Hyperbolic tangent derivative

Read the equation

The derivative of tanh(x) is 1−tanh²(x). It is largest at zero and approaches zero in the saturated tails. Reusing the forward output in the backward pass avoids an additional tanh evaluation.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

    z->backward_fn = [x, y, z]() {
        x->grad += z->grad / y->value;
        y->grad -= z->grad * x->value / (y->value * y->value);
    };
    return tape.record(z);
}

// Tanh: z = tanh(x), dz/dx = 1 - tanh^2(x)
inline GradNodePtr tanh_op(GradNodePtr x, TapeRecorder& tape) {
    double t = std::tanh(x->value);
    auto z = std::make_shared<GradNode>(t);
    z->inputs = {x};
    z->backward_fn = [x, z, t]() {
        x->grad += z->grad * (1.0 - t * t);
    };
    return tape.record(z);
}

Tanh: z = tanh(x), dz/dx = 1 - tanh^2(x)

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