[Rd] sum() and mean() for (ALTREP) integer sequences

[Duncan Murdoch]

On 02/09/2021 6:55 a.m., Viechtbauer, Wolfgang (SP) wrote:
> Hi all,
> 
> I am trying to understand the performance of functions applied to integer sequences. Consider the following:
> 
> ### begin example ###
> 
> library(lobstr)
> library(microbenchmark)
> 
> x <- sample(1e6)
> obj_size(x)
> # 4,000,048 B
> 
> y <- 1:1e6
> obj_size(y)
> # 680 B
> 
> # So we can see that 'y' uses ALTREP. These are, as expected, the same:
> 
> sum(x)
> # [1] 500000500000
> sum(y)
> # [1] 500000500000
> 
> # For 'x', we have to go through the trouble of actually summing up 1e6 integers.
> # For 'y', knowing its form, we really just need to do:
> 
> 1e6*(1e6+1)/2
> # [1] 500000500000
> 
> # which should be a whole lot faster. And indeed, it is:
> 
> microbenchmark(sum(x),sum(y))
> 
> # Unit: nanoseconds
> #    expr    min       lq      mean   median       uq    max neval cld
> #  sum(x) 533452 595204.5 634266.90 613102.5 638271.5 978519   100   b
> #  sum(y)    183    245.5    446.09    338.5    447.0   3233   100  a
> 
> # Now what about mean()?
> 
> mean(x)
> # [1] 500000.5
> mean(y)
> # [1] 500000.5
> 
> # which is the same as
> 
> (1e6+1)/2
> # [1] 500000.5
> 
> # But this surprised me:
> 
> microbenchmark(mean(x),mean(y))
> 
> # Unit: microseconds
> #     expr      min        lq     mean   median       uq      max neval cld
> #  mean(x)  935.389  943.4795 1021.423  954.689  985.122 2065.974   100  a
> #  mean(y) 3500.262 3581.9530 3814.664 3637.984 3734.598 5866.768   100   b
> 
> ### end example ###
> 
> So why is mean() on an ALTREP sequence slower when sum() is faster?
> 
> And more generally, when using sum() on an ALTREP integer sequence, does R actually use something like n*(n+1)/2 (or generalized to sequences a:b -- (a+b)*(b-a+1)/2) for computing the sum? If so, why not (it seems) for mean()?

The mean.default function looks like this:

function (x, trim = 0, na.rm = FALSE, ...)
{
     if (!is.numeric(x) && !is.complex(x) && !is.logical(x)) {
         warning("argument is not numeric or logical: returning NA")
         return(NA_real_)
     }
     if (na.rm)
         x <- x[!is.na(x)]
     if (!is.numeric(trim) || length(trim) != 1L)
         stop("'trim' must be numeric of length one")
     n <- length(x)
     if (trim > 0 && n) {
         if (is.complex(x))
             stop("trimmed means are not defined for complex data")
         if (anyNA(x))
             return(NA_real_)
         if (trim >= 0.5)
             return(stats::median(x, na.rm = FALSE))
         lo <- floor(n * trim) + 1
         hi <- n + 1 - lo
         x <- sort.int(x, partial = unique(c(lo, hi)))[lo:hi]
     }
     .Internal(mean(x))
}

So it does fixups for trimming and NA removal, then calls an internal 
function.  The internal function is the first part of do_summary, here:

https://github.com/wch/r-source/blob/f9c955fc6699a1f0482e4281ba658215c0e0b949/src/main/summary.c#L541-L556 


It is using separate functions for the mean by type.  The real_mean 
function here:

https://github.com/wch/r-source/blob/f9c955fc6699a1f0482e4281ba658215c0e0b949/src/main/summary.c#L476-L515 


makes a big effort to avoid overflows.

So I suspect the reason mean.default doesn't use sum(x)/length(x) at the 
end is that on a long vector sum(x) could overflow when mean(x) shouldn't.

So why not take the ALTREP into account?  I suspect it's just too much 
trouble for a rare case.

Duncan Murdoch