Average Error: 0.8 → 0.5
Time: 10.2s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)}\right) \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)}\right)}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)} \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)}\right) \cdot \sqrt[3]{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)}\right)}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2 \cdot \cos \phi_2, \cos \phi_1\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r49484 = lambda1;
        double r49485 = phi2;
        double r49486 = cos(r49485);
        double r49487 = lambda2;
        double r49488 = r49484 - r49487;
        double r49489 = sin(r49488);
        double r49490 = r49486 * r49489;
        double r49491 = phi1;
        double r49492 = cos(r49491);
        double r49493 = cos(r49488);
        double r49494 = r49486 * r49493;
        double r49495 = r49492 + r49494;
        double r49496 = atan2(r49490, r49495);
        double r49497 = r49484 + r49496;
        return r49497;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r49498 = lambda1;
        double r49499 = phi2;
        double r49500 = cos(r49499);
        double r49501 = sin(r49498);
        double r49502 = lambda2;
        double r49503 = cos(r49502);
        double r49504 = r49501 * r49503;
        double r49505 = cos(r49498);
        double r49506 = -r49502;
        double r49507 = sin(r49506);
        double r49508 = r49505 * r49507;
        double r49509 = cbrt(r49508);
        double r49510 = r49509 * r49509;
        double r49511 = r49510 * r49509;
        double r49512 = r49504 + r49511;
        double r49513 = r49500 * r49512;
        double r49514 = r49503 * r49500;
        double r49515 = phi1;
        double r49516 = cos(r49515);
        double r49517 = fma(r49505, r49514, r49516);
        double r49518 = sin(r49502);
        double r49519 = r49501 * r49518;
        double r49520 = r49500 * r49519;
        double r49521 = r49517 + r49520;
        double r49522 = atan2(r49513, r49521);
        double r49523 = r49498 + r49522;
        return r49523;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.8

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

  3. Applied cos-diff0.8

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

  4. Applied distribute-lft-in0.8

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

  5. Applied associate-+r+0.8

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

  6. Simplified0.8

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

  8. Applied sub-neg0.8

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

  9. Applied sin-sum0.2

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

  10. Simplified0.2

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

  12. Applied add-cube-cbrt0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))