binomial

std.elementwise.binomial · Level L4

The binomial coefficient C(n, k) = n!/(k!(n − k)!), elementwise, by the multiplicative formula on the shorter side, min(k, n − k): a While over binomial_step. C(n, 0) = 1, and C(n, k) = 0 for k > n.

C(n, k) = Πᵢ₌₁ᵏ (n − k + i)/i

Signature

binomial(n: i64[s], k: i64[s]) → i64[s]

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 math.comb(n, k) in exact rational arithmetic, on all 40 test cases.
All 176 int64 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 int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Note

Exact for every n ≤ 50: each intermediate is at most C(50, 25)·25 < 2⁵³.

Identity

Calls

binomial_step count_running

Called by

catalan

sha256:12fb4884a2047438a2045e49d015af7aad32a044f8949b87e7c14a6561400ef7

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

Control handle

Symbol: Ω:std.elementwise.binomial · Ω:binomial
Pins: sha256:b7eff1ace34b5668b99a6f295b55b29795af44d6104693a9c998ec32d76e60f4this graph and the 2 it reaches through calls
Evidence: sha256:e376f74fb83da8619b82b8d4b81f89d43b74e896f8a53ba3dd66f464875462c1the 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.