Average Error: 32.8 → 32.8
Time: 33.1s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\tan^{-1}_* \frac{\left(\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}\right)\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}{\tan^{-1} a}\right) \bmod a\right)\right|\]
\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(\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 \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}\right)\right) \cdot \sqrt[3]{\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|
double f(double a) {
        double r97 = a;
        double r98 = expm1(r97);
        double r99 = sin(r98);
        double r100 = expm1(r99);
        double r101 = atan(r97);
        double r102 = atan2(r100, r101);
        double r103 = fmod(r102, r97);
        double r104 = fabs(r103);
        return r104;
}
↓
double f(double a) {
        double r105 = a;
        double r106 = expm1(r105);
        double r107 = sin(r106);
        double r108 = expm1(r107);
        double r109 = cbrt(r108);
        double r110 = r109 * r109;
        double r111 = cbrt(r110);
        double r112 = cbrt(r109);
        double r113 = cbrt(r112);
        double r114 = r113 * r113;
        double r115 = r111 * r114;
        double r116 = r115 * r113;
        double r117 = r109 * r116;
        double r118 = r117 * r109;
        double r119 = atan(r105);
        double r120 = atan2(r118, r119);
        double r121 = fmod(r120, r105);
        double r122 = fabs(r121);
        return r122;
}

Error

Bits error versus a

Derivation

  1. Initial program 32.8

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

  3. Applied add-cube-cbrt32.8

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

  5. Applied add-cube-cbrt32.8

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

  6. Applied cbrt-prod32.8

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

  8. Applied add-cube-cbrt32.8

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

  9. Applied associate-*r*32.8

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

  10. Final simplification32.8

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

Reproduce

herbie shell --seed 2020025 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))