalar>::min)();

  // Test scalar digamma edge cases
  VERIFY_IS_EQUAL(numext::digamma(Scalar(0.0)), -plusinf);
  VERIFY_IS_EQUAL(numext::digamma(Scalar(-0.0)), plusinf);
  VERIFY((numext::isnan)(numext::digamma(Scalar(-1.0))));
  VERIFY((numext::isnan)(numext::digamma(Scalar(-2.0))));
  VERIFY_IS_EQUAL(numext::digamma(plusinf), plusinf);
  VERIFY((numext::isnan)(numext::digamma(-plusinf)));
  VERIFY((numext::isnan)(numext::digamma(nan)));

  // Test polygamma at (0, 0)
  VERIFY_IS_EQUAL(numext::polygamma(Scalar(0.0), Scalar(0.0)), -plusinf);
  VERIFY_IS_EQUAL(numext::polygamma(Scalar(0.0), Scalar(-0.0)), plusinf);

  // Test scalar erf/erfc edge cases
  VERIFY_IS_EQUAL(numext::erf(Scalar(0.0)), Scalar(0.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(-0.0)), Scalar(-0.0));
  VERIFY((std::signbit)(numext::erf(Scalar(-0.0))));
  VERIFY_IS_EQUAL(numext::erf(Scalar(10.0)), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(-10.0)), Scalar(-1.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(28.0)), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(-28.0)), Scalar(-1.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(100.0)), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erf(Scalar(-100.0)), Scalar(-1.0));
  VERIFY_IS_EQUAL(numext::erf(max_val), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erf(-max_val), Scalar(-1.0));
  VERIFY_IS_EQUAL(numext::erf(plusinf), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erf(-plusinf), Scalar(-1.0));
  VERIFY((numext::isnan)(numext::erf(nan)));

  VERIFY_IS_EQUAL(numext::erfc(Scalar(0.0)), Scalar(1.0));
  VERIFY_IS_EQUAL(numext::erfc(Scalar(28.0)), Scalar(0.0));
  VERIFY_IS_EQUAL(numext::erfc(Scalar(-28.0)), Scalar(2.0));
  VERIFY_IS_EQUAL(numext::erfc(Scalar(100.0)), Scalar(0.0));
  VERIFY_IS_EQUAL(numext::erfc(Scalar(-100.0)), Scalar(2.0));
  VERIFY_IS_EQUAL(numext::erfc(max_val), Scalar(0.0));
  VERIFY_IS_EQUAL(numext::erfc(-max_val), Scalar(2.0));
  VERIFY_IS_EQUAL(numext::erfc(plusinf), Scalar(0.0));
  VERIFY_IS_EQUAL(numext::erfc(-plusinf), Scalar(2.0));
  VERIFY((numext::isnan)(numext::erfc(nan)));

  // NaN propagates through the x = 0 shortcut of igamma and its derivatives.
  VERIFY((numext::isnan)(numext::igamma(nan, Scalar(0))));
  VERIFY((numext::isnan)(numext::igamma_der_a(nan, Scalar(0))));
  VERIFY((numext::isnan)(numext::gamma_sample_der_alpha(nan, Scalar(0))));

  // betainc propagates NaN past its a = 0 and b = 0 shortcuts.
  VERIFY((numext::isnan)(numext::betainc(Scalar(0), Scalar(0.5), nan)));
  VERIFY((numext::isnan)(numext::betainc(Scalar(0.5), Scalar(0), nan)));
  VERIFY((numext::isnan)(numext::betainc(nan, Scalar(0.5), Scalar(0))));

  if (sizeof(Scalar) < sizeof(double)) {
    // MPFR-backed Boost.Math ibeta, 100 decimal digits. The result is exp(t) with
    // t = b log(1 - x) + a' log(x) + lgamma(a' + b) - lgamma(a') - lgamma(b) + log(cf / a'), a' = a + 1, summed in
    // Scalar: |t| ~ 90, but its first five terms add up to S ~ 330 in magnitude, so t carries an absolute rounding
    // error of order S eps, which exp turns into relative error. How much of it is reached depends on the libm (the
    // result is off by 62 eps with glibc 2.43), hence S eps rather than a measured constant, plus one subnormal ulp for
    // the final rounding.
    const Scalar a = Scalar(38.325092315673828), b = Scalar(5.0568113327026367), x = Scalar(0.07500767707824707);
    const double ref = 6.17294757793914262e-40;
    const double actual = static_cast<double>(numext::betainc(a + Scalar(1), b, x));
    const double ap = static_cast<double>(a) + 1, bd = static_cast<double>(b), xd = static_cast<double>(x);
    const double S = bd * std::abs(std::log1p(-xd)) + ap * std::abs(std::log(xd)) + std::lgamma(ap + bd) +
                     std::lgamma(ap) + std::abs(std::lgamma(bd));
    VERIFY(std::abs(actual - ref) <= S * static_cast<double>(NumTraits<Scalar>::epsilon()) * ref +
                                         static_cast<double>((std::numeric_limits<Scalar>::denorm_min)()));
  }

  // A negative q sums until n + q > 9. zeta(2, -20.5) = pi^2/2 + sum_{j=0}^{20} (j + 1/2)^-2.
  {
    double ref = EIGEN_PI * EIGEN_PI / 2;
    for (int j = 0; j <= 20; ++j) ref += 1 / ((j + 0.5) * (j + 0.5));
    VERIFY_IS_APPROX(numext::zeta(Scalar(2), Scalar(-20.5)), Scalar(ref));
  }

  if (sizeof(Scalar) >= 8) {
    VERIFY_IS_EQUAL(numext::erf(Scalar(1e200)), Scalar(1.0));
    VERIFY_IS_EQUAL(numext::erf(Scalar(-1e200)), Scalar(-1.0));
    VERIFY_IS_EQUAL(numext::erfc(Scalar(1e200)), Scalar(0.0));
    VERIFY_IS_EQUAL(numext::erfc(Scalar(-1e200)), Scalar(2.0));
  } else {
    VERIFY_IS_EQUAL(numext::erf(Scalar(1e30f)), Scalar(1.0));
    VERIFY_IS_EQUAL(numext::erf(Scalar(-1e30f)), Scalar(-1.0));
    VERIFY_IS_EQUAL(numext::erfc(Scalar(1e30f)), Scalar(0.0));
    VERIFY_IS_EQUAL(numext::erfc(Scalar(-1e30f)), Scalar(2.0));
  }

  // Denormals / subnormals
  if (denorm_min > Scalar(0)) {
    VERIFY(numext::erf(denorm_min) > Scalar(0));
    VERIFY(numext::erf(-denorm_min) < Scalar(0));
  }
  VERIFY(numext::erf(min_val) > Scalar(0));
  VERIFY(numext::erf(-min_val) < Scalar(0));

  // Array / vectorized edge cases
  using ArrayType = Array<Scalar, Dynamic, 1>;
  ArrayType x(14), erf_ref(14), erfc_ref(14);
  Scalar large_val = (sizeof(Scalar) >= 8) ? Scalar(1e200) : Scalar(1e30f);
  x << Scalar(0.0), Scalar(-0.0), denorm_min, min_val, Scalar(2.0), Scalar(-2.0), Scalar(28.0), Scalar(-28.0),
      large_val, -large_val, max_val, -max_val, plusinf, -plusinf;
  erf_ref << Scalar(0.0), Scalar(-0.0), numext::erf(denorm_min), numext::erf(min_val), Scalar(std::erf(2.0)),
      Scalar(std::erf(-2.0)), Scalar(1.0), Scalar(-1.0), Scalar(1.0), Scalar(-1.0), Scalar(1.0), Scalar(-1.0),
      Scalar(1.0), Scalar(-1.0);
  erfc_ref << Scalar(1.0), Scalar(1.0), numext::erfc(denorm_min), numext::erfc(min_val), Scalar(std::erfc(2.0)),
      Scalar(std::erfc(-2.0)), Scalar(0.0), Scalar(2.0), Scalar(0.0), Scalar(2.0), Scalar(0.0), Scalar(2.0),
      Scalar(0.0), Scalar(2.0);

  ArrayType erf_res = x.erf();
  verify_component_wise(erf_res, erf_ref);
  erf_res = erf(x);
  verify_component_wise(erf_res, erf_ref);

  ArrayType erfc_res = x.erfc();
  verify_component_wise(erfc_res, erfc_ref);
  erfc_res = erfc(x);
  verify_component_wise(erfc_res, erfc_ref);

  // Array with NaN
  ArrayType x_nan(3), erf_nan_ref(3);
  x_nan << Scalar(1.0), nan, Scalar(-1.0);
  erf_nan_ref << Scalar(std::erf(1.0)), nan, Scalar(std::erf(-1.0));
  ArrayType erf_nan_res = x_nan.erf();
  verify_component_wise(erf_nan_res, erf_nan_ref);
}

EIGEN_DECLARE_TEST(special_functions) {
  CALL_SUBTEST_1(array_special_functions<ArrayXf>());
  CALL_SUBTEST_1(test_special_functions_edge_cases<float>());
  CALL_SUBTEST_2(array_special_functions<ArrayXd>());
  CALL_SUBTEST_2(test_special_functions_edge_cases<double>());
  CALL_SUBTEST_3(scalar_ndtri<long double>());
  CALL_SUBTEST_4(scalar_erf_erfc<long double>());
  CALL_SUBTEST_5(custom_scalar_erf_erfc());
  // TODO(cantonios): half/bfloat16 don't have enough precision to reproduce results above.
  // CALL_SUBTEST_4(array_special_functions<ArrayX<Eigen::half>>());
  // CALL_SUBTEST_5(array_special_functions<ArrayX<Eigen::bfloat16>>());
}