Average Error: 61.2 → 60.2
Time: 58.9s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[2 \cdot \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}}\right)\]
\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(\left(\cosh a\right) \bmod \left(a \cdot a\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)
double f(double a) {
        double r15121 = a;
        double r15122 = cosh(r15121);
        double r15123 = r15121 * r15121;
        double r15124 = fmod(r15122, r15123);
        double r15125 = log1p(r15121);
        double r15126 = pow(r15124, r15125);
        double r15127 = acos(r15126);
        return r15127;
}
↓
double f(double a) {
        double r15128 = 2.0;
        double r15129 = a;
        double r15130 = cosh(r15129);
        double r15131 = r15129 * r15129;
        double r15132 = fmod(r15130, r15131);
        double r15133 = log1p(r15129);
        double r15134 = pow(r15132, r15133);
        double r15135 = acos(r15134);
        double r15136 = exp(r15135);
        double r15137 = cbrt(r15136);
        double r15138 = log(r15137);
        double r15139 = r15128 * r15138;
        double r15140 = exp(r15132);
        double r15141 = log(r15140);
        double r15142 = pow(r15141, r15133);
        double r15143 = acos(r15142);
        double r15144 = exp(r15143);
        double r15145 = cbrt(r15144);
        double r15146 = log(r15145);
        double r15147 = r15139 + r15146;
        return r15147;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.2

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

  3. Applied add-log-exp61.2

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

  5. Applied add-cube-cbrt61.2

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

  6. Applied log-prod61.2

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

  7. Simplified61.2

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

  9. Applied add-log-exp60.2

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

  10. Final simplification60.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))