Average Error: 13.5 → 0.2
Time: 15.9s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \sqrt[3]{\cos \lambda_1}\right)\right) + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right)}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \sqrt[3]{\cos \lambda_1}\right)\right) + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right)}
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) atan2(((double) (((double) sin(((double) (lambda1 - lambda2)))) * ((double) cos(phi2)))), ((double) (((double) (((double) cos(phi1)) * ((double) sin(phi2)))) - ((double) (((double) (((double) sin(phi1)) * ((double) cos(phi2)))) * ((double) cos(((double) (lambda1 - lambda2))))))))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) atan2(((double) (((double) (((double) (((double) sin(lambda1)) * ((double) cos(lambda2)))) - ((double) (((double) cos(lambda1)) * ((double) sin(lambda2)))))) * ((double) cos(phi2)))), ((double) (((double) (((double) cos(phi1)) * ((double) sin(phi2)))) - ((double) (((double) (((double) (((double) cbrt(((double) cos(lambda1)))) * ((double) cbrt(((double) cos(lambda1)))))) * ((double) (((double) (((double) cos(phi2)) * ((double) sin(phi1)))) * ((double) (((double) cos(lambda2)) * ((double) cbrt(((double) cos(lambda1)))))))))) + ((double) (((double) sin(lambda1)) * ((double) (((double) sin(lambda2)) * ((double) (((double) cos(phi2)) * ((double) sin(phi1))))))))))))));
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

    \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  2. Using strategy rm

  3. Applied sin-diff7.2

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  4. Using strategy rm

  5. Applied cos-diff0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\]

  6. Applied distribute-lft-in0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\]

  7. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)} + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right) + \color{blue}{\sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\left(\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \sqrt[3]{\cos \lambda_1}\right)} \cdot \left(\cos \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right) + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)}\]

  11. Applied associate-*l*0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\color{blue}{\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \left(\sqrt[3]{\cos \lambda_1} \cdot \left(\cos \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)} + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)}\]

  12. Simplified0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \color{blue}{\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \sqrt[3]{\cos \lambda_1}\right)\right)} + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)\right)}\]

  13. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sqrt[3]{\cos \lambda_1} \cdot \sqrt[3]{\cos \lambda_1}\right) \cdot \left(\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \sqrt[3]{\cos \lambda_1}\right)\right) + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))