\[\log_* (1 + {\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)})\]
Test:
(log1p (pow (sinh b) (atan2 a (sin a))))
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 9.6 s
Input Error: 5.5
Output Error: 5.4
Log: ⚲
Profile: 🕒
\(\log_* (1 + {\left(e^{\log \left({\left(\sqrt[3]{\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}\right)}^3\right)}\right)}^2)\)
  1. Started with
    \[\log_* (1 + {\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)})\]
    5.5
  2. Using strategy rm
    5.5
  3. Applied add-sqr-sqrt to get
    \[\log_* (1 + \color{red}{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}) \leadsto \log_* (1 + \color{blue}{{\left(\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}^2})\]
    5.5
  4. Using strategy rm
    5.5
  5. Applied add-exp-log to get
    \[\log_* (1 + {\color{red}{\left(\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}}^2) \leadsto \log_* (1 + {\color{blue}{\left(e^{\log \left(\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}\right)}}^2)\]
    5.4
  6. Using strategy rm
    5.4
  7. Applied add-cube-cbrt to get
    \[\log_* (1 + {\left(e^{\log \color{red}{\left(\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}\right)}}\right)}^2) \leadsto \log_* (1 + {\left(e^{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{\left(\sinh b\right)}^{\left(\tan^{-1}_* \frac{a}{\sin a}\right)}}}\right)}^3\right)}}\right)}^2)\]
    5.4
  8. Removed slow pow expressions

Original test:


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