taylor_remainder

std.seq.taylor_remainder · Level L2

For a polynomial p (coefficients highest degree first), a center a, a point x and an order n: p(x), the Taylor polynomial T_n(x) of p at a, and the remainder p(x) − T_n(x), computed directly. The remainder is 0 when n ≥ deg p. A finite jet says nothing about the Taylor series of a non-polynomial function, or about analyticity.

R_n(x) = p(x) − Σ_{k≤n} p⁽ᵏ⁾(a)(x − a)ᵏ/k!

Signature

taylor_remainder(c: f64[m], a: f64[], x: f64[], n: f64[]) → f64[], f64[], f64[]

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 binomial expansion of each power about a, in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 120 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

horner poly_taylor_at

Called by

—

sha256:6de85cd3d66d058a4b35052bc9e915c1418d54b709900e5ac5c7bd86f551cef7

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.taylor_remainder · Ω:taylor_remainder
Pins: sha256:6712101ac0ead734b946700a2cc022749739b3038ade0ff611ae5a721d9a7f24this graph and the 5 it reaches through calls
Evidence: sha256:1a4a88bb14ce04bc6a70804aabaf650fe1d5263899ce4278990d222e84576087the 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.