trial_step

std.elementwise.trial_step · Level L4

One step of trial division that records the factorization as it goes. Where d² ≤ m and d divides m, divide d out and record it: the smallest prime factor, the first time; a new distinct prime, with φ multiplied by (1 − 1/d) exactly; or a repeat, which makes the number not squarefree. Where d does not divide, move to the next candidate: 3 after 2, then odd numbers. The body factor_summary runs.

d | m: m ← m/d, record d; else d ← next candidate

Signature

trial_step(m: i64[s], d: i64[s], last: i64[s], spf: i64[s], omega: i64[s], sqf: i64[s], phi: i64[s]) → i64[s], i64[s], i64[s], i64[s], 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 trial division by d, recording the smallest prime factor, ω, squarefreeness and φ in exact rational arithmetic, on all 40 test cases.
All 1,015 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

Calls

—

Called by

factor_summary

sha256:c4ef6d7f81e71f4371bf021f9e9f3f89e55195e82a64d71fce1c3321acb28bfa

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.trial_step · Ω:trial_step
Pins: sha256:85af246d72e736eaa3eafeaea3d45fb9e7e9870f7600b5097d49e7265d4ddcbathis graph alone
Evidence: sha256:fae1301f26ffa399e95bc3efa9d9c30cc639944b4b062676c00926839c20f6f5the 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.