Watch one error become a learning step.

Predict the direction, differentiate the real graph, then flip only the label. The forward prediction stays fixed; the gradient reverses.

SIGMOID + BINARY CROSS-ENTROPYAUTOMATIC DIFFERENTIATION · CPU
LOGIT · wx+b—Linear evidence before sigmoid
PREDICTION · σ(z)—Forward probability
LOSS · BCE(y,p)—Penalty for this label
ERROR SIGNAL · p-y—Shared backward signal

Chosen label 1

dW = —
dB = —
new W = —
new B = —
WEIGHT STEP—

Flipped label 0

same p = —
dW = —
new W = —
Only the target changed.
WEIGHT STEP—

Why. For sigmoid plus binary cross-entropy, the backward signal simplifies to p − y. The weight gradient multiplies that signal by x. Gradient descent subtracts the gradient, so changing the target can reverse the update even though the prediction is identical.

READY FOR AUDIT

One label flip reverses the learning step.

w ↑LABEL 1 UPDATE
0.622459SAME FORWARD PREDICTION
w ↓LABEL 0 UPDATE

The prediction does not know the target. The loss does. Backprop carries that target-relative error backward, and the optimizer follows the resulting slope.

Real single-neuron autodiff, target flips and direction-label limits

Read the explanation

This tool runs the copied TensorFlow dot J S four point twenty-two automatic differentiation library in the browser, not a backend model. The default input two, weight zero point two and bias zero point one give logit zero point five. The sigmoid is about zero point six two two four five nine. Flipping label one to zero leaves that forward probability unchanged. Two equal bars at six hundred display pixels per probability unit are about three hundred seventy-three point four seven five pixels long. This is one scalar example and one neuron, not a trained language model or general network. For an unsaturated sigmoid with binary cross-entropy, the default weight gradient is the prediction minus the label, times input two. For label one it is about minus zero point seven five five zero eight one. For label zero it is about plus one point two four four nine one nine. Gradient descent subtracts learning rate zero point one times that gradient. Starting at weight zero point two gives about zero point two seven five five zero eight versus zero point zero seven five five zero eight. The number line uses fifteen hundred display pixels per weight unit: positions are five hundred fifty and two hundred fifty around the starting position four hundred thirty-six point seven three-eight. Floating-point library values are exported; the geometry rounds for teaching. At input zero, both weight gradients multiply by zero and the weight remains zero point two. Equal bars at fifteen hundred pixels per weight unit are three hundred pixels long; the bias can still change. The source direction display uses a strict greater-than comparison and shows a down arrow for equality, while its status says direction reverses. That wording is inaccurate for zero updates. The actual TensorFlow loss clips probabilities near zero and one, so saturated derivatives can also be zero rather than the simplified textbook rule. Download preserves the full trace, engine version, limitations and real timestamp. Editing inputs without Audit leaves a stale exported trace. This tool does not implement batches, hidden layers, momentum or regularization.

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