Average Error: 37.8 → 30.1
Time: 2.7m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.8901552595312907 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\left(\lambda_2 \cdot \left(-\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \log \left(e^{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}}\right) \cdot \lambda_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r35109616 = R;
        double r35109617 = lambda1;
        double r35109618 = lambda2;
        double r35109619 = r35109617 - r35109618;
        double r35109620 = phi1;
        double r35109621 = phi2;
        double r35109622 = r35109620 + r35109621;
        double r35109623 = 2.0;
        double r35109624 = r35109622 / r35109623;
        double r35109625 = cos(r35109624);
        double r35109626 = r35109619 * r35109625;
        double r35109627 = r35109626 * r35109626;
        double r35109628 = r35109620 - r35109621;
        double r35109629 = r35109628 * r35109628;
        double r35109630 = r35109627 + r35109629;
        double r35109631 = sqrt(r35109630);
        double r35109632 = r35109616 * r35109631;
        return r35109632;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r35109633 = phi1;
        double r35109634 = phi2;
        double r35109635 = r35109633 - r35109634;
        double r35109636 = r35109635 * r35109635;
        double r35109637 = lambda1;
        double r35109638 = lambda2;
        double r35109639 = r35109637 - r35109638;
        double r35109640 = r35109633 + r35109634;
        double r35109641 = 2.0;
        double r35109642 = r35109640 / r35109641;
        double r35109643 = cos(r35109642);
        double r35109644 = r35109639 * r35109643;
        double r35109645 = r35109644 * r35109644;
        double r35109646 = r35109636 + r35109645;
        double r35109647 = 1.8901552595312907e+297;
        bool r35109648 = r35109646 <= r35109647;
        double r35109649 = -r35109643;
        double r35109650 = r35109638 * r35109649;
        double r35109651 = r35109643 * r35109643;
        double r35109652 = r35109643 * r35109651;
        double r35109653 = cbrt(r35109652);
        double r35109654 = exp(r35109653);
        double r35109655 = log(r35109654);
        double r35109656 = r35109655 * r35109637;
        double r35109657 = r35109650 + r35109656;
        double r35109658 = r35109657 * r35109644;
        double r35109659 = r35109658 + r35109636;
        double r35109660 = sqrt(r35109659);
        double r35109661 = R;
        double r35109662 = r35109660 * r35109661;
        double r35109663 = r35109634 - r35109633;
        double r35109664 = r35109661 * r35109663;
        double r35109665 = r35109648 ? r35109662 : r35109664;
        return r35109665;
}


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 1.8901552595312907 \cdot 10^{+297}:\\
\;\;\;\;\sqrt{\left(\lambda_2 \cdot \left(-\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \log \left(e^{\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)}}\right) \cdot \lambda_1\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)} \cdot R\\

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

\end{array}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

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))) < 1.8901552595312907e+297

    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 *-commutative1.8

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

    5. Applied sub-neg1.8

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

    6. Applied distribute-rgt-in1.8

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

    8. Applied add-log-exp1.8

      \[\leadsto R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\lambda_1 \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)} + \left(-\lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube1.8

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

    if 1.8901552595312907e+297 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 59.6

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

    3. Applied *-commutative59.6

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

    4. 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 simplification30.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.8901552595312907 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\left(\lambda_2 \cdot \left(-\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \log \left(e^{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}}\right) \cdot \lambda_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(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))))))