Average Error: 33.4 → 33.4
Time: 18.2s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left({\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\right)\right) \bmod a\right)}\right|\]
\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|
↓
\left|\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left({\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\right)\right) \bmod a\right)}\right|
double f(double a) {
        double r9846 = a;
        double r9847 = expm1(r9846);
        double r9848 = sin(r9847);
        double r9849 = expm1(r9848);
        double r9850 = atan(r9846);
        double r9851 = atan2(r9849, r9850);
        double r9852 = fmod(r9851, r9846);
        double r9853 = fabs(r9852);
        return r9853;
}
↓
double f(double a) {
        double r9854 = a;
        double r9855 = expm1(r9854);
        double r9856 = sin(r9855);
        double r9857 = expm1(r9856);
        double r9858 = atan(r9854);
        double r9859 = atan2(r9857, r9858);
        double r9860 = exp(r9859);
        double r9861 = 3.0;
        double r9862 = pow(r9860, r9861);
        double r9863 = cbrt(r9862);
        double r9864 = log(r9863);
        double r9865 = fmod(r9864, r9854);
        double r9866 = cbrt(r9865);
        double r9867 = r9866 * r9866;
        double r9868 = sqrt(r9861);
        double r9869 = pow(r9860, r9868);
        double r9870 = pow(r9869, r9868);
        double r9871 = cbrt(r9870);
        double r9872 = log(r9871);
        double r9873 = fmod(r9872, r9854);
        double r9874 = cbrt(r9873);
        double r9875 = r9867 * r9874;
        double r9876 = fabs(r9875);
        return r9876;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.4

    \[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  2. Using strategy rm

  3. Applied add-log-exp33.4

    \[\leadsto \left|\left(\color{blue}{\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)\right)} \bmod a\right)\right|\]
  4. Using strategy rm

  5. Applied add-cbrt-cube33.4

    \[\leadsto \left|\left(\left(\log \color{blue}{\left(\sqrt[3]{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}} \cdot e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right) \cdot e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)}\right) \bmod a\right)\right|\]

  6. Simplified33.4

    \[\leadsto \left|\left(\left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}}\right)\right) \bmod a\right)\right|\]
  7. Using strategy rm

  8. Applied add-cube-cbrt33.4

    \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}}\right|\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt33.4

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}}}\right)\right) \bmod a\right)}\right|\]

  11. Applied pow-unpow33.4

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{\color{blue}{{\left({\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}}\right)\right) \bmod a\right)}\right|\]

  12. Final simplification33.4

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{3}}\right)\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\sqrt[3]{{\left({\left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}}\right)\right) \bmod a\right)}\right|\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))