  Semiconductor quantum dot spin qubits are promising candidates for quantum
computing. In these systems, the dynamically corrected gates offer considerable
reduction of gate errors and are therefore of great interest both theoretically
and experimentally. They are, however, designed under the static-noise model
and may be considered as low-frequency filters. In this work, we perform a
comprehensive theoretical study of the response of a type of dynamically
corrected gates, namely the {\sc supcode} for singlet-triplet qubits, to
realistic $1/f$ noises with frequency spectra $1/\omega^\alpha$. Through
randomized benchmarking, we have found that {\sc supcode} offers improvement of
the gate fidelity for $\alpha\gtrsim1$ and the improvement becomes
exponentially more pronounced with the increase of the noise exponent in the
range $1\lesssim\alpha\leq3$ studied. On the other hand, for small $\alpha$,
{\sc supcode} will not offer any improvement. The $\delta J$-{\sc supcode},
specifically designed for systems where the nuclear noise is absent, is found
to offer additional error reduction than the full {\sc supcode} for charge
noises. The computed filter transfer functions of the {\sc supcode} gates are
also presented.
