newton_horner_step

std.seq.newton_horner_step · Level L0

One step of Horner's rule for Newton's form: p ← cₖ + (t − xₖ)·p, at every t. The body that newton_eval scans, from the last coefficient down.

p′ = cₖ + (t − xₖ)·p

Signature

newton_horner_step(p: f64[q], ck: f64[], xk: f64[], t: f64[q]) → f64[q]

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 ck + (t − xk)·p in exact rational arithmetic, on all 40 test cases.
All 136 float64 results inside the running error bound; the closest uses 75% 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 66% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 83%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 7.71

Identity

Calls

—

Called by

newton_eval

sha256:26e1d46dcd45c387e6776146f7036ff2049857e8090e2653865cfb8e44cd7171

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

Control handle

Symbol: Ω:std.seq.newton_horner_step · Ω:newton_horner_step
Pins: sha256:a02f6c275aaa45d346704efe2917687dab9059e0b965ec4daeb1c73ec76e2eadthis graph alone
Evidence: sha256:40ddde6844aeebce14fba670446fdd485c8bce06e665c8080f4935f135e05f2cthe 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.