Average Error: 34.3 → 34.3
Time: 1.1m
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[\sqrt{\sqrt{e^{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}} \cdot e^{\log \left(\sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right) \cdot 3}\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
\sqrt{\sqrt{e^{\log \left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}} \cdot e^{\log \left(\sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right) \cdot 3}
double f(double a, double c) {
        double r15066 = c;
        double r15067 = cosh(r15066);
        double r15068 = a;
        double r15069 = log1p(r15068);
        double r15070 = fmod(r15067, r15069);
        return r15070;
}
↓
double f(double a, double c) {
        double r15071 = c;
        double r15072 = cosh(r15071);
        double r15073 = a;
        double r15074 = log1p(r15073);
        double r15075 = fmod(r15072, r15074);
        double r15076 = log(r15075);
        double r15077 = exp(r15076);
        double r15078 = sqrt(r15077);
        double r15079 = sqrt(r15078);
        double r15080 = sqrt(r15075);
        double r15081 = sqrt(r15080);
        double r15082 = log(r15081);
        double r15083 = 3.0;
        double r15084 = r15082 * r15083;
        double r15085 = exp(r15084);
        double r15086 = r15079 * r15085;
        return r15086;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.3

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

  3. Applied add-sqr-sqrt34.3

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

  5. Applied add-sqr-sqrt34.3

    \[\leadsto \sqrt{\color{blue}{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)} \cdot \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}} \cdot \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]

  6. Applied sqrt-prod34.3

    \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\right)} \cdot \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]

  7. Applied associate-*l*34.3

    \[\leadsto \color{blue}{\sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot \left(\sqrt{\sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}} \cdot \sqrt{\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\right)}\]

  8. Simplified34.3

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

  10. Applied add-exp-log34.3

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

  11. Applied pow-exp34.3

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

  13. Applied add-exp-log34.3

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

  14. Final simplification34.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  (fmod (cosh c) (log1p a)))