Average Error: 37.2 → 28.9
Time: 4.1m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 8.059112047867875 \cdot 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]
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 8.059112047867875 \cdot 10^{+307}:\\
\;\;\;\;R \cdot \sqrt{\left(\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) \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 r6989629 = R;
        double r6989630 = lambda1;
        double r6989631 = lambda2;
        double r6989632 = r6989630 - r6989631;
        double r6989633 = phi1;
        double r6989634 = phi2;
        double r6989635 = r6989633 + r6989634;
        double r6989636 = 2.0;
        double r6989637 = r6989635 / r6989636;
        double r6989638 = cos(r6989637);
        double r6989639 = r6989632 * r6989638;
        double r6989640 = r6989639 * r6989639;
        double r6989641 = r6989633 - r6989634;
        double r6989642 = r6989641 * r6989641;
        double r6989643 = r6989640 + r6989642;
        double r6989644 = sqrt(r6989643);
        double r6989645 = r6989629 * r6989644;
        return r6989645;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r6989646 = phi1;
        double r6989647 = phi2;
        double r6989648 = r6989646 - r6989647;
        double r6989649 = r6989648 * r6989648;
        double r6989650 = lambda1;
        double r6989651 = lambda2;
        double r6989652 = r6989650 - r6989651;
        double r6989653 = r6989646 + r6989647;
        double r6989654 = 2.0;
        double r6989655 = r6989653 / r6989654;
        double r6989656 = cos(r6989655);
        double r6989657 = r6989652 * r6989656;
        double r6989658 = r6989657 * r6989657;
        double r6989659 = r6989649 + r6989658;
        double r6989660 = 8.059112047867875e+307;
        bool r6989661 = r6989659 <= r6989660;
        double r6989662 = R;
        double r6989663 = cbrt(r6989656);
        double r6989664 = r6989663 * r6989663;
        double r6989665 = r6989664 * r6989663;
        double r6989666 = r6989665 * r6989652;
        double r6989667 = r6989666 * r6989657;
        double r6989668 = r6989667 + r6989649;
        double r6989669 = sqrt(r6989668);
        double r6989670 = r6989662 * r6989669;
        double r6989671 = r6989647 - r6989646;
        double r6989672 = r6989671 * r6989662;
        double r6989673 = r6989661 ? r6989670 : r6989672;
        return r6989673;
}

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))) < 8.059112047867875e+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-cbrt1.8

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

    if 8.059112047867875e+307 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 60.8

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

    2. Taylor expanded around 0 46.8

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

  4. Final simplification28.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 8.059112047867875 \cdot 10^{+307}:\\ \;\;\;\;R \cdot \sqrt{\left(\left(\left(\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))))))