log_beta

std.elementwise.log_beta · Level L5

The logarithm of the beta function, log B(a, b) = log Γ(a) + log Γ(b) − log Γ(a + b), for a, b > 0: finite where B itself underflows.

log B(a, b) = log Γ(a) + log Γ(b) − log Γ(a+b)

Signature

log_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 log(Γ(a)·Γ(b)/Γ(a+b)) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 142 float64 results inside the running error bound; the closest uses 20% 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 5% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 13%
bit-equal to the NumPy formula in float64: 10%
largest error, in units in the last place: 79

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

—

Called by

beta

sha256:52b0123470fec70b639d52e78777e485e39cba915a195394ade8c46ff754211e

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.log_beta · Ω:log_beta
Pins: sha256:0fce43173801a3298cce9b2c293138624b3c54bd16f44e38d267c0bc6f266776this graph alone
Evidence: sha256:43fe79158277dc65bf44756b644eeca39f8b953d0d1310e01045c5e5a4a8909athe 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.