Extreme numerical values in floating-point precision in R

Question

Can somebody please explain me the following output. I know that it has something to do with floating point precision, but the order of magnitue (difference 1e308) surprises me.

0: high precision

> 1e-324==0
[1] TRUE
> 1e-323==0
[1] FALSE

1: very unprecise

> 1 - 1e-16 == 1
[1] FALSE
> 1 - 1e-17 == 1
[1] TRUE

Answer 1

R uses IEEE 754 double-precision floating-point numbers.

Floating-point numbers are more dense near zero. This is a result of their being designed to compute accurately (the equivalent of about 16 significant decimal digits, as you have noticed) over a very wide range.

Perhaps you expected a fixed-point system with uniform absolute precision. In practice fixed-point is either wasteful or the ranges of each intermediate computation must be carefully estimated beforehand, with drastic consequences if they are wrong.

Positive floating-point numbers look like this, schematically:

+-+-+-+--+--+--+----+----+----+--------+--------+--------+--
0

The smallest positive normal double-precision number is 2 to the power of the minimal exponent. Near one, the double-precision floating-point numbers are already spread quite wide apart. There is a distance of 2^-53 from one to the number below it, and a distance of 2^-52 from one to the number above it.