Average Error: 61.2 → 60.2
Time: 56.9s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\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 \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)
double f(double a) {
        double r13442 = a;
        double r13443 = cosh(r13442);
        double r13444 = r13442 * r13442;
        double r13445 = fmod(r13443, r13444);
        double r13446 = log1p(r13442);
        double r13447 = pow(r13445, r13446);
        double r13448 = acos(r13447);
        return r13448;
}
↓
double f(double a) {
        double r13449 = 2.0;
        double r13450 = a;
        double r13451 = cosh(r13450);
        double r13452 = r13450 * r13450;
        double r13453 = fmod(r13451, r13452);
        double r13454 = log1p(r13450);
        double r13455 = pow(r13453, r13454);
        double r13456 = acos(r13455);
        double r13457 = exp(r13456);
        double r13458 = cbrt(r13457);
        double r13459 = log(r13458);
        double r13460 = r13449 * r13459;
        double r13461 = exp(r13453);
        double r13462 = log(r13461);
        double r13463 = pow(r13462, r13454);
        double r13464 = acos(r13463);
        double r13465 = exp(r13464);
        double r13466 = cbrt(r13465);
        double r13467 = log(r13466);
        double r13468 = r13460 + r13467;
        return r13468;
}

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-exp61.2

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

  5. Applied add-cube-cbrt61.2

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

  6. Applied log-prod61.2

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

  7. Simplified61.2

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

  9. Applied add-log-exp60.2

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

  10. Final simplification60.2

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

Reproduce

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