Average Error: 24.7 → 24.9
Time: 24.9s
Precision: binary64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)\right)}}{\sqrt{1 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right)}^{3}}\right)\right) - {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}\right)\]
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)
↓
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)\right)}}{\sqrt{1 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right)}^{3}}\right)\right) - {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}\right)
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) (R * ((double) (2.0 * ((double) atan2(((double) sqrt(((double) (((double) pow(((double) sin((((double) (phi1 - phi2)) / 2.0))), 2.0)) + ((double) (((double) (((double) (((double) cos(phi1)) * ((double) cos(phi2)))) * ((double) sin((((double) (lambda1 - lambda2)) / 2.0))))) * ((double) sin((((double) (lambda1 - lambda2)) / 2.0))))))))), ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) sin((((double) (phi1 - phi2)) / 2.0))), 2.0)) + ((double) (((double) (((double) (((double) cos(phi1)) * ((double) cos(phi2)))) * ((double) sin((((double) (lambda1 - lambda2)) / 2.0))))) * ((double) sin((((double) (lambda1 - lambda2)) / 2.0)))))))))))))))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return ((double) (R * ((double) (2.0 * ((double) atan2(((double) sqrt(((double) (((double) pow(((double) sin((((double) (phi1 - phi2)) / 2.0))), 2.0)) + ((double) (((double) cos(phi1)) * ((double) (((double) cos(phi2)) * ((double) (((double) sin((((double) (lambda1 - lambda2)) / 2.0))) * ((double) cbrt(((double) pow(((double) sin((((double) (lambda1 - lambda2)) / 2.0))), 3.0)))))))))))))), ((double) sqrt(((double) (1.0 + ((double) (((double) (((double) cos(phi1)) * ((double) (((double) cos(phi2)) * ((double) (((double) sin((((double) (lambda1 - lambda2)) / 2.0))) * ((double) cbrt(((double) pow(((double) sin((((double) (lambda2 - lambda1)) / 2.0))), 3.0)))))))))) - ((double) pow(((double) sin((((double) (phi1 - phi2)) / 2.0))), 2.0))))))))))))));
}

Error

Bits error versus R

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 24.7

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

  2. Simplified24.7

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

  4. Applied add-cbrt-cube24.7

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

  5. Simplified24.7

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}{\sqrt{1 + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right)}^{3}}}\right)\right) - {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}\right)\]
  6. Using strategy rm

  7. Applied add-cbrt-cube24.9

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

  8. Simplified24.9

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

  9. Final simplification24.9

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

Reproduce

herbie shell --seed 2020196 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))