Average Error: 34.8 → 34.8
Time: 35.5s
Precision: 64
\[\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
↓
\[e^{\log \left(\left(\left(c \cdot c\right) \cdot \left(\frac{1}{2} + \left(\left(\sqrt[3]{c \cdot c} \cdot \sqrt[3]{\frac{1}{24}}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) + 1\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]
\left(\left(\cosh c\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)
↓
e^{\log \left(\left(\left(c \cdot c\right) \cdot \left(\frac{1}{2} + \left(\left(\sqrt[3]{c \cdot c} \cdot \sqrt[3]{\frac{1}{24}}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) + 1\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}
double f(double a, double c) {
        double r638658 = c;
        double r638659 = cosh(r638658);
        double r638660 = a;
        double r638661 = log1p(r638660);
        double r638662 = fmod(r638659, r638661);
        return r638662;
}
↓
double f(double a, double c) {
        double r638663 = c;
        double r638664 = r638663 * r638663;
        double r638665 = 0.5;
        double r638666 = cbrt(r638664);
        double r638667 = 0.041666666666666664;
        double r638668 = cbrt(r638667);
        double r638669 = r638666 * r638668;
        double r638670 = r638667 * r638664;
        double r638671 = cbrt(r638670);
        double r638672 = r638669 * r638671;
        double r638673 = r638672 * r638671;
        double r638674 = r638665 + r638673;
        double r638675 = r638664 * r638674;
        double r638676 = 1.0;
        double r638677 = r638675 + r638676;
        double r638678 = a;
        double r638679 = log1p(r638678);
        double r638680 = fmod(r638677, r638679);
        double r638681 = log(r638680);
        double r638682 = exp(r638681);
        return r638682;
}

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. Taylor expanded around 0 34.9

    \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot {c}^{2} + \left(\frac{1}{24} \cdot {c}^{4} + 1\right)\right)} \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  3. Simplified34.8

    \[\leadsto \left(\color{blue}{\left(1 + \left(\frac{1}{24} \cdot \left(c \cdot c\right) + \frac{1}{2}\right) \cdot \left(c \cdot c\right)\right)} \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt34.8

    \[\leadsto \left(\left(1 + \left(\color{blue}{\left(\sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)} \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}} + \frac{1}{2}\right) \cdot \left(c \cdot c\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  6. Taylor expanded around 0 34.8

    \[\leadsto \left(\left(1 + \left(\left(\color{blue}{\left(\sqrt[3]{\frac{1}{24}} \cdot {\left({c}^{2}\right)}^{\frac{1}{3}}\right)} \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)} + \frac{1}{2}\right) \cdot \left(c \cdot c\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]

  7. Simplified34.8

    \[\leadsto \left(\left(1 + \left(\left(\color{blue}{\left(\sqrt[3]{c \cdot c} \cdot \sqrt[3]{\frac{1}{24}}\right)} \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)} + \frac{1}{2}\right) \cdot \left(c \cdot c\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)\]
  8. Using strategy rm

  9. Applied add-exp-log34.8

    \[\leadsto \color{blue}{e^{\log \left(\left(1 + \left(\left(\left(\sqrt[3]{c \cdot c} \cdot \sqrt[3]{\frac{1}{24}}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)} + \frac{1}{2}\right) \cdot \left(c \cdot c\right)\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}}\]

  10. Final simplification34.8

    \[\leadsto e^{\log \left(\left(\left(c \cdot c\right) \cdot \left(\frac{1}{2} + \left(\left(\sqrt[3]{c \cdot c} \cdot \sqrt[3]{\frac{1}{24}}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) \cdot \sqrt[3]{\frac{1}{24} \cdot \left(c \cdot c\right)}\right) + 1\right) \bmod \left(\mathsf{log1p}\left(a\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a c)
  :name "Random Jason Timeout Test 004"
  (fmod (cosh c) (log1p a)))