Average Error: 23.9 → 13.6
Time: 33.1s
Precision: binary64
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↓
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↓
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) (((double) (phi1 - phi2)) / 2.0)))), 2.0)) + ((double) (((double) (((double) (((double) cos(phi1)) * ((double) cos(phi2)))) * ((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))))))), ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) sin(((double) (((double) (phi1 - phi2)) / 2.0)))), 2.0)) + ((double) (((double) (((double) (((double) cos(phi1)) * ((double) cos(phi2)))) * ((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) sin(((double) (((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) (((double) (((double) sin(((double) (phi1 / 2.0)))) * ((double) cos(((double) (phi2 / 2.0)))))) - ((double) (((double) cos(((double) (phi1 / 2.0)))) * ((double) sin(((double) (phi2 / 2.0)))))))), 2.0)) + ((double) (((double) cos(phi1)) * ((double) (((double) cos(phi2)) * ((double) (((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))) * ((double) (((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) (((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))))))))))))))))), ((double) sqrt(((double) (1.0 + ((double) (((double) (((double) cos(phi1)) * ((double) (((double) cos(phi2)) * ((double) (((double) (((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) (((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))) * ((double) cbrt(((double) sin(((double) (((double) (lambda1 - lambda2)) / 2.0)))))))))) * ((double) sin(((double) (((double) (lambda2 - lambda1)) / 2.0)))))))))) - ((double) pow(((double) (((double) (((double) sin(((double) (phi1 / 2.0)))) * ((double) cos(((double) (phi2 / 2.0)))))) - ((double) (((double) cos(((double) (phi1 / 2.0)))) * ((double) sin(((double) (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 23.9
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  2. Simplified23.9
    \[\leadsto \]
  3. Using strategy rm
  4. Applied div-sub23.9
    \[\leadsto \]
  5. Applied sin-diff23.3
    \[\leadsto \]
  6. Using strategy rm
  7. Applied div-sub23.3
    \[\leadsto \]
  8. Applied sin-diff13.4
    \[\leadsto \]
  9. Using strategy rm
  10. Applied add-cube-cbrt13.5
    \[\leadsto \]
  11. Using strategy rm
  12. Applied add-cube-cbrt13.6
    \[\leadsto \]
  13. Final simplification13.6
    \[\leadsto \]
Reproduce
herbie shell --seed 2020192 
(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))))))))))