Average Error: 33.4 → 33.4
Time: 27.0s
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 r12826 = a;
        double r12827 = expm1(r12826);
        double r12828 = sin(r12827);
        double r12829 = expm1(r12828);
        double r12830 = atan(r12826);
        double r12831 = atan2(r12829, r12830);
        double r12832 = fmod(r12831, r12826);
        double r12833 = fabs(r12832);
        return r12833;
}
↓
double f(double a) {
        double r12834 = a;
        double r12835 = expm1(r12834);
        double r12836 = sin(r12835);
        double r12837 = expm1(r12836);
        double r12838 = atan(r12834);
        double r12839 = atan2(r12837, r12838);
        double r12840 = 3.0;
        double r12841 = pow(r12839, r12840);
        double r12842 = cbrt(r12841);
        double r12843 = fmod(r12842, r12834);
        double r12844 = fabs(r12843);
        return r12844;
}

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-cbrt-cube33.4

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

  4. Simplified33.4

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

  5. Final simplification33.4

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

Reproduce

herbie shell --seed 2020042 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))