Average Error: 36.9 → 28.8
Time: 1.0m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 7.727116615399575 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]
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(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \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) \le 7.727116615399575 \cdot 10^{+304}:\\
\;\;\;\;R \cdot \sqrt{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_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)}\\

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

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2955569 = R;
        double r2955570 = lambda1;
        double r2955571 = lambda2;
        double r2955572 = r2955570 - r2955571;
        double r2955573 = phi1;
        double r2955574 = phi2;
        double r2955575 = r2955573 + r2955574;
        double r2955576 = 2.0;
        double r2955577 = r2955575 / r2955576;
        double r2955578 = cos(r2955577);
        double r2955579 = r2955572 * r2955578;
        double r2955580 = r2955579 * r2955579;
        double r2955581 = r2955573 - r2955574;
        double r2955582 = r2955581 * r2955581;
        double r2955583 = r2955580 + r2955582;
        double r2955584 = sqrt(r2955583);
        double r2955585 = r2955569 * r2955584;
        return r2955585;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2955586 = phi1;
        double r2955587 = phi2;
        double r2955588 = r2955586 - r2955587;
        double r2955589 = r2955588 * r2955588;
        double r2955590 = lambda1;
        double r2955591 = lambda2;
        double r2955592 = r2955590 - r2955591;
        double r2955593 = r2955586 + r2955587;
        double r2955594 = 2.0;
        double r2955595 = r2955593 / r2955594;
        double r2955596 = cos(r2955595);
        double r2955597 = r2955592 * r2955596;
        double r2955598 = r2955597 * r2955597;
        double r2955599 = r2955589 + r2955598;
        double r2955600 = 7.727116615399575e+304;
        bool r2955601 = r2955599 <= r2955600;
        double r2955602 = R;
        double r2955603 = r2955596 * r2955596;
        double r2955604 = r2955596 * r2955603;
        double r2955605 = cbrt(r2955604);
        double r2955606 = r2955605 * r2955592;
        double r2955607 = r2955606 * r2955597;
        double r2955608 = r2955607 + r2955589;
        double r2955609 = sqrt(r2955608);
        double r2955610 = r2955602 * r2955609;
        double r2955611 = r2955587 - r2955586;
        double r2955612 = r2955611 * r2955602;
        double r2955613 = r2955601 ? r2955610 : r2955612;
        return r2955613;
}

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))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))) < 7.727116615399575e+304

    1. Initial program 1.7

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

    3. Applied add-cbrt-cube1.8

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

    if 7.727116615399575e+304 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 60.2

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

    3. Applied add-cbrt-cube60.2

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

    4. Taylor expanded around 0 46.7

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

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 7.727116615399575 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))))))