Average Error: 37.4 → 29.4
Time: 2.7m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.801232010353918 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]
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 1.801232010353918 \cdot 10^{+304}:\\
\;\;\;\;R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \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}:\\
\;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\

\end{array}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r8480856 = R;
        double r8480857 = lambda1;
        double r8480858 = lambda2;
        double r8480859 = r8480857 - r8480858;
        double r8480860 = phi1;
        double r8480861 = phi2;
        double r8480862 = r8480860 + r8480861;
        double r8480863 = 2.0;
        double r8480864 = r8480862 / r8480863;
        double r8480865 = cos(r8480864);
        double r8480866 = r8480859 * r8480865;
        double r8480867 = r8480866 * r8480866;
        double r8480868 = r8480860 - r8480861;
        double r8480869 = r8480868 * r8480868;
        double r8480870 = r8480867 + r8480869;
        double r8480871 = sqrt(r8480870);
        double r8480872 = r8480856 * r8480871;
        return r8480872;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r8480873 = phi1;
        double r8480874 = phi2;
        double r8480875 = r8480873 - r8480874;
        double r8480876 = r8480875 * r8480875;
        double r8480877 = lambda1;
        double r8480878 = lambda2;
        double r8480879 = r8480877 - r8480878;
        double r8480880 = r8480873 + r8480874;
        double r8480881 = 2.0;
        double r8480882 = r8480880 / r8480881;
        double r8480883 = cos(r8480882);
        double r8480884 = r8480879 * r8480883;
        double r8480885 = r8480884 * r8480884;
        double r8480886 = r8480876 + r8480885;
        double r8480887 = 1.801232010353918e+304;
        bool r8480888 = r8480886 <= r8480887;
        double r8480889 = R;
        double r8480890 = r8480879 * r8480884;
        double r8480891 = r8480883 * r8480890;
        double r8480892 = r8480891 + r8480876;
        double r8480893 = sqrt(r8480892);
        double r8480894 = r8480889 * r8480893;
        double r8480895 = r8480874 - r8480873;
        double r8480896 = r8480895 * r8480889;
        double r8480897 = r8480888 ? r8480894 : r8480896;
        return r8480897;
}

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))) < 1.801232010353918e+304

    1. Initial program 2.0

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

    3. Applied associate-*r*2.0

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

    if 1.801232010353918e+304 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 60.4

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

    2. Taylor expanded around 0 47.3

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.801232010353918 \cdot 10^{+304}:\\ \;\;\;\;R \cdot \sqrt{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))))))