Average Error: 0 → 0
Time: 6.5m
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 r917664 = lambda1;
        double r917665 = phi2;
        double r917666 = cos(r917665);
        double r917667 = lambda2;
        double r917668 = r917664 - r917667;
        double r917669 = sin(r917668);
        double r917670 = r917666 * r917669;
        double r917671 = phi1;
        double r917672 = cos(r917671);
        double r917673 = cos(r917668);
        double r917674 = r917666 * r917673;
        double r917675 = r917672 + r917674;
        double r917676 = atan2(r917670, r917675);
        double r917677 = r917664 + r917676;
        return r917677;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r917678 = lambda1;
        double r917679 = lambda2;
        double r917680 = r917678 - r917679;
        double r917681 = sin(r917680);
        double r917682 = phi2;
        double r917683 = cos(r917682);
        double r917684 = r917681 * r917683;
        double r917685 = cos(r917680);
        double r917686 = phi1;
        double r917687 = cos(r917686);
        double r917688 = fma(r917683, r917685, r917687);
        double r917689 = atan2(r917684, r917688);
        double r917690 = r917678 + r917689;
        return r917690;
}

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 2019141 +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)))))))