Average Error: 0 → 0
Time: 6.9m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}\]
\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{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r716465 = lambda1;
        double r716466 = phi2;
        double r716467 = cos(r716466);
        double r716468 = lambda2;
        double r716469 = r716465 - r716468;
        double r716470 = sin(r716469);
        double r716471 = r716467 * r716470;
        double r716472 = phi1;
        double r716473 = cos(r716472);
        double r716474 = cos(r716469);
        double r716475 = r716467 * r716474;
        double r716476 = r716473 + r716475;
        double r716477 = atan2(r716471, r716476);
        double r716478 = r716465 + r716477;
        return r716478;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r716479 = lambda1;
        double r716480 = lambda2;
        double r716481 = r716479 - r716480;
        double r716482 = sin(r716481);
        double r716483 = phi2;
        double r716484 = cos(r716483);
        double r716485 = r716482 * r716484;
        double r716486 = cos(r716481);
        double r716487 = phi1;
        double r716488 = cos(r716487);
        double r716489 = fma(r716484, r716486, r716488);
        double r716490 = atan2(r716485, r716489);
        double r716491 = r716479 + r716490;
        return r716491;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0

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

  2. Simplified0

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))