Average Error: 33.7 → 33.8
Time: 15.4s
Precision: 64
\[\left|\left(\left(\tan^{-1}_* \frac{\mathsf{expm1}\left(\sin \left(\mathsf{expm1}\left(a\right)\right)\right)}{\tan^{-1} a}\right) \bmod a\right)\right|\]
↓
\[\left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + 1}{\left(\left({\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{2} + 1\right) - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right) \cdot \mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]
\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(\tan^{-1}_* \frac{\frac{1}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + 1}{\left(\left({\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{2} + 1\right) - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right) \cdot \mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}{\tan^{-1} a}\right) \bmod a\right)\right|
double code(double a) {
	return fabs(fmod(atan2(expm1(sin(expm1(a))), atan(a)), a));
}
↓
double code(double a) {
	return fabs(fmod(atan2((1.0 / ((pow(exp(sin(expm1(a))), 3.0) + 1.0) / (((pow(exp(sin(expm1(a))), 2.0) + 1.0) - exp(sin(expm1(a)))) * expm1((2.0 * sin(expm1(a))))))), atan(a)), a));
}

Error

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 33.7

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

  3. Applied expm1-udef36.9

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

  5. Applied flip--36.9

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

  6. Simplified33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right) + 0\right)}}{e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} + 1}}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  7. Using strategy rm

  8. Applied clear-num33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\color{blue}{\frac{1}{\frac{e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} + 1}{\mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right) + 0\right)}}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  9. Simplified33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\color{blue}{\frac{e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} + 1}{\mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]
  10. Using strategy rm

  11. Applied flip3-+33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\frac{\color{blue}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + {1}^{3}}{e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} \cdot e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} + \left(1 \cdot 1 - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} \cdot 1\right)}}}{\mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  12. Applied associate-/l/33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\color{blue}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + {1}^{3}}{\mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right) \cdot \left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} \cdot e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} + \left(1 \cdot 1 - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)} \cdot 1\right)\right)}}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  13. Simplified33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + {1}^{3}}{\color{blue}{\left(\left({\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{2} + 1\right) - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right) \cdot \mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]

  14. Final simplification33.8

    \[\leadsto \left|\left(\left(\tan^{-1}_* \frac{\frac{1}{\frac{{\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{3} + 1}{\left(\left({\left(e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right)}^{2} + 1\right) - e^{\sin \left(\mathsf{expm1}\left(a\right)\right)}\right) \cdot \mathsf{expm1}\left(2 \cdot \sin \left(\mathsf{expm1}\left(a\right)\right)\right)}}}{\tan^{-1} a}\right) \bmod a\right)\right|\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 006"
  :precision binary64
  (fabs (fmod (atan2 (expm1 (sin (expm1 a))) (atan a)) a)))