sigmoid

std.elementwise.sigmoid · Level L0

The logistic function σ(x) = 1/(1 + e⁻ˣ), between 0 and 1: the derivative of softplus, and the inverse of the log-odds. NOVA's Sigmoid primitive, which uses eˣ/(1 + eˣ) where e⁻ˣ would overflow.

σ(x) = 1/(1 + e⁻ˣ)

Signature

sigmoid(x: f64[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.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Agrees with the reference 1 / (1 + np.exp(-x)) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 128 float64 results inside the running error bound; the closest uses 15% 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 18% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 72%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 1.43

Identity

Calls

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Called by

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sha256:2ceb1b72742050002012375dfd2e9a187e000e21de0456ef053997461fcdee32

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.sigmoid · Ω:sigmoid
Pins: sha256:58e8dc26a98f99ca7fcc88c188b6437fbd5289209c05aa34216fb0a8cb49130fthis graph alone
Evidence: sha256:c2794d2a62e5abb07918a496a82d05206117f31d6493af2eb05fd845fbdb6c5fthe 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.