Average Error: 60.4 → 59.2
Time: 46.0s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\log \left(e^{\cos^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)\right)}^{\left(\log_* (1 + a)\right)}\right)}\right)\]
double f(double a) {
        double r599515 = a;
        double r599516 = cosh(r599515);
        double r599517 = r599515 * r599515;
        double r599518 = fmod(r599516, r599517);
        double r599519 = log1p(r599515);
        double r599520 = pow(r599518, r599519);
        double r599521 = acos(r599520);
        return r599521;
}
↓
double f(double a) {
        double r599522 = a;
        double r599523 = cosh(r599522);
        double r599524 = r599522 * r599522;
        double r599525 = fmod(r599523, r599524);
        double r599526 = sqrt(r599525);
        double r599527 = exp(r599526);
        double r599528 = cbrt(r599527);
        double r599529 = log(r599528);
        double r599530 = r599529 + r599529;
        double r599531 = r599529 + r599530;
        double r599532 = r599526 * r599531;
        double r599533 = log1p(r599522);
        double r599534 = pow(r599532, r599533);
        double r599535 = acos(r599534);
        double r599536 = exp(r599535);
        double r599537 = log(r599536);
        return r599537;
}


\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)
↓
\log \left(e^{\cos^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)\right)}^{\left(\log_* (1 + a)\right)}\right)}\right)

Error

Bits error versus a

Derivation

  1. Initial program 60.4

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

  3. Applied add-log-exp59.5

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

  5. Applied add-log-exp59.5

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

  7. Applied add-sqr-sqrt59.5

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

  8. Applied exp-prod59.5

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

  9. Applied log-pow59.2

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

  11. Applied add-cube-cbrt59.2

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

  12. Applied log-prod59.2

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

  13. Simplified59.2

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

  14. Final simplification59.2

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))