approximation_error

std.elementwise.approximation_error · Level L1

The errors of an approximation a to a reference value x: the absolute error |a − x|; the relative error |a − x|/|x|, defined only when x ≠ 0 (has_rel says whether it is, and rel is 0 otherwise); and whether |a − x| ≤ τ for the tolerance τ ≥ 0, as 1 or 0. Truncation, rounding and data error are different sources and are not interchangeable.

|a − x|, |a − x|/|x|, [|a − x| ≤ τ]

Signature

approximation_error(a: f64[n], x: f64[n], tol: f64[]) → f64[n], f64[n], f64[n], f64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the definitions, in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 680 float64 results inside the running error bound; the closest uses 43% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 47% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 98%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 1.18

Identity

Calls

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Called by

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sha256:e0b93249ace2bfae965354d22fe03eea08c091c75d3bcfe91468ee3ebec29107

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.elementwise.approximation_error · Ω:approximation_error
Pins: sha256:0e94b63a7241f71216a6afce954367cb06191556e3090420163f4cd2408fab9fthis graph alone
Evidence: sha256:2e8b4f49b0e94ed42deca13fe7d68fba168ead1dfd73939c57092731a938dd5dthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.