Average Error: 61.3 → 59.0
Time: 55.7s
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 r23360 = a;
        double r23361 = cosh(r23360);
        double r23362 = r23360 * r23360;
        double r23363 = fmod(r23361, r23362);
        double r23364 = log1p(r23360);
        double r23365 = pow(r23363, r23364);
        double r23366 = acos(r23365);
        return r23366;
}
↓
double f(double a) {
        double r23367 = a;
        double r23368 = 1.542338768776222e-162;
        bool r23369 = r23367 <= r23368;
        double r23370 = cosh(r23367);
        double r23371 = r23367 * r23367;
        double r23372 = fmod(r23370, r23371);
        double r23373 = 2.0;
        double r23374 = log1p(r23367);
        double r23375 = exp(r23374);
        double r23376 = cbrt(r23375);
        double r23377 = log(r23376);
        double r23378 = r23373 * r23377;
        double r23379 = r23378 + r23377;
        double r23380 = pow(r23372, r23379);
        double r23381 = acos(r23380);
        double r23382 = exp(r23372);
        double r23383 = log(r23382);
        double r23384 = pow(r23383, r23374);
        double r23385 = acos(r23384);
        double r23386 = r23369 ? r23381 : r23385;
        return r23386;
}

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))))