Average Error: 0 → 0
Time: 6.6m
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 r747450 = lambda1;
        double r747451 = phi2;
        double r747452 = cos(r747451);
        double r747453 = lambda2;
        double r747454 = r747450 - r747453;
        double r747455 = sin(r747454);
        double r747456 = r747452 * r747455;
        double r747457 = phi1;
        double r747458 = cos(r747457);
        double r747459 = cos(r747454);
        double r747460 = r747452 * r747459;
        double r747461 = r747458 + r747460;
        double r747462 = atan2(r747456, r747461);
        double r747463 = r747450 + r747462;
        return r747463;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r747464 = lambda1;
        double r747465 = lambda2;
        double r747466 = r747464 - r747465;
        double r747467 = sin(r747466);
        double r747468 = phi2;
        double r747469 = cos(r747468);
        double r747470 = r747467 * r747469;
        double r747471 = cos(r747466);
        double r747472 = phi1;
        double r747473 = cos(r747472);
        double r747474 = fma(r747469, r747471, r747473);
        double r747475 = atan2(r747470, r747474);
        double r747476 = r747464 + r747475;
        return r747476;
}

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 2019142 +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)))))))