Interfacing with External C++ Code
In rstan: R Interface to Stan

library(rstan)
knitr::opts_chunk$set(
  echo = TRUE, eval = FALSE, error = TRUE
)

Starting with the 2.13 release, it is much easier to use external C++ code in a Stan program. This vignette briefly illustrates how to do so.

Suppose that you have (part of) a Stan program that involves Fibonacci numbers, such as ```{stan eval = FALSE, output.var = "fib"} functions { int fib(int n); int fib(int n) { if (n <= 0) reject("n must be positive"); return n <= 2 ? 1 : fib(n - 1) + fib(n - 2); } } model {} // use the fib() function somehow

On the second line, we have _declared_ the `fib` function before it is _defined_
in order to call it recursively. 

For functions that are not recursive, it is not necessary to declare them before 
defining them but it may be advantageous. For example, I often like to hide the
definitions of complicated utility functions that are just a distraction using
the `#include "file"` mechanism
```{stan eval = FALSE, output.var="includes"}
functions {
  real complicated(real a, real b, real c, real d, real e, real f, real g);
#include "complicated.stan"
}
model {} // use the complicated() function somehow

This Stan program would have to be parsed using the stanc_builder function in the rstan package rather than the default stanc function (which is called by sampling and stan internally).

Returning to the Fibonacci example, it is not necessary to define the fib function using the Stan language because Stan programs with functions that are declared but not defined can use the standard capabilities of the C++ toolchain to provide the function definitions in C++. For example, this program produces a parser error by default

mc <- 
'
functions { int fib(int n); }
model {} // use the fib() function somehow
'
try(stan_model(model_code = mc, model_name = "parser_error"), silent = TRUE)

However, if we specify the allow_undefined and includes arguments to the stan_model function, and define a fib function in the named C++ header file, then it will parse and compile

stan_model(model_code = mc, model_name = "external", allow_undefined = TRUE,
           includes = paste0('\n#include "', 
                             file.path(getwd(), 'fib.hpp'), '"\n'))

Specifying the includes argument is a bit awkward because the C++ representation of a Stan program is written and compiled in a temporary directory. Thus, the includes argument must specify a full path to the fib.hpp file, which in this case is in the working directory. Also, the path must be enclosed in double-quotes, which is why single quotes are used in the separate arguments to the paste0 function so that double-quotes are interpreted literally. Finally, the includes argument should include newline characters ("\n") at the start and end. It is possible to specify multiple paths using additional newline characters or include a "meta-header" file that contains #include directives to other C++ header files.

The result of the includes argument is inserted into the C++ file directly at the end of the lines (as opposed to CmdStan where it is inserted directly before the following lines) ```{Rcpp, eval = FALSE}

include

namespace some_namespace {

using std::istream; using std::string; using std::stringstream; using std::vector; using stan::io::dump; using stan::math::lgamma; using stan::model::prob_grad; using namespace stan::math;

typedef Eigen::Matrix vector_d; typedef Eigen::Matrix row_vector_d; typedef Eigen::Matrix matrix_d;

static int current_statement_begin__;

// various function declarations and / or definitions

include "/full/path/to/fib.hpp"

Thus, the definition of the `fib` function in the fib.hpp file need not be
enclosed in any particular namespace (which is a random string by default. The 
"meta-include" stan/model/model_header.hpp file reads as
```r
cat(readLines(system.file("include", "src", "stan", "model", "model_header.hpp", 
                          package = "StanHeaders")), sep = "\n")

so the definition of the fib function in the fib.hpp file could utilize any function in the Stan Math Library (without having to prefix function calls with stan::math::), some typedefs to classes in the Eigen matrix algebra library, plus streams, exceptions, etc. without having to worry about the corresponding header files. Nevertheless, an external C++ file may contain additional include directives that bring in class definitions, for example.

Now let's examine the fib.hpp file, which contains the C++ lines ```{Rcpp, eval = FALSE} int fib(const int&n, std::ostream* pstream__) { if (n <= 0) { stringstream errmsg; errmsg << "n must be positive"; throw std::domain_error(errmsg.str()); } return n <= 1 ? 1 : fib(n - 1, 0) + fib(n - 2, 0); }

This C++ function is essentially what the preceding user-defined function in the 
Stan language
```{stan eval = FALSE, output.var="def"}
int fib(int n) {
  if (n <= 0) reject("n must be positive");
  return n <= 2 ? 1 : fib(n - 1) + fib(n - 2);
}

parses to. Thus, there is no speed advantage to defining the fib function in the external fib.hpp file. However, it is possible to use an external C++ file to handle the gradient of a function analytically as opposed to using Stan's autodifferentiation capabilities, which are slower and more memory intensive but fully general. In this case, the fib function only deals with integers so there is nothing to take the derivative of. The primary advantage of using an external C++ file is flexibility to do things that cannot be done directly in the Stan language. It is also useful for R packages like rstanarm that may want to define some C++ functions in the package's src directory and rely on the linker to make them available in its Stan programs, which are compiled at (or before) installation time.

In the C++ version, we check if n is non-positive, in which case we throw an exception. The way the MCMC samplers are configured, if there is an exception thrown and it is std::domain_error, it will treat that iteration as a rejection but as a recoverable error: set that iteration's log probability value to negative infinity and move to the next iteration. If there is a std::invalid_argument thrown, MCMC terminates. We treat these as non-recoverable, user programming errors. It is unnecessary to prefix stringstream with std:: because of the using std::stringstream; line in the generated C++ file. However, there is no corresponding using std::domain_error; line, so it has to be qualified appropriately when the exception is thrown.

The only confusing part of the C++ version of the fib function is that it has an additional argument (with no default value) named pstream__ that is added to the declaration of the fib function by Stan. Thus, your definition of the fib function needs to match with this signature. This additional argument is a pointer to a std::ostream and is only used if your function prints something to the screen, which is rare. Thus, when we call the fib function recursively in the last line, we specify fib(n - 1, 0) + fib(n - 2, 0); so that the output (if any, and in this case there is none) is directed to the null pointer.

This vignette has employed a toy example with the Fibonacci function, which has little apparent use in a Stan program and if it were useful, would more easily be implemented as a user-defined function in the functions block as illustrated at the outset. The ability to use external C++ code only becomes useful with more complicated C++ functions. It goes without saying that this mechanism ordinarily cannot call functions in C, Fortran, R, or other languages because Stan needs the derivatives with respect to unknown parameters in order to perform estimation. These derivatives are handled with custom C++ types that cannot be processed by functions in other languages that only handle primitive types such as double, float, etc.

That said, it is possible to accomplish a great deal in C++, particularly when utilizing the Stan Math Library. For more details, see The Stan Math Library: Reverse-Mode Automatic Differentiation in C++ and its GitHub repository. The functions that you declare in the functions block of a Stan program will typically involve templating and type promotion in their signatures when parsed to C++ (the only exceptions are functions whose only arguments are integers, as in the fib function above). Suppose you wanted to define a function whose arguments are real numbers (or at least one of the arguments is). For example,

mc <- 
'
functions { real sinc(real x); }
transformed data { real sinc_pi = sinc(pi()); }
'
stan_model(model_code = mc, model_name = "external", allow_undefined = TRUE,
           includes = paste0('\n#include "', 
                             file.path(getwd(), 'sinc.hpp'), '"\n'))

The sinc.hpp file reads as

cat(readLines("sinc.hpp"), sep = "\n")

The body of the first sinc function is simply its mathematical definition in the form of a ternary operator, which is sufficient when the input is a double.

The last lines of sinc.hpp are a specialization for when the input is an unknown real parameter, which is represented in the Stan Math Library as a stan::math::var. Since the derivative of the sinc function is easy to compute analytically, we extract the underlying double-precision value from the inputted stan::math::var and use that to calculate the function value and its first derivative. Then, we return the result of precomputed_gradients, whose arguments are the function value (f), the derivative of x with respect to any other parameters (x.vi_), and the first derivative of f (dfdx_). The latter two are actually std::vectors but only have one element each because there is only one unknown.

An easy way to see what the generated function signature will be is to call the stanc function in the rstan package with allow_undefined = TRUE and inspect the resuling C++ code. In this case, I first did

try(readLines(stanc(model_code = mc, allow_undefined = TRUE)$cppcode))

to see what function signatures needed to be written for sinc.hpp.

Once you go to the trouble of writing a numerically stable C++ function, we would welcome a pull request on GitHub to include your C++ function in the Stan Math Library for everyone to benefit from, provided that it can be licensed under the 3-clause BSD license and its use is not overly-specific to your particular Stan program.

The Stan Math Library is compliant with the C++14 standard but not all compilers fully support the C++14 standard. In particular, the compiler that comes with RTools does not support all C++14 features. But you can use many C++14 features, such as the auto keyword.