Average Error: 38.5 → 0.1
Time: 9.2s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sqrt[3]{{\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]
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(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sqrt[3]{{\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r99541 = R;
        double r99542 = lambda1;
        double r99543 = lambda2;
        double r99544 = r99542 - r99543;
        double r99545 = phi1;
        double r99546 = phi2;
        double r99547 = r99545 + r99546;
        double r99548 = 2.0;
        double r99549 = r99547 / r99548;
        double r99550 = cos(r99549);
        double r99551 = r99544 * r99550;
        double r99552 = r99551 * r99551;
        double r99553 = r99545 - r99546;
        double r99554 = r99553 * r99553;
        double r99555 = r99552 + r99554;
        double r99556 = sqrt(r99555);
        double r99557 = r99541 * r99556;
        return r99557;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r99558 = phi2;
        double r99559 = 0.5;
        double r99560 = r99558 * r99559;
        double r99561 = cos(r99560);
        double r99562 = phi1;
        double r99563 = r99562 * r99559;
        double r99564 = cos(r99563);
        double r99565 = r99561 * r99564;
        double r99566 = sin(r99560);
        double r99567 = sin(r99563);
        double r99568 = r99566 * r99567;
        double r99569 = 3.0;
        double r99570 = pow(r99568, r99569);
        double r99571 = cbrt(r99570);
        double r99572 = r99565 - r99571;
        double r99573 = lambda1;
        double r99574 = lambda2;
        double r99575 = r99573 - r99574;
        double r99576 = r99572 * r99575;
        double r99577 = r99562 - r99558;
        double r99578 = hypot(r99576, r99577);
        double r99579 = R;
        double r99580 = r99578 * r99579;
        return r99580;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.5

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

  2. Simplified3.7

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

  3. Taylor expanded around inf 3.7

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_1 - \lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  4. Simplified3.7

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \cdot R\]
  5. Using strategy rm

  6. Applied distribute-lft-in3.7

    \[\leadsto \mathsf{hypot}\left(\cos \color{blue}{\left(0.5 \cdot \phi_2 + 0.5 \cdot \phi_1\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  7. Applied cos-sum0.1

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  8. Simplified0.1

    \[\leadsto \mathsf{hypot}\left(\left(\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  9. Simplified0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  12. Applied add-cbrt-cube0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\sqrt[3]{\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)}} \cdot \sqrt[3]{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  13. Applied cbrt-unprod0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  14. Simplified0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sqrt[3]{\color{blue}{{\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

  15. Final simplification0.1

    \[\leadsto \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sqrt[3]{{\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{3}}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))