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 r2853928 = lambda1;
        double r2853929 = phi2;
        double r2853930 = cos(r2853929);
        double r2853931 = lambda2;
        double r2853932 = r2853928 - r2853931;
        double r2853933 = sin(r2853932);
        double r2853934 = r2853930 * r2853933;
        double r2853935 = phi1;
        double r2853936 = cos(r2853935);
        double r2853937 = cos(r2853932);
        double r2853938 = r2853930 * r2853937;
        double r2853939 = r2853936 + r2853938;
        double r2853940 = atan2(r2853934, r2853939);
        double r2853941 = r2853928 + r2853940;
        return r2853941;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2853942 = lambda1;
        double r2853943 = lambda2;
        double r2853944 = r2853942 - r2853943;
        double r2853945 = sin(r2853944);
        double r2853946 = phi2;
        double r2853947 = cos(r2853946);
        double r2853948 = r2853945 * r2853947;
        double r2853949 = cos(r2853944);
        double r2853950 = phi1;
        double r2853951 = cos(r2853950);
        double r2853952 = fma(r2853947, r2853949, r2853951);
        double r2853953 = atan2(r2853948, r2853952);
        double r2853954 = r2853942 + r2853953;
        return r2853954;
}

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 2019104 +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)))))))