As quantum computers grow more capable, simulating them on conventional hardware becomes more challenging yet more attractive since this helps in design and verification. Some quantum algorithms and circuits are amenable to surprisingly efficient simulation, and this makes hard-to-simulate computations particularly valuable. For such circuits, we develop accurate massively-parallel simulation with dramatic speedups over earlier methods on 42- and 45-qubit circuits. We propose two ways to trade circuit fidelity for computational speedups, so as to match the error rate of any quantum computer. Using Google Cloud, we simulate approximate sampling from the output of a circuit with 7 × 8 qubits and depth 42 with fidelity 0.5% at an estimated cost of $35K.