Average Error: 33.4 → 33.5
Time: 48.2s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]
\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(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|
double f(double a) {
        double r32512 = a;
        double r32513 = expm1(r32512);
        double r32514 = sin(r32513);
        double r32515 = expm1(r32514);
        double r32516 = atan(r32512);
        double r32517 = atan2(r32515, r32516);
        double r32518 = fmod(r32517, r32512);
        double r32519 = fabs(r32518);
        return r32519;
}
↓
double f(double a) {
        double r32520 = a;
        double r32521 = expm1(r32520);
        double r32522 = sin(r32521);
        double r32523 = cbrt(r32522);
        double r32524 = cbrt(r32523);
        double r32525 = 2.0;
        double r32526 = pow(r32524, r32525);
        double r32527 = cbrt(r32526);
        double r32528 = 5.0;
        double r32529 = pow(r32524, r32528);
        double r32530 = r32527 * r32529;
        double r32531 = cbrt(r32524);
        double r32532 = r32530 * r32531;
        double r32533 = r32532 * r32523;
        double r32534 = expm1(r32533);
        double r32535 = atan(r32520);
        double r32536 = atan2(r32534, r32535);
        double r32537 = fmod(r32536, r32520);
        double r32538 = cbrt(r32537);
        double r32539 = r32538 * r32538;
        double r32540 = r32539 * r32538;
        double r32541 = fabs(r32540);
        return r32541;
}

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-cube-cbrt33.4

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

  5. Applied add-cube-cbrt33.4

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

  6. Applied associate-*r*33.4

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

  7. Simplified33.5

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

  9. Applied add-cube-cbrt33.5

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

  10. Applied cbrt-prod33.5

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

  11. Applied associate-*r*33.5

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

  12. Simplified33.5

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right)} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  13. Using strategy rm

  14. Applied add-cube-cbrt33.5

    \[\leadsto \left|\color{blue}{\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}}\right|\]

  15. Final simplification33.5

    \[\leadsto \left|\left(\sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)} \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right) \cdot \sqrt[3]{\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}\right)}^{5}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}}}\right) \cdot \sqrt[3]{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}{\tan^{-1} a}\right) \bmod a\right)}\right|\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))