Random Jason Timeout Test 006

Average Error: 32.9 → 32.9

Time: 43.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.242935524573972e+38

Math

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

↓

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

C

double f(double a) {
        double r580905 = a;
        double r580906 = expm1(r580905);
        double r580907 = sin(r580906);
        double r580908 = expm1(r580907);
        double r580909 = atan(r580905);
        double r580910 = atan2(r580908, r580909);
        double r580911 = fmod(r580910, r580905);
        double r580912 = fabs(r580911);
        return r580912;
}

↓

double f(double a) {
        double r580913 = a;
        double r580914 = expm1(r580913);
        double r580915 = sin(r580914);
        double r580916 = expm1(r580915);
        double r580917 = atan(r580913);
        double r580918 = atan2(r580916, r580917);
        double r580919 = exp(r580918);
        double r580920 = log(r580919);
        double r580921 = fmod(r580920, r580913);
        double r580922 = cbrt(r580921);
        double r580923 = fmod(r580918, r580913);
        double r580924 = cbrt(r580923);
        double r580925 = r580922 * r580924;
        double r580926 = cbrt(r580915);
        double r580927 = r580926 * r580926;
        double r580928 = r580927 * r580926;
        double r580929 = expm1(r580928);
        double r580930 = atan2(r580929, r580917);
        double r580931 = exp(r580930);
        double r580932 = log(r580931);
        double r580933 = fmod(r580932, r580913);
        double r580934 = cbrt(r580933);
        double r580935 = r580925 * r580934;
        double r580936 = fabs(r580935);
        return r580936;
}

Error

Bits error versus a

Derivation

1. Initial program 32.9

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

3. Applied add-log-exp 32.9

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

5. Applied add-cube-cbrt 33.0

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

6. Taylor expanded around 0 32.9

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

8. Applied add-cube-cbrt 32.9

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

9. Final simplification 32.9

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))