beta

std.elementwise.beta · Level L5

The beta function B(a, b) = Γ(a)Γ(b)/Γ(a + b) for a, b > 0, as e^(log B). Calls log_beta.

B(a, b) = ∫₀¹ tᵃ⁻¹(1 − t)ᵇ⁻¹ dt

Signature

beta(a: f64[n], b: f64[n]) → 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.
Agrees with the reference Γ(a)·Γ(b)/Γ(a+b) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 194 float64 results inside the running error bound; the closest uses 14% 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 4% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 7%
bit-equal to the NumPy formula in float64: 7%
largest error, in units in the last place: 38

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

log_beta

Called by

—

sha256:d9c588c587d76670676650e238b4377af4eac3291fe974a3546d8670f5f45161

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.beta · Ω:beta
Pins: sha256:e98d6379151fe20590934c6b201248983885d7b13c6ae7a92acdbd53edc67f8bthis graph and the 1 it reaches through calls
Evidence: sha256:fb0306e2b378004e3befd4d4f22f048c4a807879dcc475de069121dc0a6c2853the 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.