Tolerance

Where is tol actually used?

Tolerance is not applied to the series you see at the top level. It is passed down through any nested or transformed series and used only when we check whether a value is in the series at the underlying level.

The two places where it is used are:

• Arithmetic progressions (arithmetic()): we decide if a value is in the series using the implied step index \((x - representative) / spacing\). Because of floating point, this index is rarely an exact integer (e.g. (0.3 - 0) / 0.1). A value is treated as a discrete value if that index is within tol of an integer.
• Numeric-vector-based series (plain numeric vectors or as_discretes()): we decide if a value is in the series by checking whether it is within tol of any stored discrete value.

For transformed series (e.g. 1 / arithmetic()), the queried value is mapped back to the underlying series and the same logic is applied there with the same tol; no separate tolerance is applied to the transformed values.

Example: tol with arithmetic()

Without a tolerance, values can be hard to recognize as discrete values in an arithmetic series because the implied index might be 2.999999999 instead of 3.

x <- arithmetic(representative = 0, spacing = 0.1)
has_discretes(x, 0.3)
#> [1] TRUE
has_discretes(x, 0.3, tol = 0)
#> [1] FALSE

Example: tol with an explicit numeric series

Even if a series is represented as a numeric vector, you can still run into tiny floating-point mismatches.

v <- c(0, 0.1, 0.2, 0.1 * 3)   # last entry is not exactly 0.3

has_discretes(v, 0.3)
#> [1] TRUE
has_discretes(v, 0.3, tol = 0)
#> [1] FALSE

If you want “mathematical series” behaviour for an explicit series, use tol = 0. If you want “numerical computing” behaviour (robust to round-off), keep tol positive.

Example: preserving numeric vectors

When a numeric vector is converted to a “discretes” object via as_discretes(), the root numeric vector is preserved, even if the series gets transformed.

Consider transforming a numeric vector vs. that same vector expressed as a “discretes” object.

raw <- 10^(-5:5)
raw
#>  [1] 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05
preserved <- 10^as_discretes(-5:5)
preserved
#> Transformed series of length 11:
#> 1e-05, 1e-04, 0.001, ..., 1000, 10000, 1e+05

The numeric vector gets transformed right away, whereas it remains as the base series when transformed as a “discretes” object. This has implications because errors are propagated differently.

For example, an exponent slightly off by 1e-10 results in a more significant error after transformation that may not be within the default tol:

# Exponent slightly off from `5`
q <- 10^(5 - 1e-10)
# Error magnitude after transformation
10^5 - q
#> [1] 2.302586e-05

This means that “is this value in the series?” fails for the eagerly transformed vector, but not when the vector is preserved. This is because tolerance is applied to the root series.

has_discretes(raw, q)
#> [1] FALSE
has_discretes(preserved, q)
#> [1] TRUE

Choosing a tol

• Default is usually fine: sqrt(.Machine$double.eps) is intended to smooth over tiny floating-point noise.
• Use tol = 0 for exactness: when you truly want explicit sets to behave exactly, perhaps useful in some situations when working with ‘integer’ storage mode.
• Increase tol for noisy values: if your values come from computation or measurement and you are noticing values incorrectly treated as not in the series, increase tol accordingly.