poly_taylor_at

std.seq.poly_taylor_at · Level L2

The Taylor coefficients of a polynomial at a: the coefficients of p in powers of (x − a), highest degree first, so the last is p(a), the one before it p′(a), and the k-th from the end p⁽ᵏ⁾(a)/k!. By repeated synthetic division, n passes of taylor_pass. A polynomial of degree at most n − 1 is reconstructed exactly. The reference is the closed form with binomial coefficients.

p(x) = Σₖ bₖ (x − a)^{n−1−k}, b from the end = p⁽ᵏ⁾(a)/k!

Signature

poly_taylor_at(c: f64[n], a: 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 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 bₖ = Σ_{j≤k} cⱼ·C(n − 1 − j, k − j)·a^(k − j) in exact rational arithmetic, on all 40 test cases.
All 153 float64 results inside the running error bound; the closest uses 37% 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 38% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 80%
bit-equal to the NumPy formula in float64: 75%
largest error, in units in the last place: 20

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

taylor_pass

Called by

taylor_remainder

sha256:c709f39378496b8263fd814dcc5b6cfcb95438930ae557dbf73d5272b3631a20

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.poly_taylor_at · Ω:poly_taylor_at
Pins: sha256:36d68a4cbe322219782a51ac16b8e40cef45a3a3ff5d037670d126de2a708789this graph and the 2 it reaches through calls
Evidence: sha256:e1d79df000a50030a58cc3983daecf39ff101d7460cc5730f866d7fc365c02bfthe 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.