Average Error: 33.5 → 33.6
Time: 20.1s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\tan^{-1}_* \frac{\left(\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}\right)\right) \cdot \left(1 \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
\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(\left(\tan^{-1}_* \frac{\left(\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}\right)\right) \cdot \left(1 \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|
double f(double a) {
        double r23636 = a;
        double r23637 = expm1(r23636);
        double r23638 = sin(r23637);
        double r23639 = expm1(r23638);
        double r23640 = atan(r23636);
        double r23641 = atan2(r23639, r23640);
        double r23642 = fmod(r23641, r23636);
        double r23643 = fabs(r23642);
        return r23643;
}
↓
double f(double a) {
        double r23644 = a;
        double r23645 = expm1(r23644);
        double r23646 = sin(r23645);
        double r23647 = expm1(r23646);
        double r23648 = cbrt(r23647);
        double r23649 = r23648 * r23648;
        double r23650 = cbrt(r23649);
        double r23651 = cbrt(r23648);
        double r23652 = r23650 * r23651;
        double r23653 = r23648 * r23652;
        double r23654 = 1.0;
        double r23655 = r23654 * r23648;
        double r23656 = r23653 * r23655;
        double r23657 = atan(r23644);
        double r23658 = atan2(r23656, r23657);
        double r23659 = fmod(r23658, r23644);
        double r23660 = fabs(r23659);
        return r23660;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.5

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

  3. Applied add-cube-cbrt33.5

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

  5. Applied add-cube-cbrt33.6

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

  6. Applied cbrt-prod33.6

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

  8. Applied *-un-lft-identity33.6

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

  9. Applied cbrt-prod33.6

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

  10. Simplified33.6

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

  11. Final simplification33.6

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))