Division and quotient derivatives

Read the equation

For a nonzero denominator y, the sensitivities of x/y are 1/y and −x/y². The code directly uses these expressions and has no denominator regularization. A zero denominator therefore remains outside the mathematical domain and can produce nonfinite values.

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;
    };
    return tape.record(z);
}

// Division: z = x / y, dz/dx = 1/y, dz/dy = -x/y^2
inline GradNodePtr div(GradNodePtr x, GradNodePtr y, TapeRecorder& tape) {
    auto z = std::make_shared<GradNode>(x->value / y->value);
    z->inputs = {x, y};
    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);
}

Division: z = x / y, dz/dx = 1/y, dz/dy = -x/y^2

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