f_sf

std.stats.f_sf · Level L5

The F distribution's upper tail P(F > x): I_{d₂/(d₂+d₁x)}(d₂/2, d₁/2), the same beta integral read from the other end, so that a small tail keeps its relative accuracy.

P(F > x) = I_{d₂/(d₂+d₁x)}(d₂/2, d₁/2)

Signature

f_sf(x: f64[n], d1: f64[], d2: f64[]) → 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 the beta integral I(d1/2, d2/2) from d1·x/(d1·x + d2) to 1 to 80 digits (100-digit arithmetic), on all 40 test cases.
All 105 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 1% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 9%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 19

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

anova_oneway

sha256:222a8aa7ce6fa2e4b21fffb72a57828394afba8ef72f4315b014c3795fa6c98f

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.f_sf · Ω:f_sf
Pins: sha256:a20e587e717f4d51d8231da8b1508dbf4bd22d9a891dcfb75c546a7ef4d3a120this graph alone
Evidence: sha256:6315c38be78440af470603019bcf818a4fc3b9bfa892e9e8c119b8ccd5ab4413the 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.