Two nonempirical kinetic energy density dependent approximations are introduced. First, the local tau approximation (LTA) is proposed in which the exchange energy E-x depends only on a kinetic energy density tau. This LTA scheme appears to be complementary to the local spin density (LSD) approximation in the sense that its exchange contribution to the atomization energy Delta E-x=E-x(atoms)-E-x(molecule) is fairly accurate for systems where LSD fails. On the other hand, in cases where LSD works well LTA results for Delta E-x are worse. Secondly, the tau PBE approximation to E-x is developed which combines some of the advantages of LTA and of the Perdew-Burke-Ernzerhof (PBE) exchange functional. Like the PBE exchange functional, tau PBE is free of empirical parameters. Furthermore, it yields improved atomization energies compared to the PBE approximation.