Average Error: 33.4 → 33.5
Time: 51.7s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]
\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(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\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) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\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) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\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) \bmod a\right)}\right|
double f(double a) {
        double r33689 = a;
        double r33690 = expm1(r33689);
        double r33691 = sin(r33690);
        double r33692 = expm1(r33691);
        double r33693 = atan(r33689);
        double r33694 = atan2(r33692, r33693);
        double r33695 = fmod(r33694, r33689);
        double r33696 = fabs(r33695);
        return r33696;
}
↓
double f(double a) {
        double r33697 = a;
        double r33698 = expm1(r33697);
        double r33699 = sin(r33698);
        double r33700 = cbrt(r33699);
        double r33701 = cbrt(r33700);
        double r33702 = 2.0;
        double r33703 = pow(r33701, r33702);
        double r33704 = cbrt(r33703);
        double r33705 = 5.0;
        double r33706 = pow(r33701, r33705);
        double r33707 = r33704 * r33706;
        double r33708 = cbrt(r33701);
        double r33709 = r33707 * r33708;
        double r33710 = r33709 * r33700;
        double r33711 = expm1(r33710);
        double r33712 = atan(r33697);
        double r33713 = atan2(r33711, r33712);
        double r33714 = fmod(r33713, r33697);
        double r33715 = cbrt(r33714);
        double r33716 = r33715 * r33715;
        double r33717 = r33716 * r33715;
        double r33718 = fabs(r33717);
        return r33718;
}

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-cube-cbrt33.4

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

  5. Applied add-cube-cbrt33.4

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

  6. Applied associate-*r*33.4

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

  7. Simplified33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  8. Using strategy rm

  9. Applied add-cube-cbrt33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  10. Applied cbrt-prod33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right)}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  11. Applied associate-*r*33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\color{blue}{\left(\left({\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right)} \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  12. Simplified33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  13. Using strategy rm

  14. Applied add-cube-cbrt33.5

    \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}}\right|\]

  15. Final simplification33.5

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))