Average Error: 38.5 → 28.8
Time: 10.3s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 6.9214802068712766 \cdot 10^{307}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
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 6.9214802068712766 \cdot 10^{307}:\\
\;\;\;\;R \cdot \sqrt{\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) \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}:\\
\;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r158011 = R;
        double r158012 = lambda1;
        double r158013 = lambda2;
        double r158014 = r158012 - r158013;
        double r158015 = phi1;
        double r158016 = phi2;
        double r158017 = r158015 + r158016;
        double r158018 = 2.0;
        double r158019 = r158017 / r158018;
        double r158020 = cos(r158019);
        double r158021 = r158014 * r158020;
        double r158022 = r158021 * r158021;
        double r158023 = r158015 - r158016;
        double r158024 = r158023 * r158023;
        double r158025 = r158022 + r158024;
        double r158026 = sqrt(r158025);
        double r158027 = r158011 * r158026;
        return r158027;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r158028 = lambda1;
        double r158029 = lambda2;
        double r158030 = r158028 - r158029;
        double r158031 = phi1;
        double r158032 = phi2;
        double r158033 = r158031 + r158032;
        double r158034 = 2.0;
        double r158035 = r158033 / r158034;
        double r158036 = cos(r158035);
        double r158037 = r158030 * r158036;
        double r158038 = r158037 * r158037;
        double r158039 = r158031 - r158032;
        double r158040 = r158039 * r158039;
        double r158041 = r158038 + r158040;
        double r158042 = 6.921480206871277e+307;
        bool r158043 = r158041 <= r158042;
        double r158044 = R;
        double r158045 = cbrt(r158036);
        double r158046 = r158045 * r158045;
        double r158047 = r158046 * r158045;
        double r158048 = r158030 * r158047;
        double r158049 = r158048 * r158037;
        double r158050 = r158049 + r158040;
        double r158051 = sqrt(r158050);
        double r158052 = r158044 * r158051;
        double r158053 = r158032 - r158031;
        double r158054 = r158044 * r158053;
        double r158055 = r158043 ? r158052 : r158054;
        return r158055;
}

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))) < 6.921480206871277e+307

    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

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

    if 6.921480206871277e+307 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 63.9

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

    2. Taylor expanded around 0 47.5

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

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 6.9214802068712766 \cdot 10^{307}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))))))