Average Error: 0 → 0
Time: 6.5m
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 r541502 = lambda1;
        double r541503 = phi2;
        double r541504 = cos(r541503);
        double r541505 = lambda2;
        double r541506 = r541502 - r541505;
        double r541507 = sin(r541506);
        double r541508 = r541504 * r541507;
        double r541509 = phi1;
        double r541510 = cos(r541509);
        double r541511 = cos(r541506);
        double r541512 = r541504 * r541511;
        double r541513 = r541510 + r541512;
        double r541514 = atan2(r541508, r541513);
        double r541515 = r541502 + r541514;
        return r541515;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r541516 = lambda1;
        double r541517 = lambda2;
        double r541518 = r541516 - r541517;
        double r541519 = sin(r541518);
        double r541520 = phi2;
        double r541521 = cos(r541520);
        double r541522 = r541519 * r541521;
        double r541523 = cos(r541518);
        double r541524 = phi1;
        double r541525 = cos(r541524);
        double r541526 = fma(r541521, r541523, r541525);
        double r541527 = atan2(r541522, r541526);
        double r541528 = r541516 + r541527;
        return r541528;
}

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 2019134 +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)))))))