Average Error: 0.9 → 0.3
Time: 11.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\frac{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)}^{3} + {\left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)}\right), \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right)\right)\right)}}\]
\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 \lambda_2\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}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right), \log \left(e^{\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)}\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)))), ((pow(fma(cos(lambda1), (cos(lambda2) * cos(phi2)), cos(phi1)), 3.0) + pow((cos(phi2) * (sin(lambda1) * sin(lambda2))), 3.0)) / fma(fma(cos(lambda1), (cos(phi2) * cos(lambda2)), cos(phi1)), log(exp(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 0.9

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

  3. Applied cos-diff0.9

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

  4. Applied distribute-lft-in0.9

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

  5. Applied associate-+r+0.9

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

  6. Simplified0.9

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

  8. Applied sin-diff0.2

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

  10. Applied flip3-+0.3

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

  11. Simplified0.3

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

  13. Applied add-log-exp0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020102 +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)))))))