falling_step
std.elementwise.falling_step · Level L1One factor of the falling product n(n − 1)…(n − k + 1): where i ≤ k, multiply by n − i + 1 and count. The body permutations runs.
(i, acc) → (i + 1, acc·(n − i + 1)) while i ≤ k
Signature
falling_step(i: i64[s], acc: i64[s], n: i64[s], k: i64[s]) → 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
where i <= k: i + 1, acc*(n - i + 1)in exact rational arithmetic, on all 40 test cases. - All 328 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
Identity
sha256:026ac5622f72c6a40b7b21736445b581f648348c88e6324ce707906893477123The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.
Control handle
- Symbol
- Ω:std.elementwise.falling_step · Ω:falling_step
- Pins
sha256:3137831f424e33fc5b7ca99b25ad6540214466b113439c61bb9f26c92352fb07this graph alone- Evidence
sha256:4d254555f8ac4e8cc1bec3b9a2a07fd63d12d131e4f6d403215adc50a9d4ab7bthe 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.