Distance on a great circle

Average Error: 24.0 → 24.1

Time: 58.0s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1653475 = R;
        double r1653476 = 2.0;
        double r1653477 = phi1;
        double r1653478 = phi2;
        double r1653479 = r1653477 - r1653478;
        double r1653480 = r1653479 / r1653476;
        double r1653481 = sin(r1653480);
        double r1653482 = pow(r1653481, r1653476);
        double r1653483 = cos(r1653477);
        double r1653484 = cos(r1653478);
        double r1653485 = r1653483 * r1653484;
        double r1653486 = lambda1;
        double r1653487 = lambda2;
        double r1653488 = r1653486 - r1653487;
        double r1653489 = r1653488 / r1653476;
        double r1653490 = sin(r1653489);
        double r1653491 = r1653485 * r1653490;
        double r1653492 = r1653491 * r1653490;
        double r1653493 = r1653482 + r1653492;
        double r1653494 = sqrt(r1653493);
        double r1653495 = 1.0;
        double r1653496 = r1653495 - r1653493;
        double r1653497 = sqrt(r1653496);
        double r1653498 = atan2(r1653494, r1653497);
        double r1653499 = r1653476 * r1653498;
        double r1653500 = r1653475 * r1653499;
        return r1653500;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r1653501 = phi2;
        double r1653502 = cos(r1653501);
        double r1653503 = lambda1;
        double r1653504 = lambda2;
        double r1653505 = r1653503 - r1653504;
        double r1653506 = 2.0;
        double r1653507 = r1653505 / r1653506;
        double r1653508 = sin(r1653507);
        double r1653509 = r1653502 * r1653508;
        double r1653510 = phi1;
        double r1653511 = cos(r1653510);
        double r1653512 = r1653508 * r1653511;
        double r1653513 = r1653510 - r1653501;
        double r1653514 = r1653513 / r1653506;
        double r1653515 = sin(r1653514);
        double r1653516 = r1653515 * r1653515;
        double r1653517 = fma(r1653509, r1653512, r1653516);
        double r1653518 = sqrt(r1653517);
        double r1653519 = cos(r1653514);
        double r1653520 = r1653519 * r1653519;
        double r1653521 = cos(r1653505);
        double r1653522 = r1653504 / r1653506;
        double r1653523 = cos(r1653522);
        double r1653524 = r1653503 / r1653506;
        double r1653525 = sin(r1653524);
        double r1653526 = r1653523 * r1653525;
        double r1653527 = sin(r1653522);
        double r1653528 = cos(r1653524);
        double r1653529 = r1653527 * r1653528;
        double r1653530 = r1653526 - r1653529;
        double r1653531 = r1653521 * r1653530;
        double r1653532 = r1653508 - r1653531;
        double r1653533 = cbrt(r1653532);
        double r1653534 = cbrt(r1653506);
        double r1653535 = r1653533 / r1653534;
        double r1653536 = r1653535 * r1653511;
        double r1653537 = r1653536 * r1653509;
        double r1653538 = r1653520 - r1653537;
        double r1653539 = sqrt(r1653538);
        double r1653540 = atan2(r1653518, r1653539);
        double r1653541 = R;
        double r1653542 = r1653541 * r1653506;
        double r1653543 = r1653540 * r1653542;
        return r1653543;
}

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 24.0

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

2. Simplified 24.0

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

4. Applied add-cbrt-cube 24.0

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

6. Applied sin-mult 24.0

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

7. Applied associate-*l/ 24.0

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

8. Applied cbrt-div 24.0

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

9. Simplified 24.0

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

11. Applied div-sub 24.0

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

12. Applied sin-diff 24.1

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

13. Final simplification 24.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(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))))))))))