Average Error: 39.4 → 3.6
Time: 6.5s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{3}}, \phi_1 - \phi_2\right) \cdot R\]
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(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{3}}, \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r91614 = R;
        double r91615 = lambda1;
        double r91616 = lambda2;
        double r91617 = r91615 - r91616;
        double r91618 = phi1;
        double r91619 = phi2;
        double r91620 = r91618 + r91619;
        double r91621 = 2.0;
        double r91622 = r91620 / r91621;
        double r91623 = cos(r91622);
        double r91624 = r91617 * r91623;
        double r91625 = r91624 * r91624;
        double r91626 = r91618 - r91619;
        double r91627 = r91626 * r91626;
        double r91628 = r91625 + r91627;
        double r91629 = sqrt(r91628);
        double r91630 = r91614 * r91629;
        return r91630;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r91631 = lambda1;
        double r91632 = lambda2;
        double r91633 = r91631 - r91632;
        double r91634 = phi1;
        double r91635 = phi2;
        double r91636 = r91634 + r91635;
        double r91637 = 2.0;
        double r91638 = r91636 / r91637;
        double r91639 = cos(r91638);
        double r91640 = 3.0;
        double r91641 = pow(r91639, r91640);
        double r91642 = cbrt(r91641);
        double r91643 = r91633 * r91642;
        double r91644 = r91634 - r91635;
        double r91645 = hypot(r91643, r91644);
        double r91646 = R;
        double r91647 = r91645 * r91646;
        return r91647;
}

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 39.4

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

  2. Simplified3.6

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

  4. Applied add-cbrt-cube3.6

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}}, \phi_1 - \phi_2\right) \cdot R\]

  5. Simplified3.6

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\color{blue}{{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{3}}}, \phi_1 - \phi_2\right) \cdot R\]

  6. Final simplification3.6

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{3}}, \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2020039 +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))))))