Average Error: 34.8 → 34.8
Time: 13.8s
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 r10313 = c;
        double r10314 = cosh(r10313);
        double r10315 = a;
        double r10316 = log1p(r10315);
        double r10317 = fmod(r10314, r10316);
        return r10317;
}
↓
double f(double a, double c) {
        double r10318 = exp(1.0);
        double r10319 = c;
        double r10320 = cosh(r10319);
        double r10321 = a;
        double r10322 = log1p(r10321);
        double r10323 = fmod(r10320, r10322);
        double r10324 = log(r10323);
        double r10325 = 2.0;
        double r10326 = r10324 / r10325;
        double r10327 = pow(r10318, r10326);
        double r10328 = r10327 * r10327;
        return r10328;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.8

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

  3. Applied add-exp-log34.8

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

  5. Applied pow134.8

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

  6. Applied log-pow34.8

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

  7. Applied exp-prod34.8

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

  8. Simplified34.8

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

  10. Applied sqr-pow34.8

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

  11. Final simplification34.8

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  :precision binary64
  (fmod (cosh c) (log1p a)))