Average Error: 60.5 → 59.4
Time: 45.2s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\left(\log \left(\sqrt{e^{\sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}}}\right) + \log \left(\sqrt{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]
double f(double a) {
        double r100836 = a;
        double r100837 = cosh(r100836);
        double r100838 = r100836 * r100836;
        double r100839 = fmod(r100837, r100838);
        double r100840 = log1p(r100836);
        double r100841 = pow(r100839, r100840);
        double r100842 = acos(r100841);
        return r100842;
}
↓
double f(double a) {
        double r100843 = a;
        double r100844 = cosh(r100843);
        double r100845 = r100843 * r100843;
        double r100846 = fmod(r100844, r100845);
        double r100847 = exp(r100846);
        double r100848 = log(r100847);
        double r100849 = sqrt(r100848);
        double r100850 = exp(r100849);
        double r100851 = sqrt(r100850);
        double r100852 = log(r100851);
        double r100853 = sqrt(r100846);
        double r100854 = exp(r100853);
        double r100855 = sqrt(r100854);
        double r100856 = log(r100855);
        double r100857 = r100852 + r100856;
        double r100858 = r100857 * r100853;
        double r100859 = log1p(r100843);
        double r100860 = pow(r100858, r100859);
        double r100861 = acos(r100860);
        return r100861;
}


\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)
↓
\cos^{-1} \left({\left(\left(\log \left(\sqrt{e^{\sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}}}\right) + \log \left(\sqrt{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)

Error

Bits error versus a

Derivation

  1. Initial program 60.5

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

  3. Applied add-log-exp59.6

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

  5. Applied add-sqr-sqrt59.6

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

  6. Applied exp-prod59.6

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

  7. Applied log-pow59.3

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

  9. Applied add-sqr-sqrt59.3

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

  10. Applied log-prod59.3

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

  12. Applied add-log-exp59.4

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

  13. Final simplification59.4

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))