Random Jason Timeout Test 006

Average Error: 33.4 → 33.5

Time: 26.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. a = 5.984365234656847e+140

Math

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

↓

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

C

double f(double a) {
        double r13285 = a;
        double r13286 = expm1(r13285);
        double r13287 = sin(r13286);
        double r13288 = expm1(r13287);
        double r13289 = atan(r13285);
        double r13290 = atan2(r13288, r13289);
        double r13291 = fmod(r13290, r13285);
        double r13292 = fabs(r13291);
        return r13292;
}

↓

double f(double a) {
        double r13293 = a;
        double r13294 = expm1(r13293);
        double r13295 = sin(r13294);
        double r13296 = cbrt(r13295);
        double r13297 = cbrt(r13296);
        double r13298 = r13297 * r13297;
        double r13299 = 4.0;
        double r13300 = pow(r13297, r13299);
        double r13301 = cbrt(r13300);
        double r13302 = r13301 * r13301;
        double r13303 = r13302 * r13301;
        double r13304 = r13298 * r13303;
        double r13305 = r13304 * r13296;
        double r13306 = expm1(r13305);
        double r13307 = atan(r13293);
        double r13308 = atan2(r13306, r13307);
        double r13309 = 3.0;
        double r13310 = pow(r13308, r13309);
        double r13311 = cbrt(r13310);
        double r13312 = fmod(r13311, r13293);
        double r13313 = fabs(r13312);
        return r13313;
}

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-cube 33.4

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

4. Simplified 33.4

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

6. Applied add-cube-cbrt 33.4

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

8. Applied add-cube-cbrt 33.5

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

9. Applied associate-*l* 33.5

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

10. Simplified 33.5

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

12. Applied add-cube-cbrt 33.5

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

13. Final simplification 33.5

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

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)))