Random Jason Timeout Test 006

Average Error: 33.6 → 33.7

Time: 31.6s

Precision: 64

• could not determine a ground truth for program body

  1. a = 3.2948925611475874e+88

Math

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

↓

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

C

double f(double a) {
        double r20667 = a;
        double r20668 = expm1(r20667);
        double r20669 = sin(r20668);
        double r20670 = expm1(r20669);
        double r20671 = atan(r20667);
        double r20672 = atan2(r20670, r20671);
        double r20673 = fmod(r20672, r20667);
        double r20674 = fabs(r20673);
        return r20674;
}

↓

double f(double a) {
        double r20675 = a;
        double r20676 = expm1(r20675);
        double r20677 = sin(r20676);
        double r20678 = cbrt(r20677);
        double r20679 = r20678 * r20678;
        double r20680 = cbrt(r20679);
        double r20681 = r20680 * r20680;
        double r20682 = r20681 * r20680;
        double r20683 = r20682 * r20678;
        double r20684 = expm1(r20683);
        double r20685 = atan(r20675);
        double r20686 = atan2(r20684, r20685);
        double r20687 = 3.0;
        double r20688 = pow(r20686, r20687);
        double r20689 = exp(r20688);
        double r20690 = log(r20689);
        double r20691 = cbrt(r20690);
        double r20692 = fmod(r20691, r20675);
        double r20693 = fabs(r20692);
        return r20693;
}

Error

Bits error versus a

Derivation

1. Initial program 33.6

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

3. Applied add-cbrt-cube 33.6

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

4. Simplified 33.6

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

6. Applied add-cube-cbrt 33.7

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

8. Applied add-cube-cbrt 33.7

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

10. Applied add-log-exp 33.7

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

11. Final simplification 33.7

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))