  We calculated the QCD equation of state using Taylor expansions that include
contributions from up to sixth order in the baryon, strangeness and electric
charge chemical potentials. Calculations have been performed with the Highly
Improved Staggered Quark action in the temperature range $T\in [135~{\rm MeV},
330~{\rm MeV}]$ using up to four different sets of lattice cut-offs
corresponding to lattices of size $N_\sigma^3\times N_\tau$ with aspect ratio
$N_\sigma/N_\tau=4$ and $N_\tau =6-16$. The strange quark mass is tuned to its
physical value and we use two strange to light quark mass ratios $m_s/m_l=20$
and $27$, which in the continuum limit correspond to a pion mass of about $160$
MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion
coefficients are used to estimate truncation errors of the fourth-order
expansion. We show that truncation errors are small for baryon chemical
potentials less then twice the temperature ($\mu_B\le 2T$). The fourth-order
equation of state thus is suitable for the modeling of dense matter created in
heavy ion collisions with center-of-mass energies down to $\sqrt{s_{NN}}\sim
12$ GeV. We provide a parametrization of basic thermodynamic quantities that
can be readily used in hydrodynamic simulation codes. The results on up to
sixth order expansion coefficients of bulk thermodynamics are used for the
calculation of lines of constant pressure, energy and entropy densities in the
$T$-$\mu_B$ plane and are compared with the crossover line for the QCD chiral
transition as well as with experimental results on freeze-out parameters in
heavy ion collisions. These coefficients also provide estimates for the
location of a possible critical point. We argue that results on sixth order
expansion coefficients disfavor the existence of a critical point in the QCD
phase diagram for $\mu_B/T\le 2$ and $T/T_c(\mu_B=0) > 0.9$.
