Iām guessing that what you mean by āsafeā is *Would it give the same answer as it would in exact (infinite-precision) arithmetic?* The answer, of course, is āit dependsā, but the most general answer is ānoā.

That is, suppose you are comparing two numbers `x`

and `y`

that are computed by two different floating-point algorithms, and you want a comparison function `is_same(x,y)`

that returns `true`

if you would have `x==y`

in infinite precision.

Suppose that you your algorithms are accurate to 8 significant digits. Then you could do `isapprox(x, y, rtol=1e-8)`

. Or you could do `round(x, sigdigits=8) == round(y, sigdigits=8)`

, which is almost equivalent but much slower (about 100Ć slower on my computer!).

Of course, to do this, you need to have a rough sense of the accuracy of your algorithms. If it is a single scalar operation like `0.1 + 0.2`

, then it should be accurate to nearly machine precision, but for more complicated algorithms error analysis is much tricker. The default in `isapprox`

(the `ā`

operator) is to compare about half of the significant digits in the current precision, which is reasonable for many algorithms (losing more than half of the significant digits means you have a pretty inaccurate calculation), but is obviously not universally appropriate.

Naturally, be aware that such approximate comparisons may give false positives (returning `true`

for two values that are *supposed* to be distinct in infinite precision, but differ by a very small amount).

Your suggestion, `round(x, digits=8) == round(y, digits=8)`

, is roughly equivalent to (but vastly slower than) `isapprox(x, y, atol=1e-8)`

ā an *absolute* tolerance rather than a relative tolerance. Usually, a relative tolerance is more appropriate in floating-point calculations, because relative tolerances are scale invariant.

If you want a rigorous guarantee that two values *might* be the same, you can use Interval Arithmetic and implement `might_be_same(x,y) = !isdisjoint(x,y)`

. This might give you false positives, but will never give false negatives.