piecewise_limits

std.elementwise.piecewise_limits · Level L1

Limits and continuity at a for the function equal to lm·x + lb left of a, to rm·x + rb right of a, and to V at a itself: the left limit lm·a + lb, the right limit rm·a + rb, whether the two-sided limit exists (they agree), and whether the function is continuous at a (both equal V), each decided exactly for exact inputs. The limit does not need V; continuity does.

lim_{x→a⁻} = lm·a + lb, lim_{x→a⁺} = rm·a + rb, continuous ⇔ both = V

Signature

piecewise_limits(lm: f64[], lb: f64[], rm: f64[], rb: f64[], a: f64[], V: f64[]) → f64[], f64[], f64[], f64[]

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 the two affine values at a in exact rational arithmetic, on all 40 test cases.
All 160 float64 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 float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

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Called by

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sha256:3b7aae3190b26988c4bbb85eab53093649283a685cc1f87e2fee3891a6954c1d

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.piecewise_limits · Ω:piecewise_limits
Pins: sha256:e7d84b41be6720a2bb0442e5ec2f6c96b0398e472a0c0f63003a3bc2788d94f4this graph alone
Evidence: sha256:7df218c80ff4a310aafe384fc2f351643b4ce32d66a05d88d5339e45a5a258a0the 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.