Fully functional std::complex specializations and overloads

Developers
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I’m reading your code more closely today before swapping out my implementation for yours in my application at work.

Right now, the inheritance structure I’m reading is:
std::complex<stan::math::var> inherits from stan::math::complex_base<stan::math::var>. The operations of stan::math::complex_base statically downcast their return references back to std::complex<stan::math::var>.

For reference, my inheritance structure is:
std::complex<stan::math::var> inherits from stan::math::complex<stan::math::var>, which in turn inherits from std::complex<stan::math::zeroing<stan::math::var>>>. There is also an operator std::complex() defined on stan::math::complex, which implicitly statically downcasts to std::complex for both base classes. (The extra level of inheritance together with the implicit downcasting delegates all operations back to the standard library as if they were operating on a zeroing version of stan’s types. This is why there is so little code, excluding the Jenkins autoformat misconfiguration that had touched a lot of files.)

While trying to test your code:
I was trying to build the libraries. I ended up partially following a page I wrote in the math wiki a while back, along with an example Jenkins test run. make test-headers fails with the error below. ./runTests.py fails because due to a missing Google Test library. I’m on commit 2c746c2cc9d19feb036dae474444b86d940be210 (HEAD → feature/0123-complex-var, origin/feature/0123-complex-var). Edit: Forgot to add that I’m on Windows with MSYS2 x86_64 using GCC 9.1.

g++ -std=c++1y -m64 -D_REENTRANT -Wall -Wno-unused-function -Wno-uninitialized -Wno-unused-but-set-variable -Wno-unused-variable -Wno-sign-compare -Wno-unused-local-typedefs      -I lib/tbb_2019_U8/include -O3  -I . -I lib/eigen_3.3.3 -I lib/boost_1.69.0 -I lib/sundials_4.1.0/include  -D_USE_MATH_DEFINES  -DBOOST_DISABLE_ASSERTS      -c -std=c++1y -m64 -D_REENTRANT -Wall -Wno-unused-function -Wno-uninitialized -Wno-unused-but-set-variable -Wno-unused-variable -Wno-sign-compare -Wno-unused-local-typedefs      -I lib/tbb_2019_U8/include -O3  -I . -I lib/eigen_3.3.3 -I lib/boost_1.69.0 -I lib/sundials_4.1.0/include -O0 -include stan/math/mix/mat/fun/typedefs.hpp test/dummy.cpp -o nul
In file included from ./stan/math/rev/core.hpp:49,
                 from ./stan/math/mix/mat/fun/typedefs.hpp:6,
                 from <command-line>:
./stan/math/rev/core/std_complex.hpp: In instantiation of 'std::complex<double> stan::math::value_of_rec(const std::complex<_Tp>&) [with T = stan::math::var]':
./stan/math/rev/core/std_complex.hpp:693:48:   required from 'std::complex<_Tp> stan::math::complex_asinh(const std::complex<_Tp>&) [with V = stan::math::var]'
./stan/math/rev/core/std_complex.hpp:1537:25:   required from here
./stan/math/rev/core/std_complex.hpp:229:23: error: no matching function for call to 'value_of_rec(stan::math::complex_base<stan::math::var>::value_type)'
  229 |   return {value_of_rec(z.real()), value_of_rec(z.imag())};
      |           ~~~~~~~~~~~~^~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:228:29: note: candidate: 'template<class T> std::complex<double> stan::math::value_of_rec(const std::complex<_Tp>&)'
  228 | inline std::complex<double> value_of_rec(const std::complex<T>& z) {
      |                             ^~~~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:228:29: note:   template argument deduction/substitution failed:
./stan/math/rev/core/std_complex.hpp:229:23: note:   'stan::math::complex_base<stan::math::var>::value_type' {aka 'stan::math::var'} is not derived from 'const std::complex<_Tp>'
  229 |   return {value_of_rec(z.real()), value_of_rec(z.imag())};
      |           ~~~~~~~~~~~~^~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:229:47: error: no matching function for call to 'value_of_rec(stan::math::complex_base<stan::math::var>::value_type)'
  229 |   return {value_of_rec(z.real()), value_of_rec(z.imag())};
      |                                   ~~~~~~~~~~~~^~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:228:29: note: candidate: 'template<class T> std::complex<double> stan::math::value_of_rec(const std::complex<_Tp>&)'
  228 | inline std::complex<double> value_of_rec(const std::complex<T>& z) {
      |                             ^~~~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:228:29: note:   template argument deduction/substitution failed:
./stan/math/rev/core/std_complex.hpp:229:47: note:   'stan::math::complex_base<stan::math::var>::value_type' {aka 'stan::math::var'} is not derived from 'const std::complex<_Tp>'
  229 |   return {value_of_rec(z.real()), value_of_rec(z.imag())};
      |                                   ~~~~~~~~~~~~^~~~~~~~~~
./stan/math/rev/core/std_complex.hpp:229:57: error: could not convert '{<expression error>, <expression error>}' from '<brace-enclosed initializer list>' to 'std::complex<double>'
  229 |   return {value_of_rec(z.real()), value_of_rec(z.imag())};
      |                                                         ^
      |                                                         |
      |                                                         <brace-enclosed initializer list>