\[\log_* (1 + {\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)})\]
Test:
(log1p (pow (sinh b) (atan2 a (sin a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 9.3 s
Input Error: 8.8
Output Error: 8.7
Log: ⚲
Profile: 🕒
\(\log_* (1 + {\left({\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}\right)}^3}\right)}^3\right)}^3)\)
  1. Started with
    \[\log_* (1 + {\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)})\]
    8.8
  2. Using strategy rm
    8.8
  3. Applied add-cube-cbrt to get
    \[\log_* (1 + \color{red}{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}) \leadsto \log_* (1 + \color{blue}{{\left(\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}^3})\]
    8.8
  4. Using strategy rm
    8.8
  5. Applied add-cube-cbrt to get
    \[\log_* (1 + {\color{red}{\left(\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}}^3) \leadsto \log_* (1 + {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}\right)}^3\right)}}^3)\]
    8.8
  6. Using strategy rm
    8.8
  7. Applied add-cube-cbrt to get
    \[\log_* (1 + {\left({\left(\sqrt[3]{\color{red}{\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}}\right)}^3\right)}^3) \leadsto \log_* (1 + {\left({\left(\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}\right)}^3}}\right)}^3\right)}^3)\]
    8.7

Original test:


(lambda ((a default) (b default))
  #:name "(log1p (pow (sinh b) (atan2 a (sin a))))"
  (log1p (pow (sinh b) (atan2 a (sin a)))))