Distance on a great circle

Average Error: 24.2 → 13.8

Time: 45.2s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70510 = R;
        double r70511 = 2.0;
        double r70512 = phi1;
        double r70513 = phi2;
        double r70514 = r70512 - r70513;
        double r70515 = r70514 / r70511;
        double r70516 = sin(r70515);
        double r70517 = pow(r70516, r70511);
        double r70518 = cos(r70512);
        double r70519 = cos(r70513);
        double r70520 = r70518 * r70519;
        double r70521 = lambda1;
        double r70522 = lambda2;
        double r70523 = r70521 - r70522;
        double r70524 = r70523 / r70511;
        double r70525 = sin(r70524);
        double r70526 = r70520 * r70525;
        double r70527 = r70526 * r70525;
        double r70528 = r70517 + r70527;
        double r70529 = sqrt(r70528);
        double r70530 = 1.0;
        double r70531 = r70530 - r70528;
        double r70532 = sqrt(r70531);
        double r70533 = atan2(r70529, r70532);
        double r70534 = r70511 * r70533;
        double r70535 = r70510 * r70534;
        return r70535;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70536 = R;
        double r70537 = 2.0;
        double r70538 = phi1;
        double r70539 = cos(r70538);
        double r70540 = lambda1;
        double r70541 = lambda2;
        double r70542 = r70540 - r70541;
        double r70543 = r70542 / r70537;
        double r70544 = sin(r70543);
        double r70545 = r70539 * r70544;
        double r70546 = phi2;
        double r70547 = cos(r70546);
        double r70548 = r70545 * r70547;
        double r70549 = r70548 * r70544;
        double r70550 = r70538 / r70537;
        double r70551 = sin(r70550);
        double r70552 = r70546 / r70537;
        double r70553 = cos(r70552);
        double r70554 = r70551 * r70553;
        double r70555 = sin(r70552);
        double r70556 = cos(r70550);
        double r70557 = r70555 * r70556;
        double r70558 = r70554 - r70557;
        double r70559 = pow(r70558, r70537);
        double r70560 = r70549 + r70559;
        double r70561 = sqrt(r70560);
        double r70562 = 1.0;
        double r70563 = r70562 - r70559;
        double r70564 = exp(r70544);
        double r70565 = log(r70564);
        double r70566 = r70539 * r70565;
        double r70567 = r70547 * r70566;
        double r70568 = 3.0;
        double r70569 = pow(r70544, r70568);
        double r70570 = cbrt(r70569);
        double r70571 = r70567 * r70570;
        double r70572 = r70563 - r70571;
        double r70573 = sqrt(r70572);
        double r70574 = atan2(r70561, r70573);
        double r70575 = r70537 * r70574;
        double r70576 = r70536 * r70575;
        return r70576;
}

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 24.2

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

2. Simplified 24.2

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

4. Applied div-sub 24.2

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

5. Applied sin-diff 23.6

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

6. Simplified 23.6

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

8. Applied div-sub 23.6

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

9. Applied sin-diff 13.8

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

10. Simplified 13.8

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

12. Applied add-log-exp 13.8

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

14. Applied add-cbrt-cube 13.8

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

15. Simplified 13.8

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

16. Final simplification 13.8

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

Reproduce

herbie shell --seed 2019196 
(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))))))))))