Average Error: 13.4 → 0.2
Time: 14.1s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} - \left(\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}}\]
\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 - \frac{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} - \left(\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}{\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}}
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (((sin(phi1) * cos(phi2)) * (pow((cos(lambda1) * cos(lambda2)), 3.0) - (((sin(lambda1) * sin(-lambda2)) * (sin(lambda1) * sin(-lambda2))) * (sin(lambda1) * sin(-lambda2))))) / (((cos(lambda1) * cos(lambda2)) * (cos(lambda1) * cos(lambda2))) + (((sin(lambda1) * sin(-lambda2)) * (sin(lambda1) * sin(-lambda2))) + ((cos(lambda1) * cos(lambda2)) * (sin(lambda1) * sin(-lambda2))))))));
}

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 13.4

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

  3. Applied sin-diff6.7

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

  5. Applied sub-neg6.7

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

  6. Applied cos-sum0.2

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

  7. Simplified0.2

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

  9. Applied flip3--0.2

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

  10. Applied associate-*r/0.2

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

  12. Applied add-cbrt-cube0.2

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

  13. Applied rem-cube-cbrt0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 
(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))))))