Average Error: 34.3 → 34.1
Time: 37.2s
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 r19162 = c;
        double r19163 = cosh(r19162);
        double r19164 = a;
        double r19165 = log1p(r19164);
        double r19166 = fmod(r19163, r19165);
        return r19166;
}
↓
double f(double a, double c) {
        double r19167 = 0.125;
        double r19168 = c;
        double r19169 = exp(r19168);
        double r19170 = -r19168;
        double r19171 = exp(r19170);
        double r19172 = r19169 + r19171;
        double r19173 = log1p(r19172);
        double r19174 = 3.0;
        double r19175 = pow(r19173, r19174);
        double r19176 = cbrt(r19175);
        double r19177 = expm1(r19176);
        double r19178 = pow(r19177, r19174);
        double r19179 = r19167 * r19178;
        double r19180 = cbrt(r19179);
        double r19181 = a;
        double r19182 = log1p(r19181);
        double r19183 = fmod(r19180, r19182);
        return r19183;
}

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)))