Complex-valued Softplus in C++. Useful when summing two VERY LARGE numbers.

softplus.hh

/**
 * 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 https://mozilla.org/MPL/2.0/.
 */
#pragma once
#include <complex>
#include <algorithm>

// Real softplus.
template <typename T>
T softplus(T x)
{
  using namespace std;

  return log(1 + exp(-abs(x))) + max(x, T(0));
}

// Complex softplus.
template <typename T>
std::complex<T> softplus(std::complex<T> z)
{
  using namespace std;

  T r = real(z);
  T i = imag(z);
  T logval0 = softplus(2*r);
  T logval2;
  T ci = cos(i);

  if (ci > 0) {
      T logval1 = r + log(2*ci);
      logval2 = 0.5*(logval0 + softplus(logval1 - logval0));
  } else {
      T logval1 = r + log(2*abs(ci));
      logval2 = 0.5*(logval0 + log(1 - exp(logval1 - logval0)));
  }
  T arg = asin(exp(r - logval2) * sin(i));
  if (ci < 0 && r > log(-1/ci)) {
      arg = M_PI - arg;
  }
  return std::complex<T>(logval2, arg);
}

softplus.md

To compute A + B within float32 where both A and B is ~1e100, we can do:

  log(A + B) = log(exp(a) + exp(b)) = a + softplus(b - a)

where a = log(A) and b = log(B).