Why does all.equal in R not test for differences per observations where mismatch happen by tolerance amount, rather than the aggregate mean?

Question

all.equal in R tests for differences (absolute / relative) for observations where mismatch happen, and then makes sure that it is within tolerance amount. Ideally, it should test all observations where mismatch happens by tolerance amount, and then report the differences... Why is the behavior so ?

eg In the following case, I would expect the result of all.equal to be FALSE, as the first observation in x is not equal to y

> x = rep(1, 1000)
> y = rep(1, 1000)
> x = x + 0.001
> y[1] = 2
> print(all.equal(x, y, scale = 1, tolerance=0.01))
[1] TRUE

Answer 1

The all.equal function returns TRUE if the the target and current arguments are nearly equal up to a specified tolerance, which has a default value close to 1.5e-8. The function does not return FALSE otherwise, rather it returns the mean relative ( scale=NULL ) or absolute ( scale=1 ) difference.

x = rep(1, 1000)
y = rep(1, 1000)
x = x + 0.001
y[1] = 2
print(all.equal(x, y, scale = 1)) # Omit the tolerance
[1] "Mean absolute difference: 0.001998"

mean(abs(x-y))
[1] 0.001998

If you read the help page for all.equal, you would see the reason:

tolerance   numeric ≥ 0. Differences smaller than tolerance are not reported.

Edit : Based on Aaron's assessment of the OP's source of confusion, perhaps the any function was required:

y[1] = 1
any(abs(x-y) > 0.01) # Element-wise comparison
[1] FALSE

y[1] = 2
any(abs(x-y) > 0.01)
[1] TRUE