Interval arithmetic¶

A fact reported as 1,204 with XBRL decimals="0" is a rounded amount: the true value lies anywhere from 1,203.5 to 1,204.5. A rule that adds such amounts must allow for the rounding, or it fails reports that are correct. EBA validation rules therefore use interval arithmetic, following the Eurofiling interval arithmetic functions (iaf:*, version 1.1.1).

Numbers with a tolerance¶

A Number is a decimal value with a symmetric tolerance. Number.of(value, decimals) derives the tolerance from XBRL decimals: 0.5 · 10^-decimals. INF (in any case) or None means exact; constants in rules and selection defaults are exact.

from dpmxl import Number

print(Number.of(1204, 0))
print(Number.of(1204000, -3).tolerance, Number.of("0.0498", 4).tolerance)
print(Number.of(5, "INF"))

Number(value=Decimal('1204'), tolerance=Decimal('0.5'))
5E+2 0.00005
Number(value=Decimal('5'), tolerance=Decimal('0'))

Arithmetic¶

Operation	Tolerance of the result
`a + b`, `a - b`	`δa + δb`
`a * b`	`\|a·δb\| + \|b·δa\| + δa·δb`
`a / b`	The largest deviation of the four endpoint quotients from `a / b` (skipping a zero divisor)
`-a`, `abs(a)`	`δa`
`min`, `max`	The chosen value's tolerance

a, b = Number.of(10, 0), Number.of("7.0", 1)
print(a + b)
print(a * b)

Number(value=Decimal('17.0'), tolerance=Decimal('0.55'))
Number(value=Decimal('70.0'), tolerance=Decimal('4.025'))

Arithmetic keeps 34 significant digits (IEEE decimal128) in any thread decimal context; the caller's decimal context has no effect.

Comparisons¶

Two numbers compare as intervals:

Comparison	Holds when
`a = b`	`\|a − b\| ≤ δa + δb` (the intervals overlap)
`a != b`	the intervals do not overlap
`a <= b`	`a − b ≤ δa + δb`
`a >= b`	`a ≥ b − (δa + δb)`
`a < b`, `a > b`	likewise, strictly

from dpmxl.values import compare

print(compare("=", Number.of(10, 0), Number.of("10.4")))
print(compare("=", Number.of(10, 0), Number.of("10.6")))

True
False

Switching it on and off¶

Interval arithmetic applies to selections with interval: true: {tC_01.00, r0010, interval: true}, usually set once in a with context. With interval: false, or without the option, facts are compared exactly as reported. EBA rules state the option explicitly. Each operator and function in the language reference says what it does with a tolerance.

The gap in explanations¶

When a comparison fails despite the tolerance, explain reports each side's interval and the gap: how far apart the closest values within those intervals still are. In

1,204 should equal 1,211 (1,180 + 31): difference 7, at least 5.5 given the reported precision

the left side is 1,204 ± 0.5 and the right side 1,211 ± 1, so the difference of 7 shrinks to at least 5.5. The gap is rounded down, so "at least" never overstates it.

Where the specification is ambiguous¶

The specification is ambiguous about the radius formulas for products and quotients; dpm-xl uses the ones above, which avoid false failures. The columnar evaluator uses Decimal(38, 12); see Evaluate large reports.