divided_difference_step

std.seq.divided_difference_step · Level L0

One column of Newton's divided-difference table: where the mask m is 1, dᵢ becomes (dᵢ − dᵢ₋₁)/gᵢ, with gᵢ = xᵢ − xᵢ₋ₖ; where it is 0, dᵢ stays. S shifts a vector down by one. The body that divided_differences scans, one order k per step.

dᵢ′ = mᵢ·(dᵢ − dᵢ₋₁)/gᵢ + (1 − mᵢ)·dᵢ

Signature

divided_difference_step(d: f64[n], m: f64[n], g: f64[n], S: f64[n, 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 (1 − m)·d + m·(d − S·d)/g in exact rational arithmetic, on all 40 test cases.
All 160 float64 results inside the running error bound; the closest uses 21% 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 25% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 84%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 4.91

Identity

Calls

—

Called by

divided_differences

sha256:7e25487d75039574cbe1b70c1393cc879676cca92fb4eec3cde42e662c3668bb

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.divided_difference_step · Ω:divided_difference_step
Pins: sha256:14822fd97ee6200fd9f28300e6cef1611e03e5aab637b0dfb7e7925e960c0582this graph alone
Evidence: sha256:e7a95e740a1e08afb66a143f239a7c473675afbeab98268dac7de4496a1ff85cthe 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.