Why is the Rcpp implementation in my example much slower than the R function?

Your R code seems to be more or less optimal, i.e. all the real work is done in compiled code. For the C++ code the main problem I could find is calling c1.ncol() in a tight loop. If I replace that with m, the C++ solution is almost as fast as R. If I add RcppArmadillo to the mix, I get a very compact syntax, but not faster than the pure Rcpp code. To me this shows that it can be really hard to beat well written R code:

//  [[Rcpp::depends(RcppArmadillo)]]
#include <RcppArmadillo.h>
using namespace Rcpp;

// [[Rcpp::export]]
List arma_test(const arma::vec& Delta,
           const arma::mat& delta,
           const arma::mat& rx) {
  int l = delta.n_cols;
  List c(l);

  for (int i = 0; i < l; ++i) {
    c(i) = rx.each_col() % (Delta / (1 + delta.col(i)));
  }

  return c;  
}

// [[Rcpp::export]]
List rcpp_test(NumericVector Delta, 
               NumericMatrix delta, 
               NumericMatrix rx) {

  int n = Delta.length();
  int m = rx.ncol();

  List c(delta.ncol()); 
  NumericMatrix c1;
  for(int i = 0; i < delta.ncol(); ++i){
    c1 = NumericMatrix(n, m);
    for(int k = 0; k < n; ++k){
      double tmp = Delta[k] / (1 + delta(k, i));
      for(int j = 0; j < m; ++j){
        c1(k, j) = rx(k, j) * tmp; 
      }
    }
    c(i) = c1;
  }

  return c;

}

/*** R
test <- function(Delta, delta, rx){

  const <- list()
  for(i in 1:ncol(delta)){
    const[[i]] <- rx * (Delta / (1 + delta[, i]))
  }

  const

}

n <- 50000
Delta <- exp(rnorm(n))
delta <- exp(matrix(rnorm(n * 5), nrow = n))
rx <- matrix(rnorm(n * 20), nrow = n)
bench::mark(test(Delta, delta, rx),
            arma_test(Delta, delta, rx),
            rcpp_test(Delta, delta, rx))
 */

Output:

# A tibble: 3 x 14
  expression     min    mean  median     max `itr/sec` mem_alloc  n_gc n_itr
  <chr>      <bch:t> <bch:t> <bch:t> <bch:t>     <dbl> <bch:byt> <dbl> <int>
1 test(Delt…  84.3ms  85.2ms  84.9ms  86.6ms     11.7     44.9MB     2     4
2 arma_test… 106.5ms 107.7ms 107.7ms 108.9ms      9.28    38.1MB     3     2
3 rcpp_test… 101.9ms 103.2ms 102.2ms 106.6ms      9.69    38.1MB     1     4
# … with 5 more variables: total_time <bch:tm>, result <list>, memory <list>,
#   time <list>, gc <list>

I have also explicitly initialized the output list to the required size avoiding the push_back, but that did not make a big difference. With vector like data structures from Rcpp you should definitely avoid using push_back though, since there a copy will be made every time you extend the vector.

I would like to add to @RalfStubner's excellent answer.

You will notice that we are making many allocations in the first for loop (i.e. c1 = NumerMatrix(n, m)). This can be expensive as we are initializing every element to 0 in addition to allocating memory. We can change this to the following for increased efficiency:

NumericMatrix c1 = no_init_matrix(n, m)

I also went ahead and added the keyword const where possible. It's debatable if doing this allows the compiler to optimize certain pieces of code, but I still add it where I can for code clarity (i.e. "I don't want this variable to change"). With that, we have:

// [[Rcpp::export]]
List rcpp_test_modified(const NumericVector Delta, 
                        const NumericMatrix delta, 
                        const NumericMatrix rx) {

    int n = Delta.length();
    int m = rx.ncol();
    int dCol = delta.ncol();

    List c(dCol);

    for(int i = 0; i < dCol; ++i) {
        NumericMatrix c1 = no_init_matrix(n, m);

        for(int k = 0; k < n; ++k) {
            const double tmp = Delta[k] / (1 + delta(k, i));

            for(int j = 0; j < m; ++j) {
                c1(k, j) = rx(k, j) * tmp; 
            }
        }

        c[i] = c1;
    }

    return c;

}

And here are some benchmarks (Armadillo solution left out):

bench::mark(test(Delta, delta, rx),
            rcpp_test_modified(Delta, delta, rx),
            rcpp_test(Delta, delta, rx))
# A tibble: 3 x 14
  expression     min   mean  median    max `itr/sec` mem_alloc  n_gc n_itr total_time result memory time 
  <chr>      <bch:t> <bch:> <bch:t> <bch:>     <dbl> <bch:byt> <dbl> <int>   <bch:tm> <list> <list> <lis>
1 test(Delt… 12.27ms 17.2ms 14.56ms 29.5ms      58.1    41.1MB    13     8      138ms <list… <Rpro… <bch…
2 rcpp_test…  7.55ms 11.4ms  8.46ms   26ms      87.8    38.1MB    16    21      239ms <list… <Rpro… <bch…
3 rcpp_test… 10.36ms 15.8ms 13.64ms 28.9ms      63.4    38.1MB    10    17      268ms <list… <Rpro… <bch…
# … with 1 more variable: gc <list>

And we see a 50% improvement with the Rcpp version.