Average Error: 33.4 → 33.3
Time: 44.1s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\log \left(\log \left(e^{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right)\right) \bmod a\right)}\right|\]
\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(\log \left(e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}\right)\right) \bmod a\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) \cdot \sqrt[3]{\left(\left(\log \left(\log \left(e^{e^{\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}}}\right)\right)\right) \bmod a\right)}\right|
double f(double a) {
        double r1083320 = a;
        double r1083321 = expm1(r1083320);
        double r1083322 = sin(r1083321);
        double r1083323 = expm1(r1083322);
        double r1083324 = atan(r1083320);
        double r1083325 = atan2(r1083323, r1083324);
        double r1083326 = fmod(r1083325, r1083320);
        double r1083327 = fabs(r1083326);
        return r1083327;
}
↓
double f(double a) {
        double r1083328 = a;
        double r1083329 = expm1(r1083328);
        double r1083330 = sin(r1083329);
        double r1083331 = expm1(r1083330);
        double r1083332 = atan(r1083328);
        double r1083333 = atan2(r1083331, r1083332);
        double r1083334 = exp(r1083333);
        double r1083335 = log(r1083334);
        double r1083336 = fmod(r1083335, r1083328);
        double r1083337 = cbrt(r1083336);
        double r1083338 = fmod(r1083333, r1083328);
        double r1083339 = cbrt(r1083338);
        double r1083340 = r1083337 * r1083339;
        double r1083341 = exp(r1083334);
        double r1083342 = log(r1083341);
        double r1083343 = log(r1083342);
        double r1083344 = fmod(r1083343, r1083328);
        double r1083345 = cbrt(r1083344);
        double r1083346 = r1083340 * r1083345;
        double r1083347 = fabs(r1083346);
        return r1083347;
}

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-log-exp33.4

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

  5. Applied add-cube-cbrt33.4

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

  7. Applied add-log-exp33.4

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

  8. Taylor expanded around 0 33.3

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

  9. Final simplification33.3

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

Reproduce

herbie shell --seed 2019170 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))