Average Error: 60.4 → 59.2
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[(\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) \cdot \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) + \left(-\sin^{-1} \left({\left(\log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\right))_*\]

Error

Bits error versus a

Derivation

  1. Initial program 60.4

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp59.5

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\log_* (1 + a)\right)}\right)\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt59.5

    \[\leadsto \cos^{-1} \left({\left(\log \left(e^{\color{blue}{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  6. Applied exp-prod59.5

    \[\leadsto \cos^{-1} \left({\left(\log \color{blue}{\left({\left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)}^{\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  7. Applied log-pow59.2

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}}^{\left(\log_* (1 + a)\right)}\right)\]
  8. Using strategy rm

  9. Applied acos-asin59.2

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt59.2

    \[\leadsto \frac{\pi}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}} - \sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  12. Applied add-cube-cbrt59.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}} - \sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  13. Applied times-frac59.2

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}} - \sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  14. Applied fma-neg59.2

    \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right) \cdot \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) + \left(-\sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\right))_*}\]

  15. Simplified59.2

    \[\leadsto (\color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right)} \cdot \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) + \left(-\sin^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\right))_*\]

  16. Final simplification59.2

    \[\leadsto (\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}} \cdot \frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) \cdot \left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{2}}\right) + \left(-\sin^{-1} \left({\left(\log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\right))_*\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))