Ambiguity between `fwd` and `mix` signatures for `hessian()`

[andrjohns]

Description

With the current signatures for hessian in fwd and mix, it is not possible to call the fwd implementation with double types.

mix/functor/hessian.hpp:

template <typename F>
void hessian(const F& f, const Eigen::Matrix<double, Eigen::Dynamic, 1>& x,
             double& fx, Eigen::Matrix<double, Eigen::Dynamic, 1>& grad,
             Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>& H)

fwd/functor/hessian.hpp:

template <typename T, typename F>
void hessian(const F& f, const Eigen::Matrix<T, Eigen::Dynamic, 1>& x, T& fx,
             Eigen::Matrix<T, Eigen::Dynamic, 1>& grad,
             Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic>& H)

Because the scalar type is explicit in the mix signature and templated in the fwd signature, calls to hessian with double types will always resolve to the mix implementation. This makes testing/validation (or even just use as an alternative) a bit of a hurdle.

I think any kind of fix for this would imply a breaking change (changing function names/arguments), so could be bundled in the 5.0 release (@SteveBronder)?

Current Version:

v4.8.1

[SteveBronder]

I think this makes sense to fix. Which signatures do you want to add/change? We would put this in 5.0 breaking changes

[andrjohns]

I think we want to keep the mix implementation being called by default, since it's more efficient, but add a way to force the fwd impl to be called.

I can think of two ways:

1. Rename the fwd version to hessian_fwd/hessian_fvar/etc
2. Add a boolean template parameter, defaulting to true, such that hessian<false>(...) resolves to the fwd impl

Thoughts/preferences?