Random Jason Timeout Test 006

Average Error: 32.9 → 32.9

Time: 1.2m

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 r1090814 = a;
        double r1090815 = expm1(r1090814);
        double r1090816 = sin(r1090815);
        double r1090817 = expm1(r1090816);
        double r1090818 = atan(r1090814);
        double r1090819 = atan2(r1090817, r1090818);
        double r1090820 = fmod(r1090819, r1090814);
        double r1090821 = fabs(r1090820);
        return r1090821;
}

↓

double f(double a) {
        double r1090822 = a;
        double r1090823 = expm1(r1090822);
        double r1090824 = sin(r1090823);
        double r1090825 = expm1(r1090824);
        double r1090826 = atan(r1090822);
        double r1090827 = atan2(r1090825, r1090826);
        double r1090828 = exp(r1090827);
        double r1090829 = log(r1090828);
        double r1090830 = fmod(r1090829, r1090822);
        double r1090831 = cbrt(r1090830);
        double r1090832 = fmod(r1090827, r1090822);
        double r1090833 = cbrt(r1090832);
        double r1090834 = r1090831 * r1090833;
        double r1090835 = cbrt(r1090824);
        double r1090836 = r1090835 * r1090835;
        double r1090837 = r1090836 * r1090835;
        double r1090838 = expm1(r1090837);
        double r1090839 = atan2(r1090838, r1090826);
        double r1090840 = exp(r1090839);
        double r1090841 = log(r1090840);
        double r1090842 = fmod(r1090841, r1090822);
        double r1090843 = cbrt(r1090842);
        double r1090844 = r1090834 * r1090843;
        double r1090845 = fabs(r1090844);
        return r1090845;
}

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 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))