Average Error: 33.7 → 33.7
Time: 18.9s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\left(\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\left(\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]
\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{\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)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\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) \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|
double f(double a) {
        double r12505 = a;
        double r12506 = expm1(r12505);
        double r12507 = sin(r12506);
        double r12508 = expm1(r12507);
        double r12509 = atan(r12505);
        double r12510 = atan2(r12508, r12509);
        double r12511 = fmod(r12510, r12505);
        double r12512 = fabs(r12511);
        return r12512;
}
↓
double f(double a) {
        double r12513 = a;
        double r12514 = expm1(r12513);
        double r12515 = sin(r12514);
        double r12516 = expm1(r12515);
        double r12517 = cbrt(r12516);
        double r12518 = r12517 * r12517;
        double r12519 = r12518 * r12517;
        double r12520 = atan(r12513);
        double r12521 = atan2(r12519, r12520);
        double r12522 = fmod(r12521, r12513);
        double r12523 = cbrt(r12522);
        double r12524 = r12523 * r12523;
        double r12525 = atan2(r12516, r12520);
        double r12526 = fmod(r12525, r12513);
        double r12527 = cbrt(r12526);
        double r12528 = r12524 * r12527;
        double r12529 = fabs(r12528);
        return r12529;
}

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-cube-cbrt33.7

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

  5. Applied add-cube-cbrt33.7

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

  6. Taylor expanded around 0 33.7

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

  7. Final simplification33.7

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

Reproduce

herbie shell --seed 2020064 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))