Average Error: 61.3 → 59.0
Time: 54.5s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 1.542338768776221979718923094404481642848 \cdot 10^{-162}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(2 \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}}\right) + \log \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}}\right)\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \end{array}\]
\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 \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}}\right) + \log \left(\sqrt[3]{\sqrt[3]{e^{\mathsf{log1p}\left(a\right)}}}\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 r23276 = a;
        double r23277 = cosh(r23276);
        double r23278 = r23276 * r23276;
        double r23279 = fmod(r23277, r23278);
        double r23280 = log1p(r23276);
        double r23281 = pow(r23279, r23280);
        double r23282 = acos(r23281);
        return r23282;
}
↓
double f(double a) {
        double r23283 = a;
        double r23284 = 1.542338768776222e-162;
        bool r23285 = r23283 <= r23284;
        double r23286 = cosh(r23283);
        double r23287 = r23283 * r23283;
        double r23288 = fmod(r23286, r23287);
        double r23289 = 2.0;
        double r23290 = log1p(r23283);
        double r23291 = exp(r23290);
        double r23292 = cbrt(r23291);
        double r23293 = cbrt(r23292);
        double r23294 = log(r23293);
        double r23295 = r23289 * r23294;
        double r23296 = r23295 + r23294;
        double r23297 = r23289 * r23296;
        double r23298 = log(r23292);
        double r23299 = r23297 + r23298;
        double r23300 = pow(r23288, r23299);
        double r23301 = acos(r23300);
        double r23302 = exp(r23288);
        double r23303 = log(r23302);
        double r23304 = pow(r23303, r23290);
        double r23305 = acos(r23304);
        double r23306 = r23285 ? r23301 : r23305;
        return r23306;
}

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

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

    9. Applied add-cube-cbrt61.3

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

    10. Applied log-prod61.3

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

    11. Simplified61.3

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

    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

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

Reproduce

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