newton_step

std.func.newton_step · Level L1

One Newton step for f(x) = y, elementwise: x − (f(x) − y)/f′(x), with the derivative given as the function df. The body newton loops on; it takes the loop's tolerance and does not use it. A template over f and df.

x′ = x − (f(x) − y)/df(x)

Signature

newton_step(x: f64[n], y: f64[n], tol: f64[]) → 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 witness f = softplus, df = sigmoid: x − (softplus(x) − y)/sigmoid(x) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 129 float64 results inside the running error bound; the closest uses 89% 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 40% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 47%
bit-equal to the NumPy formula in float64: 79%
largest error, in units in the last place: 3.0e+3

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

newton

sha256:a3d1840fd2e83cb3baeade12d5712bf70700c814ee03e49dc3863add3fe00ae0

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.func.newton_step[df=std.elementwise.sigmoid,f=std.elementwise.softplus]
Pins: sha256:deda9e15714e5a009752229227d712b99aabe834b90a544cf45112b948fdd10ethis graph and the 3 it reaches through calls
Evidence: sha256:b7b1d9b5f5fa2b3cd5372054f6328b6c58db4172bad81802b3d58a19852414eathe 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.