Average Error: 61.3 → 60.4
Time: 28.1s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \left(\log \left(\sqrt{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\cos^{-1} \left({\left(\log \left(\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) + \left(\log \left(\sqrt{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{\sqrt{e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
double f(double a) {
        double r4712 = a;
        double r4713 = cosh(r4712);
        double r4714 = r4712 * r4712;
        double r4715 = fmod(r4713, r4714);
        double r4716 = log1p(r4712);
        double r4717 = pow(r4715, r4716);
        double r4718 = acos(r4717);
        return r4718;
}
↓
double f(double a) {
        double r4719 = a;
        double r4720 = cosh(r4719);
        double r4721 = r4719 * r4719;
        double r4722 = fmod(r4720, r4721);
        double r4723 = exp(r4722);
        double r4724 = sqrt(r4723);
        double r4725 = log(r4724);
        double r4726 = sqrt(r4724);
        double r4727 = log(r4726);
        double r4728 = r4727 + r4727;
        double r4729 = r4725 + r4728;
        double r4730 = log1p(r4719);
        double r4731 = pow(r4729, r4730);
        double r4732 = acos(r4731);
        return r4732;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.3

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
  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 add-sqr-sqrt60.4

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

  6. Applied log-prod60.4

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

  8. Applied add-sqr-sqrt60.4

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

  9. Applied sqrt-prod60.4

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

  10. Applied log-prod60.4

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

  11. Final simplification60.4

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

Reproduce

herbie shell --seed 2020089 
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))