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 r1713634 = lambda1;
        double r1713635 = phi2;
        double r1713636 = cos(r1713635);
        double r1713637 = lambda2;
        double r1713638 = r1713634 - r1713637;
        double r1713639 = sin(r1713638);
        double r1713640 = r1713636 * r1713639;
        double r1713641 = phi1;
        double r1713642 = cos(r1713641);
        double r1713643 = cos(r1713638);
        double r1713644 = r1713636 * r1713643;
        double r1713645 = r1713642 + r1713644;
        double r1713646 = atan2(r1713640, r1713645);
        double r1713647 = r1713634 + r1713646;
        return r1713647;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1713648 = lambda1;
        double r1713649 = lambda2;
        double r1713650 = r1713648 - r1713649;
        double r1713651 = sin(r1713650);
        double r1713652 = phi2;
        double r1713653 = cos(r1713652);
        double r1713654 = r1713651 * r1713653;
        double r1713655 = cos(r1713650);
        double r1713656 = phi1;
        double r1713657 = cos(r1713656);
        double r1713658 = fma(r1713653, r1713655, r1713657);
        double r1713659 = atan2(r1713654, r1713658);
        double r1713660 = r1713648 + r1713659;
        return r1713660;
}

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 2019121 +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)))))))