Numeric representation and error budgets

The idea

Float64 stores a finite binary approximation of many decimal fractions. Arithmetic introduces rounding, so two mathematically equivalent computations need not produce identical bit patterns. Exact equality remains meaningful when identical representation is required, but measured or computed quantities often need a tolerance derived from their units and error budget. isapprox combines absolute and relative tolerances; choose them intentionally instead of hiding uncertainty behind an arbitrary large threshold.

Near zero, a relative tolerance alone may reject a small residual because there is no useful nonzero scale. An absolute tolerance can express an acceptable error in the measurement’s units. Rational values such as 1//10 represent this fraction exactly within the limits of their integer storage, but they are not a universal substitute for floating-point scientific computation. Also consider overflow, cancellation, and accumulation order.

println(0.1 + 0.2 == 0.3)
println(isapprox(0.1 + 0.2, 0.3))
println(1//10 + 2//10 == 3//10)

false
true
true

The floating-point sum is close but not exactly equal to the stored 0.3. The rational expression represents the fractions exactly here.