Average Error: 37.4 → 0.1
Time: 29.8s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_1, \left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) - \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) - \lambda_1 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}
↓
\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_1, \left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) - \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) - \lambda_1 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3847613 = R;
        double r3847614 = lambda1;
        double r3847615 = lambda2;
        double r3847616 = r3847614 - r3847615;
        double r3847617 = phi1;
        double r3847618 = phi2;
        double r3847619 = r3847617 + r3847618;
        double r3847620 = 2.0;
        double r3847621 = r3847619 / r3847620;
        double r3847622 = cos(r3847621);
        double r3847623 = r3847616 * r3847622;
        double r3847624 = r3847623 * r3847623;
        double r3847625 = r3847617 - r3847618;
        double r3847626 = r3847625 * r3847625;
        double r3847627 = r3847624 + r3847626;
        double r3847628 = sqrt(r3847627);
        double r3847629 = r3847613 * r3847628;
        return r3847629;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3847630 = phi1;
        double r3847631 = 0.5;
        double r3847632 = r3847630 * r3847631;
        double r3847633 = cos(r3847632);
        double r3847634 = phi2;
        double r3847635 = r3847634 * r3847631;
        double r3847636 = cos(r3847635);
        double r3847637 = lambda1;
        double r3847638 = r3847636 * r3847637;
        double r3847639 = lambda2;
        double r3847640 = sin(r3847632);
        double r3847641 = sin(r3847635);
        double r3847642 = r3847640 * r3847641;
        double r3847643 = r3847639 * r3847642;
        double r3847644 = r3847639 * r3847636;
        double r3847645 = r3847644 * r3847633;
        double r3847646 = r3847643 - r3847645;
        double r3847647 = r3847637 * r3847642;
        double r3847648 = r3847646 - r3847647;
        double r3847649 = fma(r3847633, r3847638, r3847648);
        double r3847650 = r3847630 - r3847634;
        double r3847651 = hypot(r3847649, r3847650);
        double r3847652 = R;
        double r3847653 = r3847651 * r3847652;
        return r3847653;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 37.4

    \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

  2. Simplified4.0

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \cdot R}\]

  3. Taylor expanded around inf 4.0

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]
  4. Using strategy rm

  5. Applied distribute-rgt-in4.0

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2} + \phi_2 \cdot \frac{1}{2}\right)}, \phi_1 - \phi_2\right) \cdot R\]

  6. Applied cos-sum0.1

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  7. Taylor expanded around inf 0.1

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right) + \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_2\right)\right) - \left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \lambda_1\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  8. Simplified0.1

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right), \lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) - \lambda_1 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  9. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \lambda_1, \left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right) - \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)\right) - \lambda_1 \cdot \left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))