Table 1 Complexity results for the eligible kernel functions

$\frac{t^2-1}{2}-logt$

$O(rlog\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

e.g., [1]

$\frac{1}{2}{(t-\frac{1}{t})}^2$

$O(r^{\frac{2}{3}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[6]

$\frac{t^2-1}{2}+\frac{t^{1-q}-1}{q-1}$, q>1

$O(q{r}^{\frac{q+1}{2q}}log\frac{r}{\epsilon })$

$O(q^2\sqrt{r}log\frac{r}{\epsilon })$

[6]

$\frac{t^2-1}{2}+\frac{t^{1-q}-1}{q(q-1)}-\frac{q-1}{q}(t-1)$, q>1

$O(q{r}^{\frac{q+1}{2q}}log\frac{r}{\epsilon })$

$O(q^2\sqrt{r}log\frac{r}{\epsilon })$

[5]

$\frac{t^2-1}{2}+\frac{e^{\frac{1}{t}}-e}{e}$

$O(\sqrt{r}{(logr)}^{2}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[6]

$\frac{t^2-1}{2}-{\int }_1^t{e}^{\frac{1}{\xi }-1}d\xi$

$O(\sqrt{r}{(logr)}^{2}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[6]

$\frac{t^2-1}{2}+\frac{e^{q(\frac{1}{t}-1)}-q}{q}$, q ≥ 1

$O(q\sqrt{r}log\frac{r}{\epsilon })$

$O(q\sqrt{qr}log\frac{r}{\epsilon })$

[7]

$\frac{t^2-1}{2}-{\int }_1^t{e}^{q(\frac{1}{\xi }-1)}d\xi$, q ≥ 1

$O(q\sqrt{r}log\frac{r}{\epsilon })$

$O(q\sqrt{qr}log\frac{r}{\epsilon })$

[6]

$\frac{t^2-1}{2}+\frac{{(e-1)}^2}{e}\frac{1}{e^t-1}-\frac{e-1}{e}$

$O(r^{\frac{3}{4}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[10]

$8t^2-11t+1+\frac{2}{\sqrt{t}}-4logt$

$O(r^{\frac{5}{6}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[19]

$8t^2-10t+\frac{2}{t^3}$

$O(r^{\frac{5}{8}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[14]

$\frac{t^2-1}{2}+\frac{6}{\pi }tan(\frac{\pi (1-t)}{2+4t})$

$O(r^{\frac{3}{4}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[17]

$\frac{t^2-1}{2}-log(t)+\frac{1}{8}{tan}^2(\frac{\pi (1-t)}{2+4t})$

$O(r^{\frac{2}{3}}log\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[21]

$\frac{p(t^2-1)}{2}+\frac{t^{-pq}-1}{q(q+1)}-\frac{pq(t-1)}{q+1}$, p ≥ 1, q>0

$O(\sqrt{r}logrlog\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[15]

$t+\frac{1}{t}-2$

$O(rlog\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[9]

$t-1+\frac{t^{1-q}-1}{q-1}$, q>1

$O(qrlog\frac{r}{\epsilon })$

$O(q^2\sqrt{r}log\frac{r}{\epsilon })$

[6]

$\frac{t^{p+1}-1}{p+1}-logt$, p∈[0,1]

$O(rlog\frac{r}{\epsilon })$

$O(\sqrt{r}log\frac{r}{\epsilon })$

[18]

$\{\begin{array}{l}\frac{t^{p+1}-1}{p+1}+\frac{t^{1-q}-1}{q-1},t>0,p\in [0,1],q>1\\ \frac{t^{p+1}-1}{p+1}-logt,t>0,p\in [0,1],q=1\end{array}$

$O(q{r}^{\frac{p+q}{q(1+p)}}log\frac{r}{\epsilon })$

$O(q^2\sqrt{r}log\frac{r}{\epsilon })$

[8]