Average Error: 0 → 0
Time: 8.3m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]
\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}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2833986 = lambda1;
        double r2833987 = phi2;
        double r2833988 = cos(r2833987);
        double r2833989 = lambda2;
        double r2833990 = r2833986 - r2833989;
        double r2833991 = sin(r2833990);
        double r2833992 = r2833988 * r2833991;
        double r2833993 = phi1;
        double r2833994 = cos(r2833993);
        double r2833995 = cos(r2833990);
        double r2833996 = r2833988 * r2833995;
        double r2833997 = r2833994 + r2833996;
        double r2833998 = atan2(r2833992, r2833997);
        double r2833999 = r2833986 + r2833998;
        return r2833999;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2834000 = lambda1;
        double r2834001 = lambda2;
        double r2834002 = r2834000 - r2834001;
        double r2834003 = sin(r2834002);
        double r2834004 = phi2;
        double r2834005 = cos(r2834004);
        double r2834006 = r2834003 * r2834005;
        double r2834007 = cos(r2834002);
        double r2834008 = phi1;
        double r2834009 = cos(r2834008);
        double r2834010 = fma(r2834005, r2834007, r2834009);
        double r2834011 = atan2(r2834006, r2834010);
        double r2834012 = r2834000 + r2834011;
        return r2834012;
}

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

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019107 +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)))))))