circular_mean

std.stats.circular_mean · Level L5

The mean direction of angles (radians): the direction of the sum of their unit vectors, atan2(Σ sin θ, Σ cos θ), in [−π, π]. The arithmetic mean of 359° and 1° is 180°; their circular mean is 0°.

θ̄ = atan2(Σ sin θᵢ, Σ cos θᵢ)

Signature

circular_mean(theta: f64[n]) → 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.
Agrees with the reference np.arctan2(np.sin(theta).sum(), np.cos(theta).sum()) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 40 float64 results inside the running error bound; the closest uses 6% 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 6% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 75%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 13

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:d4587fb9cb8405509787b983247bb09d723415c17ac2028d2050a6793f993e29

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

Control handle

Symbol: Ω:std.stats.circular_mean · Ω:circular_mean
Pins: sha256:2369678138bd046f27835f9227fe1e1df1d061e9cb12a446f4d3e759d6d58eb8this graph alone
Evidence: sha256:00d982101d339a8da875ed113e977caf63f3b8f2826de61a0155133b962e7a38the 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.