log_binomial

std.stats.log_binomial · Level L5

The logarithm of the binomial coefficient C(k + m, k), from log-gamma: log Γ(k+m+1) − log Γ(k+1) − log Γ(m+1). The coefficient itself would overflow long before its logarithm does.

log C(k+m, k) = log Γ(k+m+1) − log Γ(k+1) − log Γ(m+1)

Signature

log_binomial(k: i64[n], m: i64[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.

ki64[n]1mi64[n]Addk1AddnAddm1LogGammalg_k1Addn1LogGammalg_m1LogGammalg_n1SubtracttSubtractlogclogcf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference log(math.comb(k + m, k)), the exact integer to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 176 float64 results inside the running error bound; the closest uses 6% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
17%
bit-equal to the NumPy formula in float64
17%
largest error, in units in the last place
24

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
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sha256:dcc4a4d7b5fb47203bf0878661d6dde870aa7194425deab1e578ccdea3c05dab

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.log_binomial · Ω:log_binomial
Pins
sha256:c436d4f7ab0b629388b1b5350e0137c3f67d3e5c174d39ab3d8213bacd99f40cthis graph alone
Evidence
sha256:04b2d588f7eec3fff859f041e6a826d0d344c87d0da6463e31bbea05b53906b1the 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.