Average Error: 24.2 → 24.2
Time: 52.6s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}} \cdot \left(R \cdot 2\right)\]
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)
↓
\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_1 \cdot \cos \phi_2, \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 \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}} \cdot \left(R \cdot 2\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4158473 = R;
        double r4158474 = 2.0;
        double r4158475 = phi1;
        double r4158476 = phi2;
        double r4158477 = r4158475 - r4158476;
        double r4158478 = r4158477 / r4158474;
        double r4158479 = sin(r4158478);
        double r4158480 = pow(r4158479, r4158474);
        double r4158481 = cos(r4158475);
        double r4158482 = cos(r4158476);
        double r4158483 = r4158481 * r4158482;
        double r4158484 = lambda1;
        double r4158485 = lambda2;
        double r4158486 = r4158484 - r4158485;
        double r4158487 = r4158486 / r4158474;
        double r4158488 = sin(r4158487);
        double r4158489 = r4158483 * r4158488;
        double r4158490 = r4158489 * r4158488;
        double r4158491 = r4158480 + r4158490;
        double r4158492 = sqrt(r4158491);
        double r4158493 = 1.0;
        double r4158494 = r4158493 - r4158491;
        double r4158495 = sqrt(r4158494);
        double r4158496 = atan2(r4158492, r4158495);
        double r4158497 = r4158474 * r4158496;
        double r4158498 = r4158473 * r4158497;
        return r4158498;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4158499 = lambda1;
        double r4158500 = lambda2;
        double r4158501 = r4158499 - r4158500;
        double r4158502 = 2.0;
        double r4158503 = r4158501 / r4158502;
        double r4158504 = sin(r4158503);
        double r4158505 = r4158504 * r4158504;
        double r4158506 = phi1;
        double r4158507 = cos(r4158506);
        double r4158508 = phi2;
        double r4158509 = cos(r4158508);
        double r4158510 = r4158507 * r4158509;
        double r4158511 = r4158506 - r4158508;
        double r4158512 = r4158511 / r4158502;
        double r4158513 = sin(r4158512);
        double r4158514 = r4158513 * r4158513;
        double r4158515 = fma(r4158505, r4158510, r4158514);
        double r4158516 = sqrt(r4158515);
        double r4158517 = cos(r4158512);
        double r4158518 = r4158517 * r4158517;
        double r4158519 = exp(r4158504);
        double r4158520 = log(r4158519);
        double r4158521 = r4158505 * r4158520;
        double r4158522 = cbrt(r4158521);
        double r4158523 = expm1(r4158522);
        double r4158524 = log1p(r4158523);
        double r4158525 = r4158504 * r4158524;
        double r4158526 = r4158525 * r4158510;
        double r4158527 = r4158518 - r4158526;
        double r4158528 = sqrt(r4158527);
        double r4158529 = atan2(r4158516, r4158528);
        double r4158530 = R;
        double r4158531 = r4158530 * r4158502;
        double r4158532 = r4158529 * r4158531;
        return r4158532;
}

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. Simplified24.2

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

  4. Applied log1p-expm1-u24.2

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube24.2

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\]
  7. Using strategy rm

  8. Applied add-log-exp24.2

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\]

  9. Final simplification24.2

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}} \cdot \left(R \cdot 2\right)\]

Reproduce

herbie shell --seed 2019143 +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))))))))))