poly_multiply

std.seq.poly_multiply · Level L3

The product of two polynomials, coefficients highest degree first as horner takes them: (fg)ₖ = Σᵢ fᵢ·gₖ₋ᵢ, the finite convolution, computed as a Toeplitz matrix of g times f. Exact for integer coefficients, where every product and sum is an integer. NumPy's np.polymul.

(fg)ₖ = Σᵢ fᵢ·gₖ₋ᵢ

Signature

poly_multiply(f: f64[a], g: f64[c]) → f64[a+c−1]

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 the double loop over fᵢ and gⱼ in exact rational arithmetic, on all 40 test cases.
All 233 float64 results inside the running error bound; the closest uses 81% 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 70% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 84%
largest error, in units in the last place: 15

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

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Called by

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sha256:0d5cfda07ce41dd7cc577d1d1ea16489f173be0c2e20c8bc83b71fa75dcc4dc9

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_multiply · Ω:poly_multiply
Pins: sha256:a47b224850a68a526da3e6a2565e654c6a20be4b3068ad2f5d2d4ca0a7e7acedthis graph alone
Evidence: sha256:548719eb49151b165bf36dd167b70e72004f137dfb82d1f8d203ebde02d75a6bthe 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.