Average Error: 61.2 → 60.3
Time: 34.5s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\log \left(e^{\cos^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\log \left(e^{\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)}\right)
double f(double a) {
        double r4734 = a;
        double r4735 = cosh(r4734);
        double r4736 = r4734 * r4734;
        double r4737 = fmod(r4735, r4736);
        double r4738 = log1p(r4734);
        double r4739 = pow(r4737, r4738);
        double r4740 = acos(r4739);
        return r4740;
}
↓
double f(double a) {
        double r4741 = a;
        double r4742 = cosh(r4741);
        double r4743 = r4741 * r4741;
        double r4744 = fmod(r4742, r4743);
        double r4745 = exp(r4744);
        double r4746 = log(r4745);
        double r4747 = log1p(r4741);
        double r4748 = pow(r4746, r4747);
        double r4749 = acos(r4748);
        double r4750 = exp(r4749);
        double r4751 = log(r4750);
        return r4751;
}

Error

Bits error versus a

Derivation

  1. Initial program 61.2

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

  3. Applied add-log-exp61.2

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

  5. Applied add-log-exp60.3

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

  6. Final simplification60.3

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))