Average Error: 0 → 0
Time: 8.3m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{(\left(\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\cos \phi_1\right))_*}\]
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3603899 = lambda1;
        double r3603900 = phi2;
        double r3603901 = cos(r3603900);
        double r3603902 = lambda2;
        double r3603903 = r3603899 - r3603902;
        double r3603904 = sin(r3603903);
        double r3603905 = r3603901 * r3603904;
        double r3603906 = phi1;
        double r3603907 = cos(r3603906);
        double r3603908 = cos(r3603903);
        double r3603909 = r3603901 * r3603908;
        double r3603910 = r3603907 + r3603909;
        double r3603911 = atan2(r3603905, r3603910);
        double r3603912 = r3603899 + r3603911;
        return r3603912;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3603913 = lambda1;
        double r3603914 = lambda2;
        double r3603915 = r3603913 - r3603914;
        double r3603916 = sin(r3603915);
        double r3603917 = phi2;
        double r3603918 = cos(r3603917);
        double r3603919 = r3603916 * r3603918;
        double r3603920 = cos(r3603915);
        double r3603921 = phi1;
        double r3603922 = cos(r3603921);
        double r3603923 = fma(r3603918, r3603920, r3603922);
        double r3603924 = atan2(r3603919, r3603923);
        double r3603925 = r3603913 + r3603924;
        return r3603925;
}


\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))_*}

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 2019102 +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)))))))