Average Error: 0 → 0
Time: 8.2m
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 r8215427 = lambda1;
        double r8215428 = phi2;
        double r8215429 = cos(r8215428);
        double r8215430 = lambda2;
        double r8215431 = r8215427 - r8215430;
        double r8215432 = sin(r8215431);
        double r8215433 = r8215429 * r8215432;
        double r8215434 = phi1;
        double r8215435 = cos(r8215434);
        double r8215436 = cos(r8215431);
        double r8215437 = r8215429 * r8215436;
        double r8215438 = r8215435 + r8215437;
        double r8215439 = atan2(r8215433, r8215438);
        double r8215440 = r8215427 + r8215439;
        return r8215440;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r8215441 = lambda1;
        double r8215442 = lambda2;
        double r8215443 = r8215441 - r8215442;
        double r8215444 = sin(r8215443);
        double r8215445 = phi2;
        double r8215446 = cos(r8215445);
        double r8215447 = r8215444 * r8215446;
        double r8215448 = cos(r8215443);
        double r8215449 = phi1;
        double r8215450 = cos(r8215449);
        double r8215451 = fma(r8215446, r8215448, r8215450);
        double r8215452 = atan2(r8215447, r8215451);
        double r8215453 = r8215441 + r8215452;
        return r8215453;
}

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 2019128 +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)))))))