Average Error: 34.3 → 34.1
Time: 36.8s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[\left(\left(\sqrt[3]{\frac{1}{8} \cdot {\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right)\right)}^{3}}\right)\right)}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
\left(\left(\sqrt[3]{\frac{1}{8} \cdot {\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right)\right)}^{3}}\right)\right)}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
double f(double a, double c) {
        double r18692 = c;
        double r18693 = cosh(r18692);
        double r18694 = a;
        double r18695 = log1p(r18694);
        double r18696 = fmod(r18693, r18695);
        return r18696;
}
↓
double f(double a, double c) {
        double r18697 = 0.125;
        double r18698 = c;
        double r18699 = exp(r18698);
        double r18700 = -r18698;
        double r18701 = exp(r18700);
        double r18702 = r18699 + r18701;
        double r18703 = log1p(r18702);
        double r18704 = 3.0;
        double r18705 = pow(r18703, r18704);
        double r18706 = cbrt(r18705);
        double r18707 = expm1(r18706);
        double r18708 = pow(r18707, r18704);
        double r18709 = r18697 * r18708;
        double r18710 = cbrt(r18709);
        double r18711 = a;
        double r18712 = log1p(r18711);
        double r18713 = fmod(r18710, r18712);
        return r18713;
}

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-cbrt-cube34.3

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

  4. Simplified34.3

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

  5. Taylor expanded around inf 34.3

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

  7. Applied expm1-log1p-u34.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{1}{8} \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right)\right)\right)}}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  8. Using strategy rm

  9. Applied add-cbrt-cube34.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{1}{8} \cdot {\left(\mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right) \cdot \mathsf{log1p}\left(e^{c} + e^{-c}\right)\right) \cdot \mathsf{log1p}\left(e^{c} + e^{-c}\right)}}\right)\right)}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  10. Simplified34.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{1}{8} \cdot {\left(\mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right)\right)}^{3}}}\right)\right)}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  11. Final simplification34.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{1}{8} \cdot {\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(e^{c} + e^{-c}\right)\right)}^{3}}\right)\right)}^{3}}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  :precision binary64
  (fmod (cosh c) (log1p a)))