Average Error: 23.7 → 23.7
Time: 1.0m
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[2 \cdot \left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\cos \phi_2 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \cos \phi_1\right)}}\right)\]
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)
↓
2 \cdot \left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\cos \phi_2 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \cos \phi_1\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3655266 = R;
        double r3655267 = 2.0;
        double r3655268 = phi1;
        double r3655269 = phi2;
        double r3655270 = r3655268 - r3655269;
        double r3655271 = r3655270 / r3655267;
        double r3655272 = sin(r3655271);
        double r3655273 = pow(r3655272, r3655267);
        double r3655274 = cos(r3655268);
        double r3655275 = cos(r3655269);
        double r3655276 = r3655274 * r3655275;
        double r3655277 = lambda1;
        double r3655278 = lambda2;
        double r3655279 = r3655277 - r3655278;
        double r3655280 = r3655279 / r3655267;
        double r3655281 = sin(r3655280);
        double r3655282 = r3655276 * r3655281;
        double r3655283 = r3655282 * r3655281;
        double r3655284 = r3655273 + r3655283;
        double r3655285 = sqrt(r3655284);
        double r3655286 = 1.0;
        double r3655287 = r3655286 - r3655284;
        double r3655288 = sqrt(r3655287);
        double r3655289 = atan2(r3655285, r3655288);
        double r3655290 = r3655267 * r3655289;
        double r3655291 = r3655266 * r3655290;
        return r3655291;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3655292 = 2.0;
        double r3655293 = R;
        double r3655294 = phi1;
        double r3655295 = phi2;
        double r3655296 = r3655294 - r3655295;
        double r3655297 = r3655296 / r3655292;
        double r3655298 = sin(r3655297);
        double r3655299 = r3655298 * r3655298;
        double r3655300 = lambda1;
        double r3655301 = lambda2;
        double r3655302 = r3655300 - r3655301;
        double r3655303 = r3655302 / r3655292;
        double r3655304 = sin(r3655303);
        double r3655305 = cos(r3655295);
        double r3655306 = r3655304 * r3655305;
        double r3655307 = cos(r3655294);
        double r3655308 = r3655307 * r3655304;
        double r3655309 = r3655306 * r3655308;
        double r3655310 = r3655299 + r3655309;
        double r3655311 = sqrt(r3655310);
        double r3655312 = cos(r3655297);
        double r3655313 = r3655312 * r3655312;
        double r3655314 = exp(r3655304);
        double r3655315 = log(r3655314);
        double r3655316 = r3655305 * r3655315;
        double r3655317 = r3655315 * r3655307;
        double r3655318 = r3655316 * r3655317;
        double r3655319 = r3655313 - r3655318;
        double r3655320 = sqrt(r3655319);
        double r3655321 = atan2(r3655311, r3655320);
        double r3655322 = r3655293 * r3655321;
        double r3655323 = r3655292 * r3655322;
        return r3655323;
}

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 23.7

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

  2. Simplified23.7

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

  4. Applied add-log-exp23.7

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

  6. Applied add-log-exp23.7

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

  7. Final simplification23.7

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

Reproduce

herbie shell --seed 2019158 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))