Average Error: 61.0 → 60.2
Time: 29.6s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(2 \cdot \log \left(\sqrt[3]{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(2 \cdot \log \left(\sqrt[3]{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)
double f(double a) {
        double r3045 = a;
        double r3046 = cosh(r3045);
        double r3047 = r3045 * r3045;
        double r3048 = fmod(r3046, r3047);
        double r3049 = log1p(r3045);
        double r3050 = pow(r3048, r3049);
        double r3051 = acos(r3050);
        return r3051;
}
↓
double f(double a) {
        double r3052 = 2.0;
        double r3053 = a;
        double r3054 = cosh(r3053);
        double r3055 = r3053 * r3053;
        double r3056 = fmod(r3054, r3055);
        double r3057 = exp(r3056);
        double r3058 = log(r3057);
        double r3059 = log1p(r3053);
        double r3060 = pow(r3058, r3059);
        double r3061 = acos(r3060);
        double r3062 = exp(r3061);
        double r3063 = cbrt(r3062);
        double r3064 = log(r3063);
        double r3065 = r3052 * r3064;
        double r3066 = cbrt(r3057);
        double r3067 = log(r3066);
        double r3068 = r3052 * r3067;
        double r3069 = r3068 + r3067;
        double r3070 = pow(r3069, r3059);
        double r3071 = acos(r3070);
        double r3072 = exp(r3071);
        double r3073 = cbrt(r3072);
        double r3074 = log(r3073);
        double r3075 = r3065 + r3074;
        return r3075;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.0

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp60.1

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  4. Using strategy rm

  5. Applied add-log-exp60.1

    \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt60.1

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

  8. Applied log-prod60.1

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

  9. Simplified60.1

    \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)} + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
  10. Using strategy rm

  11. Applied add-cube-cbrt60.2

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

  12. Applied log-prod60.2

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

  13. Simplified60.2

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

  14. Final simplification60.2

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

Reproduce

herbie shell --seed 2020034 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))