layernorm

std.nn.layernorm · Level L1

Layer normalisation of one feature vector, with scale γ, shift β and a small ε. Calls variance.

(x − x̄)/√(σ² + ε)·γ + β

Signature

layernorm(x: f64[n], gamma: f64[n], beta: f64[n], eps: f64[]) → 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 a call to another library function; select it to open that function.

xf64[n]epsf64[]gammaf64[n]betaf64[n]MeanmuvariancevSubtractdAddveSqrtsdDividexnMultiplyscaledAddyyf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference (x - x.mean()) / np.sqrt(x.var() + eps) * gamma + beta to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 159 float64 results inside the running error bound; the closest uses 97% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
74%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
30

Large ulp counts appear only where cancellation drives a result toward zero; the absolute error is still inside the bound.

Identity

Calls
variance
Called by
—
sha256:94f490284675a5947627a8c808a2e6b373897eab270767319f58e142aa6b2936

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