glmath/ext_src/special_math.hpp

// Original code derived from Boost and is distributed here
// under the Boost license (licenses/boost-license.txt)
//    Copyright (c) 2006 Xiaogang Zhang
//    Copyright (c) 2007, 2017 John Maddock
// Secondary code copyright by its author and is distributed here
// under the BSD-3 license (LICENSE.md). Derived from
// stan/math/prim/fun/log_modified_bessel_first_kind.hpp

#ifndef __COSMOTOOL_SPECIAL_MATH_HPP
#define __COSMOTOOL_SPECIAL_MATH_HPP

#include "algo.hpp"
#include <boost/math/constants/constants.hpp>
#include <boost/math/tools/rational.hpp>
#include <cmath>
#include <limits>

// Taken and adapted from
// https://github.com/stan-dev/math/blob/develop/stan/math/prim/fun/log_modified_bessel_first_kind.hpp

namespace CosmoTool {

template <typename T> T log1p_exp(T x) {
  if (x > T(0)) {
    return x + std::log1p(std::exp(-x));
  }
  return std::log1p(std::exp(x));
}

template <typename T> T multiply_log(T a, T b) {
  if (a == 0 && b == 0)
    return 0;
  return a * std::log(b);
}

template <typename T> T inf() { return std::numeric_limits<T>::infinity(); }

template <typename T> T log_sum_exp(T const a, T const b) {
  if (a == -inf<T>()) {
    return b;
  }
  if (a == inf<T>() && b == inf<T>()) {
    return inf<T>();
  }
  if (a > b) {
    return a + log1p_exp(b - a);
  }
  return b + log1p_exp(a - b);
}

/* Log of the modified Bessel function of the first kind,
 * which is better known as the Bessel I function. See
 * modified_bessel_first_kind.hpp for the function definition.
 * The derivatives are known to be incorrect for v = 0 because a
 * simple constant 0 is returned.
 *
 * @tparam T common type for calculation
 * @param v Order, can be a non-integer but must be at least -1
 * @param z Real non-negative number
 * @throws std::domain_error if either v or z is NaN, z is
 * negative, or v is less than -1
 * @return log of Bessel I function
 */
template <typename T> T log_modified_bessel_first_kind(T const v, T const z) {
  using boost::math::tools::evaluate_polynomial;
  using std::log;
  using std::pow;
  using std::sqrt;
  static const double LOG_TWO = std::log(2.0);
  static const double EPSILON = std::numeric_limits<double>::epsilon();
  static const double TWO_PI = 2.0 * boost::math::constants::pi<double>();

  if (z == 0) {
    if (v == 0) {
      return 0.0;
    }
    if (v > 0) {
      return -std::numeric_limits<T>::infinity();
    }
    return std::numeric_limits<T>::infinity();
  }
  if (std::isinf(z)) {
    return z;
  }
  if (v == 0) {
    // Modified Bessel function of the first kind of order zero
    // we use the approximating forms derived in:
    // "Rational Approximations for the Modified Bessel Function of the
    // First Kind -- I0(x) for Computations with Double Precision"
    // by Pavel Holoborodko, see
    // http://www.advanpix.com/2015/11/11/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i0-computations-double-precision
    // The actual coefficients used are [Boost's] own, and extend
    // Pavel's work to precisions other than double.

    if (z < 7.75) {
      // Bessel I0 over[10 ^ -16, 7.75]
      // Max error in interpolated form : 3.042e-18
      // Max Error found at double precision = Poly : 5.106609e-16
      //                                       Cheb : 5.239199e-16
      static const double P[] = {
          1.00000000000000000e+00, 2.49999999999999909e-01,
          2.77777777777782257e-02, 1.73611111111023792e-03,
          6.94444444453352521e-05, 1.92901234513219920e-06,
          3.93675991102510739e-08, 6.15118672704439289e-10,
          7.59407002058973446e-12, 7.59389793369836367e-14,
          6.27767773636292611e-16, 4.34709704153272287e-18,
          2.63417742690109154e-20, 1.13943037744822825e-22,
          9.07926920085624812e-25};
      return log1p_exp(multiply_log(2.0, z) - log(4.0) +
                       log(evaluate_polynomial(P, 0.25 * square(z))));
    }
    if (z < 500) {
      // Max error in interpolated form : 1.685e-16
      // Max Error found at double precision = Poly : 2.575063e-16
      //                                       Cheb : 2.247615e+00
      static const double P[] = {
          3.98942280401425088e-01,  4.98677850604961985e-02,
          2.80506233928312623e-02,  2.92211225166047873e-02,
          4.44207299493659561e-02,  1.30970574605856719e-01,
          -3.35052280231727022e+00, 2.33025711583514727e+02,
          -1.13366350697172355e+04, 4.24057674317867331e+05,
          -1.23157028595698731e+07, 2.80231938155267516e+08,
          -5.01883999713777929e+09, 7.08029243015109113e+10,
          -7.84261082124811106e+11, 6.76825737854096565e+12,
          -4.49034849696138065e+13, 2.24155239966958995e+14,
          -8.13426467865659318e+14, 2.02391097391687777e+15,
          -3.08675715295370878e+15, 2.17587543863819074e+15};
      return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
    }
    // Max error in interpolated form : 2.437e-18
    // Max Error found at double precision = Poly : 1.216719e-16
    static const double P[] = {3.98942280401432905e-01, 4.98677850491434560e-02,
                               2.80506308916506102e-02, 2.92179096853915176e-02,
                               4.53371208762579442e-02};
    return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
  }
  if (v == 1) { // WARNING: will not autodiff for v = 1 correctly
    // modified from Boost's bessel_i1_imp in the double precision case
    // see credits above in the v == 0 case
    if (z < 7.75) {
      // Bessel I0 over[10 ^ -16, 7.75]
      // Max error in interpolated form: 5.639e-17
      // Max Error found at double precision = Poly: 1.795559e-16

      static const double P[] = {
          8.333333333333333803e-02, 6.944444444444341983e-03,
          3.472222222225921045e-04, 1.157407407354987232e-05,
          2.755731926254790268e-07, 4.920949692800671435e-09,
          6.834657311305621830e-11, 7.593969849687574339e-13,
          6.904822652741917551e-15, 5.220157095351373194e-17,
          3.410720494727771276e-19, 1.625212890947171108e-21,
          1.332898928162290861e-23};
      T a = square(z) * 0.25;
      T Q[3] = {1, 0.5, evaluate_polynomial(P, a)};
      return log(z) + log(evaluate_polynomial(Q, a)) - LOG_TWO;
    }
    if (z < 500) {
      // Max error in interpolated form: 1.796e-16
      // Max Error found at double precision = Poly: 2.898731e-16

      static const double P[] = {
          3.989422804014406054e-01,  -1.496033551613111533e-01,
          -4.675104253598537322e-02, -4.090895951581637791e-02,
          -5.719036414430205390e-02, -1.528189554374492735e-01,
          3.458284470977172076e+00,  -2.426181371595021021e+02,
          1.178785865993440669e+04,  -4.404655582443487334e+05,
          1.277677779341446497e+07,  -2.903390398236656519e+08,
          5.192386898222206474e+09,  -7.313784438967834057e+10,
          8.087824484994859552e+11,  -6.967602516005787001e+12,
          4.614040809616582764e+13,  -2.298849639457172489e+14,
          8.325554073334618015e+14,  -2.067285045778906105e+15,
          3.146401654361325073e+15,  -2.213318202179221945e+15};
      return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
    }
    // Max error in interpolated form: 1.320e-19
    // Max Error found at double precision = Poly: 7.065357e-17
    static const double P[] = {
        3.989422804014314820e-01, -1.496033551467584157e-01,
        -4.675105322571775911e-02, -4.090421597376992892e-02,
        -5.843630344778927582e-02};
    return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
  }
  if (z > 100) {
    // Boost does something like this in asymptotic_bessel_i_large_x
    T lim = pow((square(v) + 2.5) / (2 * z), 3) / 24;
    if (lim < (EPSILON * 10)) {
      T s = 1;
      T mu = 4 * square(v);
      T ex = 8 * z;
      T num = mu - 1;
      T denom = ex;
      s -= num / denom;
      num *= mu - 9;
      denom *= ex * 2;
      s += num / denom;
      num *= mu - 25;
      denom *= ex * 3;
      s -= num / denom;
      s = z - log(sqrt(z * TWO_PI)) + log(s);
      return s;
    }
  }

  T log_half_z = log(0.5 * z);
  T lgam = (v > -1) ? lgamma(v + 1.0) : 0;
  T lcons = (2.0 + v) * log_half_z;
  T out;
  if (v > -1) {
    out = log_sum_exp(v * log_half_z - lgam, lcons - lgamma(v + 2));
    lgam += log1p(v);
  } else {
    out = lcons;
  }

  double m = 2;
  double lfac = 0;
  T old_out;
  do {
    lfac += log(m);
    lgam += log(v + m);
    lcons += 2 * log_half_z;
    old_out = out;
    out = log_sum_exp(out, lcons - lfac - lgam); // underflows eventually
    m++;
  } while (out > old_out || out < old_out);
  return out;
}

} // namespace CosmoTool

#endif