Average Error: 61.0 → 60.1
Time: 26.5s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\log \left(\sqrt{e^{\cos^{-1} \left({\left(e^{\log \left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\log \left(\sqrt{e^{\cos^{-1} \left({\left(e^{\log \left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt{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)
double f(double a) {
        double r2725 = a;
        double r2726 = cosh(r2725);
        double r2727 = r2725 * r2725;
        double r2728 = fmod(r2726, r2727);
        double r2729 = log1p(r2725);
        double r2730 = pow(r2728, r2729);
        double r2731 = acos(r2730);
        return r2731;
}
↓
double f(double a) {
        double r2732 = a;
        double r2733 = cosh(r2732);
        double r2734 = r2732 * r2732;
        double r2735 = fmod(r2733, r2734);
        double r2736 = log(r2735);
        double r2737 = exp(r2736);
        double r2738 = log1p(r2732);
        double r2739 = pow(r2737, r2738);
        double r2740 = acos(r2739);
        double r2741 = exp(r2740);
        double r2742 = sqrt(r2741);
        double r2743 = log(r2742);
        double r2744 = exp(r2735);
        double r2745 = log(r2744);
        double r2746 = pow(r2745, r2738);
        double r2747 = acos(r2746);
        double r2748 = exp(r2747);
        double r2749 = sqrt(r2748);
        double r2750 = log(r2749);
        double r2751 = r2743 + r2750;
        return r2751;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.0

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

  7. Applied add-sqr-sqrt60.2

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

  8. Applied log-prod60.2

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

  10. Applied add-exp-log60.2

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

  11. Simplified60.1

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

  12. Final simplification60.1

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))