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 r681025 = lambda1;
        double r681026 = phi2;
        double r681027 = cos(r681026);
        double r681028 = lambda2;
        double r681029 = r681025 - r681028;
        double r681030 = sin(r681029);
        double r681031 = r681027 * r681030;
        double r681032 = phi1;
        double r681033 = cos(r681032);
        double r681034 = cos(r681029);
        double r681035 = r681027 * r681034;
        double r681036 = r681033 + r681035;
        double r681037 = atan2(r681031, r681036);
        double r681038 = r681025 + r681037;
        return r681038;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r681039 = lambda1;
        double r681040 = lambda2;
        double r681041 = r681039 - r681040;
        double r681042 = sin(r681041);
        double r681043 = phi2;
        double r681044 = cos(r681043);
        double r681045 = r681042 * r681044;
        double r681046 = cos(r681041);
        double r681047 = phi1;
        double r681048 = cos(r681047);
        double r681049 = fma(r681044, r681046, r681048);
        double r681050 = atan2(r681045, r681049);
        double r681051 = r681039 + r681050;
        return r681051;
}

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 2019138 +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)))))))