Average Error: 61.2 → 60.2
Time: 58.5s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)
double f(double a) {
        double r14842 = a;
        double r14843 = cosh(r14842);
        double r14844 = r14842 * r14842;
        double r14845 = fmod(r14843, r14844);
        double r14846 = log1p(r14842);
        double r14847 = pow(r14845, r14846);
        double r14848 = acos(r14847);
        return r14848;
}
↓
double f(double a) {
        double r14849 = 2.0;
        double r14850 = a;
        double r14851 = cosh(r14850);
        double r14852 = pow(r14850, r14849);
        double r14853 = fmod(r14851, r14852);
        double r14854 = log1p(r14850);
        double r14855 = pow(r14853, r14854);
        double r14856 = acos(r14855);
        double r14857 = exp(r14856);
        double r14858 = cbrt(r14857);
        double r14859 = log(r14858);
        double r14860 = r14849 * r14859;
        double r14861 = exp(r14853);
        double r14862 = log(r14861);
        double r14863 = pow(r14862, r14854);
        double r14864 = acos(r14863);
        double r14865 = exp(r14864);
        double r14866 = cbrt(r14865);
        double r14867 = log(r14866);
        double r14868 = r14860 + r14867;
        return r14868;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.2

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

  3. Applied add-log-exp60.2

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

  5. Applied add-log-exp60.2

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

  6. Simplified61.2

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

  8. Applied add-cube-cbrt61.2

    \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}} \cdot \sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) \cdot \sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)}\]

  9. Applied log-prod61.2

    \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}} \cdot \sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)}\]

  10. Simplified61.2

    \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
  11. Using strategy rm

  12. Applied add-log-exp60.2

    \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]

  13. Final simplification60.2

    \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))