Average Error: 39.2 → 3.6
Time: 6.6s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R\]
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(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r80509 = R;
        double r80510 = lambda1;
        double r80511 = lambda2;
        double r80512 = r80510 - r80511;
        double r80513 = phi1;
        double r80514 = phi2;
        double r80515 = r80513 + r80514;
        double r80516 = 2.0;
        double r80517 = r80515 / r80516;
        double r80518 = cos(r80517);
        double r80519 = r80512 * r80518;
        double r80520 = r80519 * r80519;
        double r80521 = r80513 - r80514;
        double r80522 = r80521 * r80521;
        double r80523 = r80520 + r80522;
        double r80524 = sqrt(r80523);
        double r80525 = r80509 * r80524;
        return r80525;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r80526 = lambda1;
        double r80527 = lambda2;
        double r80528 = r80526 - r80527;
        double r80529 = phi1;
        double r80530 = phi2;
        double r80531 = r80529 + r80530;
        double r80532 = 2.0;
        double r80533 = r80531 / r80532;
        double r80534 = cos(r80533);
        double r80535 = r80528 * r80534;
        double r80536 = r80529 - r80530;
        double r80537 = hypot(r80535, r80536);
        double r80538 = R;
        double r80539 = r80537 * r80538;
        return r80539;
}

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 39.2

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

  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 pow13.6

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

  5. Applied pow13.6

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

  6. Applied pow-prod-down3.6

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

  7. Final simplification3.6

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

Reproduce

herbie shell --seed 2020060 +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))))))