The Numerical Template Toolbox (NT2)
collection of header-only C++ libraries that make it
possible to explicitly request the use of SIMD instructions
when possible, while falling back to regular scalar
operations when not. NT2 itself is powered
by Boost, alongside two proposed
Boost libraries –
Boost.Dispatch, which provides a
mechanism for efficient tag-based dispatch for functions,
Boost.SIMD, which provides a framework for the
implementation of algorithms that take advantage of SIMD
and exposes these libraries for use with
If you haven’t already, read the
RcppNT2 introduction article
to get acquainted with the RcppNT2 package.
Computing the Variance
As you may or may not know, there are a number of algorithms
for computing the variance,
each making different tradeoffs in algorithmic complexity
versus numerical stability. We’ll focus on implementing a
two-pass algorithm, whereby we compute the mean in a first
pass, and later the sum of squares in a second pass.
First, let’s look at the R code one might write
to compute the variance.
 0.833939 0.833939
We can decompose the operation into a few distinct steps:
- Compute the mean for our vector ‘x’,
- Compute the squared deviations from the mean,
- Sum the deviations about the mean,
- Divide the summation by the length minus one.
Naively, we could imagine writing a ‘simdTransform()’ for
step 2, and an ‘simdReduce()’ for step 3. However, this
is a bit inefficient as the transform would require
allocating a whole new vector, with the same length as
our initial vector. When neither ‘simdTransform()’ nor
‘simdReduce()’ seem to be a good fit, we can fall back to
‘simdFor()’. We can pass a stateful functor to handle
accumulation of the transformed results.
Let’s write a class that encapsulates this ‘sum of
Now that we have our accumulator class defined, we can
use it to compute the variance. We’ll call our function
‘simdVar()’, and export it using Rcpp attributes in the
Let’s confirm that this works as expected…
 9.166667 9.166667
And let’s benchmark, to see how performance compares.
expr min lq mean median uq max var(small) 11.784 14.3395 16.37862 15.096 15.7225 40.346 simdVar(small) 1.506 1.7045 2.06541 1.947 2.1055 10.935
expr min lq mean median uq max var(large) 3046.597 3194.231 3278.7417 3301.6205 3323.581 3809.090 simdVar(large) 579.784 594.887 607.0411 607.9845 614.386 712.038
As we can see, taking advantage of SIMD instructions
has improved the runtime quite drastically.
However, we should note that this is not an entirely fair
Rs implementation of the variance. In
particular, we do not have a nice mechanism for handling
missing values; if your data does have any
values, they will simply be propagated (and not
necessarily in the same way that
missingness). If you’re interested, a separate example
is provided as part of the RcppNT2 package, and you can
browse the standalone source file
This article provides just a taste of how RcppNT2 can be used.
If you’re interested in learning more, please check out the