Average Error: 61.4 → 60.4
Time: 40.4s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\left(\sqrt{\frac{\pi}{2}} + \sqrt{\sin^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right) \cdot \left(\sqrt{\frac{\pi}{2}} - \sqrt{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\left(\sqrt{\frac{\pi}{2}} + \sqrt{\sin^{-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)}\right) \cdot \left(\sqrt{\frac{\pi}{2}} - \sqrt{\sin^{-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 r11637 = a;
        double r11638 = cosh(r11637);
        double r11639 = r11637 * r11637;
        double r11640 = fmod(r11638, r11639);
        double r11641 = log1p(r11637);
        double r11642 = pow(r11640, r11641);
        double r11643 = acos(r11642);
        return r11643;
}
↓
double f(double a) {
        double r11644 = atan2(1.0, 0.0);
        double r11645 = 2.0;
        double r11646 = r11644 / r11645;
        double r11647 = sqrt(r11646);
        double r11648 = a;
        double r11649 = cosh(r11648);
        double r11650 = r11648 * r11648;
        double r11651 = fmod(r11649, r11650);
        double r11652 = exp(r11651);
        double r11653 = sqrt(r11652);
        double r11654 = log(r11653);
        double r11655 = r11654 + r11654;
        double r11656 = log1p(r11648);
        double r11657 = pow(r11655, r11656);
        double r11658 = asin(r11657);
        double r11659 = sqrt(r11658);
        double r11660 = r11647 + r11659;
        double r11661 = log(r11652);
        double r11662 = pow(r11661, r11656);
        double r11663 = asin(r11662);
        double r11664 = sqrt(r11663);
        double r11665 = r11647 - r11664;
        double r11666 = r11660 * r11665;
        return r11666;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.4

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

  3. Applied add-log-exp60.4

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

  5. Applied acos-asin60.4

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt60.4

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

  8. Applied add-sqr-sqrt60.4

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

  9. Applied difference-of-squares60.4

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

  11. Applied add-sqr-sqrt60.4

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

  12. Applied log-prod60.4

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

  13. Final simplification60.4

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))