Average Error: 61.3 → 59.0
Time: 56.0s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 1.542338768776221979718923094404481642848 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(2 \cdot \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \end{array}\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\begin{array}{l}
\mathbf{if}\;a \le 1.542338768776221979718923094404481642848 \cdot 10^{-162}:\\
\;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(2 \cdot \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right)\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double a) {
        double r23446 = a;
        double r23447 = cosh(r23446);
        double r23448 = r23446 * r23446;
        double r23449 = fmod(r23447, r23448);
        double r23450 = log1p(r23446);
        double r23451 = pow(r23449, r23450);
        double r23452 = acos(r23451);
        return r23452;
}
↓
double f(double a) {
        double r23453 = a;
        double r23454 = 1.542338768776222e-162;
        bool r23455 = r23453 <= r23454;
        double r23456 = cosh(r23453);
        double r23457 = r23453 * r23453;
        double r23458 = fmod(r23456, r23457);
        double r23459 = 2.0;
        double r23460 = log1p(r23453);
        double r23461 = exp(r23460);
        double r23462 = cbrt(r23461);
        double r23463 = log(r23462);
        double r23464 = r23459 * r23463;
        double r23465 = r23464 + r23463;
        double r23466 = pow(r23458, r23465);
        double r23467 = acos(r23466);
        double r23468 = exp(r23458);
        double r23469 = log(r23468);
        double r23470 = pow(r23469, r23460);
        double r23471 = acos(r23470);
        double r23472 = r23455 ? r23467 : r23471;
        return r23472;
}

Error

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if a < 1.542338768776222e-162

    1. Initial program 64.0

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

    3. Applied add-log-exp61.3

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

    5. Applied add-cube-cbrt61.3

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

    6. Applied log-prod61.3

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

    7. Simplified61.3

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

    if 1.542338768776222e-162 < a

    1. Initial program 58.8

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

    3. Applied add-log-exp56.9

      \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification59.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.542338768776221979718923094404481642848 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(2 \cdot \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))