Average Error: 33.6 → 33.6
Time: 19.1s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\sqrt[3]{{\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right)}^{3}}\right) \bmod a\right)\right|\]
\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 r8781 = a;
        double r8782 = expm1(r8781);
        double r8783 = sin(r8782);
        double r8784 = expm1(r8783);
        double r8785 = atan(r8781);
        double r8786 = atan2(r8784, r8785);
        double r8787 = fmod(r8786, r8781);
        double r8788 = fabs(r8787);
        return r8788;
}
↓
double f(double a) {
        double r8789 = a;
        double r8790 = expm1(r8789);
        double r8791 = sin(r8790);
        double r8792 = expm1(r8791);
        double r8793 = atan(r8789);
        double r8794 = atan2(r8792, r8793);
        double r8795 = 3.0;
        double r8796 = pow(r8794, r8795);
        double r8797 = cbrt(r8796);
        double r8798 = fmod(r8797, r8789);
        double r8799 = fabs(r8798);
        return r8799;
}

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-cbrt-cube33.6

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

  4. Simplified33.6

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

  5. Final simplification33.6

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))