Average Error: 39.0 → 29.5
Time: 10.5s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 2.94628321762693042 \cdot 10^{297}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
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)}
↓
\begin{array}{l}
\mathbf{if}\;\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) \le 2.94628321762693042 \cdot 10^{297}:\\
\;\;\;\;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 \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\

\mathbf{else}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r77486 = R;
        double r77487 = lambda1;
        double r77488 = lambda2;
        double r77489 = r77487 - r77488;
        double r77490 = phi1;
        double r77491 = phi2;
        double r77492 = r77490 + r77491;
        double r77493 = 2.0;
        double r77494 = r77492 / r77493;
        double r77495 = cos(r77494);
        double r77496 = r77489 * r77495;
        double r77497 = r77496 * r77496;
        double r77498 = r77490 - r77491;
        double r77499 = r77498 * r77498;
        double r77500 = r77497 + r77499;
        double r77501 = sqrt(r77500);
        double r77502 = r77486 * r77501;
        return r77502;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r77503 = lambda1;
        double r77504 = lambda2;
        double r77505 = r77503 - r77504;
        double r77506 = phi1;
        double r77507 = phi2;
        double r77508 = r77506 + r77507;
        double r77509 = 2.0;
        double r77510 = r77508 / r77509;
        double r77511 = cos(r77510);
        double r77512 = r77505 * r77511;
        double r77513 = r77512 * r77512;
        double r77514 = r77506 - r77507;
        double r77515 = r77514 * r77514;
        double r77516 = r77513 + r77515;
        double r77517 = 2.9462832176269304e+297;
        bool r77518 = r77516 <= r77517;
        double r77519 = R;
        double r77520 = cbrt(r77511);
        double r77521 = r77520 * r77520;
        double r77522 = r77521 * r77520;
        double r77523 = r77505 * r77522;
        double r77524 = r77512 * r77523;
        double r77525 = r77524 + r77515;
        double r77526 = sqrt(r77525);
        double r77527 = r77519 * r77526;
        double r77528 = r77507 - r77506;
        double r77529 = r77519 * r77528;
        double r77530 = r77518 ? r77527 : r77529;
        return r77530;
}

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. Split input into 2 regimes

  2. if (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))) < 2.9462832176269304e+297

    1. Initial program 1.8

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

    3. Applied add-cube-cbrt1.9

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

    if 2.9462832176269304e+297 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 62.3

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

    2. Taylor expanded around 0 46.9

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 2.94628321762693042 \cdot 10^{297}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))))))