Average Error: 61.1 → 60.1
Time: 29.5s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)} \cdot {\left(\log \left(\sqrt{e^{\sqrt[3]{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
\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(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\mathsf{log1p}\left(a\right)\right)} \cdot {\left(\log \left(\sqrt{e^{\sqrt[3]{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
double f(double a) {
        double r6130 = a;
        double r6131 = cosh(r6130);
        double r6132 = r6130 * r6130;
        double r6133 = fmod(r6131, r6132);
        double r6134 = log1p(r6130);
        double r6135 = pow(r6133, r6134);
        double r6136 = acos(r6135);
        return r6136;
}
↓
double f(double a) {
        double r6137 = a;
        double r6138 = cosh(r6137);
        double r6139 = r6137 * r6137;
        double r6140 = fmod(r6138, r6139);
        double r6141 = cbrt(r6140);
        double r6142 = log1p(r6137);
        double r6143 = pow(r6141, r6142);
        double r6144 = exp(r6140);
        double r6145 = log(r6144);
        double r6146 = cbrt(r6145);
        double r6147 = r6146 * r6141;
        double r6148 = exp(r6147);
        double r6149 = sqrt(r6148);
        double r6150 = log(r6149);
        double r6151 = r6141 * r6141;
        double r6152 = exp(r6151);
        double r6153 = sqrt(r6152);
        double r6154 = log(r6153);
        double r6155 = r6150 + r6154;
        double r6156 = pow(r6155, r6142);
        double r6157 = r6143 * r6156;
        double r6158 = acos(r6157);
        return r6158;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.1

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  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-cube-cbrt60.2

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

  6. Applied exp-prod60.2

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

  7. Applied log-pow60.1

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

  8. Applied unpow-prod-down60.1

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

  10. Applied add-sqr-sqrt60.1

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

  11. Applied log-prod60.1

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

  13. Applied add-log-exp60.1

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

  14. Final simplification60.1

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))