Average Error: 34.6 → 34.6
Time: 13.6s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[{e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)}\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
{e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)}
double f(double a, double c) {
        double r14543 = c;
        double r14544 = cosh(r14543);
        double r14545 = a;
        double r14546 = log1p(r14545);
        double r14547 = fmod(r14544, r14546);
        return r14547;
}
↓
double f(double a, double c) {
        double r14548 = exp(1.0);
        double r14549 = c;
        double r14550 = cosh(r14549);
        double r14551 = a;
        double r14552 = log1p(r14551);
        double r14553 = fmod(r14550, r14552);
        double r14554 = log(r14553);
        double r14555 = 2.0;
        double r14556 = r14554 / r14555;
        double r14557 = pow(r14548, r14556);
        double r14558 = r14557 * r14557;
        return r14558;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.6

    \[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  2. Using strategy rm

  3. Applied add-exp-log34.6

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

  5. Applied pow134.6

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

  6. Applied log-pow34.6

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

  7. Applied exp-prod34.6

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

  8. Simplified34.6

    \[\leadsto {\color{blue}{e}}^{\left(\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt34.6

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

  12. Applied sqr-pow34.6

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

  13. Simplified34.6

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

  14. Simplified34.6

    \[\leadsto {e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)} \cdot \color{blue}{{e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)}}\]

  15. Final simplification34.6

    \[\leadsto {e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}{2}\right)}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  :precision binary64
  (fmod (cosh c) (log1p a)))