Quantum discord Q is a function of density matrix elements. The domain of such a function in the case of two-qubit system with X density matrix may consist of three subdomains at most: two ones where the quantum discord is expressed in closed analytical forms (Q_{π/2} and Q_0) and an intermediate subdomain for which, to extract the quantum discord Q_θ, it is required to solve in general numerically a one-dimensional minimization problem to find the optimal measurement angle θ\in(0,π/2). Hence the quantum discord is given by a piecewise-analytic-numerical formula Q=\min{Q_{π/2}, Q_θ, Q_0}. Equations for determining the boundaries between these subdomains are obtained. The boundaries consist of bifurcation points. The Q_θ subdomains are discovered in the generalized Horodecki states, in the dynamical phase flip channel model, in the anisotropic spin systems at thermal equilibrium, in the heteronuclear dimers in an external magnetic field. We found that transitions between Q_θ subdomain and Q_{π/2} and Q_0 ones occur suddenly but continuously and smoothly, i.e., nonanalyticity is hidden and can be observed in higher derivatives of discord function.