Random Jason Timeout Test 006

Average Error: 33.4 → 33.4

Time: 17.8s

Precision: 64

• could not determine a ground truth for program body

  1. a = 4.595623646030552e+209

Math

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

↓

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

C

double f(double a) {
        double r8913 = a;
        double r8914 = expm1(r8913);
        double r8915 = sin(r8914);
        double r8916 = expm1(r8915);
        double r8917 = atan(r8913);
        double r8918 = atan2(r8916, r8917);
        double r8919 = fmod(r8918, r8913);
        double r8920 = fabs(r8919);
        return r8920;
}

↓

double f(double a) {
        double r8921 = a;
        double r8922 = expm1(r8921);
        double r8923 = sin(r8922);
        double r8924 = expm1(r8923);
        double r8925 = atan(r8921);
        double r8926 = atan2(r8924, r8925);
        double r8927 = exp(r8926);
        double r8928 = log(r8927);
        double r8929 = fmod(r8928, r8921);
        double r8930 = cbrt(r8929);
        double r8931 = 3.0;
        double r8932 = pow(r8927, r8931);
        double r8933 = cbrt(r8932);
        double r8934 = log(r8933);
        double r8935 = fmod(r8934, r8921);
        double r8936 = cbrt(r8935);
        double r8937 = r8930 * r8936;
        double r8938 = fmod(r8926, r8921);
        double r8939 = cbrt(r8938);
        double r8940 = r8937 * r8939;
        double r8941 = fabs(r8940);
        return r8941;
}

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-exp 33.4

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

5. Applied add-cube-cbrt 33.4

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

6. Taylor expanded around 0 33.4

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

8. Applied add-cbrt-cube 33.4

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

9. Simplified 33.4

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

10. Final simplification 33.4

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))