Average Error: 60.9 → 60.0
Time: 49.7s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)} \cdot \frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)} \cdot \frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
double f(double a) {
        double r17637 = a;
        double r17638 = cosh(r17637);
        double r17639 = r17637 * r17637;
        double r17640 = fmod(r17638, r17639);
        double r17641 = log1p(r17637);
        double r17642 = pow(r17640, r17641);
        double r17643 = acos(r17642);
        return r17643;
}
↓
double f(double a) {
        double r17644 = a;
        double r17645 = cosh(r17644);
        double r17646 = 2.0;
        double r17647 = pow(r17644, r17646);
        double r17648 = fmod(r17645, r17647);
        double r17649 = exp(r17648);
        double r17650 = sqrt(r17649);
        double r17651 = log(r17650);
        double r17652 = log(r17649);
        double r17653 = sqrt(r17652);
        double r17654 = sqrt(r17648);
        double r17655 = r17654 / r17646;
        double r17656 = r17653 * r17655;
        double r17657 = r17651 + r17656;
        double r17658 = log1p(r17644);
        double r17659 = pow(r17657, r17658);
        double r17660 = acos(r17659);
        return r17660;
}

Error

Bits error versus a

Derivation

  1. Initial program 60.9

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp60.0

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  4. Simplified60.0

    \[\leadsto \cos^{-1} \left({\left(\log \color{blue}{\left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt60.1

    \[\leadsto \cos^{-1} \left({\left(\log \color{blue}{\left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}} \cdot \sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  7. Applied log-prod60.0

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt60.0

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \log \left(\sqrt{e^{\color{blue}{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)} \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  10. Applied exp-prod60.0

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \log \left(\sqrt{\color{blue}{{\left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)}^{\left(\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)}}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  11. Applied sqrt-pow160.0

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \log \color{blue}{\left({\left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)}^{\left(\frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2}\right)}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  12. Applied log-pow59.7

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \color{blue}{\frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2} \cdot \log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  13. Simplified60.9

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2} \cdot \color{blue}{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  14. Using strategy rm

  15. Applied add-log-exp60.0

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2} \cdot \sqrt{\color{blue}{\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)}}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

  16. Final simplification60.0

    \[\leadsto \cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}\right) + \sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)} \cdot \frac{\sqrt{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}}{2}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))