Average Error: 0 → 0
Time: 8.0m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}\]
\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(\left(\cos \phi_2\right), \left(\cos \left(\lambda_1 - \lambda_2\right)\right), \left(\cos \phi_1\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2791241 = lambda1;
        double r2791242 = phi2;
        double r2791243 = cos(r2791242);
        double r2791244 = lambda2;
        double r2791245 = r2791241 - r2791244;
        double r2791246 = sin(r2791245);
        double r2791247 = r2791243 * r2791246;
        double r2791248 = phi1;
        double r2791249 = cos(r2791248);
        double r2791250 = cos(r2791245);
        double r2791251 = r2791243 * r2791250;
        double r2791252 = r2791249 + r2791251;
        double r2791253 = atan2(r2791247, r2791252);
        double r2791254 = r2791241 + r2791253;
        return r2791254;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2791255 = lambda1;
        double r2791256 = lambda2;
        double r2791257 = r2791255 - r2791256;
        double r2791258 = sin(r2791257);
        double r2791259 = phi2;
        double r2791260 = cos(r2791259);
        double r2791261 = r2791258 * r2791260;
        double r2791262 = cos(r2791257);
        double r2791263 = phi1;
        double r2791264 = cos(r2791263);
        double r2791265 = fma(r2791260, r2791262, r2791264);
        double r2791266 = atan2(r2791261, r2791265);
        double r2791267 = r2791255 + r2791266;
        return r2791267;
}

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

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

  3. Final simplification0

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

Reproduce

herbie shell --seed 2019120 +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)))))))