  In this paper, we investigate fast algorithms in the small fraction order
regime to approximate the Caputo derivative $^C_0D_t^\alpha u(t)$ when $\alpha$
is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on
the sum-of-exponential approximation to reduce the cost of evaluating the
history part. FIR is the numerical scheme originally proposed in [16], and FIDR
is an alternative scheme proposed in [26], and we show that the latter is
superior when $\alpha$ is small. With quantitative estimates, we prove that
given a certain error threshold, the computational cost of evaluating the
history part of the Caputo derivative can be decreased as $\alpha$ gets small.
Hence, only minimal cost for the fast evaluation is required in the small
$\alpha$ regime, which matches prevailing protocols in engineering practice. We
also present improved stability and error analysis of FIDR for solving linear
fractional diffusion equations, which achieves clear dependence of the error
bound on the fraction order $\alpha$. Finally, we carry out systematic
numerical studies for the performances of both FIR and FIDR schemes, where we
explore the trade-off between accuracy and efficiency when $\alpha$ is small.
