Average Error: 39.5 → 29.6
Time: 2.8m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.85335802842088590346712744457318896647 \cdot 10^{302}:\\ \;\;\;\;\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]
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 1.85335802842088590346712744457318896647 \cdot 10^{302}:\\
\;\;\;\;\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\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)} \cdot R\\

\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 r10564464 = R;
        double r10564465 = lambda1;
        double r10564466 = lambda2;
        double r10564467 = r10564465 - r10564466;
        double r10564468 = phi1;
        double r10564469 = phi2;
        double r10564470 = r10564468 + r10564469;
        double r10564471 = 2.0;
        double r10564472 = r10564470 / r10564471;
        double r10564473 = cos(r10564472);
        double r10564474 = r10564467 * r10564473;
        double r10564475 = r10564474 * r10564474;
        double r10564476 = r10564468 - r10564469;
        double r10564477 = r10564476 * r10564476;
        double r10564478 = r10564475 + r10564477;
        double r10564479 = sqrt(r10564478);
        double r10564480 = r10564464 * r10564479;
        return r10564480;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r10564481 = phi1;
        double r10564482 = phi2;
        double r10564483 = r10564481 - r10564482;
        double r10564484 = r10564483 * r10564483;
        double r10564485 = lambda1;
        double r10564486 = lambda2;
        double r10564487 = r10564485 - r10564486;
        double r10564488 = r10564481 + r10564482;
        double r10564489 = 2.0;
        double r10564490 = r10564488 / r10564489;
        double r10564491 = cos(r10564490);
        double r10564492 = r10564487 * r10564491;
        double r10564493 = r10564492 * r10564492;
        double r10564494 = r10564484 + r10564493;
        double r10564495 = 1.853358028420886e+302;
        bool r10564496 = r10564494 <= r10564495;
        double r10564497 = exp(r10564491);
        double r10564498 = log(r10564497);
        double r10564499 = r10564487 * r10564498;
        double r10564500 = r10564499 * r10564492;
        double r10564501 = r10564500 + r10564484;
        double r10564502 = sqrt(r10564501);
        double r10564503 = R;
        double r10564504 = r10564502 * r10564503;
        double r10564505 = r10564482 - r10564481;
        double r10564506 = r10564505 * r10564503;
        double r10564507 = r10564496 ? r10564504 : r10564506;
        return r10564507;
}

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))) < 1.853358028420886e+302

    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 add-log-exp2.0

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

    if 1.853358028420886e+302 < (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2)))

    1. Initial program 63.1

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

    2. Taylor expanded around 0 47.0

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

  4. Final simplification29.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \le 1.85335802842088590346712744457318896647 \cdot 10^{302}:\\ \;\;\;\;\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))