## Background

After a brief dive into Scala, I am back to writing C++. However, I do have a much better appreciation for functional programming and recursion. I am far from an expert at either, but I am interested in increasing my programming skills. I decided to revive my blog and try to post things I find fun or interesting. I am currently reading "Effective Modern C++" by Scott Meyers and continually come across Metaprogramming online. I was poking around Stack Overflow and I found this post which asks about tail recursion in Template Metaprogramming (TMP). I thought this was interesting and decided to see if I could write the naive recursive Fibonacci number generator using TMP.

I had already written this in Scala, which looks like:

import scala.annotation.tailrec
def fib(n: Int): Int = {
@tailrec
def loop(iter: Int, prev: Int, next: Int): Int = {
if (iter >= n) prev
else loop(iter + 1, next, prev + next)
}
loop(0, 0, 1)
}
fib(10)

However, fib(10) will execute at runtime and the Java Virtual Machine occurs additional runtime overhead each time you run the program. A neat benefit of TMP in C++ is the compiler can compute fib(10) and then each invocation of the program is as simple as printing an integer. My first implementation in C++, looked like:

#include <cstdint>
#include <iostream>

namespace impl {

template<int64_t n, bool isPositive>
struct fib_impl {
static constexpr int64_t val = fib_impl<n - 1, isPositive>::val + fib_impl<n - 2, isPositive>::val;
};

template<>
struct fib_impl<1, true> {
static constexpr int64_t val = 1;
};

template<>
struct fib_impl<0, true> {
static constexpr int64_t val = 0;
};

// If calling fib<-1>::val it will try to do the recursion infinitely
// -> this template short circuits that recursion
template<int64_t n>
struct fib_impl<n, false> {
static constexpr int64_t val = -1;
};

} // namespace impl

template<int64_t n>
struct fib {
static_assert(n >= 0, "Error: fib can't be called with a negative integer");
static constexpr int64_t val = impl::fib_impl<n, (n >= 0)>::val;
};

int main() {
//    static_assert(fib<-1>::val); // This will fail.
//    static_assert(fib<10>::val == 55); // Make sure it works at compile time!
std::cout << fib<91>::val << '\n';
return 0;
}

I want the interface of fib to accept only a positive integer, therefore we abstract away whether, or not, the integer is positive with impl::fib_impl. In this implementation, you need 3 template specializations. Two are the termination conditions: 0 and 1; the other provides protection from an infinite recursion when you give a negative number to fib. Even though you get an error from the static_assert(fib<-1>::val), the compiler still tries to create infinite templates. Luckily, your compiler will protect you from creating literally infinite templates (GCC 7.2.1 allowed 900 to be generated, use -ftemplate-depth=<value> to change it). This implementation isn't tail recursive because the recursion isn't in the tail position. The recursive call,

fib_impl<n - 1, isPositive>::val + fib_impl<n - 2, isPositive>::val

is shaped like recursive_template(...) + recursive_template(...), but must look like: recursive_template(...) to be tail recursive. You can verify this by modifying the Scala code. In C++, I believe the only way to find out if tail recursion is actually applied is looking at the assembly for loops. Unfortunately, this is done at compile time and you can't review the compile time assembly (to my knowledge). The tail recursive implementation is:

#include <cstdint>
#include <iostream>

namespace impl {

template <int64_t n, int64_t prev, int64_t next, bool isPositive>
struct fib_impl {
static constexpr int64_t val = fib_impl<n - 1, next, prev + next, isPositive>::val;
};

template <int64_t prev, int64_t next>
struct fib_impl<0, prev, next, true> {
static constexpr int64_t val = prev;
};

template <int64_t n, int64_t prev, int64_t next>
struct fib_impl<n, prev, next, false> {
static constexpr int64_t val = -1;
};

} // namespace impl

template <int64_t n>
struct fib {
static_assert(n >= 0, "Error: fib can't be called with negative numbers!");
static constexpr int64_t val = impl::fib_impl<n, 0, 1, (n >= 0)>::val;
};

int main() {
//    static_assert(fib<-1>::val); // This will fail.
//    static_assert(fib<10>::val == 55); // Make sure it works at compile time
std::cout << fib<91>::val << '\n';
return 0;
}

Great, now the recursive call is in the tail position. Additionally, we only need 2 template specializations. The one where n = 0 and the infinite template recursion protection for negative integers. I compiled both versions with GCC 7.2.1 using the C++11 standard (which is necessary for constexpr) 10 times and measured the average compile time. It was essentially the same (about 0.2s). The tail recursive version has a major downside though: it overflows a int64_t faster than the non-tail recursive version. The largest value of n for the non-tail recursive version was 92 and for the tail recursive version was 91. The reason for this is because the template recursion for fib<92>::val contains a prev + next which would contain a value to large to fit in int64_t.

This code was an academic exercise, but I think it is neat. This is my first experience with TMP and I am very interested to learn more. Feel free to message me, or follow me, on Twitter with constructive criticism or for future blog posts. Some of the blog posts I have planned:

• Setting up a stateful advanced research computer using Warewulf to provision compute nodes
• Using free cloud resources to write a streaming data application in Python
• Using boost::multi_array in a boost::mpi C++ program for matrix and vector computations
• Writing a reaper to manage situations where Slurm doesn't kill user processes on compute nodes
• Building a proposal system into an existing Slurm installation