Average Error: 33.6 → 33.6
Time: 34.0s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]
\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(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\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) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\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) \bmod a\right)}\right|
double f(double a) {
        double r17629 = a;
        double r17630 = expm1(r17629);
        double r17631 = sin(r17630);
        double r17632 = expm1(r17631);
        double r17633 = atan(r17629);
        double r17634 = atan2(r17632, r17633);
        double r17635 = fmod(r17634, r17629);
        double r17636 = fabs(r17635);
        return r17636;
}
↓
double f(double a) {
        double r17637 = a;
        double r17638 = expm1(r17637);
        double r17639 = sin(r17638);
        double r17640 = expm1(r17639);
        double r17641 = atan(r17637);
        double r17642 = atan2(r17640, r17641);
        double r17643 = fmod(r17642, r17637);
        double r17644 = cbrt(r17643);
        double r17645 = r17644 * r17644;
        double r17646 = cbrt(r17639);
        double r17647 = r17646 * r17646;
        double r17648 = r17647 * r17646;
        double r17649 = expm1(r17648);
        double r17650 = atan2(r17649, r17641);
        double r17651 = fmod(r17650, r17637);
        double r17652 = cbrt(r17651);
        double r17653 = r17645 * r17652;
        double r17654 = fabs(r17653);
        return r17654;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.6

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

  3. Applied add-cube-cbrt33.6

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

  5. Applied add-cube-cbrt33.6

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

  6. Final simplification33.6

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

Reproduce

herbie shell --seed 2019195 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))