Average Error: 34.4 → 34.3
Time: 40.4s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[\left(\left(\sqrt[3]{\sqrt{{\left(\cosh c\right)}^{3}}} \cdot \left(\sqrt{e^{c} + e^{-c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
\left(\left(\sqrt[3]{\sqrt{{\left(\cosh c\right)}^{3}}} \cdot \left(\sqrt{e^{c} + e^{-c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
double f(double a, double c) {
        double r20705 = c;
        double r20706 = cosh(r20705);
        double r20707 = a;
        double r20708 = log1p(r20707);
        double r20709 = fmod(r20706, r20708);
        return r20709;
}
↓
double f(double a, double c) {
        double r20710 = c;
        double r20711 = cosh(r20710);
        double r20712 = 3.0;
        double r20713 = pow(r20711, r20712);
        double r20714 = sqrt(r20713);
        double r20715 = cbrt(r20714);
        double r20716 = exp(r20710);
        double r20717 = -r20710;
        double r20718 = exp(r20717);
        double r20719 = r20716 + r20718;
        double r20720 = sqrt(r20719);
        double r20721 = 0.5;
        double r20722 = sqrt(r20721);
        double r20723 = r20720 * r20722;
        double r20724 = r20715 * r20723;
        double r20725 = a;
        double r20726 = log1p(r20725);
        double r20727 = fmod(r20724, r20726);
        return r20727;
}

Error

Bits error versus a

Bits error versus c

Derivation

  1. Initial program 34.4

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

  3. Applied add-cbrt-cube34.4

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

  4. Simplified34.4

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

  6. Applied add-sqr-sqrt34.4

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

  7. Applied cbrt-prod34.4

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

  8. Taylor expanded around inf 34.3

    \[\leadsto \left(\left(\sqrt[3]{\sqrt{{\left(\cosh c\right)}^{3}}} \cdot \color{blue}{\left(\sqrt{e^{c} + e^{-c}} \cdot \sqrt{\frac{1}{2}}\right)}\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  9. Final simplification34.3

    \[\leadsto \left(\left(\sqrt[3]{\sqrt{{\left(\cosh c\right)}^{3}}} \cdot \left(\sqrt{e^{c} + e^{-c}} \cdot \sqrt{\frac{1}{2}}\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  :precision binary64
  (fmod (cosh c) (log1p a)))