Average Error: 38.8 → 28.6
Time: 36.2s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 4.605055345730565045055879951608156666024 \cdot 10^{305}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
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 4.605055345730565045055879951608156666024 \cdot 10^{305}:\\
\;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}{\lambda_1 + \lambda_2} + \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 r83661 = R;
        double r83662 = lambda1;
        double r83663 = lambda2;
        double r83664 = r83662 - r83663;
        double r83665 = phi1;
        double r83666 = phi2;
        double r83667 = r83665 + r83666;
        double r83668 = 2.0;
        double r83669 = r83667 / r83668;
        double r83670 = cos(r83669);
        double r83671 = r83664 * r83670;
        double r83672 = r83671 * r83671;
        double r83673 = r83665 - r83666;
        double r83674 = r83673 * r83673;
        double r83675 = r83672 + r83674;
        double r83676 = sqrt(r83675);
        double r83677 = r83661 * r83676;
        return r83677;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r83678 = lambda1;
        double r83679 = lambda2;
        double r83680 = r83678 - r83679;
        double r83681 = phi1;
        double r83682 = phi2;
        double r83683 = r83681 + r83682;
        double r83684 = 2.0;
        double r83685 = r83683 / r83684;
        double r83686 = cos(r83685);
        double r83687 = r83680 * r83686;
        double r83688 = r83687 * r83687;
        double r83689 = r83681 - r83682;
        double r83690 = r83689 * r83689;
        double r83691 = r83688 + r83690;
        double r83692 = 4.605055345730565e+305;
        bool r83693 = r83691 <= r83692;
        double r83694 = R;
        double r83695 = r83678 * r83678;
        double r83696 = r83679 * r83679;
        double r83697 = r83695 - r83696;
        double r83698 = r83697 * r83686;
        double r83699 = r83678 + r83679;
        double r83700 = r83698 / r83699;
        double r83701 = r83687 * r83700;
        double r83702 = r83701 + r83690;
        double r83703 = sqrt(r83702);
        double r83704 = r83694 * r83703;
        double r83705 = r83682 - r83681;
        double r83706 = r83694 * r83705;
        double r83707 = r83693 ? r83704 : r83706;
        return r83707;
}

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))) < 4.605055345730565e+305

    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 flip--1.8

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

    4. Applied associate-*l/1.8

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

    if 4.605055345730565e+305 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 63.5

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

    2. Taylor expanded around 0 46.4

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

  4. Final simplification28.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) \le 4.605055345730565045055879951608156666024 \cdot 10^{305}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \frac{\left(\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)}{\lambda_1 + \lambda_2} + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))))