Average Error: 33.4 → 33.5
Time: 41.8s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(e^{\log \left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{6}\right)} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)\right|\]
\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(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(e^{\log \left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{6}\right)} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)\right|
double f(double a) {
        double r30161 = a;
        double r30162 = expm1(r30161);
        double r30163 = sin(r30162);
        double r30164 = expm1(r30163);
        double r30165 = atan(r30161);
        double r30166 = atan2(r30164, r30165);
        double r30167 = fmod(r30166, r30161);
        double r30168 = fabs(r30167);
        return r30168;
}
↓
double f(double a) {
        double r30169 = a;
        double r30170 = expm1(r30169);
        double r30171 = sin(r30170);
        double r30172 = cbrt(r30171);
        double r30173 = cbrt(r30172);
        double r30174 = 6.0;
        double r30175 = pow(r30173, r30174);
        double r30176 = log(r30175);
        double r30177 = exp(r30176);
        double r30178 = r30177 * r30172;
        double r30179 = expm1(r30178);
        double r30180 = atan(r30169);
        double r30181 = atan2(r30179, r30180);
        double r30182 = exp(r30181);
        double r30183 = log(r30182);
        double r30184 = fmod(r30183, r30169);
        double r30185 = fabs(r30184);
        return r30185;
}

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-cube-cbrt33.5

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

  7. Applied add-cube-cbrt33.5

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

  8. Applied associate-*r*33.5

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

  9. Simplified33.5

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

  11. Applied add-exp-log59.5

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

  12. Applied add-exp-log59.5

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

  13. Applied pow-exp59.5

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

  14. Applied add-exp-log59.5

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

  15. Applied prod-exp59.6

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

  16. Applied prod-exp59.5

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

  17. Simplified33.5

    \[\leadsto \left|\left(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(e^{\color{blue}{\log \left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{6}\right)}} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)\right|\]

  18. Final simplification33.5

    \[\leadsto \left|\left(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(e^{\log \left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{6}\right)} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)\right|\]

Reproduce

herbie shell --seed 2019209 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))