Average Error: 61.2 → 58.7
Time: 46.7s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right) \le 5.372690074837192250710590172430025646122 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}\right)\\ \end{array}\]
\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}\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right) \le 5.372690074837192250710590172430025646122 \cdot 10^{-8}:\\
\;\;\;\;\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}\right)\\

\end{array}
double f(double a) {
        double r17121 = a;
        double r17122 = cosh(r17121);
        double r17123 = r17121 * r17121;
        double r17124 = fmod(r17122, r17123);
        double r17125 = log1p(r17121);
        double r17126 = pow(r17124, r17125);
        double r17127 = acos(r17126);
        return r17127;
}
↓
double f(double a) {
        double r17128 = a;
        double r17129 = cosh(r17128);
        double r17130 = r17128 * r17128;
        double r17131 = fmod(r17129, r17130);
        double r17132 = log1p(r17128);
        double r17133 = pow(r17131, r17132);
        double r17134 = acos(r17133);
        double r17135 = 5.372690074837192e-08;
        bool r17136 = r17134 <= r17135;
        double r17137 = exp(r17131);
        double r17138 = sqrt(r17137);
        double r17139 = log(r17138);
        double r17140 = r17139 + r17139;
        double r17141 = pow(r17140, r17132);
        double r17142 = acos(r17141);
        double r17143 = exp(r17132);
        double r17144 = log(r17143);
        double r17145 = pow(r17131, r17144);
        double r17146 = acos(r17145);
        double r17147 = r17136 ? r17142 : r17146;
        return r17147;
}

Error

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if (acos (pow (fmod (cosh a) (* a a)) (log1p a))) < 5.372690074837192e-08

    1. Initial program 61.6

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

    3. Applied add-log-exp58.9

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

    5. Applied add-sqr-sqrt58.9

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

    6. Applied log-prod58.9

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

    if 5.372690074837192e-08 < (acos (pow (fmod (cosh a) (* a a)) (log1p a)))

    1. Initial program 60.8

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

    3. Applied add-log-exp58.5

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

  4. Final simplification58.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right) \le 5.372690074837192250710590172430025646122 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log \left(e^{\mathsf{log1p}\left(a\right)}\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))