Average Error: 33.6 → 33.6
Time: 18.4s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
\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{\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 r19519 = a;
        double r19520 = expm1(r19519);
        double r19521 = sin(r19520);
        double r19522 = expm1(r19521);
        double r19523 = atan(r19519);
        double r19524 = atan2(r19522, r19523);
        double r19525 = fmod(r19524, r19519);
        double r19526 = fabs(r19525);
        return r19526;
}
↓
double f(double a) {
        double r19527 = a;
        double r19528 = expm1(r19527);
        double r19529 = sin(r19528);
        double r19530 = expm1(r19529);
        double r19531 = atan(r19527);
        double r19532 = atan2(r19530, r19531);
        double r19533 = fmod(r19532, r19527);
        double r19534 = fabs(r19533);
        return r19534;
}

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-cube-cbrt33.6

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

  4. Taylor expanded around 0 33.6

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

  5. Final simplification33.6

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))