Average Error: 1.0 → 0.2
Time: 11.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) \cdot {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) \cdot {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2))))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return (lambda1 + atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) + (cos(lambda1) * sin(-lambda2)))), (((fma(cos(lambda1), (cos(lambda2) * cos(phi2)), cos(phi1)) * pow(fma(cos(lambda1), (cos(lambda2) * cos(phi2)), cos(phi1)), 2.0)) + pow((cos(phi2) * (sin(lambda1) * sin(lambda2))), 3.0)) / fma(fma(cos(lambda1), (cos(phi2) * cos(lambda2)), cos(phi1)), fma(cos(lambda1), (cos(phi2) * cos(lambda2)), cos(phi1)), ((cos(phi2) * (sin(lambda1) * sin(lambda2))) * ((cos(phi2) * (sin(lambda1) * sin(lambda2))) - fma(cos(lambda1), (cos(lambda2) * cos(phi2)), cos(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 1.0

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

  3. Applied cos-diff1.0

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

  4. Applied distribute-lft-in1.0

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

  5. Applied associate-+r+1.0

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

  6. Simplified1.0

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

  8. Applied sub-neg1.0

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

  9. Applied sin-sum0.2

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

  10. Simplified0.2

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

  12. Applied flip3-+0.2

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

  13. Simplified0.2

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

  15. Applied cube-mult0.2

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

  16. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{2}} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}\]

  17. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\frac{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) \cdot {\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{2} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))