Average Error: 33.7 → 33.7
Time: 16.7s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\sqrt[3]{{\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right)}^{3}}\right) \bmod a\right)\right|\]
\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(\sqrt[3]{{\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right)}^{3}}\right) \bmod a\right)\right|
double f(double a) {
        double r17628 = a;
        double r17629 = expm1(r17628);
        double r17630 = sin(r17629);
        double r17631 = expm1(r17630);
        double r17632 = atan(r17628);
        double r17633 = atan2(r17631, r17632);
        double r17634 = fmod(r17633, r17628);
        double r17635 = fabs(r17634);
        return r17635;
}
↓
double f(double a) {
        double r17636 = a;
        double r17637 = expm1(r17636);
        double r17638 = sin(r17637);
        double r17639 = expm1(r17638);
        double r17640 = atan(r17636);
        double r17641 = atan2(r17639, r17640);
        double r17642 = 3.0;
        double r17643 = pow(r17641, r17642);
        double r17644 = cbrt(r17643);
        double r17645 = fmod(r17644, r17636);
        double r17646 = fabs(r17645);
        return r17646;
}

Error

Bits error versus a

Derivation

  1. Initial program 33.7

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

  3. Applied add-cbrt-cube33.7

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

  4. Simplified33.7

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

  5. Final simplification33.7

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

Reproduce

herbie shell --seed 2019356 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))