Average Error: 61.2 → 60.3
Time: 53.2s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \left(2 \cdot \log \left(\sqrt[3]{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
\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 \cdot a\right)\right)}}\right) + \left(2 \cdot \log \left(\sqrt[3]{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
double f(double a) {
        double r19819 = a;
        double r19820 = cosh(r19819);
        double r19821 = r19819 * r19819;
        double r19822 = fmod(r19820, r19821);
        double r19823 = log1p(r19819);
        double r19824 = pow(r19822, r19823);
        double r19825 = acos(r19824);
        return r19825;
}
↓
double f(double a) {
        double r19826 = a;
        double r19827 = cosh(r19826);
        double r19828 = r19826 * r19826;
        double r19829 = fmod(r19827, r19828);
        double r19830 = exp(r19829);
        double r19831 = sqrt(r19830);
        double r19832 = log(r19831);
        double r19833 = 2.0;
        double r19834 = cbrt(r19831);
        double r19835 = log(r19834);
        double r19836 = r19833 * r19835;
        double r19837 = r19836 + r19835;
        double r19838 = r19832 + r19837;
        double r19839 = log1p(r19826);
        double r19840 = pow(r19838, r19839);
        double r19841 = acos(r19840);
        return r19841;
}

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-sqr-sqrt60.3

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

  6. Applied log-prod60.3

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

  8. Applied add-cube-cbrt60.3

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

  9. Applied log-prod60.3

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

  10. Simplified60.3

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

  11. Final simplification60.3

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))