Average Error: 0 → 0
Time: 7.9m
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 r2458468 = lambda1;
        double r2458469 = phi2;
        double r2458470 = cos(r2458469);
        double r2458471 = lambda2;
        double r2458472 = r2458468 - r2458471;
        double r2458473 = sin(r2458472);
        double r2458474 = r2458470 * r2458473;
        double r2458475 = phi1;
        double r2458476 = cos(r2458475);
        double r2458477 = cos(r2458472);
        double r2458478 = r2458470 * r2458477;
        double r2458479 = r2458476 + r2458478;
        double r2458480 = atan2(r2458474, r2458479);
        double r2458481 = r2458468 + r2458480;
        return r2458481;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2458482 = lambda1;
        double r2458483 = lambda2;
        double r2458484 = r2458482 - r2458483;
        double r2458485 = sin(r2458484);
        double r2458486 = phi2;
        double r2458487 = cos(r2458486);
        double r2458488 = r2458485 * r2458487;
        double r2458489 = cos(r2458484);
        double r2458490 = phi1;
        double r2458491 = cos(r2458490);
        double r2458492 = fma(r2458487, r2458489, r2458491);
        double r2458493 = atan2(r2458488, r2458492);
        double r2458494 = r2458482 + r2458493;
        return r2458494;
}

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 2019124 +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)))))))