Average Error: 61.2 → 60.2
Time: 26.4s
Precision: 64
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\]
↓
\[\frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]
\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)
↓
\frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\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)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}
double f(double a) {
        double r4820 = a;
        double r4821 = cosh(r4820);
        double r4822 = r4820 * r4820;
        double r4823 = fmod(r4821, r4822);
        double r4824 = log1p(r4820);
        double r4825 = pow(r4823, r4824);
        double r4826 = acos(r4825);
        return r4826;
}
↓
double f(double a) {
        double r4827 = atan2(1.0, 0.0);
        double r4828 = r4827 * r4827;
        double r4829 = 0.25;
        double r4830 = 1.0;
        double r4831 = -r4830;
        double r4832 = a;
        double r4833 = cosh(r4832);
        double r4834 = r4832 * r4832;
        double r4835 = fmod(r4833, r4834);
        double r4836 = exp(r4835);
        double r4837 = log(r4836);
        double r4838 = log1p(r4832);
        double r4839 = pow(r4837, r4838);
        double r4840 = asin(r4839);
        double r4841 = cbrt(r4840);
        double r4842 = 6.0;
        double r4843 = pow(r4841, r4842);
        double r4844 = r4831 * r4843;
        double r4845 = fma(r4828, r4829, r4844);
        double r4846 = 0.5;
        double r4847 = 2.0;
        double r4848 = pow(r4832, r4847);
        double r4849 = fmod(r4833, r4848);
        double r4850 = pow(r4849, r4838);
        double r4851 = asin(r4850);
        double r4852 = fma(r4827, r4846, r4851);
        double r4853 = r4845 / r4852;
        return r4853;
}

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-exp60.2

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

  5. Applied acos-asin60.2

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

  7. Applied flip--60.2

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

  8. Simplified60.2

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}}{\frac{\pi}{2} + \sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\]

  9. Simplified60.2

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\color{blue}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}}\]

  10. Final simplification60.2

    \[\leadsto \frac{\mathsf{fma}\left(\pi \cdot \pi, \frac{1}{4}, \left(-1\right) \cdot {\left(\sqrt[3]{\sin^{-1} \left({\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)}\right)}^{6}\right)}{\mathsf{fma}\left(\pi, \frac{1}{2}, \sin^{-1} \left({\left(\left(\cosh a\right) \bmod \left({a}^{2}\right)\right)}^{\left(\mathsf{log1p}\left(a\right)\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  :precision binary64
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))