  Composite pulse sequences, which produce arbitrary pre-defined rotations of a
qubit on the Bloch sphere, are presented. The composite sequences contain up to
17 pulses and can compensate up to eight orders of experimental errors in the
pulse amplitude and the pulse duration. Composite sequences for three basic
quantum gates -- X (NOT), Hadamard and arbitrary rotation -- are derived. Three
classes of composite sequences are presented -- one symmetric and two
asymmetric. They contain as their lowest members two well-known composite
sequences -- the three-pulse symmetric SCROFULOUS pulse and the four-pulse
asymmetric BB1 pulse, which compensate first and second-order errors,
respectively. The shorter sequences are derived analytically, and the longer
ones numerically (instead by nesting and concatenation, as mostly done
hitherto). Consequently, the composite sequences derived here match or
outperform the existing ones in terms of either speed or accuracy, or both. For
example, we derive a second-order composite sequence, which is faster (by about
13\%) than the famous BB1 sequence. For higher-order sequences the speed-up
becomes much more pronounced. This is important for quantum information
processing as the sequences derived here provide more options for finding the
sweet spot between ultrahigh fidelity and high speed.
