Distance on a great circle

Average Error: 24.5 → 14.3

Time: 47.1s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r80290 = R;
        double r80291 = 2.0;
        double r80292 = phi1;
        double r80293 = phi2;
        double r80294 = r80292 - r80293;
        double r80295 = r80294 / r80291;
        double r80296 = sin(r80295);
        double r80297 = pow(r80296, r80291);
        double r80298 = cos(r80292);
        double r80299 = cos(r80293);
        double r80300 = r80298 * r80299;
        double r80301 = lambda1;
        double r80302 = lambda2;
        double r80303 = r80301 - r80302;
        double r80304 = r80303 / r80291;
        double r80305 = sin(r80304);
        double r80306 = r80300 * r80305;
        double r80307 = r80306 * r80305;
        double r80308 = r80297 + r80307;
        double r80309 = sqrt(r80308);
        double r80310 = 1.0;
        double r80311 = r80310 - r80308;
        double r80312 = sqrt(r80311);
        double r80313 = atan2(r80309, r80312);
        double r80314 = r80291 * r80313;
        double r80315 = r80290 * r80314;
        return r80315;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r80316 = R;
        double r80317 = phi1;
        double r80318 = cos(r80317);
        double r80319 = lambda1;
        double r80320 = lambda2;
        double r80321 = r80319 - r80320;
        double r80322 = 2.0;
        double r80323 = r80321 / r80322;
        double r80324 = sin(r80323);
        double r80325 = r80318 * r80324;
        double r80326 = phi2;
        double r80327 = cos(r80326);
        double r80328 = r80325 * r80327;
        double r80329 = r80328 * r80324;
        double r80330 = r80317 / r80322;
        double r80331 = sin(r80330);
        double r80332 = r80326 / r80322;
        double r80333 = cos(r80332);
        double r80334 = r80331 * r80333;
        double r80335 = sin(r80332);
        double r80336 = cos(r80330);
        double r80337 = r80335 * r80336;
        double r80338 = r80334 - r80337;
        double r80339 = pow(r80338, r80322);
        double r80340 = r80329 + r80339;
        double r80341 = sqrt(r80340);
        double r80342 = 1.0;
        double r80343 = r80342 - r80339;
        double r80344 = cbrt(r80324);
        double r80345 = r80344 * r80344;
        double r80346 = exp(r80324);
        double r80347 = log(r80346);
        double r80348 = cbrt(r80347);
        double r80349 = r80345 * r80348;
        double r80350 = r80349 * r80318;
        double r80351 = r80350 * r80327;
        double r80352 = r80351 * r80324;
        double r80353 = r80343 - r80352;
        double r80354 = sqrt(r80353);
        double r80355 = atan2(r80341, r80354);
        double r80356 = r80355 * r80322;
        double r80357 = r80316 * r80356;
        return r80357;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 24.5

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

2. Simplified 24.5

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

4. Applied div-sub 24.5

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

5. Applied sin-diff 23.9

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

6. Simplified 23.9

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

8. Applied div-sub 23.9

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

9. Applied sin-diff 14.2

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

10. Simplified 14.2

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

12. Applied add-cube-cbrt 14.3

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

14. Applied add-log-exp 14.3

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

15. Final simplification 14.3

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

Reproduce

herbie shell --seed 2019195 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* 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))))))))))